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The Generalization of Paraproducts and
the Full T 1 Theorem for Sobolev and
Triebel-Lizorkin Spaces

By'

Kunchuan Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics
Spring 1996

Dissertation Advisor: Michael Frazier

ABSTRACT

The Generalization of Paraproducts and
the Full T 1 Theorem for Sobolev and
Triebel-Lizorkin Spaces

By
Kunchuan Wang

We study the boundedness of generalized Calderon-Zygmund operators acting on
Sobolev and, more generally, Triebel—Lizorkin spaces of arbitrary order of smoothness.
We are able to relax the assumptions T(:1:7) = 0 and/ or T*(:r7) = 0, which have been
required in earlier results by other authors. To do this, we consider generalized
paraproduct operators. We obtain the sharpness of our assumptions for some special
cases. Using the same technique, we also obtain some sharper results for “norming”

and “molecular” families.

To My Dad, My Mom And My Family

iii

Acknowledgments

With great pleasure, I would like to express my sincere gratitude to all those who
assisted and helped me while I was studying and doing research for this dissertation.
First and foremost I am deeply grateful to Michael Frazier, my thesis advisor, for
his helpful comments, for his encouragement, patience and hospitality, and for his
guidance and assistance in my research and writing of this dissertation. He helped
me to learn not only how to do research in mathematics, but also how to write
mathematical papers using proper English. Also he supported me financially to attend
the A.M.S. 900th meeting at DePaul University, Chicago, IL, and gave me a chance to
present my results in the Special Session on Extensions and Applications of Harmonic
Analysis: Spaces of Homogeneous Type and Wavelet Analysis. The education I
received under his guidance will be most valuable in my future career.

I wish to express special thanks to members of our study groups. Xiaodi Wang
and Lin Xu were in our first study group when I was preparing for the qualifying
and preliminary examinations. There was a wavelet study group, which was led by
Michael Frazier and whose members included Xiaodi Wang, Shangqian Zhang and
others.

I would like to express my appreciation to David Yen, the former Director of
Graduate Studies for me giving an assistantship to support me financially.

I would also like to express my appreciation to Sheldon Axler, Joel Shapiro,
William Sledd and Zhengfang Zhou for serving on my Doctoral Committee.

I am forever indebted to my family: my parents and my sisters, especially my
parents, who supported me financially and in spirit.

iv

Finally but not least I would like to express my appreciation to the people who
developed ETE‘XI. With this convenient typesetting, I could type my dissertation
by myself. There are many people who assisted me, but are not mentioned here. I

apologize to them.

Michigan State University Kunchuan Wang
Spring 1996

 

1W document preparation system was developed by Leslie Lamport as a special version of
Donald Knuth’s TEX program for computer typesetting. TEX is a trademark of the American
Mathematical Society.

Table of Contents

CHAPTER 1. Introduction and Preliminary Results 1
1.1 Introduction .............................................................. 1
1.2 Definitions and Preliminary Results ...................................... 17

CHAPTER 2. The Full T1 Theorem for Triebel-Lizorkin Spaces

With a = 0 26
2.1 Introduction ............................................................. 26
2.2 Key Lemmas ............................................................. 27

2.3 Main Theorems: Boundedness of Paraproduct and Calderon-Zygmund Opera-
tors ...................................................................... 33

CHAPTER 3. Paraproducts and the Full T1 Theorem

for L3 and 13?" 37
3.1 Introduction ............................................................. 37
3.2 Key Lemmas ............................................................. 38
3.3 Main Theorems .......................................................... 43

vi

CHAPTER 4. Applications to Molecular and Norming Families 52

4.1 Introduction ............................................................. 52
4.2 Applications of Results in Chapter 2 ...................................... 54
4.3 Applications of Results in Chapter 3 ...................................... 56

CHAPTER 5. Sharpness - 60
5.1 Some Sharpness Results Related To ESQ .................................. 60
5.2 Some Sharpness Results Related To 13;?" .................................. 71

CHAPTER 6. Conclusions and thure Research Plans Related To

This Topic 78
6.1 Conclusions .............................................................. 78
6.1.1 Conclusions Related To Fl?" ......................................... 78
6.1.2 Conclusions Related To Ff? ......................................... 80
6.2 Banach Frames: Molecular and Norming Families ......................... 81
6.3 Applications of Families of Molecules ..................................... 82
6.3.1 Pseudo-differential Operators of Type 1,1 ............................ 82
6.3.2 Off—diagonal Cases .................................................. 82
6.4 Boundedness of Some Matrices ........................................... 83
6.5 Weighted Cases .......................................................... 84
Bibliography 86

vii

Chapter 1

Introduction and Preliminary

Results

“There is only one thing you should do. Go into yourself. Find out the
reason that commands you to write; see whether it has spread its
roots into the very depths of your heart; confess to yourself whether
you would have to die if you were forbidden to write. This most of
all: ask yourself in the most silent hour of your night: must I write?
Dig into yourself for a deep answer. And if this answer rings with a
strong, simple “I must,” then build your life in accordance with this
necessity; your whole life, even into its humblest and most indifferent

hour, must become a Sign and witness to this impulse.”

— RAINER MARIA RILKE, Letters to a young poet (1903 —- 1908).

1 .1 Introduction.

The main purpose of this dissertation is to extend the celebrated “full” T1 theorem
in [8]. We study the boundedness of generalized Caldero'n-Zygmund singular integral
operators acting on Sobolev and, more generally, Triebel-Lizorkin spaces of arbitrary
order of smoothness instead of the usual LP spaces. We are able to relax the as-

sumptions T(:c7) = 0 and/or T*(:r'7) = 0, which have been required in earlier results

by other authors (e.g., [23], [24], [26], [11], [17], and [40]). To do this, we consider
generalized paraproduct operators (see (1.47) below for definition). We obtain the
sharpness of our assumptions for some special cases. Using the same technique, we
also obtain some sharper results for “norming” and “molecular” families.

For a more detailed description of the evolution of the theory of singular integral
operators of Calderén-Zygmund type or more generally singular integral operators,
one may refer to [27] or [16]. We prefer to motivate our study by emphasizing some ex-
amples. Let us start in one dimension R1. The Hilbert transform H of an appropriate
function f is defined by
z _1_ and,

(1.1) Hf(x) = 1 / f—“‘—y)dy

7'7 R 3! 7r R 37 - y
where the integrals are in the principal value sense. In the classical development
of harmonic analysis, the Hilbert transform occurs naturally. To see this let, for
simplicity, f be a real-valued function in LP(R), 1 S p < +00 and consider the

integral

 

(1.2) F(z) = i f“) dt.

7r R z — t
By a standard argument, one can show that F is analytic in the upper-half plane

R2+ = {x + iy : x E R, y > 0}. Let 2 = :1: + iy. Then we can separate the real and

imaginary parts of F (2) as

_ _1_ y 1 w—t
F(Z) — WLf(t)($—t)2+y2dt+an(t)(x—t)2+y2dt

= f*Py($)+if*Qy($)

 

 

where Py(:r) = fixgiy is the Poisson kernel in 1R21L and Qy(:r) = iP-i? is called the
conjugate Poisson kernel in R1. It is well-known that when f 6 L”, 1 S p < +00,
f * Py(:c) —-> f(:z:) a. e. as y —> 0, by properties of approximate identities. Also
u(:r, y) = Py at f (as) is harmonic, so it solves the Dirichlet problem in upper-half plane.
We don’t want to lead readers to this direction. What we want to show readers are

some questions related to singular integral operators. Let us observe one behavior of

the conjugate Poisson integral. One boundary property of this is when y —+ 0, we can

see that f*Qy(:r) —> Hf(x). So if Hf(.r) makes sense then F(x+iy) -—> f(:r)+in(:c)
as y —) 0. In other words, the Hilbert transform plays the role, on the boundary, of
the harmonic conjugate operator. There is a similar study of such theory on the unit
disk and unit circle, see [19] for details. One natural question is: how are the size or
smoothness of the real part f(:1:) on the boundary related to the size or smoothness
of the imaginary part H f (:13)? For example, if f E L”, does this imply Hf 6 L”? For
p = 1 or p = +00 this is false, but for 1 < p < +00 it is true. This classical result was
shown by M. Riesz, using complex function theory. This approach is inappropriate
for more general convolution operators.

Since the Hilbert transform is not bounded on L1, the space { f E L1 : Hf 6 L1}
is a proper subspace of L1. It is known as the Hardy space H1. Many theorems in
analysis that hold for L" for 1 < p < +00, but fail for L1, continue to hold for H1.
There is even a satisfactory space H p for 0 < p < 1.

In the previous discussion we only considered questions in one dimension. How can

this whole stucture be extended to n-dimensional spaces? The natural generalizations

of the Hilbert transform are the Riesz transforms R], j = 1, - - - , n, given by
(1.3) mm = K.- * f(x) = w: / fiance — y)dy

where the constant wn is the area of unit ball in Rn“. Basically the Riesz transforms
play the role in R1“ that the Hilbert transform does in R1. To see this, let f E L2
and fj E L2 for j = l, ---, n. Let uo(;r,y) = Py =0: f(:r) be the Poisson integral of f
and uJ-(a:,y) = P,, * fj($), j = 1, ~--, 72 be the Poisson integrals of fj respectively.
Note that a: E R" and y > 0. Then necessary and sufficient conditions that

(1'4) szij3 j=13°Han
is that the generalized Cauchy-Riemann equations hold, i.e.

n
011-
——J —0, an

j:0 3:13,-

8_ui-_Q£t£ j¢kwithxozy.

(1.5) 8:13,, — dxj’

Thus the role of the analytic function u + it) on R: with boundary values f + z'H f

is taken by the generalizd Cauchy-Riemann system (110,111, - - -,u,,) with boundary

values (f,R1f,---,R,,f).

Regarding LP boundedness, it turns out (by more general results of Calderon-
Zygmund stated below) that for l < p < +00, the Riesz transforms are bounded on
LP(R"). Moreover for 0 < p S 1, we can find appropriate substitutes for L”(R"),

namely the real-variables Hardy spaces H P (Rn). For example we can define
Hl=<[f:f€Ll and ij€L1,j=1,---, n}.

Next let us consider a more general form of the singular integral operators men-
tioned before. In [4], Calderon and Zygmund introduced what are now called the clas-
sical Calderon-Zygmund singular integral operators, which are of convolution type,
i.e. of the form
(1-6) Tf($) = R K03 — y)f(y)dy = K * f(:v)-

There are various types of assumptions placed on K; for example one may assume

that

(i) K(:L') = Q(a:)|:r|“", where 9(a) is homogeneous of degree 0, i.e. Q(t:1:) 2 (1(27)
for all t > 0,

(ii) 9 satisfies the cancellation condition [5%, Q(a:)da(a:) = 0, where do is the

induced Lebesgue measure on the sphere 5"”, and

(ziz) 0 satisfies “Dini type” condition:

1
(1.7) If sup |Q(r) — Q(.r')| = 10(6), then / 6_lw(6)d6 < +00.
Ix—x’lsé 0
Izl=lr’l=1

An even more general set of conditions considered by the Calderon-Zygmund

school is that
(a) |K($)| S Clair”,
(b) fr<lrl<R K(:c) = 0, for 0 < r < R, and

(C) ”((35 - y) — K($)| S C|y|‘|x - yl‘""‘ when |$| 2 Clyl

for some 5, c, C > 0. For a complete treatment of this topic, one may refer to
[34]. The main object is to show that such an operator is bounded on LP when
1 < p < +00. There is no analogue of the complex variables approach of Riesz for the
Hilbert transform. Instead, Calderon and Zygmund developed powerful real-variable
methods for this problem. The strategy of the proof is following. The first step
is proving the L2 case which mainly depends on the tools 'of convolution, Fourier
transforms, and the Plancherel theorem. Note that they are basic results in harmonic
analysis in R”. The second step is the L1 case. For this case there is no boundedness
in the strong sense, but such an operator is bounded in a weak sense, i.e., T is of
weak type 1,1. Then applying an interpolation theorem and the previous two cases,
one shows the Lp-boundedness for 1 < p < 2. Finally by a duality argument, one
proves the remaining cases (see [34] for more details).

Next let us consider non—convolution cases, for example, pseudo-differential oper-
ators, Calderon’s k—th commutators and double layer potentials. Before we illustrate
this type of example, recall the definition of Fourier transform. For a function 90, let

95(6) be the Fourier transform of (,0 defined by

(1.8) «5(5) = [name-ridge.

Now let us begin with pseudo—differential operators. A pseudo—differential operator is

formally defined by

1

(1.9) T.f(:c) = W Ana(x,€)e““f(€)d€

for some symbol a. Let us observe some extreme cases. If the symbol a is independent

of x, a(:r,§) = m(§), then T, is a Fourier multiplier operator:

A

(1.10) (mm = moans);

while if a is independent of 5, a($,£) = g(:z:), then Ta is a pointwise multiplication

operator:

0-“) (Taf)(1') = 9($)f($)-

More generally, if L is the partial differential operator (with possibly non-constant

coefficients) given by
(1.12) Mac) = 2 WWW),
IWISN

by Fourier inversion L is a pseudo-differential operator of the form (1.9) with a(:c, E) =
ZI’YISN a,(a:)(i§)7. Because of this, pseudo—differential operators play a central role in
the study of partial differential equations. In general we need to put some assumptions

on a(x, f) to have a reasonable operator. For example, we say a 6 5%, if
(1-13) lDsza($,€)l S Cm(1+l€|)"”+'”'-

Then the kernel K(:r, y) of Ta is given by

1
(270"

It was shown in [35] that if the symbol a(:2:,£) belongs to SR1 then

 

(1.14) Kay) = / a(a=,§)e“"x'y’d€.

0-15) IDEDZK(x.y)l s an — yl-n—Im-M

for all 6 and 7. Although such a kernel is of Calderon-Zygmund type, it is not
necessarily bounded on L2 in general, as shown in [35]. What conditions should we
add so that such operators will be bounded? See Theorem 1.1.1 below for answer.
For a more general symbols, see Chapter 7 [35] for details. In [41], the author gave
some history of this type of operators and some further references. Let us notice
that the proof in [41] depends heavily on the theory of the go-transform developed by
Frazier and Jawerth in [12—14]. To prove boundedness, it is sufficient to show that
such an operator maps “smooth atoms” into “smooth molecules”, or equivalently it
maps smooth molecules into smooth molecules. This is a powerful method to prove
that an operator is bounded. We won’t give the definitions of smooth atoms and
molecules here. Interested readers may refer to [12—14] or Chapter 4.

Now let us turn our attention to another example, the Cauchy integral. Let A be

a Lipschitz function, i.e. |A(a:) — A(y)| S Clr — y] for at, y E R1 with a = A’ E L°°.

Consider the graph F = {z(:z:) = a: + iA(:z:) : :1: E R} C_:_ C and define the Cauchy
integral of f on I‘ by

(1.16) we) = —1—./ -——1—f(z(y))z’(y)dy.

2m R 2(y) — z(:r)

We can define an analytic function F (2) related to C A f such that when 2 approaches
to the graph F, F (2) approaches C A f . When we observe the properties for Cauchy
integrals and those for the Hilbert transform, their behavior is similar. Note that
there is a connection between the Cauchy integrals and the Calderén commutators,

defined below. To see this, suppose IIA’ I] Loo < l, and write

(2(y) - 206))"1 = (y - w + z'(A(y) - AWN—1

 

 

 

= 1 i (4A0? : :(afly

— (I?
y [:20

since |(A(y) — A($))/(y - 93H S IIA’llLoo < 1-
Thus, setting g(y) = f(z(y))z’(y), CAf(;z:) can be expanded in terms of the

Calderén commutators, which are of the form

(1.17) T.,.g(x)= [R (A(‘”)‘A(y))k 1 g(y)dy,

r—y x—y

 

 

for k 2 1, where a = A’ and A is a Lipschitz function on R1. For further comments on
these operators, one may refer to [5] or [39]. Note the first order term Ta,og(:r) is just
the Hilbert transform. However for k _>_ 1 and A not constant, T0,), is not translation
invariant. Nevertheless the singularity of the kernel of T0,], is of the same order as for
the Hilbert transform.

For higher dimensions, the operator analogous to C A is the double layer potential
on a Lipschitz domain (I with boundary 5'. This example is related to solving the
Dirichlet problem: Au = 0 in Q and u = f on S, where f is a given function on
S and A is the Laplace operator. For w E (I, let ’D f (to) represent the double layer
potential of f. As w converges to a: E S, ’Df(w) converges to %f(:r.) + Tf(.r) where

in local coordinates

__ -1 A(0r) - A(y) - (a: - y) - VA(y)
(”8) m“) ‘ w" A (la: — yr + (Am — A<y>>2><n+1v2f (”“3“

for wn the area of the unit sphere in R". For a Lipschitz domain (I, we have |]A’]] Loo <

 

+00. One can obtain LP estimates for the solution of the Dirichlet problem on Q if
one can prove the LP boundedness of T, which is a generalized Calderon-Zygmund
operator. This is a deep problem in analysis, which we don’t consider at this moment.
The interested reader may refer to [5] or references given there for more details. Our
point is merely to exhibit that many non-convolution singular integral operators arise
naturally in harmonic analysis.

Now we give the definition of the operators that we discuss in this article. As
in [17], we consider operators with sufficiently smooth kernels. We say that K(:r, y)
is a (generalized) Calderon-Zygmund kernel of smoothness (l + 5), denoted by K E
CZKU + 5) for l 6 N0 = NU {0} and 5 6 (0,1), if K restricted to Q = {(x,y) E
R" x R" : .7: at y} is a function with continuous partial derivatives in the variable as

up to order I which satisfy, for some constant C > 0,

(1.19) IDfK(x,y)I s Clx—yl‘”""' for lfll s z,
and
(1-20) lDfK(:v,y) - DfK(x',y)l S Clx - yl‘"""‘l$ - x'le,

for ]fl| = l and 2|:r — :r’] g |:1: — y] (where the subindex 1 stands for derivatives in the
first variable, x). Note that we write K 6 C Z K (I) when the kernel K satisfies (1.19)
only. The kernels of (1.16), (1.17), (1.18) satisfy conditions of this type.

We say that T is a Calderén-Zygmund operator of smoothness (l + 5), denoted by
T E C Z 0(1 + 5), if T is a continuous linear operator from S into 8' whose Schwartz
distributional kernel K (cc, y) E C Z K (I + 5). For more detailed treatment of this see
[17].

Next we consider a condition known as the “weak boundedness property” (WBP)

in [8]. For 0 E D(R") (where D is the class of C°°-functions with compact support),

t > 0 and z E R", let 0f(x) = t‘"0((:c — z)/t). If T : 8(R") —> S’(R") is linear and
continuous, we say that T 6 WE P if for every bounded subset B of ’D, there exists
C = C(B) such that

(1-21) |(T(95),nf)l S CV",

for all 0 and 17 6 l3, 2 E R", and t > 0. Here (--,) represents the bilinear pairing
between S’ and 8. See e.g. [8] or [27] for further background.

Note that by a standard Calderon-Zygmund argument [34], one can show that T is
LP-bounded, for 1 < p < +00 when T is Lz-bounded. For convolution operators, the
L2-boundedness is a simple application of Plancherel’s theorem. But for the general
(i.e. non-convolution) cases, that is strongly nontrivial. The following result of David

and Journé was a breakthrough.

Theorem 1.1.1 (Tl-Theorem, [8]) Suppose T, T“ E CZO(5) for some 5 > 0.
Then T extends to be bounded on L2 if and only if T1, T‘l E BMO and T 6 WBP.

The meaning of T1 and T‘l is discussed in [17] or [8].

The necessity of the conditions in Theorem 1.1.1 can be obtained by a standard
argument. For example, when T is Lz-bounded, it maps L°° to BM 0. So T1 6 B M O
and hence T‘l E B M 0, by a duality argument. To prove the other direction of the full
T1 theorem, David and Journé first showed that the theorem is true for the reduced
case, i.e., when T1 = T‘l = 0. Then they proved the boundedness of paraproduct
operators (see definition (1.45) below), and used this to obtain the “full” result.

It is time to say a few words on function spaces. Beginning with the L" spaces,
there are generalizations from different points of view. The most useful generalizations
are the Hardy spaces H 7’, the Sobolev spaces LS, the Bessel potential spaces Lg, the
Hélder classes, the Zygmund classes, the Besov spaces Bf,” and the Triebel—Lizorkin
spaces F15". In this article, we mainly focus on the homogeneous and nonhomogeneous
Triebel-Lizorkin spaces F5“ or F5". In fact this is a generalized class of spaces which

includes many of the key examples above as special cases. In particular, we have the

following identifications (see [42], [14], [16]): L‘D z F32 for 1 < P < +00, Hp z F132

10

for0<p§1,Aa'~VF;°°,BMOzF§oz,LngfforkeNandl<p<+00,and
L; z F1,“2 for oz > 0 and 1 < p < +00. Thus by dealing with the Triebel-Lizorkin
spaces we can give the unified treatment of a wide range of function spaces which
appear quite dissimilar at first.

To describe this unification of function space theory, we would like to discuss
the history of Littlewood-Paley theory. This theory was originated by Littlewood
and Paley in the 19303 in the context of analytic functions on the unit disk and
Fourier series. It was generalized to the real-variables context in R" by Stein and his
colleagues starting in the 19503. As before we denote by R1“ the upper-half plane
with boundary R", i.e. R1“ = {(zr,t) : x 6 R", t > 0}. Stein and his school consider
a kernel «pt which is either t2; (scalar-valued) or thP (vector-valued). Note that in
either case we have cpt(x) 2: t‘"<p(:r/t) and fgo(:r)d:c = 0. Then for a function f, set
F (x,t) = f * Lpt(:r) and define the g-function of F associated with (,0 by

(1.22) g(F)(w) = (f IF<x,t)Iz%)l/2 = (/°° If * w<w>|2%)m.

By a standard argument and the results of classical Calderon-Zygmund singular in-
tegral operators, one can show that if f E L” for 1 < p < +00, then g(F) is in L‘D
with |]g(F)|le z Ilflle. For 0 < p S 1, {f : g(F) 6 LP} is the real Hardy space H”.

This suggests a more general approach. We say a function (,0 on R" is a Littlewood-

Paley function if it satisfies (1.23—-1.25):

(1.23) [99(3)] 3 C(1 + ]:1:|)‘"“ for some 5 > 0,
(1.24) / 99(x)d:r = 0,

(1.25) [90(33 — y) — 99(x)]d:r S Clyl‘s, y E R", for some 6 > 0.
Rn

Note that condition (1.23) implies that «,0 is in L1 and so condition (1.24) makes
sense. Also note that any function in S with vanishing integral is a Littlewood-Paley
function. For such a function (,0, let cpt(:r) = t’"cp(:r/t) as usual and for f 6 LP, put
F (2:,t) = f * 90¢(x) and define the Littlewood-Paley g-function of F associated with

ll

90 by (1.22). Then one can show that the norm equivalence holds under a certain
nondegeneracy condition on <0. Note that for other function spaces, the condition
(1.24) should be changed to a higher vanishing moment condition which depends
on the function space in question. As mentioned in [16], each approach leads to a
Calderon formula. This in turn led to the go-transform identity (see the definition
in Section 2) and the orthonormal wavelet decomposition of Meyer. All of these
approaches can be used to characterize function spaces mentioned above (see [16] or
[42—43] for details).

The European school, led by Peetre and Triebel, chose to modify the approach
above, replacing the integral with a sum and replacing the kernel formed from the
Poisson kernel by one with compactly supported Fourier transform. For this purpose
here and later, select a function (,0 E 8(R") satisfying suppcfi Q {E : % S [E] S 2}, and
|c0(§)] 2 c > 0 if g S [6] S 3. For V E Z, let 90,,(23) = 2""g0(2":1:). (Note that this is
a different dilation notation from the one above for (pt; this new convention will be
used throughout this dissertation).

Next we give the definition of the function spaces in question. For a E R, 0 <
p S +00, 0 < q S +00 and f E S’(Rn), define the homogeneous Triebel-Lizorkin

spaces via the norms

l/q
(1-26) llfllpgv E (Z (2"“lsou * f0“) ,

VEZ LP

ifp < +00, and

1/q
+00
(127) llfllpggz sgp IQI“/ Z (2”“lsou*fl)"
dyadic Q u=—logze(c2)

2u+l

To motivate the definition for p < 00, consider ( 1.22). Think of fooo as Zuez 2,,
and note that each interval [2”,2"+1] has the same dt / t measure. Thus g(F(a:)) is
analogous to (Zr/62 |f*g0,,|2)1/2. Hence llfllfigi’ is analogous to ||g(F)]|Lp, which, as we
noted above, is equivalent to ”film for 1 < p < +00 and “film for 0 < p S 1. Similar
interpretations of the classical Littlewood-Paley theory suggest the other function

space equivalences noted above.

12

Let us go back to operators. Several authors (e.g. [23], [24], [26], [11], [17] and
[40]) have obtained results on the boundedness of generalized Calderon-Zygmund
operators, for spaces of functions or distributions, other than LP. These results are
all under the assumptions T(:r7) = 0 and/or T*(a:'6) = 0 for some certain fl and '7
depending on spaces. Thus these results correspond to the reduced version of the
David-Journé result (Theorem 1.1.1) under the assumptions T1 = T*1 = 0.

Here are the versions of these “reduced case” results for the Triebel-Lizorkin

spaces. Let J = n/ min(1,p, q).

Theorem 1.1.2 ([17] Theorem 3.1) Leta < 0 and 0 < 19,61 S +00. IfT : 8 —> 5’

satisfies
(a) T e WBP,
(b) T e CZO(6) where 0 < 6 <1,
(c) T“ e CZO([J — n — or] + p) where (J — a)* < p <1,
(d) T1 = 0,
and
(e) T133") = 0 for lrl S [J — n - a],

then T extends to a bounded operator on F59.

Theorem 1.1.3 ([17] Theorem 3.7) Leta Z 0 and min(p, q) 2 1. IfT : S —> 5'

satisfies
(a) T E WBP,
(b) T e CZO([a] + 6) where a“ < 6 <1,
(5) T“ E CZO(p) where 0 < p <1,

(4) T01”) = 0 for a” lvl S [a];

13

and
('e) T" =0 ifoz=0,

then T extends to a bounded operator on Ff".

Theorem 1.1.4 ([17] Theorem 3.13) Leta Z 0 and min(p,q) < 1. UT :8 —> 8’

satisfies
(a) T E WBP,

(b) TECZO([a]+6) where a" <6< l ifJ—n—a<0 andmax(a*,J*) <6<l
ifJ—n—QZO,

(c) T“ 6 CZO([J — n] +p) where J'“ < p <1,
(<1) T0”) = 0 for a” lvl S [a],
('e) T"(x") = 0for all m g [J—n —a] ifJ—n —a 2 o,
(f) IDngKt'v, y) - 0503133, y’)| S Cly — y’lplrv - yl‘("+""+'”'+”) for 213/ - y’l S
Ix - yl z'f I7! S min([Jl - n, [0]) and lfl + 7| = [J] - n,
and

('9) 1050mm y) - Dtmmz y)! s Ola: — w’lplx — yl“"+""+“'+‘” for 21a: — m’l .<_
la: — yl if M = [a] and lfil = (111— n —[a1)+,

then T extends to a bounded operator on Fpaq.

Note that the meaning of T(x”) and T*(x7) is discussed in [17]. Before we go
any further, we would like to make a comment. The proofs of Theorems 1.1.2 —
1.1.4 depend on the method we mentioned before, i.e., the authors showed that such
operators map smooth atoms into smooth molecules (see [17] for details).

With these results in hand, we will generalize the T1 theorem by relaxing the

conditions T(x") :2 0 and/or T*(xfi) = 0. For the case a = 0, we obtain

14

 

Theorem 1.1.5 Let T, T“ E CZO(5)flWBP for some 5 > J“, where J :2 ”

min(1,p,q) .

Let U = U5>o F3720. Then T extends to a bounded linear operator on F1?" if

(a) q = 2a 1 < P < +00, T16 F302 and T1 6 F302,.

('b)1Sq<2,1<p<+00,T16F°q

00’

and T‘l 6 U;
(c) 2 < q s +00,1< p < +00,T1€ U, and T16 Fgg’;

(d) ;,.',‘.—1<qS2 J—<p<+oo,T1€F°q

’ n+1 oo)

andT*l=0;
('e) 2<qS+00, $<pSL T16U,andT*1=0.
(f) 2 SqS +00, 19: +00, T1=0, andT*1€ F33”;

or

(a)1<q<2,p=+oo,T1=0,andr*1eU.

Part (a) is the David-Journé result, which is included here for comparison. It
exhibits a particular role played by the index q = 2, which we will also see below.
Note that case ((1) includes H P x F:2 for n"? < p S 1. Also we will see (Proposition
5.1.2 below) that the theorem is sharp for Flog, ;% < q S 2.

For q = 2, we have F?2 m H1 and F302 % BMO , and these parts of (d) and (f)
follow from classical results and the David-Journé result (see, e.g. [7], p.29, Remark
3).

For the boundedness of T on F1?" for general a, we need to assume that the kernel
of the operator T satisfies more conditions, as stated in the definitions above and
the conditions below. Under these, we obtain Theorems 1.1.6, 3.3.2, 3.3.3 and 3.3.4.
Here we only quote Theorem 1.1.6 which deals with the case oz > 0, 1 S p < +00,
1 S q S +00. Theorems 3.3.2, 3.3.3, and 3.3.4 provide sufficient conditions for
boundedness in the remaining cases, namely 01 < 0, 1 S p < +00, 1 S q S +00 in
Theorem 3.3.2, 0 2 0, min(p,q) < 1 in Theorem 3.3.3 and a < 0, min(p,q) < 1 in
Theorem 3.3.4.

15

Theorem 1.1.6 Let a > 0. Assume that T : S —) 8’ satisfies
(a) T E WBP;
('b) T E CZO([a] + 6) where a“ < 6 <1;
(5) T“ E CZO(p) where 0 < p <1;
and
(d) T(x”’) = 0 for all Ivl S a — n/p.
Then T extends to a bounded linear operator on Egg if one of following cases holds.-

(5) a¢Z, lSqS+00,1Sp<+00,andforalli=1,2,~--,2"—l,

(1'28) {<T(($ _ $Q)7)al/)g))}Q E f:7(a_h|) when a " n/p < [7’] < a;

(f) a E N, 1S q S 2, l Sp < +00, andforalli=1,2,~--,2" — 1, {1.28) holds

and

(1.29) {<T((:c — $0)“),¢g))}0 61:: when m = a;

(g) 06 N, 2 <qS +00, 1Sp< +00, andfor alli = 1,2,---,2"—l, (1.28) holds

and

(1.30) {me — worwétno e we = woofzzsr when m = a.

We should compare this result with Theorem 1.1.3. The assumptions (a—c) are
the same. Regarding (d), it was shown in [17] Theorem 3.25, that T(x") = 0 for
[7] < a —n / p is a necessary condition for the F519 boundedness of T. In [17], however,
it is assumed that T(x7) = 0 for [7] S [0:]. Thus our improvement consists of replacing
this with the weaker conditions in (1.28 — 1.30) for a — n/p < [7] S [a]. For a > 0,

one might at first expect conditions on the function space T(x"’) belongs to. This

 

16

would be equivalent, by Theorem 1.2.1, to conditions on the sequence space that
{(T(x7),z,/)(Q”)}Q belongs to (in place of (1.28-1.30)). However, for 7 75 0, this is
not natural because the translation invariance of the function space is reflected in the
structure of the wg)’s but not in T(x"). This leads us to (1.28 — 1.30). In other words,
from the sequence space perspective, these conditions are natural generalizations of
the T1 6 BM 0 condition, even through, for 7 at 0, they cannot be formulated as
single condition for T(x7). Similar remarks apply to Theorems 3.3.2, 3.3.3, and 3.3.4,
in comparison with Theorems 1.1.2 and 1.1.4.

For a > 0, 1 S p S +00, and 0 < q S +00, one definition of the inhomogeneous
Triebel—Lizorkin space F:q(R”) is simply F1?" 2 Fl?" 0 L”, with

(1-31) llfllpgv = llfllpgv + llfllLr

(see e.g. [16], pp.42—3). For the case 1 < p < +00 and q = 2, FPOr2 is (equivalent to) the
Bessel potential space L2 (as in [34], p.134). In particular, when 01 = k 6 N and l <
p < +00, F152 is the usual Sobolev space L” = {f 6 LP : DVf E Lpfor [7] S k} (see
e.g. [16], p.42). Here 7 = (71, - - - ,7") is a multi-index in N3 and D" = 31;],— - - - 5%}
By (1.31), Theorems 1.1.6 and 3.3.3, and the David-Journé result, we immediately

obtain the following result for the inhomogeneous spaces Fpo‘q (including the Sobolev

and Bessel potential case L2 9: Frifl)’

Theorem 1.1.7 Suppose oz > 0, l < p < +00, and T satisfies the assumptions of
Theorem 1.1.6if1 S q S +00 or those of Theorem 3.3.3 if 0 < q < 1. Then T

extends to a bounded operator on F5”.

As far as we know, this is a new result even in the simplest case of L2 % F2“2 for

kEN.

To show the boundedness in these theorems, we shall prove boundedness results for
generalized paraproducts. In [15], Frazier and Jawerth defined paraproduct operators
via the cp-transform identity, which was defined by Meyer in [27] via the wavelet

decomposition (and by other authors earlier using the continuous Littlewood-Paley

 

17

notation). We can use the characterization of the function spaces FFQ in terms of its
coefficients with respect to the expansion in terms of either the w-transform or the
Meyer’s wavelets decomposition (see [14], [27] or [16]) to reduce the qu-boundedness
for a paraproduct to the boundedness of a certain matrix on sequence spaces f1?"
associated to FPO”. We will see that this matrix factors as a product of a diagonal
matrix and a matrix which just fails to satisfy the boundedness criterion known as
almost diagonality (see [14] or [16]). By analyzing the behavior of each factor, we
obtain the boundedness of paraproducts in Theorems 2.3.1 and 3.3.1, respectively.
The main results of this dissertation are Theorem 1.1.5 in Chapter 2 and Theorems
1.1.6, 3.3.2, 3.3.3 and 3.3.4 in Chapter 3.

Corresponding to the proof of the boundedness of paraproducts, there are some
applications to the theory of molecules and norming families (see (4.1) and (4.2)
below for definitions or [15] for more details,) which were introduced by Frazier and
J awerth in [15]. The organization of this article is as follows: First we give background
in Chapter 1. In Chapter 2, we prove the full T1 Theorem for F3". We deal with
the more general case of F1?” in Chapter 3. Next we obtain some applications to
“molecular” and “norming” families in Chapter 4. We will give a discussion of the
sharpness of some of the results of the previous chapters in Chapter 5. Finally, we
will state some problems related to this dissertation and outline our research plan

related to this topic.

1.2 Definitions and Preliminary Results.

For convenience, we will use the following terminology. Let R be the set of all real
numbers, Z the set of all integers, N0 the set of all nonnegative integers, R“ n-
dimensional real space, 5 the Schwartz space of all smooth, rapidly decreasing func-
tions, and 8’ its dual, the set of all tempered distributions on R”. Let q’ be the
conjugate index to q, i.e. q’ = q/(q — 1) for 1 S q S +00. Also for any x E R, let
[x] be the integer part of x, x“ = x - [x] and x+ = max(0,x). For a multi-index

’7 = (71,72: ' ° ' ’71:) E N3, let [7] = 2:21:17,- be the trace of 7.

18

We say that a cube Q Q R" is a dyadic cube if Q = Quk = {x E R" : 2"“k, S
x.- S 2“’(k,- +1), i = 1,2,---,n} for some V E Z and k = (k1,k2,---,k,,) E Z". Let
[(Q) = 2'” be the side length of this cube and xQ = 2“’k the “left corner” of Q. Let
'P be the class of polynomials on R", and let suppcp denote the closed support of (,0.

In what follows, we can use either the cp-transform identity (see [11] or [13])
or the expansion in terms of Meyer’s wavelets ([29]). To describe the g0-transform
expansion, we choose a function (,0 as before. Then there exists a function w E S
satisfying the same conditions, namely suppzfi Q {E : % S [E] S 2} and [10(6)] 2 c > 0
if 3 S [5] S- 3, such that also ZoezW‘Z’Q—VO = 1 for 5 75 0. For Q = Quk,
let cpq(x) = [Ql‘iw(2"x — k) and similarly for 1%. Then we have the Lp-transform
identity

(1.32) f = Z<Lm>¢o
a

(see [12] or [13]). Here and through this paper, 2Q means that the sum is taken over
all dyadic cubes Q.

Meyer’s wavelets {wq}Q dyadic on R are of a similar nature, but with suppz/i g {C :
35} S [{I S 83”}. They are constructed so that {Il’QlQ dyadgc is a complete orthonormal
family in L2(R). So we obtain

(1.33) f= Z< f.¢QW
Q

in R. In R" there are 2" — 1 functions {1% l 221-1 such that {108”},9 form a complete
orthonormal family 1n L2(R"). So we have

2"-1

(1.34) f = Z ZWS’W
i=1 Q

For simplicity we work with (1.32), but we emphasize that Meyer’s wavelets could
be used equally well in our development. Moreover, they will be needed for Proposi-
tions 5.1.1 and 5.2.1, showing the sharpness of our results.

Here we define the sequence space f5"? that corresponds to Fi‘j'q via either the (,0-

transform identity or the Meyer’s wavelet decomposition. For 0 < p S +00, 0 < q S

19

+00, 01 E R and a sequence 3 = {SQ}Q, let XQ denote the characteristic function of

Q, and define

1/(1
(1-35) ”3”}? E (2(lQl—l’2'a’"|SQ|X0)q) ,
Q

LP

ifp < +00, and

1/9
(1.36) ”5”qu E Sl}1)p([P[—1/ ZUQl—l/z-a/nlSQlXqu)

P 09’
where the sums and the supremum, respectively, are taken over all dyadic cubes in R".

Let fag be the space of all complex sequence 3 = s d adgc such that 3 mi < +00.
p Q C? y h,

Theorem 1.2.1 ([13] Theorem I, or [14] Theorem 2.2) Suppose a E R, 0 <
p,q S +00, and the functions (,0 and w are as before. Given f E S’/’P, let sQ =
(f, (pQ), for each dyadic cube Q (so that f = 2Q(f,(,0Q)¢Q = ZQ sQwQ). Then
f 6 F59 if and only if the sequence 3 = {sQ}Q E ff", and we have ]]f]]F:q z ||s[[f-;q,
(i.e., there exist C1,C2 > 0 such that Cillsllqu S llfllfigq S Cg]]8]]j;q). Also, for any

sequence t = {tQ}Q, we have

(1.37) ll Zto‘hollrfl 5 Clltllfs’”
Q

with C > 0 independent oft.

For Meyer’s wavelets {¢(i)}:l-l, we have

2"—l

(1.38) ”fuse x Z Ils“’||;;q,
i=1

where 38’ = (f, 1&8”) and s“) = {sg’}Q. The inequality “S” follows from the molecular
theory (e.g., Theorem 3.5 in [14]), while the other inequality follows from the theory

of norming families (Theorem 3.7 in [14]).

20

Theorem 1.2.2 Suppose a E R, 1 S q S +00, 3 = {Sq}Q E ff“ andt = {tQ}Q E
fo‘oaq’. Then

(1-39) l<s,t>| = lgsot—Ql S Ca.q||8||j;~v ° lltllf-ogaqi-

For 1 S q < +00, this estimate is part of the duality ( .10“)* z fog“, (see [14],
Theorem 5.9). For q = +00, we do not have duality, but this inequality follows from
the discrete version of the Carleson measure lemma (see [27], pp. 335-336 or [15], p.
411), applied to [QIO/nH/th and [QI'a/"‘1/2sQ in place of Ag and Sq. Note that for
q = 2 and a = 0, this is equivalent to the difficult half of the H l — BM 0 duality.

We say that a matrix A = {an}Q,p is almost diagonal for ff" if there exists 5 > 0

such that
(1.40) 80ng [anl/wqp(€) < +00,

where

“'41) W“) : (hay(1+malcli(;)filo)))—J-EX

min { (%) ("M/2 , (2%) (n+e)/2+J—n}

 

for J = n/ min(1,p, q).

Theorem 1.2.3 ([14], Theorem 3.3) An almost diagonal matrixfor qu is bounded

on ff“.

We write X —-> Y if X and Y are quasi—normed spaces and the identity is a

continuous map of X into Y.

Theorem 1.2.4 (Imbedding Theorem) (a) For a E R, 0 < p S +00, and
0 < ql S q2 S +00, we have

' 091 ' aqz 'aqi 'an
Fp —-)Fp and fp —>fp .

 

21

(b) (Generalized Sobolev Imbedding Theorem) For (11 > 02, 0 < p1 < p2 <

+00, 01 — :7 = 02 — 1,12 andO < q1,q2 S +00, we have

' 0191 ' 01202 .0191 'a2q2
FPI _>sz and fm —> P2 '

Part (a) follows trivially from the imbedding €91 —> 8‘”. The F case of (b) is
well-known, and can be found e.g. in [42], p.129. The f case can be proved by an
analogous argument, or can be obtained from the F case by using Meyer’s wavelets
{wg)};,q. To see this, fix i 6 {1,2, . . . ,2" — 1}. Given 3 E 15:11:“, let f = 2Q sng).
Then, by Theorem 1.2.1,

(1.4?) ||S|ljg,m R5 llflll'js,“2 S Cllfllr~,:',wl % CllSlljggqx-

Theorem 1.2.5 (Vector-valued maximal inequality, [10]) Let

(1.43) Mf(:c) = sup -1— |f(y)|dy

r>0 7‘" B(x,r)
denote the Hardy-Littlewood maximal function of f. Suppose 1 < p < +00 and
1 < q S +00. Then

00 l/q 00 1
(1.44) (21mm) so... (Em-1°)

LP ’ LP

/9

A relatively simple proof of this can be found in [35], Section 2.1.

For the reduced step, David and Journé used paraproduct operators to prove
the full T1 theorem (see e.g. [27] for background). Let (I) E S and f (I) = 1. For
V E Z, let <I>,,(x) = 2""<I>(2"x), and for Q = ka, let <I>Q(x) = 2“’"/2<I>,,(x — 2’”) =

[QI'1/2<I>($—£'(15—;3-), as usual. For g 6 F30” we define the paraproduct operator Hg by

(1.45) 1w) = Z<o.oo>IQI-W<f.<1>o>wo
Q

22

Theorem 1.2.6 ([8], [27] or [15], Theorem 9.1) Suppose g E BMO and 1 <
p < +00. Then Hg E CZO(1) fl WBP, H91: 9, H31: 0, and

(1-46) lngfllLP S Cpllgllsmollfllu

for all f E L”.

We will generalize this theorem to FF?" in Theorems 2.3.1 and 3.3.1. For our
purposes in Chapter 3, we need to generalize the paraproducts. By Lemma 3.2.1
below, for every m E No and each multi—index 7 with [7] S m, there exists a function
(1)“) E S with fxB<I>(7)(x)dx = 63;, for [6] S m. Define (1)8) as usual by (D8) =
[Ql'l/2<1>(1)((x — xQ)/€(Q)). For g E 15.13"”, we define the paraproduct operator I15”

of order 7 by

(1.47) H§"’(f) = Demon-t- "<f.<1>8)>1bo-

Lemma 1.2.7 In fact, for g E Fl3"°° and m E No, we obtain 11;"), (1157))" E
020(m + 1) 0 WP. 119’s!) = 96.3., for 1.81 s m and (H§”’)‘(:c") = o, for
one.

Proof. Observe that the kernel of Hg“) is

 

1%1

(1.48) K<x.y>=2<o,oo>IQI-%- Wows).

Q

The facts that Hgl)(xt3) = 65,, for [)8] S [7] and (11(7));(xfi) = 0 are easy and similar
to the argument in Theorem 1.2.6, by the choice of the function (PM. Also, since

<l>(7),1,b 6 S, we have

(7) . "1 2 M UH
(1.49) |<I>Q (cc)|, |¢Q($)ISCIQl ’ (1+ ((62) l

and

(7) -12- n Irv-ml _,,_.o|-1
(1.50) |D3<I>Q(:v)l,119346201)“lel ’ W (1+ 3(6)) l

23

for 0 < [6] S m +1. Since g E Flgl’oo, IQ[_1’2—hl’nl(ga90Q)l S C < +00. Then

I (. y )|< CZIWw )-| ld’o( )I
Q
Fix x,y E R", and select 14) E Z so that 2“’° S [x — y] < 2“’°+1. Let

“115 5°? 2 (1+'2Io1'>'"’1(+'—-o')

I/=-00 ((Q):2""

 

and

 

=2 2 (+w'xo31'>’(1+'+I—')

V-Vo +1 ((Q)= 2‘"

Then [K(x,y)| S C(L(x,y) + M(x,y)). To estimate L, simply use (1 + €(Q)[y —
xQI)‘”"l S 1, and note that

[33-le -n-—l ~ —n-l
g(Q) ) ~keZZ"(1+lkl) so.

 

(1.51) 2 (1+

((Q)=2""
So L(x,y) S CZ:‘3___OO 2"” S C2"°" S Clx — y]‘", as desired. For M(x,y), note

by the triangle inequality that every dyadic Q belongs to either A = {Q dyadic :
[x — xQI Z [x — y]/2} or B = {Q dyadic: [y — xQI Z [x — y[/2}. Thus

...,2. (1+ [halal—"‘1 (1+ IyIIQinl—H s C (liwi’Iln—l

 

 

 

since (1 +£(Q)‘l[x —xQ]) Z C(1+€(Q)‘1]x —y[) 2 C€(Q)‘1]x — y] and using (1.51).

Symmetrically, we have the same estimate for 21(0):?” ; hence
Q68

M($,y) S C Z 2””2‘”("+1)[$_y]-n—1

II=V0+1

= Clx — yl""‘l Z 2‘” s Clx — yl‘".

u=uo+1

This gives (1.19) with B = 0. We obtain [DfiK(x, y)[ S Clx — y]"“w| and hence
(1. 19) withl= m+1 and 0 < [S] S m+1, and (1.20) with [2 m+1 and [,3]: m+1
in the same way from (1.50). The only difference 18 that the factor 2"" in L and M

24

is replaced by 2"(”+[m). For L this easily gives the estimate. The corresponding term

for M is

0 Z 2V(n+lfil)2-v(n+lfil+1)[$ _ yl—n—lfiI—l S 0],, _ yl-n-lm.

V=Vo+1

So we obtain II?) E CZO(m + 1). Also, by the same argument, we have (HWY E
C Z 0(m + 1).
To show II?) E WBP, suppose 0,17 E D. Simple size estimates (or see Lemma
B.2 in [14]) give
|(o:,q>gv>)l s Ct-n/z min{l?nl/12/2, [3’32] (1+ m]:{;($QQ)[t}) n 1,
where C is determined by the bounded subset of D that 0 and n are assumed to

 

 

belong to. We have the same estimate for [($62,775)] when t < [(Q), but in the case

[(Q) S t we obtain more, namely
Kwanfll S Ct_""13(Q)1+"’2(1+1—llz — well—"’1,

because f 11);) = 0 and r] is smooth (see e.g. Lemma B.1 in [14]; we warn the reader
that the assumption j Z k in the statement should be j S 10). Fix t and pick 11 E Z
so that 2"‘“"1 < t S 2‘“. Since g E 513'“, we have [(9,90Q)| S ClQlilJ'tr’?l (e.g. by
Theorem 1.2.1), so

I<H§1>0:.n:>I s 02I<0:,<I>I;’>II<¢o,n:>I
Q

u
0: Z 2111(1+21'Iz—.+oQ|)-11-l

v=—oo ((01:2-1

+ c z 2 (ll)

V=p+1 ((Q):2""

C 2”: 2""+Ct’"’1 Si 2‘”

V=—00 V=u+l

02-1111 + 01-11-1271 3 or",

|/\

|/\

l/\

as required. U

 

25

Lemma 1.2.8 Another well-known fact is the following: ifT E CZO(E) O WBP,
then T1 E Fgm.

Proof. To show this, it is enough, by the smooth atomic decomposition of
F?1([14]) and duality, to show [(T1,aq)] S ClQll/z, for aQ a smooth atom for Q
i.e. aQ E D, suppaQ g 3Q, faq 2 0, [D'VaQI S quQl’l/z‘th For this, choose
a function 17 E 5 satisfying n(x) = 1 if x E 5Qoo and n(x) = 0 if x E 7Qoo, and let
nQ(x) = [QI‘1/2n(%5—)Q), as usual. Then by definition (see [8])

(T1100) = (WV/2117762104?)‘l‘<1 — lQll’zflQITiaol-

By WBP, I<Tno,ao>I s 0. Also 11.- e 50, IT‘aQ(:v)| s CIQI‘1’2(1+ €(Q)“Ix —
qu)’""(see e.g. [11], p.175). So

|(1-IQII’2UQ,T*GQ)| S CIQII’2-

 

Chapter 2
The Full T 1 Theorem for
Triebel-Lizorkin Spaces With a = 0

A GOOD NOTATION SHOULD BE UNAMBIGUOUS,
PREGNANT, EASY TO REMEMBER;

IT SHOULD AVOID HARMFUL SECOND MEANING, AND
THE ORDER AND CONNECTION OF SIGNS SHOULD

SUGGEST THE ORDER AND CONNECTION OF THINGS.

— GEORGE POLYA, HOW TO SOLVE IT (1951)

2.1 Introduction.

We follow the method of [8] in the proof of Theorem 1.1.5. We use paraproducts
to obtain the full result from the reduced results in [11], [17], and [40]. We use the
characterization of the function space F29 in terms of its coefficients with respect to
the expansion in terms of either the (,0-transform or Meyer’s wavelets (see [14], [29],
or [16]) to reduce the ng-boundedness of the paraproduct to the boundedness of a
certain matrix on the sequence space f1?" associated to Fig”. We see that this matrix
factors as a product of a diagonal matrix and a matrix which just fails to satisfy the
almost diagonality criterion (see [14] or [16]) for fgq. We see that the second of these

matrix factors maps )2?" into another space (e.g., ff” for (a), (b) and (d) in Theorem

26

 

27

1.1.5), while the diagonal factor maps from this space into fgq.
We show the main theorem in Section 2.3 which follows from the result of bound—
edness of paraproducts. To prove the last, we need some lemmas which are proved

in Section 2.2. The results of this chapter are based on [44].

2.2 Key Lemmas.
Let f = 2P stp. Then, by (1.45),

1W) = Z<g.soo>IQI-%<Zsoop.<bo>wo
Q

P

ZlgmollQl’i (Z(¢P,¢Q)SP) the-

Q P

As noted in the introduction, the matrix (g, (pq)|Q]‘i((bp, (DQ) associated to 119 fac-

tors trivially. The next lemma deals with the diagonal factor.

Lemma 2.2.1 Let c = {eq}Q be a sequence and let Tc be the diagonal operator taking
a sequence 3 = {SQ}Q to Tcs, where (TCs)Q = [Ql‘icqu.

(a) For (1,5 E R, 0 < q2 S (11 S +00 and q2 S p S +00, Tc : fig“ —+ fl“+5)"'2 is
~5, g; ’ . .
bounded if and only ifc E foo(q2 ) 92, and [[c[[ ”$21,qu ZS equivalent to the operator

foo

norm [[Tclljgu _)jp(a+¢),q2.

(b) For 0,5 E R, S E (0,1) and 0 < p,q S +00, TC : fa/‘Z/fi —> [50+th is bounded

p

. . '5,q/(l-fi) ~ . .
if and only zfc E fp/(l-fi) , and [[clljDIHLIfi—f) ~ [ITCIIIS/Z’fi-rfl°+"’q’

Remark. When p = +00, part (b) follows from part(a) with q1 = B and q2 = q.

Proof. First note that since Tla) : 3Q ——> [Ql‘fisQ satisfies [|T(°)s][qu = ”Sllfgqa it
is sufficient to treat the case where a = 0 = 5.
'OI(%)Iq2

' C" ’2 . . '0
Proof of (a) if part. Suppose c E 00 . We first cla1m that To . ftp"1 —->
f2?” is bounded. For the proof, let .9 E 153291, and let tQ = |Q]i"g2z[3Q[q2 and 7Q =

 

28

IQI1'"1"ICQI"1- Then

1 O

”tllj"'i"2‘ _—_ /(Z(IQI‘1|Q[%—3§|8qu,xq)%)3""
1 Q

[(2 (|Q|—%ISQ|XQ)QI)H"q2_ — [[3 “c120"l
Q

and
011-92)
- 1-22 (43—) "1
“7” ”1%" : sup (IPI 1[112:(|Q"|Q|2 ’1’ICQ|""XQ)ql "2)
foo PQCP
(21):.” 1/(%)'
=W(/z W )
PQCP
= ”cl“ “1,,
Thus

IITsII11... — f2(IQI-1IQI-1IcoIIsono)"
Q

2 IQI1-11IcoI11IsoI11 = Z Itollwl
Q

Q

s lltlljog - 111,211

”11 |"" on |"|C|" 0,21,.”

by Theorem 1.2.2 and the identities above.
We remark that for the case q1 = +00 , (%)’ = 1 and so this means c E £32. The

proof of this case is as above, with the 8‘“ norm replaced by the supremum as usual.

Second, we claim that TC : f§g1 —-> f2? is bounded. Let s E 1531. Then

q_l2_
lchslljgoo = sup (WI-1] 2(IQI-1IQI2IcoIIsono)")

PQCP

= s1;p(|P|“ Z (lQlii‘gilselq’) (IQI‘Ih'tIcoI1))"

QEP

29

32.

Q1
_ -11
3 Ship |P| ‘(ZIQP ’lSQI“)

ogP
mgr %
1.. 11 '22. El '.
(Z IQI (111)1ICQI‘11’“)
C29"
5 ||3||jggl -ch,...%y..
For Q2 < p < +00, by the claims and an interpolation theorem (here Proposition

8.1 and Theorem 8.2 in [14] are sufficient), TC : ff“ —-) fl??? is bounded.

Proof of ('b) “2' ” part: By the remark above, we only need to consider 0 < p < +00.
Let 0 < q < +00 and suppose c E fffffl‘gf”. Let s 6 fig". Then

Muir}... = / (2(IQI‘llcqllsolxo)")q
Q
= f (E (ml-111m)" (IQI‘1ICQIXQ)")"
Q
5.2

q

|/\

[(EQ: (|Q|‘%|3QIXQ)§)
x (Z (|QI"%lcqlxq)Ugfi)(l—fi)§

Q

s / (;(|Q|’1|1leo)")"
x / (2(IQI-1IcQIxQ)”"1’)" 1""

Q

B

||5||"}0,q/f3 ' ||C||;0,q/(l-fi) '
10/5 p/(l-fi)

When q = +00, a similar argument as above gives us that if c E "Effhm, then

Tc : "Eff, -—> fl?” is bounded and the operator norm is less than or equal to ”all f'o/o? 13f
P 1-

We postpone the “only if” part of this lemma to Section 5.1, since it is not needed

for the proof of Theorems 1.1.5, 4.2.1 and 4.2.2. B

30

Corollary 2. 2.2 Suppose 0 < p S +00, 0 < q2 S ql S +00 and p S ql. If

€’(%2L)' ' ' a 5 . .
c E f;” qz ,then Tc : fg‘“ —> f; + M2 23 bounded wzth operator norm bounded by a

 

 

multzple of ”c

ft (—L)’ 92

foo

Proof. As above, we can assume a = 5 = 0. Suppose first that 1 < p, q}, (12 <

+00. Then our assumption gives q; S q; and q; S 19’. Note that (%)’q1 = (%)’q2.
l

.0.(3i*)’qi . . . ,
Hence 6 E foo ‘11 , so by Lemma 2.2.1 (a), Tc : f3?2 —-> f3”. By duality (see Remark

5.11 in [14]), we obtain the result in this case, since clearly T: 2 T5 where E = {Eq}Q.

Now suppose min(p,q1,q2) S 1. Pick 7' > 0 such that §,q7‘,9,% > 1. (If a =
+00, a/r lS interpreted as +00). Let c E 50%),” and s 6 f3“. Let t = {tQ}Q
and '1: {7g}Q where tQ = |Q|5‘5|sQ|" and 7Q = IQI5‘5ICQI’. Then a calculation

shows that ||t||% off‘" —— —||s||f- =||c|| jug) ,2. Let [:2 {hQ}Q where

                     

;0 0°(-1)’ qzlr

hQ- —- |Q|5 5|(T: s)Q|— — |(T, t)Q|. Then by the case previously treated

                 

 

unsung” -—-

                 

< C, ”All“: 21%)! 92/,- '

jp/r fP/r

:: (7'II3IIj391 'IICde:;(%:)’¢2’
as required. D

We remark that Lemma 2.2.1 (a) and Corollary 2.2.2 are stated in more generality
than we will actually need. In fact we only use the cases q; = +00 in Corollary 2.2.2

and q2 = 1 in Lemma 2.2.1 (a):

(2.1) If c 6 f2: then Tc : f3“) -—> 15:", for 0 < p,q S +00.
(2.2) If c 6 fig, then Tc : ff" -—> f2” forl S p,q S +00.

We now turn to the other factor of the matrix associated to the paraproduct.

Lemma 2.2.3 Let (I) E 8 satisfy f<I> = 1. Define (DQ as usual, i.e. ‘I>Q(:c) =
IQ ’5<I>(5—-—Q ((Q) ). Let gqp = (WADQ) for all dyadic P and Q. Let G = {ng}Q,p.

31

(a) For 0 < p < +00, G: f3? —-) f§°° is bounded.

(b) For a < 0, 0 < p S +00, and 0 < q S +00, G is almost diagonal, hence

bounded on ff”.

Proof of (a). Lets 6 £92. Then f = 2,3310sz E H" and ”flal S C||s||f32, by
Theorem 1.2.1. (Here we identify Hp with L” as usual when 1 < p < +00.) Also,

(GS)Q = ZQQPSP = 2 ($10,262)” = <ZSP¢Pa¢Q>
P

P P

= (12%) = IQI%f*(i’z/($Q)a

~

where [(Q) = 2"", <I>(:r) = <l>(—-:r), and <l>,,(a:) = 2""<l>(2":1:), as usual. So
sup lei-moan s m),
xEQ

where f“ is the non—tangential maximal function of f with respect to (l), since Ix —
mql S C€(Q) = C - 2‘” for x E Q. (The aperture of the non-tangential region must

be taken sufficiently large). Thus
IIGSngoo = IISIEIEIQITKGSMHILP

3 IIf‘IILp S C'Ilflal S CIISIIjgz,

by the boundedness of the non-tangential maximal function from H p to LP (see [35]

p.91).

Proof of (b). For [(P) S [(Q), by Lemma 8.1 in [14] p.150, we have, since
I x7¢p(:c)d:c = 0 for all 7,
6(a))“ ( Ira — my“ (“my"“vw‘”
,<I> g C (— 1 + —— _
M” Q” 4P) ((6?) m2)
for some 6 > 0, where C depends on J only.

For ((62) < [(P), by Lemma B.2 in [14] p.152, we obtain

n/2
III/)PflQH —<- C (1 +£(Pl—1I5EQ - well—re (Ci-$9

= c<i—2§>“<1+——'wz(;¢'>"“<——>

32

So choosing 5 = —2a, we obtain the result. [I]

__ .5,» _ -2(1-m.q/(1—6)
Remark: U = U5>0 Fn/g — U0<B<1 Iva-fl)

By Theorem 1.2.4 (a), U 2 U0<o<1 F;(l—fi).q/(1—B)

”(143) . On the other hand if f E U
then there exists 6 > 0 such that f 6 F373;). By Theorem 1.2.4 (b), there exists

0 < H < 1 such that 5 > n(1 -— fl)“, and f E F‘EU-mo/(l-fi)

p/(l-fi) D

Lemma 2.2.4 Let <I> E S satisfy f<I> = 1, and let an = ($13,904)) for all dyadic Q
and P. Let A = {aQP}Q,p. For 1 < p S +00, A : ff,” —> fl,” is bounded.

Proof. This follows by duality (using Theorem 1.2.2 for the case p = +00) from
Lemma 2.2.3(a), since (,0 and 1b are interchangeable by definition. Because of its

intrinsic interest, we also include the following proof for 1 < p < +00. Let s E fgl.

Then, by Theorem 1.2.1,
z ||{<Z SM}
jgz P

IIASIIjg2 = {Zi‘DPa‘PleP}

P

 

 

 

 

'02
fp

22

ZSPQP

P

 

 

 

 

Lp
It is easy to see that |<I>p| S C(M(|P|‘15XP)°)%, for 0 < a < 1, where M is the

Hardy-Littlewood maximal function. So

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ZSPQP S ZISP‘I’PI
P LP P L?
1 a l a E
_<_ 0 (Z (M (IPI'5|8p|xp)) )
P LP/0
_1.
s 0 DH zlsplx. =CII3IInga
P LP
by Theorem 1.2.5. I]

Also there is a dual result to Lemma 2.2.3(b): If 0 < p S +00, 0 < q S +00 and
a > J — n, where J = n/ min(l, p, q), as usual, then A is almost diagonal, hence is

bounded on f5”. Since we don’t explicitly use this result, we omit the proof.

33
2.3 Main Theorems: Boundedness of Paraproduct
and Calderon-Zygmund Operators.
Recall that for g 6 F350, the paraproduct operator Hg is defined by (1.45).

Theorem 2.3.1 (’o)1f0 < q s 2, 0 < p < 00 and g 6 F300, then 1191 = g, 1131 =
0, Hg 6 CZO(1)fl WBP, and

 

”fluorite s Cllgllrgg - “into, ;
for all f e ng.
(b) If2 < q S +00, 0 < p < +00 andg E U = Ubon/‘go then H91 29, H31 2
0, Hg 6 020(1) n WBP, and t

IIHg(f)||p;3v S Cllgllpgf; IIfIIfi'gqa

for all f E I?" and for all (5 > 0.

Remarks: (i) Proposition 5.1.1 below gives the sharpness of Theorem 2.3.1(a).
(ii) Since 15:2 m L” for 1 < p < +00 and F302 z BMO, the case q = 2 in part (a)
yields Theorem 1.1.5.

Proof of Theorem 2.3.1. For (a), note that F3: —i> F30”. For (b), from
the definitions it is easy to obtain f3)? —> fig”. Then by Theorem 1.2.1, we obtain
F5}? -+ F30”. Hence by Lemma 1.2.7, 119 E CZO(1)fl WBP, H91 2 g and 1131 = 0.
So we only need to show that Hg is bounded.

Let f 6 Foo. Then, by the go—transform identity, f = E (f, (pp ibp. Thus,
p P

applying Hg, we have

Ho(f) = Z<g,soo>IQI-%<f,<1>o>to
Q

Sumner-‘2' Z<Ltoe><te¢o>to
Q

P

34

Let "IQ = <g,eo>, ’7 = we, on = we), and e = {spite Then, by Theorem
1.2.1, IIfIIp'gq a: IISIIqu' So again by Theorem 1.2.1,

IIHgUlllp'gq s CllholQl’iZSp<¢p,<I>o)}||f-3q

P

cunaeuf-ge

where G = {($P,¢Q)}Q’p.
For (a), we have, by Theorem 1.2.4 (a), Lemma 2.2.3 (3.) and (2.1),

.0 t .02 G .0 T .
qu —> fp —) fpoo 4‘) qu

provided 7 E fgg. By Theorem 1.2.1, this happens exactly when 9 6 F331. So

||H9(f)||p'gq S Cllgllpgg ' IIfIIp'fi‘I-
For (b), by Theorem 1.2.4 (b), Lemma 2.2.3 (b) and Lemma 2.2.1 (b), we have,

forany0<fl<1,

- t' --2(l-B).q/fi G '—E(1‘5)tQ/B T -
fgqfifP/E _>fp/l; Aft?)

-2 1— , 1—
provided 7 E 3:13))“ 3). But by Theorems 1.2.1 and 1.2.4,
IIVII -p(1-fi).q/(1—e) *5 Ilgll .g(1—o),q/(1_e) S CIIgIIFeoo
fp/(l—fi) Fp/(I—fi) n/6
ifd > n(1— fl)/p. So

“HHUHIF‘O‘? S CII’YII -%(1—a).q/(1—;9) ' ”3”qu
p fp/(l-fi) p

s Client-3,? - “intege-

Note that the adjoint of H}, is given by

(2.3) H7.(f) = Z <h.eo>IQI-‘/2<f,eo><1>o
Q

35

Theorem 2.3.2 (o) If 2 S q S +00, 1 < p S +00 and h E Ffoq', then th =
h, 117,1: 0, Hh, H)“, E CZO(1)fl WBP, and

Ilnzunltge s Cllhlltgg' - lune...

for all f e if].
('b)SupposelSq<2,1<p<+ooor1<q<2, p=+ooandh€U=
U5>0F,f'/‘j,°. Then 11,,1 = h, 11;;1 = 0, 11h, n; e CZO(1)fl WBP and

||H2(f)||1egq S Cllhllpggy ' ||f||pge,

for all f E 13;?" and for all 5 > 0.

Proof. By Theorem 2.3.1 and duality, these results follow immediately. CI

Note again that part (a) of Theorem 1.1.5 is the David-Journé result, which is
included there for comparison. It demonstrates a particular role played by the index

q = 2, which we will also see below. Note that case ((1) includes H” % F192 for

_"_

n+1
. oq

For q = 2, we have P,” as H1 and F302 z 8M0 , and these parts of (d) and (f)

< p S 1. Also we will see (Proposition 5.1.2 below) that the theorem is sharp for

follow from classical results and the David-Journé result (see, e.g. [7], p.29, Remark

3).

Proof of Theorem 1.1.5 Let g = T1, h 2 T1 and S = T — Hg — I12. By
Theorem 2.3.1, 5' E 020(5) 0 WBP, 51 = 0 and 3‘1 2 0. By Theorem 1.1.3, 5' is
bounded on F59. So it suffices to prove Hg and H}; are F199 bounded.

(a) For q = 2, H9 and H}; are bounded on F32 since 9, h 6 F302, by Theorems 2.3.1
(a) and 2.3.2 (a).

(b) If T1 = g 6 F2: for l S q < 2 then 119 is bounded on 13:)", by Theorem 2.3.1
(a). If T1 = h E U then, by Theorem 2.3.2 (b), 2 is bounded on FIE”.

(c) This follows from (b) by duality, or directly by Theorem 2.3.1 (b) and Theorem
2.3.2 (a).

36

(d) If T1 = g 6 F23, then, by Theorem 2.3.1 (a), 119 is bounded on if" and
117,: 0 since h = 0.
(e) If T1 = g E U, then, by Theorem 2.3.1 (b), I19 is bounded on 13;)", and I17, = O.
(f) This follows from the p = 1 case of (b) by duality, or directly from Theorem
2.3.2 (a).
(g) This follows from (6) when p = 1 by duality, or directly by theorem 2.3.2 (b).
Cl

Chapter 3
Paraproducts and the Full T 1

Theorem for LE and 11.7}qu

Quite aside from format and style,
mathematical writing is supposed to say something.
Put another way: the number of ideas divided by the

number of pages is supposed to be positive.

—— J.L. KELLEY, Writing Mathematics (1991)

3.1 Introduction.

This chapter is a continuation of Chapter 2. The difference is that we deal with arbi-
trary smoothness index 01 E R. Let T be a generalized Calderon-Zygmund operator
(CZO) (see [8] or Chapter 1 for definition.) David and Journé’s fundamental result
([8]) states that T is LP-bounded for 1 < p < +00 if and only if T1, T’l E BMO
and T E WBP (that is, T has the weak boundedness property). In [17] and [40], a
generalization of this result to other function spaces, including Hardy, Sobolev, and,
more generally, Triebel—Lizorkin spaces, was obtained under the assumptions that
T(:r") = 0 and/or T*(:r7) = O for monomials 3:7 up to a certain degree depending on
the space. In Chapter 1, we obtained boundedness of T on Triebel-Lizorkin spaces

with smoothness index 0 under weaker assumptions than T1 = O and T‘l = 0. In

37

38

this chapter we obtain similar results for general smoothness index a. In particular
we include the case of the Sobolev spaces.

Our main theorems, Theorems 1.1.6, 3.3.2, 3.3.3 and 3.3.4 are stated and proved
in Section 3.3. In order to prove these results, we need to prove some key lemmas in

Section 3.2. The work of this chapter is based on [45].

3.2 Key Lemmas.

To define a generalized paraproduct operator of order 7, where 7 is a multi-index, we

need the existence of a function (PM satisfying certain conditions.

Lemma 3.2.1 Let m > 0 be given. Then there exists a real-valued function 0 E
5(R") satisfying (raj) : fxfid(:r)d:r = 6gp for |fi| S m, where 65.0 = 1 iffi = 0 and
5e,o=0 iffl#0o

Proof. Select a real-valued h E S satisfying f h(a:)d:r = 1, and let
_1)7
g,(:1:)= LTD'VMJI).
Then go 6 8 satisfies fg0(a:)dx = 1. Now for |fl| = 1, let C3 = fxfig0(;r.), and set
91(55) = 90 — 2: WWW)-
|fl3|=1
Then fg1(:r)d:r = 1 and fmfig1(a:)da: = 0 for lfll S 1, fl aé 0. Continuing this way

(for |fl| = 2, let Cg = fxfig1(a:)d:r, and let g2(.1:) = g1(:r) — Zlfilfl eggg(:c), etc.), we
obtain that g,- has the properties required of 0 if j 2 m. D

Fix a _>_ 0 and 7 a multi—index with |7| S a. By Lemma 3.2.1, there exists 0 E 5
such that f9 = 1 and (3&0) = 0 if 0 < Ifll S a. Let Q”) = t—ibh—IDW. Then,
if lfll S a and ,6 74 7, (r3,<I>(7)) = 0, while (2:7,<I>(”)) = 1. With this (1)”) and for
g 6 Fill”, we define, analogous to the discrete paraproduct operator in [27], the
generalized paraproduct operator I137) of order 7 by,

(3.1) mm) = Z<garo)lQ|“/2"7”"(fa eg’wo
Q

 

I!

39

where (1)8)(2: )2 |Q| l/2<I>(1’)(11,7635?)

Observe that the kernel of II?) is

(3.2) KIe,y)=Z<g,eo>IQI 2 WET—(y ItoIe )
Q

Lemma 3.2.2 If g 6 Fig”, then for any 1 e N, my), (HP): 6 020(1) n WBP,
Hgl)(a:fi) = grip”, for |,B| S I7] and (Hgl))’(:rfi) = O for all fl. Furthermore, the kernel
of H3") satisfies (f) and (g) of Theorem 3.3.3 below.

The proof is similar to the standard case 7 = 0 in [44]; see Lemma 1.2.7 in Section

1.2 for a discussion.

We observe in (3.11) below that the matrix corresponding to 1'1?) can be factored
as a product of two matrices. One of these factors is diagonal and the other just fails
to be almost diagonal for ff". In the following we deal with the boundedness of these

factors.

’ Lemma 3.2.3 For a multi-index 7 and a sequence d = {dq}Q, define the diagonal

matrix operator T57) by
_l_lll
(T§W)S)Q : IQI 2 n SQan

for any sequence 3 = {3Q}Q. For a], a; E R, 0 < ([2 S ql S +00, and 0 < p2 S p; <
+00, ifd E fez—a1+l7|q2(q1/q2) then Tdm:f;1'ql —> fszf’q? is bounded and

(P1/P2)’

IIT§"eII,;ee < CIIdII,.._.,..,..-..I..I.o IIe sumo,

122(91 /P2)

for all s e jgm.

Proof. Without loss of generality, we may assume 011 = (12 = 0. The case p1 = p2

follows from Lemma 2.2.1(a) and Corollary 2.2.2. Now suppose p; < p1. Let s 6 f1?!”1

40

and set 6 = p1 /p2, p = ql/qg. Applying Hblder’s inequality twice, we have

1

IlTl”’sll,-g,e = II (2: (IQI-ueolxo)” (IQI-2-L721Idono)”) 2 no.

Q
22.
_1 91 q‘
s 2(IQI 2Isono)
Q
L
P2
___]1_[ 920' ”p
><(Z (IQI2 ldolxo) )
Q
s “SH/gin -IIdII,g,,..I,
where qu’ = q2(§-;)’ = ,—-‘f‘_";’, and 1925’: P2(§- ;)’ = 5%;- D

We remark that Lemma 3.2.3 is stated in more generality than we will actually

need. In fact we only use the following cases:

(3.3) If d E f02_a‘+hl’ ,q2 then Ty): .a‘°° —> fgm is bounded, for 0 < (12 S +00

P2(P1/P2)’ P1

and0<p2<p1<+oo.

(3.4) Ifd E fag—alflllq‘then Th): fal‘“ ——> fgzl is bounded for 1 S ql S +00

P2(P1/P2)'

and0<p2Sp1<+oo.

(3.5) For 0 < 9 <1 ide le)7|(41-n(01)— 0—)/p 9/(1— 9) then T(7):f:/—6n(1—6)/p.q/0 _) ff’q

is bounded.

Remarks (1). For p1 = +00, the lemma is still true. The proof is analogous to
the proof of duality theorem in [14] pp.76—77.

(2). In some cases these results are sharp. For example, the case (3.5) is sharp
by Lemma 2.2.1(b). Also (3.4) is sharp when l < q2 < +00, 1 < p2 < p1 < +00
and p1 = png, which follows from the same proof of Lemma 2.2.1 (a); and (3.3) is
sharp when either (1) p1 2 p2 and q; < p; < +00, or (2) p1 = +00, or (3) q; + 00, or
(4)1 < qz < +00, 1< p2 < p1 < +00 and ([2 =2 p2(p1—1)/(p1——p2). The case (1)

41

and (3) follow from Lemma 2.2.1 (a) and (b), respectively. For the case (2), define a
sequence 3 by sQ= IQII/2 for all dyadic Q. Then IIS sllf'ooo— — 1 and

IlTi”’s||,-2;2- — “(ZUQI‘I “I/"lQll/I’ldolxo)‘; )l/rlIm— — Ildll I...
Q

Also the case (4) is a special one of (3.3). E]

In the following, we regard a matrix D = {de}Q,p as an operator acting on a
sequence {Sp} p, yielding a sequence {2P desp}Q. One criterion for boundedness
for such a matrix on ff” is “almost diagonality,” defined in [14] pp.53—-55 or in Chapter
1.

Lemma 3.2.4 Leta > 0, 7 a multi-index with I7I S a and (1)”) be the function as
in {3.1). For each dyadic cube P and Q, let g8}; = (ibp,<1>8)), for w as in (1.32). Let
GI"): {98)}QJ2. Then

 

(a) for 0 < p < +00, GIT) : fIflJ —) fIAII’OO is bounded;

(b) for fl < 0, 0 < p < +00, and 0 < q S +00, Ch) is almost diagonal and hence

bounded on ffI'I'YI’q.

Proof of (a). Let s 6 film and let f: ZPsplPI I“YI/"(D“’z/))p :2!) 5120711”).

Then, since (072/01: is a smooth molecule for P,

1/2
_L_11_l 2
llfI|H2SCll(Z(IPI2 "ISPIXP)>- IIo=cII3II,,..,..

P

Let 5(1) 2 P(—a:), where 0 is chosen such that (PM: IN ——,D”6I for (PM as in (3.1).
By integration by parts, f0(x)d:1: = f$7<I>(7)(a:)d:r = 1.
Now if €(Q) = 2"”, then

 

 

1 I’ll Ill
,3 IQI . («lemme

(GI'YISM = Z(¢P,¢8I>3P=Z —
P

P

: CIQIrIZSPthtQ) = CIQII%‘<f,0o>

P

1 111 ~
= C|Q|2+"f*9u($o)2

42

Let f“ be the non-tangential maximal function of f with respect to ii. Note that
Ia: - xQI S C€(Q) = C2‘” for a: E Q. (The aperture of the non-tangential region
must be taken sufficiently large). Thus,

528 IQI‘1/2"”'/"I(GI“”S)QI 3 He)

 

and
(7) _ 4-111 (7)
”G Slljhlm — || SUP IQI 2 " |(G SlQIIILP
2 Q=r€Q I
s CIIrIIo s Cllfllm s Cllsllf-pm.
Proof of (b). Since ka<I>8I(:c)d:c = 0 for IkI < I7I, by Lemma B1 in [14], when
[(C2) < ((1’), I
p I2'-+|’7| _ ""
(3.6) |<Ibp,<1>S’I>IS CI (2%) (1+€(P)"l:vo —xp|) *

2 CA (@)H+IWI (1+ qu _xPI)-A (“_QZ)Z¥Q ,
[(1’) [(1’) [(P)
where /\ > J = n/ min(1,p, q). By Lemma B1 in [14], when €(P) S [(Q),
g B+Ivl _ -J—e (1 P 23—‘+J-n
(3.7) News)» 3 CI (2%) (1+ ————'””‘;(Q;’P') (IQ—I) ,
since fakwp(:r)da: = 0 for all k.
By (3.6) and (3.7), CI") is almost diagonal and hence by Theorem 1.2.3, GI”) is

bounded on fpfi‘LIII’q. C]

Lemma 3.2.5 Fix a > 0. Let 7 be a multi-index with I7I S a and let (PM as
in (3.1). Let AI”) be the matrix {(le,ng)}Q,p, for it“) as in {1.32) and fori =
1, 2,---, 2" — 1. Let 0 < p,q S +00 and J = n/min(1,p,q). Iffl > J—n then AIII

is almost diagonal and so bounded on fig—III”.

Proof. Fix i E {1, 2,---, 2" —1}.When €(P) S [(Q), let [(P) = 2““ and
((Q) = 2‘” (i.e. 1/ S [1). Since IerIZI(x)d:c = 0 if IkI < |7I, applying Lemma B.1 in

 

43
[14], we have
I<<I> ”we; M .<_ CI2-<~-"><'i'+"/2> (1 + 2"Ieo — eeIr‘
0 (@)FM (1 + M) a (@Y/W,
If(P) [(C2) 4Q)
Soiffl>J—nand)\>J,then

(328) |<<I>I3II¢SIH S 0 (II—29M“ (1+ I—quié—I’f—I) +8 GIL?)

When €(Q) < €(P), let ((Q) = 2‘“ and 3(P) = 2‘” (i.e. 1/ < u). Since

I :cI‘ch(:I:)d:r = 0 for all k, applying Lemma B.1 in [14] again, we have

By (3.8) and (3.9), we obtain the result. I]

3.3 Main Theorems.

First we give some results for paraproduct operators of order 7.

Theorem 3.3.1 Leta Z 0 and let 7 be a multi-index with M S a. Let (PM be as in
(3.1), and defined 11;") by (3.1). Then 11;”) e CZO([oz]+1)fl WBP, (ng”))*(ek) = o
for all k, Hymn”) = g, Hymn") = 0 for IkI S M and k # 7, and III,” is bounded on
F59 if one of following cases holds

(a) 0 < q S +00, 0 < p < +00, (1 — n/p < |7I < [a], and g E Pf/q(a_hl),'

(b) (1 ¢ Z, O < q S +00, 0 < p < +00 , a—n/p < M = [a], andg E 17:70., where

01"” = a -— [a];
(c) aENU{0}, 0<qS2, 0<p<+oo, I7I=a, andgEng;
0+"(1'9l/P Q/(1'9)

('d)aENU{0}, 2<qS+OO, 0<P<+Oot I7I=atandg€1F—/(10)
forsome0€(0,1).

44
Furthermore, for (a), (b), and (c) we have following estimate
(7) . . , .

”Hg (f)l|p,?q S Cllgllrgfikm, ”film?“

for all f 6 P5”. For (d) we obtain
IIHIIIUNIFy S CIIQIIFSIII'IIEWPIq/U-9) ' llfllrng
for all f 6 F52.
Proof. In all cases, our assumption on g implies g E FIJI“. By the definition of

2 I137), the choice of (PM, w, 90 and (3.1), we have II?) E CZO([a] + 1) fl WBP. Also,
(IISII)"(:1:I°) = 0, since

 

Q|_1_111
(3.10) (HIM =<Z g,eo>I 2 among),
Q

and f$k¢Q($ )dat- — 0 for all k. By the choice of (PM, we have ( '24)?) = IQII/“hI/n
6km for IkI S I7I, hence IIIII(:L”) = g and III,II(:I:Ic ) = 0 if IkI S I7I and k # 7. To
complete the proof, we need to show that III,” is bounded. Let f 6 F5”. Then, by
(1.32),

f : Z<fa$0p>wp : Z SPIPP:
P

P
where sp = (f, (pp). Then, by Theorem 1.2.1, IIfIngq z IISIIjg‘I-

Applying III”) to f, we obtain

(3.11) HIM) = ZIg.eo>IQI-%-I%I (Z<¢p,¢8’>sp) Ibo-

Q P

Let do = (QISOQII d = {da}QI 981)» - (I/IPI‘I’SIIII and GIII— — {985w Them by
Theorem 1.2.1, ”11m f)” F3. ~ ”7;,”0 rely-1,...

For (a) and (b), choose p1 > p with a — n/p = |7| — n/pl. Then, by Theorem
1.2.4 (b), Lemma 3.2.4 (a) and (3.3),

T(7)

f'aq__) ‘ f|7|I2GI7I f|7|I00 faq
p

 

 

45

provided d E fag n/(a_ and

I7I)’

(7) (7) . . . .
IITI G sue .<. CIIdII,;;,a_,,, Ilsllfng
for all s E f,‘,"’. Hence

IIHW )Ilfigq <CIIgIII-Ie; llfllegq,

/(0- -|7D

since 9 6 Flu/1047').
For (c), by Theorem 1.2.4 (a), Lemma 3.2.4 (a) and (3.3),

T(‘7)

fr —‘> f5? 3’1 fswLI is
if d e fgoq, and
IIH§”(f)lle;2 e IITr’GWeIII-II s CH‘IIIISJ - Melt-:2
S CIIQIIFg? - llfllrng
by Theorem 1.2.1, for all f 6 P13".

For ((1), by Theorem 1.2.4 (b), Lemma 3.2.4 (b) and (3.5),

'0, i 'a—n —0 , 0 CI”) 0 71—9 0 Td a
qu __) fp/0 (1 )/P<1/ __) fp/9 (1" )/PI9/ __))fp q

is bounded ifd e fZ/IfII, elm/‘1'") ,and

IIH§”I(f)IIIe;2 e2 IITI"G<1>2II,-;IsCIIdII,;,III,;IIII.IIII—II-IIeII,-;,~I
s cIIgIII;,I;III9).III...III-II - llfllpgeI

by Theorem 1.2.1, for all f 6 P50. D

Remark. If we define the paraproducts via Meyer’s wavelets, i.e.

2"—1
(3.12) Hm =ZZ(89,II>I IQI-2-1<f,<1>8)>¢g’,
i=1

then Theorem 3.3.1 still holds replacing III") with HI?) by the same proof.

We now state Theorems 3.3.2, 3.3.3, and 3.3.4 which, along with Theorem 1.1.6

stated in Section 1.1, are some of our main results.

46

For the case min(p,q) Z 1 we obtain Theorem 1.1.6 (0: > 0) and Theorem 3.3.2
(o < 0). Recall that q’ is the conjugate index to q, i.e. q’ = q/(q- 1) for 1 S q 3 +00.
Also for any x 6 1R, let [x] be the integer part of x, x" = a: — [.13] and IL‘+ = max(0, :12)

Note that the meaning of T(:r”’) and T13”) is discussed in [17] and [40].

Theorem 3.3.2 Let a < 0. Assume that T : S —) 8’ satisfies
(a) T E WBP;
('b) T 6 020(6) where 0 < 6 <1;
('c) T" E CZO([—a] + ,0) where (—a)* < p <1;
and
(d) T*(:1:7) = 0 for '7' S -a — n + n/p.
Then T extends to a bounded linear operator on FPO" if one offollowing cases holds:
('e) a¢Z, ISq.<_+oo, 1<p<+00, and
“(IE - $Q)TIT¢8))}Q E F::/l]a+|7|)a
when —a — n + n/p < I7I < —a, for all i = 1,2, ...,2" — 1;
(f) aEZ, 2Sq£+oo, 1<p<+oo, and
{<(33 — $Qll Tll’q) ”Q E LII/((7, Mm)
when —a — n + n/p < IVI S —a, for alli = 1,2, ...,2" — 1;
('g)aEZ,lgq<2,1<p<+oo,
{<(iv — $62)”, Tibglllo E f_n/’(a+|.,l),
when —a — n + n/p < M < —a, and
{(($ - 1m)7 Tth) )}E 14—0) =U fn/6 M600

6>0

when |7| : —a, for all i = 1,2,...,2” — 1.

47

For the case min(p, q) < l, we obtain Theorem 3.3.3 (0 Z 0) and Theorem 3.3.4
(01 < 0).

Theorem 3.3.3 Let a Z O and min(p, q) < 1. Let J = n/ min(p, q). Assume that
T : S —> 8’ satisfies

(a) T E WBP;

('6) T E CZO([a]+6), where a" < <5 <1 if J—n—a < 0 and max(a"‘,J*) < 5 <1
fiJ—n—azm

('c) T“ E CZO([J — n] + p), where J" < p <1;
(d) T(a:") = 0 for all M S a — g;
(e) T*(:z:'7) = O for all Iryl S % — a — n;

(f) ngDlK(x,y)-DQDYK($,2)I S CIy-zlplx-y|’("+"’+"'+”’ for 2|y-zl S Ix-yl
if lvl S min([Jl - n, [0]) and lfi + 7| = [J] - n;

and

Ig) IDé’DIKIm) — D€DIKIz,yII s Clo: - zlplrc — III-("HM”) for 2|:v — zI 3
la: -- yI, |7l=l01l and lfll = ([J] - n - [a])+-

Let u(m,I) = U5>1f.:;3%°°. Then T extends to a bounded linear operator on F5“? if

one offollowing cases holds:

(h) 0<q32 andfor all i6{1,2,--~,2"—1},

{<T(($ - $62)”), $330 6 f:l(a-Ivl)’

when a — n/p < [7] S a and
{<(a: — xQIWIS’IIQ e uIIII, J — n/p)

when n/p—n—a<l7|SJ—n—a;

48
(i) 2<qS+oo, 0<p<1andforall i€{1,2,~~,2"—1},

IITIIsc — martigbn e u(|7|.0)
when a — n/p < |7| S a.
We will show in Corollary 5.2.5 that condition (e) in Theorem 3.3.3 in necessary if

a > 0 and 0 < q < 1. Also under assumptions (h) for 0 < q S 1 or (i), the condition

(e) is necessary.

Theorem 3.3.4 Let a < 0, min(p, q) < 1 and u(m,l) = U5>1f.:;';5’°°. Assume that
T : S —> 5' satisfies

(a) T E WBP;

('b) T e 0Z0(6) where 0 < 6 <1;

(c) T“ E CZO([J — n — a] +p) where (J — a)"‘ < p <1;

(4) T193"): 0 for l'Il S 71/17 - n - a;
and

(e) for all i = 1,2, ...,2" — 1,

{<(w — 1762)”, Tig’IIQ e U(I7l, J — n/p)
for n/p—n—a< |7| <J—n—oz.
Then T extends to a bounded linear operator on 3:10.
The proof of all of these theorems depends on the following lemma, which explains

the significance of the assumptions (1.28 — 1.30) in Theorem 1.1.6, and the similar

conditions in the other theorems.

49

Lemma 3.3.5 Suppose on, (12, )8], ,82 > 0, T E WBP, T E CZO(fll), and T“ E
CZO(flg). For 0 S I7I S (11, assume

2% 1
(3.13) —2 ZITI (:1: — eQ) twang) e Fgglm.
For 0 S I7I S 02, assume
2fl_
i=1 Q<

For (7| S a = max(a1,ag), define (PM as in {3.1). Let
(3.15) s = T— 2 fig) .. 2 (i153).

OSI'YISGI OSI'YISOz
Then S E CZO(fiQflWBP, 5" E CZO(flflflWBP, S(:v") = 0 for 0 S [7| S
ahand 5*(x'7) = 0 for 0 S (’7) S (12. Also, if the kernel ofT satisfies (f) or (g) of
Theorem 3.3.3, then so does the kernel of 5'.

Proof. Since g, E 1313"”, by Lemma 3.2.2 we have 115:) E CZO(flQflWBP.
Similarly, we have (mgr E CZO(fll) fl WBP. Hence this holds for S as well. Also
by Lemma 3.2.2, if the kernel of T satisfies (f) or (g) in Theorem 3.3.3, then so does
the kernel of S. For Ikl S [a], we have, by the orthonormality of the {¢g)}Q,j

<T((rc — war). I?) = Igwg’I.

By the definition of (1)”) and fig) for 0 S I7I, lkl S [(11] we have

<fi§1’((:v — $Q)k),¢g)> Ig.,ng’>IQI-‘/2-h'/"<Iw — mar, <98»
= (9’7, $81)“

Also, by Lemma 3.2.2, (fi),?)*((x — $Q)k) = 0 for all k. Thus for 0 S Ikl S [m]

(5((53-9301)Il)g))=<gk,¢g))- Z Ig,,ig)>a,,.=o.

OSIWISIOII

Expanding as" = (:1: — mg + xq)k, this implies (5(xk), it?) = O for all Q, j, which, by
(1.38), implies S(:c") = 0

50

A similar argument for 5" gives 5* E CZO(flg)flWBP and S*(a:") = O for
0 S '71 S 02- E]

Proof of Theorem 1.1.6. Define S as in (3.15), with 01 = a, h, = 0 for all 7
and g, as in (3.13). Then the proof of Lemma 3.3.5 show that S satisfies all conditions
of Theorem 1.1. 3, so S is bounded on F “9 To obtain the boundedness of T, we need
to show that Hm) IS bounded on F “q for all (7| S a. Note that ((1) implies g7 = 0 for
)7] S a — n/p, and hence Hg) = O. For a — n/p < M S a, the assumptions (1.28 —
1.30) yield, by Theorem 1.2.1, conditions on g, which, by Theorem 3.3.1, imply the

boundedness of fig) . [:1

Proof of Theorem 3.3.2. By duality and Theorem 1.1.6, we have the results.
[:1

For the remaining theorems, we require a lemma on the adjoint of the paraproduct

operator.

Lemma 3-3-6 Let a > 0, 0 < q < min(p,1I. Suppose h. e U(|7I,n/q — n/p) =
USN/kn“) Eggs“ when n/p — n — a < I7I S n/q - n — 0. Then (fig?) is bounded

.0“?
on Fp .

Proof. Fix 7 such that n/p — n — a < I7I S n/q — n — a. Letc g”) =
(thbgl), C(m) = {an)}Q and A?) = ((4)9), 8))}Q,p for i = 1,2,...,2" — 1. QThus,

2"-1 2"-l

for all f E Ff", f: g 2( f,¢g ))i,bg)= Z Z Sng)’ Hfllpaq’” ~ 2:1— I ||s(‘)||f’;:q

i=1

and, proceeding similarly to (3.8) but with (1133]): we obtain

II (WW n|,,,~ ~ 2 llAf"

For i = 1,2,...,2" — 1, choose p1< min(p,q) and a — n/p = fl — [7| — n/pl, for
5 > a + Ifyl. Then, by Lemmas 3.2.3 and 3.2.5,

 

 

 

Tl’l) '
09 C('_“’)) B— l IIAq 3- Mg ‘ °aq
fp fp1 " —-+ f),1 -—> fp

 

51

ifc("fl) E ff/Tgfz-hl) which gives, by Theorem 1.2.1, that h, E [hf/23623371 Q U([7[, n/q).

C]

Proof of Theorem 3.3.3. Define g7, h7 and S as in Lemma 3.3.5, with al 2 a
and a2 = J — n — a. By Lemma 3.3.5, S satisfies all conditions in Theorem 1.1.4,
hence S is bounded on F13". Also, by (h), (i), (1.38), and Theorem 3.3.1, 11:73, is
bounded on F5“? for each 7 with [7| S a. We are done if q 2 p since h7 = 0 for
[7| S n / p— n — a, by (e), which included (i). To complete the proof, the boundedness

of (fih)

hm)“ whenever n/p — n — a < [7| S J — n — a in (h) follows from Lemma 3.3.6.

C]

Proof of Theorem 3.3.4. Define S as in (3.15), with g7 = O, h1 defined by
(3.14), and 02 = J — n — a. By the proof of Lemma 3.3.5, and Theorem 1.1.2, S is
bounded on Ff”. By (d—e) and Lemma 3.3.6, so are all the (fi[‘{l,,)* for 0 S [7| S

J—n—m D

Chapter 4
Applications to Molecular and

N orming Families

Suppose you want to teach the “cat” concept to a very young child.
Do you explain that a cat is a relatively small,

primarily carnivorous mammal with retractile claws,

a distinctive sonic output, eta? I’ll bet not.

You probably show the kid a lot of different cats,

saying “kitty” each time, until it gets the idea.

To put more generally,

generalizations are best made by abstraction from experience.

Ralph P. Boas, Can We Make Mathematics intelligible? (1981)

 

4. 1 Introduction.

In [15], Frazier and Jawerth considered the following question: What are the most
general families {mQ}Q dyadic for which (4.1) or (4.2) below holds? They introduced
the following terms: Consider a function space X on IR” and the associated sequence
space Y. The family {mQ}Q is called a family of molecules for X if there exists C > 0
such that

(“1 H ZSQmQHX S CHSIIYI for all 8 = {8010 E Y,
Q

52

53

and {mq}Q is a norming family for X if there is a constant C > 0 such that

(42) [[{ffalelQllY S Cllfllx, for all f E X-

It is natural to ask: Why are we interesting in such families? Note that if T is a
continuous linear operator from 8 to S’, then T extends to be a bounded operator on
X if and only if {TwQ}Q is a family of molecules for X, where {wQ}Q is the family in
(1.32). Thus, conditions guaranteeing that (4.1) holds result in a way of analyzing T
in terms of the action of T on the canonical functions {wq}Q. Note that conditions
(4.1) and (4.2) are dual to each other. In this chapter we consider examples for
X = F5": and Y = fgq.

Let us recall the definition of smooth molecules defined by Frazier and Jawerth
in [14]. If K,N E {—1,0,1,2,3,~-}, 5 6 (0,1], 0 > N + n, and Q is dyadic, let us

recall that a function mg is a smooth (N, K, 6, U)—molecu1e for Q if

(4.3) fxlmQ(:c)d$ = 0

if I7I S N,

(4.4) ImQIxII s IQI‘1/2(1+ €(Q)“|:r — reel)“
ifx e IR",

(4-5) [Dlmdfll S |Q|_%_l%l(1+ ((621—va - 330D”

ifx E R" and [7| S K, and

_r_m_e
(4-6) ID”mQ($) — DlmQ(y)| S lQl 2 " "ll“ - yl‘S

x sup (1+ “(2)—1|“? — Z _ “IUD-a

|Z|S|$-y|
if [7| 2 K and :r, y 6 1R". By convention, (4.3) is void if N = —1 and (4.5)—(4.6) are
void if K = —1.
In [14] Theorem 3.5, it is shown that if each mQ satisfies (4.3) — (4.6) with K 2
[a], K + a > a, a > max(J,J — a), and N 2 [J — n — a], then for any sequence
3 = {3Q}Q, condition (4.1) holds for X = Ff" and Y = jig". Also in [11] Theorem

54

3.7, it is shown that if each mQ satisfies (4.3) — (4.6) with N _>_ [a], K 2 [J — n —
a], K + a > J — n — oz, and a > max(J,n + a), then {mQ}Q is a norming family for
F59.

The matrix boundedness results we obtain in Chapters 1 and 2 have applications
to the theory of molecular and norming families for F5" (see [14], [16], or [15]) to
relax condition (4.3). In Section 4.2, we will obtain results, Theorems 4.2.1 and 4.2.2
for F12". We can replace the moment zero condition by something weaker.

Note that in Proposition 5.1.3, we show that Theorem 4.2.1 is sharp in the case
2 S q S +00 and 1 < p S +00 (including LP z F32, 1 < p < +00). This result
is well-known for L2 z F202; see [38], where this result and its connection to the T1
theorem are discussed. Similarly, in Proposition 5.1.4, we show that Theorem 4.2.2
is sharp in the case 0 < q S 2.

In Section 4.3, we deal with the question for more general smooth index a. We
obtain results, Theorems 4.3.1 and 4.3.3. An analogous comment can be made for
F5“ instead of FS". We can get some sharpness results in Propositions 5.2.2 and 5.2.3.

See the remarks below. The results of this chapter are based on [44] as well as [45].

4.2 Applications of Results in Chapter 2.

In this section we plan to describe some results relating to the boundedness of matrices

obtained in Section 2.2 or the proof of boundedness of a paraproduct or its adjoint.

Theorem 4.2.1 Suppose for each dyadic cube Q, mQ is a function satisfying (1.4)
and (4.6) with K = 0. Let CQ 2 me, and set c = {cQ}. Let u = U5>o f3)? . If

(a) 1<pS+oo, 2SqS+oo anchfg’I,
or
('b)1<p<+oo,1Sq<2orp=+oo,1<q<2,anchu,

then {mQ}Q is a family of molecules for FB".

55

Proof of Theorem 4.2.1. Let s 6 f1?" and let f = 2Q Squ, for any family
{mQ}Q satisfying all conditions in the theorem. Then, f = fl + f2, where f1 =
20 sQ(mQ -- [QI‘icQQgfl and f2 = 2Q [Ql‘icQsQQQ where (I) 6 8 with f<I> =1.

Clearly there exists C independent of Q such that (mQ — [Ql'icQ¢Q) is a smooth
molecule for Q up to a universal constant C since cQ = f mq, so by Theorem 3.5 in
[14], [[flllfifiq S Cl|s||j§q. So to complete the proof, we need to show that [|f2||F:q S
CIISIIfgq.

By Theorem 1.2.1,

_.1_
||f2llrgq z <Z|P| 1’CIDSP‘I’PIIOQ>
P

 

 

'0
If

= ZIPI’iCM‘PPIIOQW
P

= IIATc(S)||qu-

0
If

 

 

 

 

Now note that (AT0)“ = TC*A"' = TEA“. But since Ip and 2,1) are interchangeable
by their definitions, we can replace A“ with G. As in the proof of Theorem 2.3.1,
TEG : f3”, —+ £39,, so ATC : f3? —-) ff”. Cl

Remark. Similar reasoning gives a direct analysis of 11’;1 as follows. Now let

co = (h, IpQ) and for f 6 Fg", let sQ = (f, IOQ). Then by Theorem 1.2.1,
||f||figq % ”8”ng and
IIHifllrgq w llZCPIPI'1’2(<I>p,Ioo)SP||;gq
P

z [[ATcsllf'gq.

Then e.g. in the case 2 S q S +00 and 1 < p S +oo, we have (using (2.2), Lemma
2.2.4, and Theorem 1.2.4 (a)) the factorization

0 TC 0 A o ' 0
fl?" —> ft” -—> ff,” -L> fS",

ifc e fgg’, i.e. h e Fgg’.

56

Theorem 4.2.2 Suppose for each dyadic Q, mQ is a function satisfying (4.4), (4.5)
and (4.6), for K 2 [J — n], K+ If > J and a > J, where J : n/min(1,p,q). Let
cQ = me and set c = {eq}Q. Also let u = U3>off/’;o. If

(a) 0<p<+oo, 0<q§2,andcefga,
07'
(b) 0<P<+OO, 2<CIS+OO anchu,

then {mQ}Q is a norming family for FIE”.

Proof of Theorem 4.2.2. Let 77m = [Ql‘I/chéq, where (I) E 8 satisfies f <1) 2
1. Then there exists C independent of Q such that {(mQ — ThQ)/C}Q is a family of
smooth (0, K, II, a)-molecules for ng. By Theorem 3.7 in [14], it is a norming family
for F12". So to complete the proof, we need to show that {High is a norming family,
or equivalently (see [15], Proposition 1.1) to show that {(wp, r’i’zQ)}Q,p is bounded on
qu- Note that (Mafia) = IQI‘l/chWPI‘I’Q) = (TcG)QP-

Exactly as in the proof of Theorem 2.3.1, we obtain “TCGSngq S C ”3” fig, com-

pleting the proof. D

4.3 Applications of Results in Chapter 3.

As in [15], we say that a family of functions {mq}Q is a norming family for F59 if

|l{(f.mQ)}Ql|j;-a s Cllfllpgv

for all f 6 F59. In [14] Theorem 3.7, it is shown that if each mQ satisfies (4.3) - (4.6)
with NZ [(1], K > [J—n—a], K+6> J—n—a, anda >max(J,n+a), then
{mQ}Q is a norming family for Ff“. We would like to relax (4.3) in the case that
(4.3) is not void, i.e. for a 2 0.

57

Theorem 4.3.1 Let a 2 0, 0 < p < +00 and 0 < q S +00. Suppose that {mQ}Q is
afamily offunctions satisfying (4.4) - (4.6) with K 2 [J—n—a], K+I§ > J—n—a,
and a > max(J,n + a), where J = n/ min(1,p, q). Suppose fomQ(x)dx = 0 for
[7| S a — n/p, for all Q. For each dyadic cube Q, let cg) = f(x — xQ)"mQ(x)dx, for
[7| S [a] and ch) = {68)}0' Also let u(l) = Ug>o 575.00. Then {mq}Q is a norming
family for F1?” if one offollowing cases holds:

(a) 0: ¢ Z, ch) 6 f:7(a-l7|) for all a — n/p < [7| S a;

(b) a E Z, 0 < q S 2, ch) 6 f:7(0-|7|) when a — n/p < [7] < a and C(7) E fgoq

when [7| = a;
('c) a E Z, 2 < q S +00, c”) E f:7(a_h|) when oz—n/p < [7| < a and ch) 6 u(oz)

when [7| = a.

The proof of Theorem 4.3.1 is based on the following.

Lemma 4.3.2 Suppose mQ satisfies (4.4) — (4.6) for some K, 6, and a. Let N Z 0
be given. For [7| S N, let (1)”) be as in (3.1), satisfying fxfi<I)(7)(x)dx = 6,9,, for all
[5| S N. Let

(4.7) m0 = Z |Q|-1/2-|r|/"cg’<1>g),
|7|SN

where cg) = f(x — xQ)"mq(x) with 0(7) = {c8)}Q E szl’oo. Then there exists a
constant A > 0, A independent of Q, such that iiiQ/A satisfies (4.3) —- (4.6).

Proof. For each multi-index k with [kl S N,

(a: — xQIkaQIxIdx = In — wo)kmo(fc)div
/ /

— [Ix — ear [2 IQI‘1/2“'”'/”cg”<1>8’(w) d.

|7|SN
:: C(k) Z (387)57]:
|7l.<_N
— C(k) cg) 0

58

This implies that friQ satisfies (4.3). Since cm 6 1513'“ and (1)”) E S, the conditions
(4.4) — (4.6) follow. Cl

Proof of Theorem 4.3.1. Define mg as in (4.7) but with N replaced by [a]. In
all cases (a) - (c), we have ch) 6 fill“. Then by Lemma 4.3.2, for some constant C,
{7725; / C }Q satisfies the sufficient conditions noted above to be a norming family. To
complete the proof, we only need to show that, for [7| S a, {[QI'I/z'IVI/"cg)<1>gl)}q
is a norming family. Writing f = 2,, spwp with [[f][F:q z ”8“qu by Theorem 1.2.1,

this is equivalent to showing that

{Z 3131/”),[Q[_l/2—hl/nc(Q7)¢g)} S Cllsllfgqa
Q qu

P

i.e. that TmGh) is bounded on F59. But this is exactly what is done in the proof of

C

Theorem 3.3.1, under the same conditions on the indices. D

In [14] Theorem 3.5, it is shown that if each mQ satisfies (4.3) — (4.6) with K 2
[a], K + a > a, 0 > max(J,J — a), and N Z [J - n — a], then for any sequence
3 = {Sq}Q, condition (4.1) holds. In the next result, we show that the condition (4.3)

can be relaxed. Obviously this is only meaningful if J — n — a Z 0, so that N Z 0.

Theorem 4.3.3 Suppose that {mq}Q is a family offunctions satisfying (4.4) — {4.6)
with K 2 [a], K+o > (1 undo > max(J,J—a), where J = n/ min(1,p, q). For each
dyadic cube Q and [7| S N, where N = [J — n — a], let cg) = f(x — xQ)7mQ(x)dx
and ch) = {cgfl}q. Let u(m, I) 2 U5), fjfsm’oo. Then {mq}Q is a molecular family
for F1?” if one offollowing cases holds:

(a) a > 0, 0 < p < +oo,0 < q S min(1,p),c(7l = 0 when [7| S n/p — n — a and
ch) 6 u(|7|,J — n/p) when n/p — n — a < [7| S N;

(b) aSOanda¢Z,1<p<+oo,1SqS+oo, ch)=0when|7|Sn/p—n—a,

I

and ch) 6 f::lla+l7|) when n/p — n — a < [7| S -a;

59

('c} a is a non-positive integer, 1 < p < +00, 2 S q S +00, ch) = 0 when

[7| S n/p — n — a and ch) 6 f_a’q' when n/p — n — a < [7| S —a;

-n/(a+l7|)

('d) or is a non-positive integer, 1 < p < +00, 1 S q < 2, ch) = 0 when [7| S

n/p—n—a, ch) 6 f—nfl;+|7|) when n/p—n—a < [7| < —a and 6(7) 6 u(—a) =

'—-a 6,
U¢S>0 fn/5+ °° when [7| 2 —a.

Proof. Note that we can get (b), (c) and (d) by duality from Theorem 4.3.1.
So it remains to prove part (a). Since ch) 6 le"°°, by Lemma 4.3.2, {iiiQ}Q sat-
isfies conditions (4.3) — (4.6), up to a universal constant, as noted above, which
is a family of molecules for Ff]. To complete the proof, we need to show that
{|Q|"1/2"7V"c8)<1)8)}q is a molecular family for F1?" for every [7| S N. By Theorem
1.2.1, this is equivalent to showing that for s = {SQ}Q E fg’q, then ((2,, splPI‘i”J%1
cglég”, z/JQ)}Q E fzf’q, or equivalently to show that AMTCW) is bounded on fz‘f", for all
[7] S N. For (a), the same proof as Lemma 3.3.6 gives the boundedness of AMTE),
for all [7| S N C]

Chapter 5

Sharpness

My most nearly immortal contributions to mathematics are

an abbreviation and a typographical symbol.

I could never believe that I was really its first inventor

The symbol is definitely not my invention — it appeared in
popular magazines (not mathematical ones) before I adopted it,
but, once again, I seem to have introduced it into mathematics.
It is the symbol that sometimes look like Cl,

and is used to indicate an end, usually the end of a proof.

It is most frequently called the “tombstone,”

but at least one generous author referred to it as the “halmos.”

—— Paul R. Halmos, I want to be a Mathematician: An Automathography In Three
Parts (1985)

5.1 Some Sharpness Results Related To ng.

We first raise a question: how sharp are (b), (c), (d) and (f) in Theorem 1.1.5?
For this, we need to define the paraproduct via Meyer’s wavelets {¢(‘)}f;l‘1, dis—

cussed in Section 1.2. Let

2"-1

(51) fing) = Z ZIg.I§5’>IQI'%—<f,<1>e>tg’.
Q

i=1

60

61

Here we select (1) so that supp<1> Q Q00, in addition to the usual assumption <1) 6 S
and f (I) = 1. By the same argument, Theorem 2.3.1 holds with L1,, in place of Hg.

Proposition 5.1.1 Let 0 < p S +00 and 0 < q S +00. If 11,, is bounded on F39
then 9 6 F337.

Proof. Fix N > n (m — 1). Choose a function 0 E 8(R") with 0(x) =
on Q00, 0(x) = 0 ifx 9? 3Q00 and fx70(x)dx = 0 for all multi-indices 7 with [7| S N.
(It is not difficult to show that such a 9 exists.) By the molecular theory ([14], p.56)
it follows that 0 6 F39. For each dyadic cube P, define 0P by
(I: — (DP

0”(x)=0( ,(—P—))

Then (0P,<I>Q)=: f¢Q(x )dx- — |Q|2, for all Q C P. By (5.1) and (1. 38), we obtain,

for0<p<+oo

IIfi IxPI IIIng ~ 2 ||IIgg.xI IIQI-xIxP in»

Z (l (X (|Q|‘%I<g,wgl)llQl’i|<0P’¢Q>|X:q> l ”

i=1 QQP

= Z (f1)(§(|Q|"il<g.¢3’)|xq)q)i)

i=1

     

IV

A
P

since (9P,<I>Q) = [Qli if Q Q P.
By the translation invariance and the dilation properties of F1?" (see e.g. [42] p.239

(7)),
”PIE-xx = IPIFIIOIIF-gx = CIPIx.
So
2n-1 I
~ _ __ a _
[[Hgllpgan-gq 2 CZSngPl 1/P(Z( [Q12 KW )>lxq)q)q)p
i=1 QCP

 

 

2n—1 .
g C 2 [[(g, 1145))
i=1

Cllgllpgg,

22

62

by Corollary 5.7 in [14], followed by (1.38). A similar argument gives, for p = +00,
“9’3“ng = C and

2n—1
IlngIlp-ggrxxg 2 CZsup(IPI1/Z(IQIxIIIgg,x""qIIxx))

PQCP

.1.

R

Cllgllpgg-

C]

Remark (1). Suppose q < 2, and let 9 E BMO \ F337. Then T = I19 satisfies
T1 = g E BMO, T‘l = O, and T E CZO(E) fl WBP, but T is not bounded on F1?"

for any p < +oo.

Remark (2). Supposen — < q_ < +oo, n—+1 < p < +00, T E CZO(e )fl WBP
for some 5 > J", where J =m1n(1,p, q) and T 18 bounded on F19". Let h = T‘l.
Suppose also that H; 6 C Z 0(5) 0 WBP and IT}; is bounded on F30. Then it follows

that g = T1 6 F33.

To prove this, let W = T — 11;. Then W*1 = 0 and W 6 020(8) (1 WBP.
By Lemma 1.2.8, we have g = W1 6 F30”. Thus, Lemma 1.2.7 implies that 119 6
020(1) n WBP. Let s = W — 11,. Then s 6 020(5) 0 WBP, and $1 = 0 = 91.
So by Theorem 1.1.3, S is bounded on F12". Since W is also, we obtain that 119 is FS"
bounded. By Proposition 5.1.1, 9 6 F33.

Remark (2) indicates that T1 6 F f: is the right condition on T1 in Theorem 1.1.5,
part (b) and ((1), hence by duality that T"'1 6 F33 is the right condition in (c) and
(f). In particular, the sharpness of Theorem 1.1.5 ((1) for p = 1 is followed.

Proposition 5. 1. 2 Let— < q S 2 and T E CZO(e )fl WBP for some 5 > J",
where J = n/ min(1,q). Then T is bounded on qu if and only if T1 6 F8: and
T“1 = 0.

Proof. The “if” is part of Theorem 1.1.5 ((1). For the only if part, note that
F10" —-) F,"2 m H 1 by Theorem 1.2.4 (a). Therefore for each dyadic cube Q, TwQ E

63
Flo" Q Hl (since wQ 6 Flog), so

<T*1,xQI = (1.7"on = [ms = 0,

since H 1 functions always have mean 0. So T" = 0.

Now Remark (2) implies that T1 6 F33. D

As mentioned in Section 4.1, we have the sharpness of Theorem 4.2.1 when q 2 2

and of Theorem 4.2.2 when 0 < q S 2.

Proposition 5.1.3 Let 1 < p S +00 and 1 < q S +00. Suppose that for each
dyadic Q, mQ satisfies {4.4) and (4.6) with K = 0. Suppose that {mq}Q is a family
of molecules for Fig". Let cQ = me and c = {cQ}Q. Then c E fgg’.

Proof. This can be obtained by a duality argument from Proposition 5.1.1.
Namely, suppose that for {mq}q given as stated, we set mg = [Ql'i'cQQQ Then
mg satisfies (4.4), (4.6) with K = O and fmq = Cq, so, up to a universal constant,
{mQ — iiiQ}Q satisfy (4.3 — 4.6) with N = O and K = 0. Hence by the usual smooth
molecule result ([14], Theorem 3.5), for all s 6 f2”, we have

[I ; SQ(mQ _ 771(2)“ng S CIISIIqu.

Applying our assumption, we obtain
(52) II Zxxxxxuxgx s CIIxIIng.
Q
Let g = 20 EQwS), for w“) the first of Meyer’s wavelets. For any f 6 F19”, let
38) = (f, wgl). Then by (1.38), ||f||F£q z 2:14 [[S(i)[[f'gq. Then by (5.1) (applied to

~

Hg) and (5.2),
||(fig)*f|lxgx = II ZxS’x’xQIIng s Cllsmlljgv S CIIfIII-xx-
Q

So (L19)* : F?" —-> ng. This implies that I19 : Flag, —> F133,, and so, by Proposition
5.1.1, g = 1:191 6 F23”. Hence c E fgg', by Theorem 1.2.1. Cl

64

Proposition 5.1.4 Let 0 < p < +00, and 0 < q S +00. Suppose that for each
dyadic cube Q, mQ satisfies (4.4), (4.5) and (4.6). Also suppose that {mQ}Q is a
norming family for Fig". Let cQ = me and c = {eq}Q. Then c E fgg.

Proof. For each dyadic Q, let 7714; = [Ql‘l/ZcQ<I>Q. Then, by Theorem 3.7 in [14],

{mQ — mQ}Q is a norming family. So

|| {(fIfiloilq [|qu S Cllfllrgq

for all f 6 FE". Let g = 2Q cng), where w“) is the first of Meyer’s wavelets.

Observe that
2"-1

fingI = ZZIg,¢3’>|Q|“/2<f,<l>o>¢8’
i=1 Q

= :(fifil'Qlit’S):
Q
Thus ||f1,( f)|| ,3. g cm f, aqua,“ ,3. g 0||f||F-£q. So ii, is bounded on pp and by
Proposition 5.1.1, 9 6 F23. Hence, by Theorem 1.2.1, c 6 fg’g. C1

Next we give an example which does not map fgl to f)?" boundedly.

Proposition 5.1.5 Let 0 < p,q < +00 and [Al 2 {[(QP,99Q>[}Q,}D. Then [AI does
not map ff," to f2” boundedly.

Proof. For m E Z+, let [3 = {P dyadic : P Q 4Q00, [(P) = 2"“} and let
3}: = [Pli if P E B and sp = 0 if P ¢ 8. Observe that, for each Q and P with
€(Q) = 2‘” and €(P) = 2"",

((I’PI 9062) = 2'("_m)”/2‘1’ * &V_m(2m(xQ " $13))»

where fix) 2 95(—x), and @(x) = 2jnfo'(2jx), as usual.
Also [Islljgi = [I ZQEB XQHLP = [[X4Q00[[Lp = C. Thus, for each Q with t’(Q) = 2'”,

IIAIon = ZIIxmoIIIPIx

PEB

= IQli Z I» .. a-mI2meo — prII.

PEB

65

If m is sufficiently large, we claim that, for Q Q Q00 and I(Q) = 2‘” with m/3 S
V S 2m/3,

ZI¢*‘Pv-(2171m($Q_xP))I>C

PEB

where C is independent of l/ and m.
Suppose that the claim holds for a moment. Let V = {V E Z : m/3 S 1/ S 2m/3}.
Then, for m sufficiently large,

IIIAISII’};. = / (;(|Q| x (|A|8)qu))

2
q

QN

IV

/ Z Z (IQI . (IAI8)QXQ)q

uEV l(Q)= 2-"
QQQoo

E
q

IV

C/ZZXQ

VEV l(Q)= ‘2‘”
QCQoo

m 2 2
2 C “)"=CIIxII-o-mx.
Qoo(3 fp

So |||A|||j31_,f-3q Z CmixZ for all m sufficiently large, and the result follows.

Proof of Claim: Let g = <1) * (fiwm. Then supp g Q {5 : [5| S 2”‘m+1} Q

{5 : [5| S 2}. Choose a function 7 E 5 satisfying 7(5) 2 1 if [5| S 2 and supp
7§fi=m<rlembMflSm

deI = Z (I2xI-n flngIIoiktdt).—xx

kEZ"

= Z g(k)e"““",

1:62"

by Fourier inversion. Since 7(5) = 0 for [5| 2 7r, substitution yields

g(x) = g * 7(3) = (Q7)V(x) = Z g(k)7($ -

kEZ"

66

Hence

I... IngIIdx s )3 lg(k)| / |7(:v — k)ldx = Z [g(k)|-[|7||L1.

keZ" keZ"
Then
IlgIILI-Il'Illgt 3 Z |g(k)|= Z IgIk)|+ 2 IngII.

keZ" |k|32m |k|>2m

We estimate the last term first. Using, e.g., Lemma B.2, p.150, in [14], for NI > 0,

Z [g(kll S CM2(V_m)" Z (1+2V-m|k|)—M—n

|k|>2m |k|>2m

z CM2("-m)" / (1+2V-m|x|)"M”"dx
Iszzm

= CM f (1+ |y|)M‘"dy
I342?"

22

GMT-"M.

Note that for |y| = 1, |(I,B',,_m)i(2"‘my)| = |gb(y)| 2 C > 0, by definition of go. Also
<I>(0) = f (I) = 1, and (I) is continuous. Since V — m S —m/3, for m sufficiently large

we have |<I>(2"’my)| > C > O, C independent of V. Hence for such an m and y,

A

IIgIIL1 : [[(I) * Pv-mIIL‘ 2 [(Q) * (Pu-mlauwmyll

I<i>I2"-myII - I$I_m(2”’my)l

2 0 > 0.
So ||g||L1 - [[7||Zl1 2 C1 > 0 for some C1, independent of V. Thus
0 < 01_<_ Z |g(k)| + 0M - 27M.
IkIszm
For all 1/ Z m/3, 2"’M S 2""M/3. Pick m sufficiently large such that CM - 2""M/3 S

C1/2. Then
2) WC)! 2 01/2.

lkls2'"

67

Hence

2 [(1) * Pv—mf2m($Q _ mPllI Z Z [(1) * fiu—m(k)l
PEB |k|S2m
2 01/2.

[:1

This shows that we cannot replace q = +00 by any q < +00 in the remark after
Lemma 2.2.4 with a = 0; so, by a duality argument, we cannot replace 1 by any q > 1
in Lemma 2.2.3 (b) with a = O and p > 1. So we have C : ff,” —) fl?” is bounded
but [GI does not map f'pz to f3°° boundedly. This is the first example we know where
the mapping properties of some matrix B are known explicit stronger than those of
[B | So Lemma 2.2.3 (a) shows the special significance of the case q = 2: by using
the non-tangential maximal function, a sharper result is obtained for C than |G|. We
do not know how to get better results for C than [GI on f1?" if q 76 2 (if they exist).

Proposition 5.1.5 shows the optimality for our purposes of Lemma 2.2.3 for |G|.

In Section 2.2, we promised to give the other direction of proof of Lemma 2.2.1.

Finally, we give these special examples in this section.

Proof of Lemma 2.2.1 (a) (for =>). Suppose Tc : fgxl —> fIP+P)'P2 is bounded.

As in the other direction, we can reduce to the case a = e = 0.

Case 1. p = +00 and q2 < ql < +oo.
First suppose that c is such that cQ = 0 except for finitely many Q. Define a
sequence 3 by sQ = [Qli'thI—qei[CQIQ2/(91-92) for each Q. Then, letting r = ql/qg,

1

31—
IsIn. = subs-1: IQll‘q"2|solq‘)

09°

_1_
(11
: 5:13p (IPI-l Z [Qll-r 02/2lchr <12)

02”

 

68

and

l

E
lchslljgox = sgp (|P|“ Z IQ|1”"PICQIPPISQI"P)

09’

L
92
= 8‘2? ('1’ I“ Z |Q|1"“P/2lcql’ )

QEP

— IISIlf-ggl -|chI,-g,r'o-

Hence IICIIjgsr’qz S IITcIIjggxnjggx-
Now drop the finiteness restriction on {Cq}Q. Note that truncating c does not

increase the norm of Tc. So by the finite case, we obtain, for all P,

1
7—

'92
IPI“ Z IQIP""PPPIon”e Slchlljgel—Ijgr-
QQP
qfinite

So taking the sup over all finite subsets and then the sup over all P, we have the

result.

Case 2. qg Sp < +00 and q2 < q1< +00.

As in Case 1, it suffices to prove the result under the assumption CQ 2 0 for all but
finitely many Q. For each P dyadic, define a sequence 3’3 by (3P)Q = [QIF-2 qlq-qz X
[eqqu/(ql‘qzl if Q Q P and (3P)Q = 0 otherwise. Then, letting r = ql/qg as above,

 

Lil
p _l 7,qu (I2ql
lls In. = / (:(IQI xIonx.) )
P 09"
L _1_
_1 r!” in W J-r
s / 2(IQI xlcon.) IPx
QEP

by Hélder’s inequality. Also,

IITcsPIIjgxx = fp<Z(IQI‘IICQII(SP)QIXQ)Q2)

QCP

69

= t(;.IIxI-Iew"'”)i
’P [(2) (IQI-xIonx.)"‘”)%

l

 

2 llspllf-gx -|P

 

 

QQP
Since
_1 1"92
j Z (IQI xlcon.)
PQEP
(P 'Q2 :2; gpz
_ -1 ,.
s. UPI—mp / (ZUQI den.) ) ,
P QEP
,1
P -I’ -1 ""2 ”2 P
llTes In. zIPI .. Z (IQI 2Icon.) HS ”,3...
PQQP
So

7"92 7%;
_ _l
IITCIIIgonfge 2sgp(IPI ‘ [P 2 (IQI 2Icon.) ) = Henge...

QEP

Case 3. q1 = q2. (This implies (,9:ng = 00.)

For each dyadic cube P, define a sequence 3” by (sP)Q = [Q|%_13 if Q = P and
(sp)Q = 0 otherwise. (Interpret this when p = +00 as (3P)Q = |P|i if Q = P and
0 otherwise.) Then in either case ”319”,}ngl = 1. Thus ||TcsP[|f-3,.,2 = [Pl—ilcfl for

ngpS+ooandso

_1
IICIIf'gooo : 3qu [P] 2 [cp| S ||Tc|[f-3q1_,f-3q2.

I
Case 4. q1 = +00. (Then (3%) q2 = qg.)
For each dyadic cube P, define a sequence 3’3 by (sP)Q = [Qli if Q Q P and
IxPIo = 0 otherwise. Then IISPIIjgooo =1 and Ilsplljgoo = II sup. IQI-xIIsPIoIXQIIo =

[Htifp<+m.

70

Whenp=+oo,
IlTesPIlf-gge = sup (IRI /Z(IQ|1|CQII(8P)QIXQ)”E)
RQCR
(1.—Ii
2 IPI“/ Z( (IQI xIcoIXQ)“
QCP

and, when q2 S p < +00,

IITcsPlljgo = (/ (:(IQI 1lchl|(s”)QI>a,)‘”)in)
Q

= (L (q; (IQl-xlcqlxq),,) 5;) it},
2 [PI—W (/ Z (IQI‘iIconQIWfl

P 09”

= IISPIljgoo(IP|"/ Z( IQI xIonx.) ).,

PQCP

L
<12

1

Therefore, taking the sup on P,

”TCIIIiw—IIS” 3 ”6“qu-

Proof of (b) ( =>). The case p = +00 follows from (a) with q] = q/fl and q2 = q.

So assume p < +oo. Again we can assume a = 5 = 0 and CQ 2 0 for all but finitely

many Q. Suppose T: fob ,q/fi —> f2? is bounded. Let s be the sequence defined by
sQ = [Ql2 miLfiilcQIITgi for each Q. Then, for 0 < q < +00,

2
2
_1 74—1—13) q e/(l—e)
s -, = 2 c , _
II Its/(gm (/ (23(IQI Ion.) )) ||||

Q

and

IITCSIIqu = (/ (Z(IQI“IceIIonx.)")q)
Q

71

l

E p
_I u—xe " I I-..
[(2 (IQI 2ICQIXQ) ) =IICIIf'qu/I13m

Q P/(l-fi)

— [ISIIqu/fi ° [ICIIf'mq/II—m-
P/fi p/(l—B)
With the same sequence 3, when q = +00, we still have the same result. CI

Remark. Corollary 2.2.2 is sharp for p, qz > 1. Because Tc : f2“ —+ f3”, T5 :

'Oq; 'Oql '0o(%)'92 .
fp, —) fp, . By the sharpness of Lemma 2.2.1 (a), we have c 6 f00 , smce

ngI'q; = (37%-

5.2 Some Sharpness Results Related To Ffiq.

Here we would like to see how sharp our main theorems in Chapter 3 are. For this

purpose, we need to recall the paraproducts defined in (3.12) via Meyer’s wavelets
{¢(i)}?;;1, where (1)”) E 8 satisfies all properties as in (3.1) for fixed a, as well as
supp (1)”) Q Q00. Our first result shows that part (c) of Theorem 3.3.1 is sharp for
11;"). This suggests that condition (6) in Theorem 1.1.6 and condition (8) in Theorem

3.3.3 are the best possible for the case [7| = oz.

Proposition 5.2.1 Let a Z 0, 0 < p S +00 and 0 < q S +oo. Let 7 be a multi-
index with [7| S a. If H?) is bounded on Fl?" then g E F112”.

Proof. Fix N > J - n — a, where J = n/ min(1,p, q). Similarly to Proposition
6.1 in [44], choose a function 17 E S with g(x) = 1 on Q00, g(x) = 0 if x ¢ 3Qoo
and kan(x)dx = 0 for all multi-indices [kl S N + [7|. Let h(x) = x7n(x). By the
molecular theory ([14], p.56) it follows that h E Ff”. For each dyadic cube P, define

M” by
hp”) = h (ma—Pip) '

Then, by the translation invariance and the dilation properties of F5” (see e.g. [42]

p.239 (7)),

 

_2 1....2
. = CIPI» .,

'Ul"

IIhPIIrgq = IIhIIrgq - IPI

72
Also, ifQ Q P,
'7
x—xp 1 111 _111
((113,281)) = f (m) PSIW)“ = IQI2+ " IPI "

Thus, by (3.12) and Theorem 1.2.1, ifO < p < +00,

 

 

2n—1
~ 1' _1_Lu
lump) x : {<xxIIIxI 2 new}. ..
i=1 p

 

 

 

 

l

 

 

 

2"- 1 5 E
_1_9_ 2' _1_-I11 q
2 (“ng (IQI 2 "|(g,¢l;))lQ| 2 "Ilhp,¢3))lxe)) )
1:1
2" 1 2 1% 2_-_l_l 1
= /(Z|QI x-%I<g,gx ""IIxo)) -IPI . ‘Pllhpllqu-
i=1 QCP
So, if [7|Sa,
~(7)
[Hg FSq—IF‘S"
m . :- P
2 CZsup |P[‘l/ (Z (IQI_-_"I(9a1l)Q)>IXQ))
i=1 P P QEP
2"—1

R

e 2 II {<g,xg’I}Q ”ng... e 0 Hang...
i=1

By a similar argument, when p = +00, IIhPIIF°q= ClPl'n and

I <>.lIF-, >0: (|P|"/ Z( IQI-x Mex IIxo)") IPILimlthllxgox-

a1—

             

 

 

 

PQQP
So if [7| S a, as above we obtain
~(7) .
H. mm, 2 C llgllxgx-

[:1

Next we consider the sharpness of the conditions on cl”) in Theorems 4.3.1 and

4.3.3.

73

Proposition 5.2.2 Let a Z 0, 0 < p, q S +00. Suppose for each dyadic cube
Q, 772;; satisfies (4.4) — (4.6)forK > [J—n—a], [(+6 > J—n—a anda >
max(J,n + a), where J = n/ min(1,p, q). Fix 1: with lkl S oz. Also suppose c8) =
f(:c — xq)7mq(x)da: = 0 if7 75 k and M S a. If {mQ}Q is a norming family for Fri”
then C(k) = {63)}42 E fl’é'”.

Proof. Let (PM E 8 satisfies all conditions in (3.1) and supp (1)“) Q Q00. Also let
77m = mg — IQI‘l/2’IkI/"cg‘)<1>gc). Then there exists a constant C independent of Q
such that {Tim / C }Q is a family of smooth ([a], K, 6,o)-molecules and so a norming
family for if", as discussed before. Thus {IQI'l/Z‘IkI/"c$)¢g)}g is a norming family
which implies that (see the proof of Proposition 1.1 in [15]) {(wg), IQI-l/z‘IkI/"cgm
@g))}q,p is bounded on f5”, where w“) is the first of Meyer’s wavelets. Let g =
20 cg‘hpg). Then fig“ is bounded on F59. By Proposition 5.2.1, 9 E Fifi” or
equivalently, by Theorem 1.2.1, C(k) E jg”. [:1

Remark. When Ikl = 0: 6 Z, 0 < q S 2 and {mq} satisfies the other assumptions
of Proposition 5.2.2, Theorem 4.3.1 (b) shows that {mq}Q is a norming family for
Fig if and only if 60:) E fgg.

Proposition 5.2.3 Let a S 0 and 1 < p, q S +00. Suppose {mq}Q is a family of
function satisfying (4.4) - (4.6) with K 2 [a], K + 6 > 01, and 0‘ > n — a. Fix It with
He] S —a. Also suppose cg” = f(:v — $Q)7mQ(a:)dx = 0 if’y 79 k and I7I S —a. If

{mQ}Q is a molecular family for 13;?" then 60‘) = {6%)}(3 E flgl’q’.

Proof. By Proposition 5.2.1 and a duality argument (see [15] p.389), we have the

result immediately. [3

Remark. When Ikl = —a E NU {0}, 2 S q S +00 and {mQ}Q satisfies the other
hypotheses of Proposition 5.2.3, Theorem 4.3.3 (c) shows that {mQ}Q is a family of

molecules for FPO" if and only if C(k) E fo‘oaq'.

74

Now we consider condition (e) in Theorem 3.3.3. Recall that a p is a smooth atom
for P1?" associated to the dyadic cube P if supp up 9 3P, f x7ap(:c)d:c 2: 0 for M S L
and IDVap(:1:)| S IPI‘1/2”I7I/" if M S [04+ + 1, where L > J -— n — a. Note that, as

usual J = n/ min(1,p, q).

Lemma 5.2.4 Let a Z 0 and assume n/p — n — a Z 0. For each dyadic P, let ap
be a smooth atom for 1.7;“? associated to P. Suppose T E CZO([a] + 5) fl WBP and
:1" e CZO(LI 0 WP If{(T((:v — wows I}Q 6 fix" for n/p — a < I’YI s a for
i=1, 2,---, 2" —1 and T(:c7) = O for M S n/p — a, then xBTap E L1 whenever

IfilSn/p—n—a.

Proof. Define S as in (3.15) with 011 = a, h, = 0 for all 'y and g, as in (3.13). Note
that we assume supp (1)“) Q Q00. Then, by Lemma 3.2.4, 5' E CZO([a] + 6) fl WBP
and 5(337) = 0 for M S 0. Thus, by (3.21) and (3.22) in [17], the cancelation
properties of up and S E CZO([a] + e), we have xfiSap E L1 for |fi| < n/p — n — a.
To complete the proof, we need to show that $3H(Z)ap 6 L1 for Ifll S n/ p — n — a
and oz — n/p < M S 0. Observe that if 3P 0 Q = 0 then (ap,<I>8)) = 0. So

2"-l

fi;:)ap(wI = 23 Z <9...IS’IIQI-1/2-'~'/"<ap,QS’IIIS’M

i=1 QzPQBQ

2"—1

+ Z Z (9.,wS’IIQI‘l/z'hwap,ig’Iwg'RxI

i=1 Q:Qg3P

f1($) +f2($)~

Then

2"—1

HacfifgIILl _ 0p: 2' (W >HQl—1/2— |q|/n(:Q__Dl/2 IQIl/2

i=1 Q: QC3P

/\

S CPIPII/2 < +00,
since g, 6 F43“. Also if P g 362 we can see

Ilwflwgnlu = IQll/2/ |(Ivo +€(Q)y)¢“)(y)ldy S CIQII/“Ifi'”,

75

since qu| S CIx—glaQ) + pr — qu S Cp€(Q). If ((62) _—_. 2‘” > [(P) = 2‘“, by

n/2+L
Lemma B. 1 in [14], then |(ap,<1>8))| S C' (§%%)n Hence
2n-1 n/2+L
_ n g P l 1,91
IIxfiflIIL. s C 22 2II< me )lIQI ”2 W (2%) IQI2+ ,.
i=1 Q:PC3Q
u
S CP 2 2V(L-IfiI) < +00,
since 9, E FoIgI’l ——> FAZIOO, so |(g,,wg))||Q|'1/2‘I7I/n S C. So xfiTap 6 L1. [:1

Corollary 5.2.5 Suppose that T satisfies assumptions in Lemma 5.2.4 and T is
bounded on F59, for a > 0. Then T*(xfi) = 0 for [BI S n/p — n — a.

Proof. To see this, note that if f E H“ 0 L1, then fxfif(x)dx = 0 whenever
lfll S n/pl — n and xfif 6 L1. Since T is bounded on F15", Tap 6 F59 —> F1912 w H“,
where n/pl = n/p — 0:. So (T*(x5),ap) = (xfi,Tap) = 0, by Lemma 5.2.4. Since this
is true for all P, we get T*(xfi) = 0. [II

This shows that the assumption (e) in Theorem 3.3.3 is necessary in the case
a > 0 and 0 < q S 1, since, by Proposition 5.2.1, if T is bounded on Ff" then

6 F43"? —> F43“. In Theorem 3. 3. 3, under assumptions (h) for 0 < q S 1 or (i),
we have {(T((x —- xQ)7),¢8 )>}Q E fill“ for n/p — a < h] S a, i = 1, 2,---,2“ — 1.
Also if T is bounded on F5“? then, by Theorem 3.25 in [17], T(x7) = 0 whenever
[7] < a — n/p. So, in addition T(x") = 0 when I7I = a — n/p, we have T*(x’6) = 0
for Ifil S n/p — n -— a, by Corollary 5.2.5.

Finally we give an example to show that the condition 9 E F 7‘: 7(0 hl) does not

guarantee that Hg7)(xա) or ޤ7)(xk ) will be in FS/q(a_ Ikl) whenever k > "y. This leads

to an example of paraproduct operator that is unbounded on Ff.

Lemma 5.2.6 For a > 0, I7] < a and 0 < q < +00, there exists a function g E

Fag

n/(a_ M) such that whenever I: > 7 and Us] < 01, fig7)(xk) ¢ F‘s/(04H).

76

Proof. Fix It > 7. Choose a sequence of dyadic cubes {Qm}§=1 with ((Qm) =
2"" such that me = xq, for all m, l, where xQ is defined in Section 1.2. Define a

12+1

sequence 3 = {sQ}Q by sQ = ‘ MIQmI if Q = Qm and sQ = 0 otherwise.
Let Em = Qm — Qm+1. Note the {Em}?,,°=1 are disjoint. Then, by Lemma 2.7 [14],

00

_1_2 _ a- n 1 I11
IISIIj:7(a—l1|) z ”(:0le 2 nm 2( MN IQmI2+"XEmqu/QIILn/(o-Ivl)
m=l
oo (a—I’VIl/n
. (2W)
m=l

oo (a-IWII/n
= (Z m"2) < +00.
m=l

Let g = SO “2%)”, where (M1) is the first of Meyer’s wavelets. Then 9 E Pym-bl)

and IIgIIF°7(a_ _M) S CIISIILSfiPhD' By (3.12), writing xk = (x — xQ + xq)k leads to

~ k
Hg”)(x") = CI’HI :(g,¢§,1))¢q(” = (123033) _—.
7 Q

So

~07) k 'a z ’0
”Hg (‘1' )IIFnha-Ikn CIISIIf"7(°-'*')

co (a-IkI)/n
Z m-2(a—le)/(a-|kl)|Qm|-Z—3Ifi lle)
m=l

m=l

oo (0-|k|)/
Z m-2(a-h|)/(a-Ikl)2mIk—vl/(a-Itch)

m=1

0° (a-|k|)/n
= C (2 m-2(a-lwl)/(a-IkI)IQm|—1.55%)
+00

Proposition 5. 2. 7 Let a > 0 and fix I: with |k| S a. For 0 < q < +00 there exists
a function g 6 F” n/(a_ I7I) for ’7 < k such that 111.12,, (:c“) is unbounded on F”, where
a - ”/12 Z Ikl-

77

Proof. Let h be the function chosen in Proposition 5.2.1. For each dyadic P,
define hP(x) = h((x — xp)/€(P)). Then, as shown in Proposition 5.2.1, we have
IIhIIrgq = CIPll’P'°/" and if Q g P. (hP.<I>8"I = IQI‘/2+'*'/"IPI-'k'/”. Choose {son
and g as before in Lemma 5.2.6. Then ޣ7)(xk) = Cg. Hence

”I: _1_151 k
HE: . (hPI=CZsQIQI 2 n<hP,<I>g)I 8’,
Q

and

~ _1_m
In?” (house 2 II{SQIQ|2 "(th¢gC)>}IIqu

119W)

1

C( 2 (IQmI—%—%ISQmIIPI—l§)pIQmI);

QmEP

l
a—|k|__1_ P E
= Z IQmI-%-I%‘IsQ-I (151) IPIH

1

2 0(2 (IQmI-%-”‘3IsQ-I)”) IIhPIIrsv

QmSP

IV

since a — Ikl Z n/p. So

l/p
~ I: _1_151 P
IIH%;~I(xk)IIF;q—.ng 2 C (2 (|le 2 . ISQmI) ) = +0..

Qm

Chapter 6
Conclusions And Future Research

Plans Related To This Topic

Two pages of final manuscript - - -

Although they look like a first draft,

they had already been rewritten and retyped — like

almost every other page — four or five times.

With each rewrite I try to make What I have

written tighter, stronger and more precise,

eliminating every element that is not doing useful work.
Then I go over it once more, reading it aloud, and am always

amazed at how much clutter can still be cut.

—- WILLIAM ZINSSER, On Writing Well (1990)

6.1 Conclusions.

6.1.1 Conclusions Related To F199.

Here we raise some questions.

First, Proposition 5.1.1 indicates that the condition T1 6 F3: in Theorem 2.3.1

is sharp for all q S 2. For q > 2, the class U is almost certainly not sharp. What

smaller space (or class) could replace U ? For q S 2, we used G : fpz —-> f3”; for

78

79

q > 2, we are led to the space u by Lemma 2.2.3 when looking for something larger
than f3” that G could map ff" into. Since nothing is larger than 00, this takes us

out of the a = 0 case if we stay within the Triebel scale.

A second question arises from an attempt to understand the matrices A (Lemma
2.2.4) and G (Lemma 2.2.3) better by considering the matrices |A| and IGl, whose
entries are the magnitudes of the corresponding entries. It turns out that [Al and [G I
are bounded from f3] into f3” for 1 < p < +00, and this is sharp in the sense that if
1 < q < +00, [AI is not bounded from 15:1 into f2?" and |G| is not bounded from fl?"
into f3”. Thus the sharp results in Lemmas 2.2.3 (a) and 2.2.4, which are specific
to the case q = 2, depend in some way on cancelation within the matrices G and A.
Although this is not surprising in principle, this is the first natural example we know,
in Littlewood-Paley theory, of a matrix B whose mapping properties are known to
be better than those of IE I In any case, we ask: if q 79 2, what are the sharp results
for A and G, similar to those in Lemmas 2.2.3 (b) and 2.2.4?

Third, we have seen a connection between the T1 theorem and the molecular
theory. For example, the condition T‘l 6 F3: is equivalent to c = {Cq} E ng,
where cQ = f szQ = (1,Tz/)Q) = (T‘1,¢Q). Of course this just says that the family
{TwQ}Q satisfies the moment condition in Theorem 4.2.1. Is there a similar condition
on families of molecules that corresponds to the condition T1 6 F3: or T1 6 U?

This question is related to the question of characterizing T by its action on atoms
or molecules. For example, we know (see, e.g. [14]) that T 6 CZO(e) fl WBP with
T1 = 0 = T‘l if and only if T maps “smooth atoms” into smooth molecules, which
is also equivalent to mapping smooth molecules into smooth molecules. This last
condition (with the more general notion of a family of molecules, as in Theorem 4.2.1)
cannot characterize, for example, the full David-Journé class, i.e., T 6 020(8) 0
WBP such that T1,T*1 E BM 0. This follows from the result of Lemarié that the
composition of two such operators is not of this type (i.e., the class is not an algebra
under composition, see [25]). Still, there may be a sufficiently general notion of a

family of molecules such that T maps smooth atoms or the family {z/JQ}Q into such a

 

80

family if and only if T satisfies the conditions in one of the parts of Theorem 1.1.5.
Finally, what are the consequence of the factorization of Hg and II; that we have
seen? Here the results of Pisier [31] may be helpful. If we look at all operators
bounded on one space, X, that factor through another space, Y, this class obviously
forms a two-sided ideal in the space of bounded operators on X. Can we use this
to find an interesting algebra of bounded operators on 13;?" containing those bounded
by the full T1 theorem (Theorem 1.1.5)? This would be especially interesting for
LP. By Lemarié’s observation noted above, this would take us out of the class of

Calderon-Zygmund operators.

6.1.2 Conclusions Related To F5”.

We conclude by pointing out a few questions that remain open. First we consider the
conditions on the operator T in our results. For example, condition (d) in Theorem
1.1.6 is necessary for I7I < a — n/p, by Theorem 3.25 in [17]. Is T(:r") = 0 when
M = a — n/ p necessary? Proposition 5.2.1 suggests that (1.29) in Theorem 1.1.6 is
the best possible. How sharp are the other cases, i.e., when a — n/ p < I7I < a? Let
g., be defined by (3.13). By Proposition 5.2.1, g.y belongs to 1.7.13” if T is bounded on

1.7:". On other hand, condition (f) in Theorem 1.1.6 says that g, belongs to FS/q(a—|1|)'
There is a gap between these two spaces. What is the best condition we can get?
Next let us look at condition (e) in Theorem 3.3.3. In view of Corollary 5.2.5 we may
ask, how sharp is condition (e) in Theorem 3.3.3 when q > 1?

Also, many of the questions in the conclusions to previous subsection can be for-
mulated in the context here as well. For example, can we characterize the class of
bounded operators in our theorems in terms of their behavior on smooth atoms or on
smooth molecules? Can we imbed these classes in explicit algebras of bounded opera-

tors? What further information does the factorization of the generalized paraproduct

operator in Theorem 3.3.1 give us?

81
6.2 Banach Frames: Molecular and N orming
Families.

In this section, let us consider a question related to Hilbert frames described in [2],
which are generalizations of orthonormal bases in Hilbert spaces. For a more detailed
treatment, see references given there. For a Hilbert space H, we say that a sequence
{:rn} in H is a frame for H if there exist two constants Cl, C2 > 0 such that for all

:r E H,
(6-1) Clllxllz E Z |(x,xn>l" S C2||fc||2,

where the norm of :1: E H is ”x” = (x,:c)1/2. C1 and C2 are the frame bounds and a
frame is tight if C1 = Cg. A frame is exact if it is no longer a frame whenever any
one of its elements is removed.

Let us define the frame operator 5 of the frame {:rn}n by
(62) SM = flaw...

for a: E H. It can been shown that S is (topologically) invertible and we can recover

mEHby

(6.3) :1: = 2C5, S-lxn):rn = Z(:r,:rn)5_l:rn,

for a: E H. Notice that S is self-adjoint, so is S '1. This formula is analogous to the ap-
transform and there is a norm equivalence. To analysize linear operators, this property
is sufficient. A question is arisen: Can we have the same stucture for Banach spaces?
Especially, can we construct Banach frames (which need to be defined) for Triebel-
Lizorkin spaces such that there is a generalized go-transform identity? Of course, it
should be equipped with the appropriate norm equivalence between function spaces
and their corresponding sequence spaces.

When we observe the definition of molecular families for F59 in (4.1) and norming
families for 13;?" in (4.2), they have a similar norm equivalence as frames for Hilbert

spaces. Can we establish the whole structure for Banach spaces using molecular

82

and norming families? When we consider a more general space F:q(fl), where Q is
a manifold, we may not assume that the Fourier transform can be treated in such
a manifold. In this case, if we can construct generalized frames, i.e., molecular and
norming families, here, then we could study some properties of a manifold via Banach
frames, for example the theory of Calderon-Zygmund singular integral operators on
manifolds. In [2], Benedetto and Walnut constructed Gabor frames for spaces other

than L2. But they used Fourier transform to obtain such result.

6.3 Applications of Families of Molecules.

6.3.1 Pseudo-differential Operators of Type 1,1.

In [21], Hormander gave a necessary and sufficient condition for pseudo-differential
operators of type 1,1 to be bounded on the Bessel potential spaces (including Sobolev
spaces). In [41], Torres extended the continuity of pseudo-differential operators of
type 1,1 to the inhomogeneous Triebel-Lizorkin spaces F59. In fact, Torres gave a
sufficient condition that is stronger than necessary but easier to check. It is well-
known that F:2 coincides with the Bessel potential spaces Lg, for 1 < p < +00. We
can extend Theorem II in [41] via the theory of families of molecules obtained in
Theorem 4.3.3. It is natural to ask that how sharp our results are in the case of Li,
for k E N and 1 < p < +00. Since Hormander’s condition is not easy to check, can
we obtain an easier condition than Hormander’s condition or an equivalent condition
in our terminology? This is our second research plan, one application of theory of

molecular and norming families.

6.3.2 Off-diagonal Cases.

For the next topic, let us consider an off-diagonal case of singular integral opera—
tors. In [20], Hofmann focused on Calderon-Zygmund type singular integral opera-
tors which map LP to Lq for 1 < q S p < +00. Can we use our machinery, i.e., the

theory of molecular and norming families (Theorems 4.3.1 and 4.3.3), to extend these

83

results to spaces other than LP, e.g., Bessel spaces L3 or more generally inhomoge-
neous Triebel-Lizorkin spaces Ff"? So far we know that this problem is related to
pointwise multiplication operators. In other words, can we extend the boundedness
of pointwise multiplication operators to inhomogeneous Triebel-Lizorkin spaces 17,?" ?
In [1], there are some results related to this topic and some references. In [42—43],
there are some results for ff”, but those results are under stronger conditions than

what we need. Can we improve those?

6.4 Boundedness of Some Matrices.

In Section 6.1.1, we raise some questions related to the boundedness of matrices.
Here we plan to explore on this topic. In [14—15], Frazier and J awerth considered the

following conditions:

(6-4) sup ”DI—1’2: |Q|V2|an| < +00
P
Q
(5-5) SUP lQl’l/ZZ IPIIIZIGQPI < +00
Q P
(6.6) sup Wei“ 2 lPll/zlaopl < +oo
Po PQPO Q fOOO
and
(6-7) suplQol'1 { Z Ilelanl} < +00.
Q° QSQo p ,9...

Let M ,- denote the collection of all matrices A such that IA] is bounded on f? for all
1 S p, q S +00. In [15] Corollary 8.2 said that a matrix A belongs to M,- if and only if
A satisfies (6.4) — (6.7). Also they commented that M j is a lattice, M f' is closed under
taking adjoints, and M,- is an algebra of bounded matrices on fl)", 1 S p,q S +00,
which contains the almost diagonal matrices that correspond to the reduced David-
Journé class. It is natural to ask whether Mf- contains all matrices corresponding to

Calderon—Zygmund singular integral operators. Or equivalently whether M j contains

84

all matrices corresponding to paraproduct operators, since we know that any gener-
alized Calderon-Zygmund singular integral operators can be rewritten as a sum of an
element in the reduced David-Journé class and two paraproduct operators as we did
in Chapter 2.

There is a related question, namely whether we can obtain similar results for (32".

Here we can get some conditions similar to (6.4) - (6.7):

(6.8) SUP: SUP “DI—”2 Z IQll/2laQPl < +00
“£2 ueZ ‘(P)=2“‘+" £(Q)=2-u

and

(6-9) SUP: SUP IQI"1/2 Z: |P|1/2|an| < +00-
”ez uez ((Qlfl—W'" ((P)=2-v

We let Mi; denote the collection of all matrices A such that A is bounded on (32" for
all 1 S p,q S +00. (For the definition of (339, see [14] or [16]). Then we have the
result: A matrix A belongs to Mi; if and only if A satisfies (6.4), (6.5), (6.8) and (6.9).
Similarly, Mi; is a lattice, M6 is closed under taking adjoints, and M6 is an algebra

of bounded matrices on hog 1 S p,q S +00, which contains the almost diagonal

p ,
matrices that correspond to the reduced David-Journé class. When T is a generalized

Calderon-Zygmund singular integral operator, does {(T’t/Jp, goq)}Q,p belong to M5?

6.5 Weighted Cases.

Finally, can we extend our results and questions above to weighted spaces? In this
situation, what weights shall we consider? Let us consider, for example, the full T1
theorem on weighted Triebel-Lizorkin spaces. We say 212 is a weight if w 2 0 and w
satisfies the “doubling” condition (see e.g. [35] for the definition). Define the weighted

space Fpaq(w) via the norm

l/q
Ilfllfignw) E (Z (zualsou * f|)") ,

”62 m...)

85

for 0 < p < +00. Similarly, define the weighted sequence space f;q(w) by

1/9
||S||j;q(w) a (2(IQI'1/2'°/"ISQ|XQ)") ,

Q um)
if 0 < p < +00. Of course when p = +00 we need to modify the definition. There are
some results analogous to Theorems 1.1.6, 3.3.2, 3.3.3, 3.3.4 and 1.1.7. A question
arises: What is the best condition on weights we can consider? We know that the
theory is satisfactory for AP weights. But since we only consider dyadic cubes in R",
can we use some condition weaker than the AP condition?
For the other questions we note in this chapter, we can also ask what the weighted

version of each one is.

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