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' '15: . $3.1 . .1" , .7 ms lllllllllllllllllllllllIllllllllllllllllllllll 3 1293 0182 This is to certify that the dissertation entitled BOUNDEDNESS OF INTEGRAL OPERATORS IN THE UPPER-HALF SPACE WITH CARLESON MEASU S presente by Naim Sai ti has been accepted towards fulfillment of the requirements for Ph. D. degree in Mathematics 7374”“ f €441 Major professor Date 08/ 18/97 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State University PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINE return on or before date due. MAY BE RECAU£D with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1m WWW.“ BOUNDEDNESS OF INTEGRAL OPERATORS IN THE UPPER-HALF SPACE WITH CARLESON MEASURES By Naim Saiti A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT BOUNDEDNESS OF INTEGRAL OPERATORS IN THE UPPER-HALF SPACE WITH CARLESON MEASURES BY Naim Saiti In this text we study the boundedness of the integral operator T, as a mapping between the Lebesgue spaces: T : LP(X, dm) —> LP(X+,du) for p > 1, where X is a space of homogeneous type with a doubling measure m, and p is a Carleson measure on X +, upper-half space over X. In Chapter 1 we study the case when the kernel of the operator T is admissible. It is known that for such a kernel the operator T is dominated by the Hormander Maximal Function, H, pointwise. Therefore it is bounded whenever H is. The case when the kernel of the operator T has a singularity is studied in Chapters 2 and 3. In Chapter 2 we prove that if the operator T satisfies the Calderon-Zygmund conditions and if the trace of the operator T, the operator To, is bounded for some p0 > 1, and if T and T0 are related by the formula: |Tf(a:, t) - T0,tf($)l S CHf(rc,t) for every (int) 6 X+,f E 0000. then T : LP°(X,dm) —> Lp°(X+,dp) is bounded. In Chapter 3 we give sufficient conditions for the boundedness of the operator T when p = 2, and X = R". In Chapter 4 we have given an application of the theory of singular integral oper- ators to the generalized Tent Spaces. DEDICATION To my father Isa Saiti, who has not got formal education, but has strongly believed in power of it. iii ACKNOWLEDGMENTS I wish to thank Dr. William T. Sledd for his help and guidance. Without his encouragement, mentoring, and unselfish efforts this thesis would have never been possible. I would also like to thank the members of my committee for their involvement. iv Contents Introduction 1 Boundedness of Integral Operators with Admissible Kernels 1.1 Definitions and Covering Lemmas .................... 1.2 Sets with the Tent Structure ....................... 1.3 Admissible Kernels ............................ 1.4 Vector-Valued Inequalities ........................ 1.5 Weighted Vector-Valued Inequalities in IR" ................ Singular Integral Operators 2.1 The Ruiz-Torrea result .......................... 2.2 The operator T 0 .............................. 2.3 Singular Integral Operators on X + ................... 2.4 Singular Integral Operator on Atomic Spaces for 0 < p g 1 ...... 2.5 Examples ................................. 2.6 Cauchy Kernel on C" (n > 1) ...................... 2.7 Cauchy-Szego Kernel on the Heisenberg Group ............. 5 5 9 16 18 21 22 23 37 46 50 3 Singular Integral Operators on Euclidean Space 4 Applications to Tent Spaces BIBLIOGRAPHY vi 54 7O 77 Introduction In the course of proving his famous corona theorem, L. Carleson, (see [4]) character- ized those positive finite Borel measures ,u on the unit ball U in the complex plane C such that 1/p (/U lflpdu) s Cllfllm, for every function f in the Hardy Space H” ( 0 < p < oo ), showing that this holds if and only if the measure )1 satisfies the property 115 S C ( 1 — s), for every set S of the form 5:55:30 2 {reiozs S r < 1,00—7r(1—s) §9<90+7r(1—s)}. Such a measure is called a Carleson measure, and such sets S Carleson sets. In this text we will study under what conditions an integral operator T is bounded as a linear mapping between the Lebesgue spaces T: L”(X,dm) —>L”(X+,du), p> 1. (0.1) over a space of homogeneous type X, supplied with a doubling measure m, where X + denotes upper-half space over X, and u a Carleson measure. A space of homoge- neous type, first defined by R.R. Coifman and C. Weiss (see [7]), is a generalization of Euclidean space, with the doubling measure being the Lebesgue measure. The technique of working in such a space may be quite different, due to, for example, not having dyadic cubes. This particular difficulty is overcome by using Calderon’s covering lemma (see [1]). 2 It is a well known fact (see [11]) that the nontangential maximal function N f and a Carleson measure )1 are related by the formula ”({f > A» _<_ C|{Nf > A}| is true for every A > 0 iff p is a Carleson measure. There is also another maximal function, the Hormander maximal function, H, that is related to a Carleson measure, p, by H : Lp(dm) —> L”(dp)is bounded for every p > 1 iff ,u is a Carleson measure. The proof of both results indicates that the distribution sets for both nontangential maximal function and the Hormander maximal function have a similar structure. We call such sets the sets with the tent structure. Using the properties of such sets, and the LP-boundedness of corresponding trace operators, the restrictions of the operators mentioned above to the space X, we prove the above statement about the Hormander maximal function, and we give an interesting generalization of it. The significance of the Hormander maximal function is that it estimates integral Operators with so-called admissible kernels pointwise (see [12]). Therefore, such in- tegral operators are LP-bounded whenever the H6rmander maximal function is. An important example of a convolution operator with admissible kernel is the Poisson transform on R". The last two sections of Chapter 1 are dedicated to the vector-valued inequalities for the Hormander maximal function, of the type studied in [9]. We use the vector- valued inequality proved by C. Fefferman and E. M. Stein (see [9]), and the approach developed in Chapter 1 to give a more elegant proof of the result proved by F. J. Ruiz and J. L. Torrea (see [18]), that, essentially, is a generalization of the Fefferman—Stein result. At the end of the chapter, we give some applications of the vector-valued theory in spaces of homogeneous type to the weighted vector-valued inequalities in 3 R". In that way we obtain the boundedness of the vector-valued convolution operators with admissible kernels. In Chapter 2, we study integral operators whose kernels may not be absolutely integrable. The boundedness of such operators as mappings BUR") —> LP(R"), was first studied by A. P. Calderon and A. Zygmund (see [2] and [3]), and for that reason, they are called the Calderon— Zygmund operators. In this text we will use the name singular integral operator, and study under what conditions such an operator, T, is bounded as a linear mapping between the Lebesgue spaces in the formula (0.1) for every p > 1. An important example of a singular integral operator in the upper-half space is the operator that assigns the complex-conjugate ii of the harmonic function u = B =1: f, to the function f on the real line. Singular integral operators whose range consists of functions with the domain in the upper-half space were studied by F. J. Ruiz and J .L. Torrea (see [18], [19], [20], [21], [22]). We start Chapter 2 by stating their main result. Using the good-A approach used by R.R. Coifman and C. Fefl'erman in [5], we have obtained a stronger conclusion, but used more assumptions, most of them are about the restriction of T f (x, t) to the hyperplane X. At the end of the chapter we give several examples of such singular integral op- erators in Euclidean space, as well as the examples of convolution operators with the Cauchy kernel on the unit ball in C", and the Cauchy-Szego kernel on the Heisenberg Group. In Chapter 3 we study the boundedness of the operator T : L2(R", dz) ——) L2(R1+1, do) when p is a. Carleson measure on R1“. The whole chapter is based on the ideas and techniques developed in [10]. We use the decomposition of an L2-function into the sum of smooth atoms (by the virtue of the Calderon formula), and we will prove that if a singular integral operator satisfies certain cancelation properties (that play the role of T1 and T*1 conditions), it will map a smooth atom into an equivalent of a smooth molecule in L2(R1+l,dp), which implies the L2-boundedness. In the last chapter, Chapter 4, we use the theory of singular integral operators to make a conclusion about the boundedness of the linear mapping T: L”(R", dz) —> Tfldu), for every 1 < p, q < 00, when u is a Carleson measure on R1“. The spaces T:(d,u) are generalized version of the tent spaces defined by Coifman, Meyer and Stein (see [6]). The main technical tool in this chapter is Theorem 4.1 proved by F. J. Ruiz and J. L. Torrea (see [21]), which makes it possible to apply the vector-valued theory of singular integral operators to the new situation. Throughout the whole text we will use the notion of accumulative constant, which means that C will represent a constant, not necessarily the same in each two consec- utive appearances. Chapter 1 Boundedness of Integral Operators with Admissible Kernels In this chapter we study the LP-boundedness of a so—called non-singular integral op- erator, or an integral operator with admissible kernel in the upper-half space supplied with a Carleson measure. Such an operator is dominated by the Hormander maxi- mal function pointwise, and therefore is bounded whenever the Hormander maximal function is. We begin by introducing spaces of homogeneous type. 1.1 Definitions and Covering Lemmas A space of homogeneous type was first defined by R. R. Coifman and G. Weiss (see [7]). First, we define pseudometric. Definition: Let X be a set. A map p : X x X —> [0, 00) is called a pseudometric if and only if it has the following properties: (i) p(:1:,y) > 0 if and only ifsr 75 y. (ii) p($, y) = My, 3) for all 23,31 E X- (iii) There exists a constant hp 2 1 such that for all 11:, y, z E X we have 0(13, 2) S kp(p($, y) + My, 2)). 5 6 In case hp 2 1, a pseudometric p is a metric and will be denoted by d(x, y). The quasi-ball B(x, r), with the center at the point x, and radius r = r(B), is the set B(x,r) = {y E X : p(x,y) < r}. Definition: A space of homogeneous type is a topological space X endowed with a pseudometric p such that: (a) The family {B(x,r) : x E X,r > 0} is a basis for the topology on X. ( b ) There exists a Borel measure m on X which is a doubling measure, i. e. there exist a constant C > 0 so that for every x E X, and r > 0 m(B(x, 2r)) 3 Cm(B(x,r)). Remark: It has been proved (see [7]), that properties (a) and (b) imply the following: There exists a number N E N such that for any x E X, and for each r > 0, the quasi-ball B(x, r) contains at most N points x, with p(x,-, 2:3) > r/2. In this text we will use the notation: [B] = m(B). Examples: Obviously every finite dimensional metric space, supplied with any doubling measure, is a space of homogeneous type. A more interesting example is: n X =R" p(x,y) =Zlm-yl’" p.- > 1- i=1 In Chapter 3 we will give two interesting examples of spaces of homogeneous type: the unit ball in C" with the non-isotopic metric p, and the Heisenberg Group on Rx Cn‘l. As we can see, there are many similarities between Euclidean space and the spaces of homogeneous type. One of the differences is that on a space of homogeneous type we may not have features like dyadic cubes. Consequently, tools like the Whitney decomposition theorem, or the Calderon-Zygmund Lemmas cannot be used. The covering lemma, which follows is due to Calderon, (see [1]), is the main tool in bridging this difficulty. We use the following version of Calderon’s lemma. Lemma 1.1 Let E be a subset of X and {B(x,r(x))}x€E a covering of E, such that the radii {r(x)}z€E are uniformly bounded. Then there exist a (possibly finite) sequence of disjoint quasi-balls {B(x,-,r(x,-))}‘f;l such that for every indexi x,- E E, and E C U,B(x,~,Kr(x,-)), where constant K depends only on the space X. Moreover, for every quasi—ball B from the covering there exists an indexi so that B C B(x,~, K r(x,)). The constant K from the previous lemma is called the space constant. We will use the notation B" = KB, where cB denotes the quasi-ball concentric with B so that r(cB) = cr(B). We conclude this section by listing the definitions of some of basic maximal func- tions and Carleson measure. The upper-half space X +, over a space of homogeneous type X, is defined by X+ = {(x,t) : x E X,t Z 0}. The tent over the point (x, t) E X + is the set T(x,t) = {(y,s) E X+ : p(y,x) + s g t}, and the tent over a quasi-ball B C X, centered at the point x E X, and with the radius r, is T(B(x,r)) = T(x,r). A Carleson square over the quasi-ball B in X, is the set S(B) = {(x,t) €X+zx€ B,O gt< r}. 8 A measure )u. is called a Carleson measure on X + if and only if there exist a constant Cu > 0 so that for every quasi-ball B in X we have #(S(B)) S. CuIBI- Notice that this implies that the measure u is a Carleson measure if and only if u(T(B)) _<_ CulBl, for every quasi-ball B in X. An important example of a Carleson measure on X + is the measure defined by p(E) = [E m X] for every E C X+, called the projection measure on X, which tells us that the measure m can be viewed as a Carleson measure. For f : X —+ C, a locally integrable function with respect to the measure m, we define the Hardy-Littlewood maximal function by Mf(x)=sup-|—ll-3—|/B|f|dm xeX, where the supremum is taken over all quasi-balls B 3 x; and the Ho'rmander maximal function by: 1 Hf(x,t)=supl—l7l/B|f|dm tZO, now, the supremum is taken over all quasi-balls B 3 x, such that the radius r(B) Z t. For a Borel measure B in the upper-half space X + we define the generalized Hormander maximal function by 1 H F x,t : su —/ Fd , fl ( ) B:T(B)g(x,t)IB(T(B)) T(Ii')| I fl where we define 377.1(7)) [T(B) [Fldfl = 0 when ,6(T(B)) = 0. In the special case, when the measure fl is the projection measure on X, H 5 becomes the ordinary Hormander maximal function. In Euclidean space IR", with Euclidean distance, and m being ordinary Lebesgue measure, it is more convenient to define the Hormander maximal function as Hf(x,t)=sup|—$-|/;2|f|dx tZO, where the supreme is taken over all cubes Q 3 x, such that €(Q) Z t (where €(Q) represents the side-length of the cube Q). The Hormander maximal function defined as above is equivalent to the Hormander maximal function defined as a supreme over the balls in R", since Lebesgue measure is a doubling measure. We conclude this section by giving the definition of a nontangential maximal function in the upper-half space X +. For given a > O, we define the upward-pointing cone with the vertex at the point (x, t) and the aperture a, by Fa(x,t) = {(y,s) E X4r : p(x,y) < a(s — t)}. Let F : X + ——> C be a measurable function, and a > 0. Then the function NaF(:v,t) = sup |F(y,s)|, (y,s)EFa (3") represents the nontangential maximal function of the function F. The function NaF(x) = NaF(x, 0) is the traditional nontangential maximal function, used in most texts in harmonic analysis . 1.2 Sets with the Tent Structure We will see that the distribution sets for each of the maximal functions we have intro- duced so far have a similar structure. We will say such sets have the tent structure. Once we show that the distribution set of a certain subadditive operator in the upper- half space X + has the tent structure, we can easily obtain the boundedness of such an operator, as a mapping Lp(X, dm) —> L”(X+, du) 10 when ,u is a Carleson measure, provided that the restriction of this function on X is a bounded operator. Definition: A set E C X + has the tent structure if and only if (x,t) E E implies that T(x, t) C E. The following theorem provides us with the most important property of the sets with tent structure on spaces of homogeneous type. Theorem 1.2 Let Borel set E C X + be a set that has the tent structure, and let u and V be Borel measures on X +, such that u(T(B')) S 11(T(B)), for every quasi-ball B in X. Then HE) S 1413). Proof: For fixed N > 0, let EN 2 En {(x,t) E X+ : t g N}. If (x,t) 6 EN, then T(x,t) C E”, and consequently B(x,t) C «(EN), where 7r(EN) denotes the projection of the set E” on X. If we do that for every point (x,t) 6 B”, we have obtained a covering of the set r(EN) C X by a family of quasi-balls, {B (x, t)}(,,,t)E EN, whose radii are uniformly bounded. Applying the Calderon covering lemma, we obtain a sequence of pair-wise disjoint quasi—balls, {3,}, such that for every quasi-ball B from the original covering there exists an index i so that B C Bf. Thus, we conclude that E” C U,T(B{). To finish proving the theorem we proceed as follows: ”(EN) ”(,U T( B:)) < Z: “(T The assumption of the theorem, the facts that {B,-} are disjoint sets, and that T (8,) C E", imply s. Dues.» 3 var”). i 11 To complete the proof of the theorem, we let N —> 00. I Let us consider the set E; = {HgF > A}. Let (x,t) E XJr and (y, s) 6 T(x,t). If B is such a quasi-ball that T(B) 3 (x,t), then clearly T(B) 3 (y, 3). Thus, we have established: (y, s) E T(x,t) implies H3F(IL‘, t) S H3F(y, s) (1.1) for every locally integrable function F : X + —-> C. This fact implies that if H 3F (x, t) > A, then H3F(y, s) > A, for every (y, s) E T(x,t). Consequently, EA is a set with the tent structure. The set F; 2 {MP > A} also has the tent structure, which is due to the fact (y, s) E T(x, t) if and only if (x, t) E F1(y, s). which implies that if (x,t) E F; and (y,s) E T(x,t), then B(x,t) C I‘l(y,s) which produces the statement (y, s) E T(x,t) implies N1F(x,t) g N1F(y,s), (1.2) for every locally integrable function F. The above facts imply the following corollary of Theorem 1.2. Corollary 1.3 If u and 1/ are two Borel measures on X +, such that u(T(B’)) S V(T(B)), 12 for every quasi-ball B in X, then there exists a constant C > 0 so that ”({HsF > M) S V({H/3F > 3}), and u({N1F > A}) s CV({N1F > A}), for every A > O, and locally integrable function F on X +. Corollary 1.3 implies the boundedness of the Hormander maximal function as a mapping: H: LP(X,dm) —+ Lp(X+,du) for every p > 1, when ,u is a Carleson measure. We can show it in the following way. Let the measure B be the projection measure. Then the generalized Hbrmander maximal function becomes the ordinary Hormander maximal function. In the case when u is a Carleson measure, and u the projection measure on X, Corollary 1.3 yields u({Hf > A}) S C|{Mf > CHI, because the restriction of the Hormander maximal function on X is the Hardy- Littlewood maximal function. Using the fact that the Hardy-Littlewood maximal function is bounded as a mapping M : I}’(X, dm) ——> If (X, dm) for every p > 1, and a standard argument, we conclude that there exists a constant Cp > O (that depends only on p) such that ||Hf||LP(X+,du) S CpllfHLP(X,dx) for every f E Lp(Xid$), is true for every p > 1. For p = 1, H is a weak—type 1-1 bounded operator; in other words, there exists a positive constant C so that for every A > 0 and f E L1(X, dx) C ”(NH > M) _<_ X||f||L1(X,dx)- 13 A function F : X + —) IR+ is said to be of horizontal bounded ratio if and only if there exists a positive constant A p such that F(x,t) g ApF(y,t) whenever p(x,y) < t. Lemma 1.4 Iffl is a doubling measure on X4", then HgF is of horizontal bounded ratio. Proof: Let us fix t > 0, and let p(x, y) < t, and let B be a quasi-ball containing the point x, so that r(B) 2 t. Then B" 3 y, and also r(B”) 2 t, which means (y,t) E T(B“). Thus 1 fl(T(B"‘)) 1 ”W 5 mm» fi(T(B*)) mm fl(T(B)) T(B) S CflHflF(yit)° lFldfi If we take the supreme of the left side of the above inequality over all quasi-balls B containing the point x, so that r(B) 2 t, we obtain HgF(III, t) S CngF(y, t), which proves the lemma. We define the vertical maximal function for a function F : X + —) R+ as F’(x,t) = sup F(x,s). sZt Obviously, for every (x,t) E X+ and a > 0 we have F*(x,t) g NaF(x,t). If the function F is of horizontal bounded ratio, then the converse is also true; in other words we have that N1F(x,t) g AFF*(x,t), for every (x,t) E X+. The following lemma shows that there is an interesting relationship between the Hormander maximal function and the nontangential maximal function. Before we state the lemma we introduce the notation AOOF(x) = N1F (x, 0). 14 Lemma 1.5 If p > 1, then there exists a positive constant Cp (that depends only on p) such that for every f E L”(X, dm) lleo(Hf)|lLv(x.dm) S Cpllfllmxam). Proof: If B is the projection measure on X, then [3 is a doubling measure and the corresponding Hormander maximal function is of horizontal bounded ratio. Hence Aoo(Hf)($) S C(Hf)*(:v,0) = CMf($)~ Now, we use the fact that for any p > 1, the Hardy-Littlewood maximal function is LP bounded, to complete the proof of the lemma. 1.3 Admissible Kernels Admissible kernels were first defined by W. T. Sledd and S. Gadbois (see [12]). The main property of integral operator whose kernel is an admissible function is that it is dominated by the Hbrmander maximal function. Usually, such kernels are absolutely integrable; so we do not call such operators singular. The most important example of such an operator is the Poisson transform on R". First, we define constants |B($, 2k+1tl| Ck = sup , k 2 0. (x,t)EX+ [B(fb‘, 15)] Definition: A function (I) : R+ —+ [0,1] is an admissible function if and only if it satisfies all of the following properties: (a) (0) =1 and (1) > 0, (b) (I) is monotone decreasing, and 15 (C) 2 qu)(2k) < 00. k=0 We define the kernel K by K _ Will) (”5’”) " fx <&%—’2)dm’ where is an admissible function, and the corresponding integral operator that) = [X K(x,y,t)f(y)dm(v)- The following lemma, from [12], solves the problem of the LID-boundedness of any integral operator with admissible kernel. Essentially, the lemma tells us that such an operator is bounded whenever the Hormander maximal function is bounded. Lemma 1.6 There exist a positive constant C so that I’Cf(cv, t)| S CHf(x,t) for every (x,t) E X+, and for every continuous function f with compact support in X. Example: Poisson kernel. Let X = IR", p(x,y) = Ix — y], and let m be Lebesgue measure on R". In this case Ck = 2(k+1)". Let Cn — (1+sz)l2fl' 9(8) Then (I) is an admissible function because 2,, 722:)??- < 00. The resulting admissi- 1+2 ble kernel K is the Poisson kernel on R" cut Kx,,t=Px, = n,. ( y) t( y) (t2+|$_y|2)+ Lemma 1.6 yields the following pointwise estimate for the Poisson transform of a function f. |(Pt * f)(x)| g CHf(x,t) for almost every (x,t) E (an)+ = R1“. (1.3) 16 As a consequence of Lemma 1.5, and Lemma 1.6 we have that for any admissible kernel K there exists a positive constant 0,0, that depends only on p > 1, so that for every f E LP(X,dm) ”Aoo(K * f)||L9(X,dm) S CpllfllLP(X,dm)- 1.4 Vector-Valued Inequalities The LP-boundedness of the Hardy-Littlewood maximal function is a well-known fact, established in the first half of this century. However, its vector-valued version, the inequality of type {/<2(Mf.(x))q)5dx}” _<_ 0 (fl: If.(x)r)5dx}’. (1.4) k k ( p, q > 1, ) was established only in 1972 by C. Fefferman and M. Stein (see [9]). The objective of this section is to prove the vector-valued version of the LP- boundedness theorem for the Hfirmander maximal function in the settings of the space of homogeneous type when the measure in the upper-half space X + is a Car- leson measure. As a consequence of the theorem we conclude that the vector-valued integral operator with admissible kernels IC : L§q(X,dm) —) L§q(X+,du), is bounded for every 1 < p, q < 00. We begin by stating the main result, that was proved by F. J. Ruiz and J. L. Torrea (see [18]). Theorem 1.7 Let u be a Carleson measure on X1“, and 1 < p,q < 00. Then there exists a constant C > 0 so that for every vector-function f (x) = (f1(x), f2(x),...) 17 such that {f(E,c [fk|q)idm}i < oo, {f(ZWfqufid/AF s 0 [fig lfqu)idm}%- (1.5) k In the course of proving this result F. J. Ruiz and J. L. Torrea considered three different cases: p = q, p < q, and p > q. The case p = q was trivial, and in the case p < g F. J. Ruiz and J. L. Torrea adapted the argument used by M. Stein and C. Fefferman ( see [9]) to the new situation successfully. The key ingredient in the proof of the theorem in the case p > q is the following maximal operator. For cp E L”(X+,du), we set Aq} and {2mm >Aq} I: k is a set with the tent structure, which is provided by the virtue of the formulas (1.1) and (1.2). Hence, Theorem 1.8 follows from the Theorem 1.2. Now, by considering a special case of the previous theorem and by using formula (1.4), we obtain the result proved by F. J. Ruiz and J. L. Torrea in the following way. Let the measure ,6 be the projection measure. Then the operator H5 becomes the ordinary Hormander maximal function. If u is a Carleson measure and V the projection measure on X, having in mind that H f (x, 0) : M f (x), from Theorem 1.8 we deduce “(20115:)”; ”LP(X+,du) .<_ 0qu 2(Mfqu)3 ”LP(X,dx)- k k Applying the Fefferman-Stein’s result and formula (1.4) to the last inequality, we obtain the statement of Theorem 1.7. 1.5 Weighted Vector-Valued Inequalities in IR“. In this section we apply the vector-valued theory in a space of homogeneous type to Euclidean spaces. A non-negative function w on R” is an Al-weight if and only if there exists a constant C > 0 so that w(x xd)x < C inf wx IQI/Q ( l for every cube Q in IR". The last condition can also be interpreted as M w(x) S Cw(x) for almost every x E R". 19 We define the weighted Hb'rmander maximal function on R" as wa(x,t) = sup ”Tu—(1Q) [Q [f(x)|w(x)dx t Z 0, where the supreme is taken over all cubes Q 3 x, €(Q) 2 t and w(Q) = fQ w(x)dx. (If w(Q) = 0, then we set iii—Q? fQ If(x)|w(x)dx = 0. ) The following lemma provides two important properties of A1 weights. Lemma 1.9 If w 6 A1, then there exists a constant C > 0 so that for every function f E L1(R", dw) fl L1(R",dx), and every (x,t) E R1“ we have Hf($at) S Cwa($at), and (the doubling condition) w(62“) S CMQ). where Q" represents the cube concentric to Q, but €(Q‘) = 38(Q). For the proof see [24]. A positive measure [1 on R1“ belongs to the class C1 (w) if and only if #(S(Q)) sup —— S Cw(x) for almost every x E R", er IQI for every cube Q in R". Notice that the last condition implies that the measure u is a Carleson measure over the space of homogeneous type endowed with the measure w, which is a doubling measure when w 6 A1, because new» _u(S(Q)) IQI in W _1_ no) " IQI w(Q)sc..{,Eg ()}|Qlw(Q)so, due to the fact that [Q] ian w s fQ w. 20 Let w 6 A1, and u E C1(w). On the space of homogeneous type (R”, w), the corre- sponding Hormander maximal function is the weighted Hormander maximal function Hw. We apply Theorem 1.7 to this situation to obtain the inequality (fa: mex, swam}; s 0 {fig lfk($)|q)5w($)dz}p. k which is true for every vector function f (x) = ( f1 (x), f2(x), ..) such that 1 {/(; lfk($)lq)5w(x)dx}p < 00. Applying Lemma 1.9 to the last formula we obtain the following corollary. Corollary 1.10 If w 6 A1 and u E C1(w), then for every vector-valued function f (x) we have 1 {fix}; Hm. ovum. v)” s 0 {fl}: lfk(x)l")iw(rr)dx] . It '6"- when p,q >1. Chapter 2 Singular Integral Operators In this chapter we study integral operators whose kernels may not be absolutely inte- grable. We will call such operators singular integral operators or Caldero’n-Zygmund operators. The most important example of a singular integral operator in the upper- half space is the harmonic conjugate Operator in the upper-half complex plane. Pre- cisely, we consider a real-valued continuous function f that has compact support in the real line, and define the harmonic function u(x,t) = P, * f (x), the Poisson transform of f. The harmonic conjugate of u, the function a, is given by ~ _1 °° x—y ’U.($,t) _ H100 (x—y)2+t2f(y)dy’ taken in the principal value sense. (The integral above does not have a singularity when t > 0 and at 00 since the function f is compactly supported. Thus, u(x, t) = filimrno flaky», (xi—3h f (y)dy.) This gives us the idea to consider the operator T that assigns the function a to function f, i. e. T f 2 ii. Since we are going to study the Operator T in a more general context of the space of homogeneous type, the harmonicity of the function T f will not play any role. The main result of this chapter concerns a singular integral operator in the upper half space whose trace in X is a singular integral Operator. 21 22 2.1 The Ruiz-Torrea result The following result, due to F. J. Ruiz and J. L. Torrea (see [18]), is proved in a much more general context than we need right now. In order to state the vector-valued version of this theorem, we need to introduce the following notation. If E and F are Banach spaces, then L1;(X+, du) = {f : X+ —+ F; f is Borel measurable and f + “fugdp < 00}, x L’Z;(X,dx) = {f : X —> E; f is Borel measurable and/X ||f]|’gdm < 00}, and let L(E, F) denote the set of all bounded linear Operators from E to F. Let the kernel K be a continuous map KszXx[0,oo)\{x=y,t=0}—>L(E,F) such that there exist positive constants C and 6 so that Cp(y. y’)‘ (2:, y)‘|B(w, p(x, y) + t)|’ “K(fiyflil — K(x,y’atllIMEF) S p whenever p(x, y) +t > 2p(y, y’). Let f be a continuous function with compact support in X, let B be a quasi-ball that contains the support Of f, and x E X a point such that x E X \ 28. Then, for t Z 0 we set u(x,t) = / K(x,y,tvummv), (2.1) where the integral symbol represents the vector integral. Notice that f (y) E E is a vector, and K (x,y,t) is a linear mapping E —> F. Therefore the expression K (x, y, t) f (y) E F represents a vector. (Recall, (see [16]), that for a vector function g : A C C -—> F over a Banach space F, we say that L, gdm = v, where v E F is a vector, if for every linear functional A : F —) C we have f A A(g)dm = A(v).) Now, we state the theorem 23 Theorem 2.1 Let ,u be a Carleson measure on X +, and let T be a vector-valued sin- gular integral operator that satisfies the assumptions above, and that has a continuous extension to L‘g’(X,dm), i. e. T: L’g’(X,dm) —> L’,’§’(X+,du) is bounded for some p0 > 1. Then: T is a weak type 1-1 bounded operator; namely there exist a constant C > 0, so that for every A > 0 C ,U.{($,t) E X+ Z ”Tf(113,t)”p > A} S :“fllLlEULdmh for every f E Lg(X,dm), and T is an LID-bounded operator for every p E (1,p0], in other words there exists a constant Cp > 0 (that depends only on p) so that llellL;(X+,dp) S CpllfllLyxurn), for every f E L’1’3(X, dm). In this chapter we will use the scalar case of Theorem 2.1. The vector case of the theorem will be used in Chapter 4. 2.2 The operator To In the special case when the measure u is the projection measure on X, we denote the corresponding singular integral Operator by To. In other words we set To f (x) = T f (x, 0). In the next section we are going to deal with the LID-boundedness of the Operator T, which will depend heavily on the properties of its trace, the Operator To. In this section we are going to provide some information about the operator To. In this chapter we restrict ourselves to the Operators T, so that the restriction of T f (x, t) to X, the expression To f (x), is an Operator represented by the formula (2.1) 24 for every x E X, not only for those x E X \ 23, where B C X is the quasi-ball that contains the support Of the function f. For x 6 2B we have to explain the meaning Of the representation formula (2.1). In this chapter we will assume that the kernel K is a continuous mapping KszXx [0,oo)\{(x=y,t=0}—>C, that satisfies the following two conditions. There exist positive constants C and 6, so that for every x,x’, y E X(x yé y), and t Z 0 we have C IK(x.y.t)I s I B(wmy) + t)l’ (2.2) and _ x. x _ x, Cp(x,x’)‘ mow) K< ,y.t)l+lK(y. .0 KO. ’msp(x.y)‘|B(x.p(x.y)+t)l’ (2.3) whenever p(x, y) > 2p(x,x’) + t. The condition (2.2), imposed on the kernel K, is needed for the L2-boundedness Of corresponding singular integral Operator on Euclidean space R" (see [23]). However F. J. Ruiz and J. L. Torrea did not use this condition when proving their theorem. An integral Operator whose kernel K satisfies conditions (2.2) and (2.3) is called a CalderOn-Zygmund, or singular integral operator. Condition (2.2) also insures that the integral in the following definition makes sense Let K be the function described as above. For any 3 > 0 we set To,.f($) = / K(x.y.0)f(y)dm(y). p(x,y)» when f is a continuous function with compact support in X. We define the maximal singular integral operator in the hyperplane, the Operator T 3* , by the formula To#f(x) =sup|T0,,f(x)| xEX. s>0 25 If the operator :r,t : LP°(X, dm) —> LP0(X, dm) is a bounded Operator for some p0 > 1, the Operator To can be defined by TOf : lim T0,sfa s—>0 where the limit is taken in the L!"0 (X, dm)-sense (see [23]). The same assumption insures the existence Of the adjoint Operator T3, defined by Tof : lflTofifa (where the limit is also taken in the Lq°(X, dm)-sense) is bounded as a mapping T; : L"°(X,dm) —> Lq°(X,dm), where qo is the conjugated exponent to p0, i. e. pic + qio = 1. Moreover, T5 is the same type of Operator as T 0, whose kernel is K (y,x,0), where K represents the complex conjugate to the function K. The following theorem is a simple consequence Of Theorem 2.1. Theorem 2.2 Let To be the operator associated with the operator T that satisfies the scalar version of the representation formula (2.1), whose kernel K satisfies the conditions (2.2) and (2.3) with t 2 0. If the operator T0# : LP°(X, dm) —> LP°(X, dm), is bounded for one p0 > 1, then the operator To : L”(X,dm) —-> L”(X,dm) is bounded for every p > 1. When p = 1, the operator To is a weak-type 1-1 bounded operator; that is, there exist a positive constant C so that for every A > 0, and every continuous function f, with compact support in X, C [{33 E X 3 [Toflxll > AH S :llfllmxumy 26 Proof: To prove Theorem 2.2, we consider two different cases p 3 p0 and p > p0. When p 3 p0, Theorem 2.1 applied to To gives that the Operator To : LP(X, dm) —) L” (X, dm) is bounded for every p E (1, p0], and weak-type 1-1 bounded when p = 1. In order to prove the case p > p0 we need to consider the adjoint Operator to To. The operator T; is Lq-bounded, for every q 2 go, where qo is the conjugate exponent to p0, as an adjoint to a bounded operator. On the other hand, since the Operator T 5‘ is of the same type as To, Theorem 2.1 implies that it is Lq—bounded for every q E (1, qo]. Thus, T“ is Lq-bounded for every q > 1, which implies the LP-boundedness of the operator T for every p > 1. The following result (see [23]) provides a connection between the singular integral Operator T, and the corresponding maximal singular integral Operator on R”. Lemma 2.3 Let K (x, y, 0) satisfy the conditions (2.2) and (2. 3) with t = 0, and for every A > 0 and x,y E R" we have K(Ax, Ay,0) = A’"K(x, y, 0). Then, there exists a positive constant C so that for every f E C8°(R"), and almost every x E R" T0#f($) s CM(Tof)(~’v) + CMf(x). where M represents the Hardy-Littlewood maximal function. For the proof of the lemma see ([23], page 67). Remark: Lemma 2.3, combined with Theorem 2.2 implies that if the operator To is LP(R")-bounded for one p0 > 1, then T0# is LP(R" )-bounded for every p > 1, due to the LP(R")-boundedness property of the Hardy-Littlewood maximal function for every p > 1. 27 2.3 Singular Integral Operators on X + In this section we are going to focus on the problem Of boundedness Of the singular integral Operator as a mapping: T:Lp(X,dm)—>LP(X+,du) forp> 1, when p is a Carleson measure. Unlike the Operator To, the Operator T does not have a nice adjoint. Therefore, we cannot tell whether T is a LP-bounded Operator for p > p0, by using Theorem 2.1. This difficulty can be overcome by using the good-A principle argument (see [5] and [24]). In this section we assume that the operator T, whose trace, To, is represented by the formula (2.1), and the conditions (2.2) and (2.3) with t = 0 , imposed on the kernel K, are satisfied. SO, T can be any operator whose range consists of functions with the domain in X +, so that T f (x, 0) = To f (x), and To satisfies the properties mentioned above. Additionally, we want the Operators T and To tO be related in the following way. There exists a positive constant B so that [Tf(x, t) — T0,tf(x)| g BHf(x, t) for every x E X, and t > 0 (2.4) for every continuous function f with compact support in X. Now, we state the main result Of this chapter. Theorem 2.4 Let u be a Carleson measure on X +,and let T and To be the operators defined as above which satisfy (24) If T0# : Lp°(X, dm) —> LP°(X, dm) is a bounded operator for some 1 < p0 < 00, then the operator T: L”°(X, dm) —> LP°(X+, du) 28 is bounded; in other words, there exist a positive constant Cpo > 0 (that depends only on p0) so that llellLP0(X+.du) S Cpollflle°(X,dm) for every f E LP°(X,dm). Proof: Without loss of generality we can assume that f is a continuous function with compact support in X, since the set CO(X) is dense in L1"0 (X). Let us fix a A > 0. We want to estimate the u—measure Of the set A = {(x,t) E XJ’Il71"f($,t)|> 3A.Hf($.t) S 7A}. where 0 < 7 < 1 is a fixed number to be specified later. We set QA={xEX:T3¢f(x)>A}. If ”y is small, then property (2.4) yields (x,t) E A; implies x 6 (2;, (2.5) because 3A < |Tf(x,t)| s ITo,tf(x)| + |To,.f(x) — Tf(x.t)|. s Tatum) + BHf(x, t) S ma) + 87A. SO if we choose 7 so small that 3 — B7 > 1, claim (2.5) holds. (More explicitly, we need '7 < 2 / B, but we may need '7 to be even smaller; so we will not specify its upper bound for now.) The set (I; is open and bounded. Let the family Of quasi-balls {B(x, r(x))}xen,, where r(x) is chosen so that B(x, r(x)) C (I; but B“ (x, r(x)) 0 {ti sé (ll, be a covering of 9» (Recall that B" represents the quasi-ball with the same center as B, but r(B") = K r(B) where K is the space constant from the CalderOn covering Lemma.) Using the covering lemma we Obtain a sequence Of pairwise disjoint quasi-balls {Bj} 29 such that Q; C LLB}. Moreover, for every index j there is a point w, E B;, so that To#f(wj) S A- The set R), = {M f > 7A} is also Open and bounded. Then, in the same way as in case of 9x, we Obtain a sequence of pairwise disjoint quasi-balls {Dk}, D, C RA, such that R; C UkD;, and accordingly, for every index k, there is a point xk E D; so that Mf(fL'k) S 7A. We set PJ— 2 (B; x [0,00)) 0 A), and P; = (D; x [0, 00)) 0 AA. Formula (2.5) implies A; C Uij. Therefore, in order to estimate p(AA) we need to estimate u(Pj) for each j. Let (xo,t0) E Pj (which means x0 E B; ) for some index j. If BJ- 0 R,\ = (b, then we have found a point x0 E B; such that M f (x0) 3 7A. If BJ- n R,\ ¢ 0, then x0 E 81- F] D; for some index k. But, for every index k there is a point xk E D, such that Mf(IEk) S 7A. We distinguish two cases: r(Bj) S r(Dk) and r(Bj) 2 r(Dk). In the case r(Bj) Z r(Dk), which implies D; C B3“ (because x0 E B,- H D; # (b), we set f(x) = f(IL‘)XB;Ie + f(a7)XX\B;It : f1($) + f2($)v where Bf denotes the quasi-ball in X concentric to B], and r(Bf) = 20K kpr(BJ-). (K is the space constant from the covering lemma, and k,, the constant from the condition (iii) in the definition of pseudo-metric.) Notice that the set B; contains points 21), (T 0# (w) < A), and xk. The formula P]: P;- r1({le1|> A.Hf S 7A} 0 {le2| > 2A,Hf 3 7M) .—_ le U p32, 30 shows us that in order to estimate p(Pj), it suffices to estimate each of u(PJ-l), and u(PJ-z) separately. Let P; = 1310 (B; x (t < Ar(B,-)) u B; x (t 2 Ar(BJ-))) = P,“ 0 10,12, where A = 40K k3. Since the measure u is a Carleson measure we have MP?) 5 #(S(ABj)) s CIBjI. We will show that when 7 > 0 is small, then P,-12 = (0. Let (x, t) E P312. Using conditions (2.4), (2.2), and the fact that H f (x, t) 3 7A, we conclude A< le1(:v.t)l s lTo,tf1(x,t)l+BvAS / ) |K(x.y.0)||f1(y)ldm(y)+BvA, p(x,!) >t which implies (1—Bry)A=bAt} [B(IZI, p(IB, y))l |f(y)|dm(y) But the integral on the left is 0, which yields a contradiction, if 1 — 37 > 0, which happens when 7 is small. We will show that Bf fl {p(x, y) > t} = (b, using the following argument. Let c be the center Of the quasi-ball Bj. Then p(x, y) S kp(p(x. 6) + p(y, 6)) S kp(p(x. C) + 20Kka(BJ-)). since y E Bf. The fact that p(x, y) > t 2 Ar(Bj) implies p(x,y) s k. (p(x, c) + ”Tue, w) = we. c) + gm, :1) Hence 2kpp(x. 6) 2 p(x, y) > 40Kk§T(Bj). which yields p(x, c) 2 20Kk,,r(B;), 31 which contradicts the fact that x E B}, so the set over which we integrate must be empty. Now, we claim that P]2 = (l), for 'y sufficiently small. Condition (2.4) implies that for any (x, t) E P].2 we have le2($,t)| S [T0,tf2(33)| + B’M SO, we need to estimate [T0,tf2(x)|, when (x, t) E P12. Recall that w,- E B; is such a point that To# f (w,) < A, which implies T x =/ Kx,y,0 fy dm y l O.tf2( )l (X\Bf)n{y:p(x,y)>t} ( l ( ) ( ll < f K 21):, ,0 dm _ ( \ f)fl{y:p(x.y)>t} ( J 3/ )f(3/) (yl‘ + (,(,,. WM, 21, 0) — Kay, 0)||f(y)|dm(y) = I + 11, (Notice that in the second integral we integrate over a larger set.) Let us estimate the term I, first. Let 17 = 21ka,r(Bj). Ift > 17 and p(x, y) > t, then using the property (iii) of pseudometrics we conclude p(cj, y) _>_ 20K k,,r(B,-) = r(Bf) (where c, is the center of the quasi-ball Bj), which implies B? C B(x,17). Hence, when t > n we have I = [{y:p(x,y)>t} K(wj’ y, O)f(y)dm(y)l I< / KM. 31. 0)f(y)dm(y)’ {y=p(wj .y)>t} +./ K 111', ,0 dm {y=p(w,-.y)st}n{y:p(z,y)>t}I ( J y )Hffilll (y) +./ K 10', ,0 dm {y=p(wj.y)>t}n{y:p(x,y)gc}I ( J y )Hflyll (3/) 32 : [T0,tf(wj)| + A + B. If y E {y : p(wj, y) g t} n {y : p(x, y) > t}, then the property (iii) of pseudo-metrics implies ct S t_ 2Kkkpr(Bj) S p(wj.y) < t. p for some 0 < 1. Employing the property |K(w,-,y,0)| S C|B(wj,p(wj,y))|"1 we Obtain that 1 A t} 0 {y = p($,y) S t}. then t < p(wj,y) < 2kpt, and in the same way as in the case of the term A we Obtain B s C inf Mf(x) g CMf(xk). 3:68:11t Thus, when t > n we have Ift S 77, then I < T w- +C B ~, 10-, ‘1 d _ I ..n ,)l {ymmwl (w. p( . y))| |f(y)| m(y) +0 IB(w.-,p(wj.y))|‘llf(y)ldm(y) S A + E + F. {y=p($.y)St}\{y:p(wJ' .y)>n} If y E {y = p(wj.y) S v} \ 3}" then on = 19Kr(Bj)p(wj. y) < n- 33 (The last claim is true, because whenever a E B}, and b E X \ Bf , the property (iii) Of pseudo-metric implies p(a, b) _>_ W949 _>_ 19K r(Bj), where c is the center of p the quasi-ball 8,. In our case w,- E B; and y E X \ Bf.) Notice that Bf C B(wj, 77), therefore, using the same argument as when estimating the term A we conclude E g C inf Mf(x) g CMf(xk). # 2:68]. If y E {y : p(x,y) _<_ t} \ {y : p(wj,y) > 77}, then 77 S p(wj’y) < 2kp7la which yields F g C inf Mf(x) g CMf(x,,). # xEBJ. When we summarize the last four results we have that for t > 77 as well as for t g 17 we have In order to estimate the term II we use the assumption (2.3) (Notice that 2p(x, Ulj) S p(x,y), because x, w, E B;, and y E X \ Bf.) to get _ fxw,t p(x, y)‘lB(x. p(x, y))| |f(y)ldm(y) T . . oo |f(y)ldm(y) S (K (3.7)) "go [,Mlgfwmgf p(x,y)e|B(x,p(x,y))|. Since p(x, y) 2 2’"20Kr(BJ-) on 2’"“BJ# \ 2me, we have °° 1 S C(K’"(B”))‘m:=. |B(x,2mr)I2mer'f(y)'dy' |B(x.2'"+‘r(B,#))l Usmg the fact that I B(x,2mr(BJ#))| S C, where C is the doubling constant for the measure m, we get II 3 CM f(xk) Z 2““ 3 07A, m=0 34 because xk E B; C 2'"B]# for every index m. We have proved |Tf2(x,t)| S A + C'yA + BryA when (x, t) E P}. Now, we choose 7 to be so small that (C + B)'y < 1, (Now we can fix 7 > 0.) to get that the set PJ-2 is empty. If r(Bj) g r(Dk), then B; C D}: (Recall x0 E B,- 0 D; 94 0.) and we write: f($) : f(x))(Df + f(33)Xx\D:It = f1($) + (2(1)) The same argument as in the other case, (now xk, wj E D; ) leads to u(Pt) S Clekl- Let .1. = {j e N: B; c of}. Then #(UJEJkPj) S 0MB)? x [000) r) AA), which implies “(A05 2: #(le‘fZMUjean) j€(Uka)° k S 2 u(Pj)+Zu(Df> out + |{Mf > on). (2.6) To finish proving the theorem we proceed as follows. / lelp°du = C /°° A”°“u({Tf > WW 3 c [0” APO-mum > mm + c [0” APO-WT: > 3w: 5 7A})dx\. 35 here, we have used the formula A C (A n B) U BC. Then (2.6) yields 3 0(7) (f(Hf)”°du + [(throdm + ((Mf)"°dm) . Finally, using the fact that the Operator H : L”0 (X, dm) —+ L” (X +, du) is bounded (since p0 > 1), the assumption Of the LP°-boundedness of T0#, the Lp0 boundedness Of the Hardy-Littlewood maximal function for any p0 > 1, and the fact that 7 is fixed, we Obtain that the last line is dominated by C||f lli‘imxem), which completes the proof of the theorem. Remark 1: In case when p0 = 1 we consider the case when the operator T0# is weak type 1-1 bounded. If all the assumptions of Theorem 2.4 are satisfied then the Operator T is also a weak—type 1-1 bounded operator. Remark 2: We can also prove the vector-valued version of Theorem 2.4. We generalize the Operator T in the same way as we did in Theorem 2.1, and also we can define the corresponding vector-valued Operator To. Essentially, in the proof Of Theorem 2.4 we need to replace the absolute value brackets by one Of the norms: ||.||E, ||.|lF, or ||.||L(E,p) (including the absolute value brackets in the definition Of the HOrmander maximal function), and instead of D" spaces we consider their vector version, either L1,; or L52. The integrals in the proof of Theorem 2.4 now become vector-integrals. The following lemma is a useful tool when checking if condition (2.4) is satisfied. 36 Lemma 2.5 Let X be such a space of homogeneous type. Suppose there exists a positive integer d such that |B(x,r)| = Crd for every x E X and r > 0. Then, condition (2.4), for the operators T and To, associated with the kernel K, is satisfied if there exists a positive constant C such that I 1 WW” " K(x’y’ 0“ S C (W " (p(x,y) + 00*) ’ (2'7) for every x, y E X andt Z 0, such that p(x, y) > t. Proof: TO prove the lemma we proceed as follows. ma, t) — use» 3 / |K(:v. y. t) — K(x, y. 0)l|f(y)|dm(v) p(x,y)>t + / |K(:v, mum/Mme) = 11+ 1.. p(x,y)St Property (2.2) yields the following estimate for the term 12. I I g C/ 2 B(x,t) |B($,p($, y) + t)' If (y)ldm(y) 1 S Cm B(x,t) lf(y)|dm(y) S CHf(JSJ) In order to estimate the term II, we use (2.7) and some elementary computation. First, the binomial formula implies (p(x, y) + t)“ - p(x, y)"l t p(x,y)d S Cm. y) ’ whenever p(x, y) > t. Thus, we have t |f(y)| 11 S C‘/P($vyl>t p(x,y) |B($,p($,y) + tll m(y) : of: t |f(y)| dm(y) 1/2k-It 1. For p E (0,1] fixed, we define a p-atom as a Borel measurable function on X whose support is in a quasi-ball B, Ix a = 0, and [a] S C ]B|"1/”. Let Ck be constants defined as in Section 1.3. If A > 0 is the doubling constant for the measure m, i. e. for every quasi-ball B C X |B(z,2r)| g A]B(z,r)|, then 0,, 3 Ah. Theorem 2.6 Let T be the operator defined as above, with the constant e as in the condition (2.3), and let p E (0,1] be such a number that the series 2,, C,:—“’2""5’c converges. Then there exists a positive constant C, that depends on the space X, p, e, and the measure 71, such that for every p-atom a we have / |Ta(x, mum, t) g o. 38 Proof: Let B = B(z, r) be the quasi-ball associated with the p-atom a. Hélder’s inequality applied tO the integral T t pd t [T(ml u(x. )I use ). using the conjugate exponents p0 / p and 1/(1 — p/po), yields P/Po S ([71?)ITa(x,t)|”°du(x,t)) (#(T(B*)))1—P/Po. The LPO-boundedness of T implies s C (/ ...(y)lpodm(,))”/”° (u(T(B*)))1’p/”° . n: 1“fl/Po s CIBI””’°“(u(T(B‘)))“”/"° = c [_“(Ylbl ”) , which is dominated by a constant due to the fact that u is a Carleson measure. When x E X \ B“, the property f a = 0 implies ITa(x. t)| = I/Bma, y. t) — K(x, 2. t))a(y>dm|(p(x, z) + t)" Let us define sets Ak = {(x,t) E X+ : 2k_1r(B*) g p(x, z) +t < 2kr(B“)}, 39 for k = 1, 2, . Then, X+ \T(B") is a disjoint union Of the sets {Ak},‘:‘_’__1. Thus Ta x,t pd x,t = / Ta x,t pd x,t. (mm: < )l M ) )_: Akl ( )l u( > The formula (2.8) implies (Mar. 0 2. p(z. 2) + t)lp(p(:v. 2) + trp' Ta x,t 1r’d x,t _ ll gmm Using the facts that Ak C T(B(z,2’°r(B‘))), and p(x,z) +t > 2"“1r(B") whenever (x,t) E A,“ we get B(z, 2"7"(B“))))|2"B"|1"’ —cpk |2kBt|pl2kB¥|l-prp£ ' Ta x,t pd x,t < Crp‘ B ”‘1 M L+\T(B‘)| ( )l M )_ I l I; We have also used the fact thatl—zl,,2—f—,BT;,J,—l S A. The facts that u is a Carleson measure, and m is a doubling measure, imply °° 1 g 00,, Z of, “Pb-Wk, k=l since the series above is convergent by the assumption Of the theorem, the expression on the left side of the formula above is dominated by a constant, which completes the proof Of the theorem. I. Remark: In case when X = R", we have that Ck = 2". Then the assumption about the convergence of 2,c Cfi'p2‘P‘k becomes 6 > i- — 1. 2.5 Examples Let X be Euclidean space R", with Euclidean distance being the metric, provided with the following measure. Let w : R" —+ [0, 00) be a bounded homogeneous function with the degree of homogeneity 0, such that inf w 2 c > 0. Then, the function w is 40 bounded, and also an A1 weight, since w(x) fiéwhflx _<_ C S C = Cw(x). C Therefore, the measure w(E) = f3w(x)dx E C R" is a doubling measure, and moreover it is comparable to Lebesgue measure on R“; i. e. there exist positive constants c and C such that for every cube Q in R" we have CIQI S w(Q) S CIQI. where |Q| represents the Lebesgue measure of the cube Q. Also, it easy to see that such a weight w belongs to the Muckenhoupt class A, for any p > 1. We define T f as the principal value of Tf(et) = / K($,y.t)f(y)w(y)dy. n for every function f E Cf,” (Rn). The integral above makes sense, because the function w does not create a new, nor eliminates any of the existing singularities. Moreover, if the kernel K satisfies K (Ax, Ay,0) = A“"K(x,y,0) for every A > 0 and x, y E R", and if we assume the boundedness of the operator To : LP(R", dw) —> LP(R", dw) for some p > 1, then the operator T3"6 is bounded for every p > 1, because w E Ap (Muckenhoupt class, see [15]) for every p > 1. 41 Harmonic Kernels Harmonic Conjugate in the Complex Plane As we know the complex conjugate to the harmonic function u(x, t) = P, at: f (x) is given by the formula 1 co _ Wat) = —7; I... (x f 21).}; fife/My for f e cr(R"). where the integral is taken in the principal value sense. Motivated by that fact, we set k(x) = ififi, that produces the kernel 1 1 x—y __ “3—9 :_ K(x,y,t)—tk( t ) r(x—y)2+t2’ which for t = 0 induces k0(x) = 11—3, the kernel Of the Hilbert transform, that is a homogeneous function with the degree of homogeneity—1. It is a well known fact that the corresponding maximal singular integral Operator T0# ; LP(R, dx) —> LP(R, dx) is bounded for every p > 1. A singular integral Operator in the upper half-plane is defined by Tf($.t) = 71;]: (x f52y+t2f(y)dv for f E 08°(1R"). taken in the principal value sense. The operator T satisfies all the assumptions in the definition of the singular integral operators on R1. The assumptions (2.2), and (2.3) (with 6 = 1) are fulfilled because 2 2 , |k($)| < —— and |(Vk)($)| = |k (I)| S (1_+—|$—|)—2' _ (1 + lxl) Thus, we only need to check if the condition (2.4) is satisfied. Elementary computa- tions yield NITRIC, t) - To,tf($)l = l/ (iXbet — 3,7375) f(flC — yldyl 42 Is] t2 _ lylSty2 l t2|f( y)l y+ M), |y|(y2 I t2)|f( y)| y < B(wmx — y)|dv + h», Bum: — with s Hf(x, t). _ lylSt Therefore, the condition (2.4) is satisfied. Thus, the Operator T: L”(R,dx) —+ L”(R2+,du), is bounded for every p > 1, and weak type 1-1 bounded, when u is a Carleson measure. Harmonic Conjugates of Functions in Several Variables. In his attempt to define the conjugate functions to the function u(x,t) =P,*f(x) xER“,t20,fEC§°(R”), in several variables, J. Horvath (see [13]) considered the integrals u(x,—w) . ' 7t :/ n 1 d =1727°ua ; w(x ) Rn (|x—y|2+t2) i2 f(y) 3/ J n taken in the principal value sense, where x, — y, denotes j-th coordinate of x — y = (x, — yl, ..., xn — yn). Therefore, we will consider the kernels CnIL'j (|x]2 + 013‘” kj($) = x = (x1,x2, ...,xn), 7': 1,2, ...,n, that each produces the kernel . _ Cn(-Tj - yj) KJ($: y: t) _ (l2? _ y|2 + t2)9_3-_17 so that kj'0(x) = cnxj|x|""‘1, is the kernel of the Riesz transform R]- ( j = 1, 2, ..., n). Notice that each km is homogeneous with the degree —n. We will consider the following singular integral Operator _ Cum — 312') w Tf($.t) — / (Ix _ y|2 +t2)1,_1f(y) (y)dy. 43 Since the measures w(E) and IE I are comparable for every set E C R", u is a Carleson measure with respect to w. Let us check if the Operator defined by T f = u], for any fixed j E {1, 2, ..., n}, is a singular integral operator on R1“. The elementary formula %(a + b)2 S a2 + b2 g (a + b)2 for a, b 2 0, yields 2”21 (n+ 2)2"_+‘21 . < _ . < ..______ IkJ($)I — (le + 1)" and IVkJ($)I — (ISEI + 1)n+1’ which imply (2.2) and (2.3). Now, we check if the property (2.4) holds. lTo,tf(rr) — Tf(x, t)| s [W (M261?!) _._ w(x - y)|w(y)dy+ m n +/ cflijI[(IyI2 + t2) 2 _ IyI +1] If(.’L‘ __ y)Iw(y)dy mzt |y|"+‘(|y|2 + at? If IyI > t, then elementary calculations yield p_+_1 calyjll(|y|2+t2) 2 - lyln+ll < Cnl(|y|+t)"“ - lyl'ml _ S cn(2n — 1 t: |y|"+1 lyl" ) this, with the boundedness of w, implies IT0,tf($) - Tf(x, t)| S 0/in P¢(y)|f($ - y)|dy + 0/]le Pt(y)|f($ - y)ldv = 0 / Baum: — play, which proves that the property (2.4) holds. Now, we need to consider the boundedness of the operator T0# : mu", dw) —+ L”(R", dw). First, notice that the kernel k(x, y, 0) = kj(x—y)w(y) satisfies k(Ax, Ay, 0) = A‘"k(x, y, 0), which implies that the corresponding Operator To satisfies the weak boundedness prop- erty (see [10]), and due to the Oddness of kg in both x and y, both conditions T01 = 0 44 and T51 = 0 are satisfied, which makes the Operator To : LP(R", dx) —+ LP(R", dx), bounded for every p > 1. Lemma 2.3 implies that the Operator T? : LP(R", dx) —> LP(R",dx) is bounded, and using the Muckenhoupt theory we can conclude the boundedness of the operator T3It : L”(R",dw) ——> LP(R",dw), because w E Ap for every p > 1. Therefore, the Operator T : L”(R", dw) —> L”(R1+1,du), is bounded for every p > 1 and weak-type 1-1 bounded. In the case Of a singular integral operator whose kernel is harmonic, and when w(x) = 1 we may Obtain the LP-boundedness Of the Operator T, in the following way. It is a well known fact that for the complex conjugate ii. Of a harmonic function ‘11. = Pt * f7 u(x,t) = (P, * ’H * f)(x), where 71 represents the Hilbert transform. When n > 1, J. Horvath proved that uj(x,t) = (P, at R]- * f)(x) 7': 1,2, ...,n, where RJ-f is the Riesz transform Of the function f in R". The formula (1.3) implies IITfIILP(R1+1,du) = “Pt * Rj * fIILP(R1+l,du) S CIIRJ' * fIILP(R",dm)a and the LP-boundedness Of the Riesz transform ( p > 1 ) yields IITfIImegHep) S CIIfIILP(R",dm)- (Notice that in the case n = 1 the Riesz transform becomes the Hilbert transform). 45 Examples — Non-harmonic Kernels In case n = 1, the kernel _ sign(x)IxI°‘ K(x,t) — |$|a+1+ta+1 3: 6 WW 2 0. is harmonic only for oz = 1; for a 96 1 the kernel is not harmonic. We claim that the corresponding convolution Operator T satisfies all the assump- tions in the definition of the singular integral Operator for a convenient a. It is easy to check that conditions (2.2) and (2.3) are satisfied for any a > 0. To show that property (2.4) holds, according to Lemma 2.5 we need to determine if ' a _1_ _ szgn(x)IxI < t —— he >t, x |e|a+1+ta+1 —IxI(IxI+t) W “le . to 1 Wthh follows from Iml°+1+t°+1 S 1,11,, which is true whenever [x] > t > 0. The corresponding Operator To is the Hilbert transform, whose maximal singular integral Operator is LID-bounded for every p > 1, thus the singular integral Operator T : L"(R", dx) —> L”(R1+1,du), corresponding to the kernel K is bounded for every p > 1 and weak-type 1-1 bounded, when ,u is a Carleson measure. When n > 1, let (2 be an odd (that is Q(—x) = —Q(x) ), bounded, and homoge- neous function on R“, with degree Of homogeneity 0 ( i. e. for every A > 0 we have {1(Ax) = f)(x) ), such that C IVQ(x)I g m for every x E R", x 76 0. Let T be the convolution Operator associated with the kernel Q(x) K($,t) = w. 46 It is a well-known fact that the convolution Operator T0 with the kernel K (x,0) has Lp—bounded maximal singular integral Operator for every p > 1 (see [23]). The function K clearly satisfies condition (2.2) because (I is a bounded function. Using the formula ax _ 563(le +t)" — 911(Iar|+t)"‘lfl ll‘l 0x,- — (IxI + t)?" ’ boundedness of f1, and IVQ(x)| _<_ 1% we obtain C K t < ———— 'V (“3’ l' - Lp(R1+l,du), defined as above, is bounded for every p > 1 and weak-type 1-1 bounded when u is a Carleson measure. 2.6 Cauchy Kernel on C” (n > 1) Let X = S be the unit sphere in C", p(x, y) = Il—(x, y)|1/2 (where (x, y) = 22=1xkyk) be the non-isotopic metric on S, and o be the rotation invariant measure on S. When n > 1, we have o(B(x,r)) x r2" where B(x,r) = {y E S : p(x,y) < r}, (2.9) (The symbol >< means there exist positive constants c and C so that for every r > 0 we have or” g o(B(x,r)) S Cr2n.) 47 The role of X + will be played by the closed unit ball U in C". The fact that 0 g t _<_ 1 makes almost no difference. We take t2 + r2 = 1, where r = IzI, so that S is obtained when t = 0. Before we proceed, let us list some of the properties of the non-isotopic metric (For the proof Of these properties and the formula (2.9) see [17]) : (a) For every x, y, z E (7, where (7 is the closed unit ball in C" we have p(x, y) S p(x, 2) + p(Z. y)- (b) p is a metric on S. (c) For every 0 S r S 1, and x,z E S we have p(rx,z) 2 I1— r(x,z)I1/2 Z \/1—_——7_'. (d) For every 0 g r S 1, and x E S we have p(rx,x) = x/I——_r. (e) IfU is a unitary map, i. e. (Ux,Uy) = (x,y), then p(Ux, Uy) = p(x, y). and for every x E S there is a unitary map U such that Ux = e, = (1,0,0, 0..). (f) For every 0 g r 5 1, and x,z E S we have p(x,z) + \/1 — r 3 . p(rz. 2) 2 The last property follows from (a), (b), and (c) as follows. p(x, z) + v1 — r _<_ p(rx,x) + p(rx,x) + \/1 — r S 3p(rx,z). 48 For every 0 S r S 1, and x, y E S we define the Cauchy kernel as C,(x, y) = C,,(l — r(x,y))"" where r2 + t2 = 1. The Cauchy kernel is a complex valued function and IC,(x, -)I is not absolutely in- tegrable. Therefore the kernel is not admissible. We will prove that the Operator represented by the formula Tf(w) = / Ct($.y)f(y)d0(y). is a singular integral operator. First, we check if for every 0 S r S 1, and x, y E S |Ct($.v)| S C0(B(flc,)0(av.y) + 0)“- The definition of C,, property (f), the fact \/1 — r X t, and property (2.9), imply ICt($. y)| = p(rx. 30’2" S C(pbv. y) + 0‘2" = 00(B($.p(x. y) + t))’1- Next, we check if for every 0 g r g 1, and x, y E S such that 2p(x, y) g p(x, z) +t (r2 + t2 = 1) it follows that x z _ Z p(x,y)2 IC‘( ’ ) C,(y, )I S C'or(B(x,p(x,z) + t))(p(x,z) + t)2' To show that the above inequality holds, we employ the formula a." — b" = (a — b) 22:1 an’kbk'l, to get lCt(x. z) — as, 2,, = r|-2<"-'°>p. k=1 Property (f) implies S CTICK — y. z)| X(ptt. 2) + t)‘2‘"""(p(y. 2) + 0’2““)- k=l 49 Recalling that p(y, z) +t Z p(x, z) + t — p(x, y) 2 %(p(x, z) + t), we Obtain = CTI<$ - y.Z)|(p($.Z)+t)'2("+1). and now we use property (2.9) , again, to conclude = CTKI - y.Z)| 0(B($.P($,Z) +t))"1(p($.z) +15)”. so we need to prove that I(x —- y, zll S Cp(x, y)2. To show it, we use property (e) to restrict ourselves to the case z = e,. Then I(x — y, z)I = Ix, — y1I, where x1,y1 E C are the first components of x, y E C". Now, the desired inequality follows from P($,y)4 — [<37 — 11,2)? = I1 — $1.771I2 — I131 — yll2 =(1-|$1I2)(1-lv1|2) 2 0. Then, we check if the singular integral Operator T, associated with the Cauchy kernel, satisfies the property ITf(x. 15) - To,tf($)| S BHf(x, t)- Using C,(x, y) = C,,(rx, y) we Obtain lCt(x. y) - Catt. y)| S C |1 - Tl| t, and then the proof goes the same as in Lemma 2.5. Now, let us check if the restriction of the Operator T to the space X, the Operator To, has its maximal singular integral Operator bounded. Property (2.4) implies ITo,,f(x)I S ITf(x,t)I + BHf(x,t) for every x E X,t _>_ 0. 50 When we take the supremum of both sides of the last inequality, with respect to t 2 0 we Obtain To#f($) S sup ITf(x. t)| + BMf($). :20 where M, represents the Hardy-Littlewood maximal function on S. Thus T0#f(33) s Ne(Tf)(x) + BMW). where Na represents the non-tangential maximal function on the unit ball. Theorem 6.3.1 (page 99, [17]), tells us that the Operator Na(T) is an LID-bounded operator for every p > 1, and the LID-boundedness Of the Hardy-Littlewood maximal function is a well known fact. Therefore the operator T0# : L"(S, do) —> L”(S', do), is bounded for every p > 1. Hence, the Operator T: L”(X,do) —> Lp(X+,du), where u is a Carleson measure on the closed unit ball, U in C" (n > 1), is bounded for every p > 1 and weak-type 1-1 bounded. 2.7 Cauchy-Szegfi Kernel on the Heisenberg Group Let X = R x C"‘1 , and let a be Lebesgue measure on R2n‘l. If we write an element x E X as x = (x1,x2), where x, E R, and x2 E 0‘“, then for x,y E X (y = (y,,y2)) we define operation 0 on X as 1‘ ° 3/ 2‘ (171+ 311+ 2%(x2,y2),x2 + 92), where (x2,y2) denotes the scalar product in 0‘“. (X ,o) is a group, with (0,0) being the neutral element, and (—x1,—x2) the inverse to (x1,x2). We define the 51 pseudometric, p on this group by _ 1/2 p(x, y) = 7(23 o y 1) where 70?) = (lxllz + 1x214) . The pseudometric p is invariant under the group action which means, that for every x,y,z E X, we have p(x,y) = p(xoz,yoz). It has been proved, see [14], that o(B(x,r)) x r" where B(x,r) = {y E X : p(x,y) < r} (2.10) We define the Cauchy-Szegb' Kernel on X by —n u(x, y) = c. (t + Ix. — yer — 21x. — y.— 2%.. ye») , for every x, y E X and t 2 0. The Cauchy-SzegO Kernel is invariant under the group action, which means that for every x, y, z E X and t 2 0, we have C,(x,y) = C,(x o z,y o 2). Let us prove that the integral Operator defined by the Cauchy-Szegéi kernel is a singular integral Operator on X +. First, we check if there exists a positive constant C such that C 0(B($. p(x, y) + t))’ CARI) y) S for every x, y E X and t Z 0. Knowing that the Cauchy-SzegO Kernel is invariant under the group action, it suffices to prove the claim for y = 0. , —n —n/2 IC,(x,0)| = C,, It + Ix2I2 + leI = C,, I(t + Isz2)2 + x1] —n/2 —n/2 S C,, It2 + ngI4 + xr‘l’I = C,, It2 + 7(x)2 S 2"celt + 7($)|‘" = C0(13(=17.)0(II=.0)+t))'1. 52 the last equation holds due to formula (2.10). Next, we verify that 1/2 Iona, 2) — C.(y. 2)) < ”(1" y) - Co. Thus, in order to prove the formula (2.11) we need to prove Ike)? — 1m? — )(y. — ml 3 Cp($,y)1/2(P($,0) +101”. Simple computations yield I|y2|2 - 1x212 - 21y. — ml 5 le2|2 -|y2|2I+lx1— m. S I152 — y2I(|$2I+Il/2I)+I$1— 3/1— 23($2.y2)I + [2301326132 — .712”- (In the last line we have used the fact that 8(x2, x2) = 0.) Using that ng — y2| S 003,101” '2 (I552 — 312'4 — I171— 111— 23(32,92>I2)1/4, 53 Ix, - 111 - 23 t, and then the proof goes the same as in Lemma 2.5. Thus, we have proved that the convolution Operator associated with the Cauchy- SzegO kernel is a singular integral Operator. Chapter 3 Singular Integral Operators on Euclidean Space In this chapter we study under what conditions imposed on the Carleson measure 71 and the kernel K, the singular integral operator T: L2(R",dx) —> L2(R1“,dp), is bounded. Euclidean space, R" is a space of homogeneous type with the quasi-metric being Euclidean distance, and Lebesgue measure being the doubling measure. The kernel K is still assumed to be a continuous map KzR" XR" x [0,oo)\{x=y,t=0}—>C, satisfying conditions (2.2) and (2.3). Additionally, we assume that the kernel K satisfies the following two cancelation properties. Kx, ,td =0, 3.1 L($,R)\B(x’r) ( y )y ( ) for everyR>r>0,t20,andeR"; and Kilt, ,th’,z,td x,t =0, 3.2 /S(B(y.R))\S(B(y.p)) ( y ) ( )“( l ( l for every y E R", Ix’ — z] > p, t 2 0, and R > p > 0. The first cancelation property is supposed to play the role of the T1-condition together with the weak 54 55 boundedness property, while the second plays the rOle Of T‘l-condition, that requires T‘(const) = 0, but now we ask for more. T*

0 or Ix —yI > 0, due to (2.2). When it is not the case, we define T f (x, 0) as in the previous chapter. Using the Calderén formula, whose proof is based on the properties Of the Fourier transform on R", it is established that any function f E L2(R" , dx) can be decomposed into the sum of the smooth atoms (see [10]). We define a smooth atom as follows. Definition: A function aQ E C8°(R") is a smooth atom in L2(R", dx) associated with the dyadic cube Q if it satisfies: (a) aQ is supported inside the cube Q”, (Recall that Q" denotes the cube concentric with the cube Q such that I(Q‘) = 3€(Q).), (blfao=0. (c) IDIaQ(x)I S c,€(Q)‘I7I“"/2 for every multi-index 7, and x E R". The following theorem (see [10]) tells us that any function f E L2(R", dx) can be written as an infinite linear combination Of smooth atoms. 56 Theorem 3.1 For any f E L2(R",dx), there exists a sequence {Sq} E [2, and a sequence of smooth atoms {aQ}, such that f = 2:3an and IIfIIL2(R",dx) = C: ISQI2, Q Q where the convergence of the series is taken in the L2-sense, and the summation is over the family of all dyadic cubes in R". For the proof see [10]. Now, we are ready to prove the main result Of this chapter. Theorem 3.2 Let u be a Carleson measure, and let T be a linear operator repre- sented by the formula (3.3), that satisfies conditions (2. 2), (2. 3), and the cancelation properties (3.1), and (3.2). Then the operator T : L2(R", dx) ——) L2(R1“,d,u), is bounded. Proof: The proof of this theorem is an adaptation of the technique developed in [10], applied to this new situation. The proof of the theorem is rather technical, and it will use several lemmas. Before we state and start proving the lemmas, let us fix dyadic cubes P and Q, let xp and xQ denote the center Of the cube P and Q, respectively, and let ap and ac, be smooth atoms associated with cubes P and Q. Lemma 3.3 Let ,u be a Carleson measure, and e > 0. Then there exists a constant C > 0 depending only on ,u and 6, so that for every cube Q, and every point z E R", duh. t) f I I s CIQI. n+1 3-2 “+5 R4" (1 + «th) 57 Proof of Lemma 3.3: Let A, = {(x,t) e 11:“ :2k_1€(Q) < Ix — zI +t g 2kt(Q)}, for k = 1, 2, Then, R1“ is a disjoint union of the sets {1462.10, where A0 = {(x, t) E R1“ : Ix — z] + t S €(Q)}. Thus / du(x,) t) ZZ/(l du( x ,t) RI“ (l-I— Ix— 2"” )n-l-e Al: +|$_ zI+t )n+e' 5(0) 3(0) On A0, we use the estimate (”Pt—(212' 1,)” H S 1 to get de t) < :1: 2+ n+e — ”(140) f"°1(+LJ-Z(e) ) Since 71 is a Carleson measure, we conclude that dut’rt) /. (+ +J—J—:;(z.:)"+“c“'Q" On each A,“ (k = 1, 2, ...), the estimate 1 < I x z n+5 — x—z "+6, (1+ “4—me ) (L—Lereft) the fact that A, c T(B(z, 2"£(Q))), and L373): > 26- 1 when (x, t) e Ak, imply / du(x, t) <2] d_u__( x, t) Rn+l T B |$_ zI-l-t n+6 A (2112-1) n+6 \ ( )(1+ «0) l " _ u(T B(12(2"€(Q)))) “Cm Q)": |B((.z 2"€(Q))l2’°‘ ' Using the fact that u is a Carleson measure, I(Q)" = [Q], and that the series 22:, 2"“ converges, we conclude that the last line is dominated by CCulQl, 58 which proves the lemma. Lemma 3.4 Ix — xQI + t ((62) ITaQ(x, t)I S Cl(Q)"‘/2 (I + ) for every (x, t) E R1“. Proof of Lemma 3.4: Let (x,t) E S (2Q*). In this case we have Ix—xQI+t 1+ 2(0) S 11, and consequently IIII—Ilth 'l-t)—n_¢E _ _ 1+ 211 " ‘. ( ((62) Using the cancelation property (3.1) we Obtain lTao($. t)| = (WW, K(x, y. t)_ 1 tO conclude _/ Ix—xQI+t -.-. ITaeIx.t)Isc7IQI ”(1+ “,2, ) , which proves the lemma. Lemma 3.5 Let a = min{e/2,1}. Then ((Q) x {(1 + Ix I(SIHI—n—(‘L (1+ ly 22,51,144} for every x,y E R", andt 2 0. ITaeIx.t) — Tao(y.t)l s CIQI-W ('z ‘ 3‘") x Proof of Lemma 3.5: If Ix — yI > I(Q), then Lemma 3.4 applied tO each term on the left side of the inequality, and the fact that 1 < (fily’, imply the lemma. Therefore, let Ix — y] < I(Q). If (x, t) E R1“\S(CQ), where c = 4\/T_t is chosen so that supp(aQ) C B(xq, §€(Q)). (Notice that we could have taken c = 3%, but we may need c to be a little bit larger.) We have ITaQ($,t) — T0001, t)| = I/Q_[K($, 2,15) - K(y. ZatllaQ(Z)dZ - For chosen x,y and t we have that Ix — y] S I(Q) S %(Ix — zI + t) is true for every 2 E Q‘, which enables us to use property (2.3), which together with the fact IaQI S CIQI‘l/2 implies that Ix—yI‘ —1 T ,t —T ,t < / (2d. | 00(33 ) ac201 )|_C q-(Ix—zI+t)"+‘|Q| Z 60 Forevery zE Q‘ we have Ix-xQI+t S Ix—zI+Iz—xQI+t S 2(Ix—zI+t) (The last inequality holds because if (x, t) E R1“ \S(cQ), then either [2 — xQI S Ix — zI or t Z Iz — xQI.) which implies that the above quantity is S ClQll/2|€B - y|‘(l$ - xol + 10"”. =7)”('éet')‘('xzfsl”)”“- Since 177?? < 1 and a < e, we have _12 Ix—yI 0' Ix—qu+t ”he lTao(x.t)-Tao(y.t)|SClQl /( ac») ( ,(Q) ) . The fact Ix — xQI +t Z €(Q) yields I$-$QI+t I(IJI—quI-l-t ) IJZ—qu-l-t I(Iy—xQI+t ) «62) Z2 M) H "d ((Q) 23 2(a) H’ (The last inequality is true since 1 + Wmefgjlfl S 1 + 'x_$QI(+CI‘;“yI+‘ S 2 + W _ 3%.); so we conclude that ('“efsl”)‘""‘30{(l+'“I:.'“)’"“+(”(2:5)“)"1’ which proves the lemma if (x, t) E R1“ \S(cQ). In the case Ix — yI < £(Q), and (x,t) E S(cQ), we have that Ix — xQI S §€(Q), Iy — qu S (g + 1)l(Q), and t S cl(Q). SO, it is easy to conclude that there exists a positive constant 6 = (1 + §)'"“ + (2 + g)‘"“ such that I$-$Q|+t)_n—€ ( Ill—le+t)—n—€ (1+ 8(6)) + 1+ «62) 26' SO all we need to prove is a x __ a —1/2 [93—3100 IT e( ,t) Ta(y.t)|SC‘IQ| (,(,,) . 61 Due to the fact supp(aQ) C Q" C B(x, ct(Q)), we can write TaQ(x,t) — TaQ(y,t) = /|x—z| 1, we estimate the integral in the last formula in the following way r"dr. oC(Q) _ 0° _ 1_ f r ‘dr 1, yields [can ‘Idr— Cln (6%)) )SC ((62) Ix- yl 2I$ - yI I17 — yI, and _ a: - III a III 3 C Q 1/2(I——) I I I I ,(Q) If 6 <1, we have d(Q) _ j r ‘dr g ceIQ)1- 2lav-III which yields III C “/2 I551"). I I: IQI ( (Q) Having in mind that III—of < 1, and a < e, we Obtain 111 < C ‘1/2(I—-—xyl) I I _ IQI ,(Q) Finally, we estimate the term IV. The cancelation property (3.1) implies IIV|= Ime) II / (y. z.t)dz — / K(y.z.t)dz ZISCHQ) Iy-ZISd(Q) 63 s CIQI‘”2 K ,td, f. (v.2)z where A: AIUAQ = {zER" : Iz—xI Sc€(Q) and Iz—yI > c€(Q)}U{z E R": Iz — x] > c€(Q) and I2 — yI S c€(Q)}. The estimate (2.2) implies I/K (,)Sy,ztdz =Cln(1+Ix—yI) golf-1” ceIQ)+Ix—yI 1 C/ r‘ dr c[(C2) 613(Q) ((Q) I because c > 1. Similarly ceIQ) _, S C r dr cl(Q)-I-'I=-y| K td IL (IN. ) z _ Ix-yl ) liv-yl _Cln(1+e€(Q)-lz—yl SC€(Q)’ because c€(Q) — Ix — y] 2 €(Q). When we put all Of those estimates together we get —1/2I$ — III WI 3 CIQI ——,(Q) . which due to the facts that or < 1 and IZ—QIII < 1 yields: 17- yl IVSCQ—1/2(I— y)0, I I I I ,(Q) which completes the proof Of the lemma. I The following lemma is the key ingredient in the proof of Theorem 3.2. Lemma 3.6 Let a = min{e/2,1}, €(P) S €(Q), and let u be a Carleson measure on R1“ . Then I / Tap(x, QWth g c (II—SW (1+ W)“ 64 Proof of Lemma 3.6: The property (3.2) implies / Tap(x, areQIa, nah = Tap(x, t)(TaQ(x, t) — TaQ(xp, t))du. (To see that the property (3.2) implies that f Tap(x,t)mdu = 0, we notice that Lemma 3.3 and Lemma 3.4 imply f ITaI2du S C, where the constant C does not depend on the atom a. If we define Tpa(x,t) = f K p(x,y,t)a(y)dy, where Kp(x, y, t) = K(x, y, t)XS(B(y,%))\S(B(y,p))1 we have that Tpa —-> To as p —> 0, where the convergence is in the L2(du)-sense. Thus, to prove the claim it suffices to show that for every p > 0 we have f Tpap(x, t)Wdu = 0. It follows directly from the condition (3.2) and the Fubini Theorem, since all the functions K p, up, and ac, are bounded over a compact set.) Let A = B(xp,3€(Q)). If x E A, then I$P - IQI S I75 — $QI + I35 — $PI S I13 — IEQI +3€(Q) + t. for every t 2 0. Hence 6 IiE-fi'J‘QI‘ft _IH Imp-WI _IH (II IIQ) I SI” 2(6)) l ' Lemma 3.4, applied to the first factor in the integral below, Lemma 3.5 applied to the second factor in the integral below, together with the simple fact (1+ W)“"" S __Q_ 1(0) (1 + Image) I)-—n"€ and the above estimate, yield /( )Tap($,t)(TaQ($,t) — TaQ(xp,t))d,, s CIPI—1/2x s A x Ix—pr+t ”he —1/2(I$—$PI)a ( IIEP-117QI)—n_E (see (I I IIP) I IQI IIQ) I‘I IIQ) ‘I’I' Elementary calculations yield 65 —1/2 -1/2 I-TP - 1’3(.?I)_n_E (@)a x s CIPI IQI (1+ ——,(Q) ,(Q) I.(1+”2:12)“(2.3“)“1- 12321)“ S (1+ lII—Zfllfl)“ leads to The estimate ( ((1,) ((P) —1/2 —1/2 Il'P-mQI -n—E(§(_P))ax SCIP' 'Q' (II €(Q) l eIQ) I517 — $PI +t —n—e+a 1 d . x [111“ I I [(P) II Since a < 6, Lemma 3.3 applied to the integral above, implies —1/2 -1/2 IxP-xQI)—n—£(€@)a SCIPI IQI (1+ ,(Q) ,(Q) IPI. which proves the lemma in the case (x, t) E S (A) In the case (x, t) E R1“ \S(A), which means Ix — pr +t > 3€(Q), we have 1+I33—113PI‘I't IIB-l‘PI +t€(Q) 1|x—x.l+t€(Q) §@ ((1’) ((Q) ((P) 2 ((Q) ((P) 2((P)I which implies 1+ Ix—pr+t 1(1 Ix—pr+t) I(Q) ((1’) 2 ((62) ((P) Using the last inequality and Lemma 3.4, (having in mind that (x, t) E R1“ \S(A)) we Obtain _ ((P) "+6 Ix — pr + t ”he / _ ITap(x,t)I SCIPI ”(462) (1+ QQ) ) . (3.4) Thus S Tap(£L‘, t) (Taq($, y) — T0003?) tlld“ [11:“\SIA) Tap($,t)TaQ(.’L‘p, t)dfl| = I + II. T ,tT ,td / (w(x ) aQ($ )#I+ R1+I\S(A) [IRZII\S(A) 66 Let us estimate II first. The inequality (3.4), applied to the first factor in the integral II, and Lemma 3.4 applied to the second factor in the integral II, yield —1/2 —1/2 w>n+e( I5’7P‘1’3QI)—n_E ”SCIPI IQI (IIQ) II eIQ) X x 7 (1 + '1 775] + t) ”H... Lemma 3.3 applied to the integral above produces _1/ —/ fl n+e ImP-xQI —n—c IISCIPI IIIQI ”(,(Q) 1+-————-,(Q) IQI- Since a < e and %I% S 1, we have which is the desired estimate for II. In order to estimate I, we need to consider the sets B={(x,t) ER1+I\S(A):2(Ix—xpl+t) > pr—xQI} and C = {(x,t) E R1.“ \S(A) 22(I1E — (lipI + t) S pr — QIQI}. The inequality (3.4), applied to the first factor in the integral bellow, and Lemma 3.4, applied to the second factor in the integral bellow, yield __ Z P n+e fBTap(x,t)TaQ(x,t)d,uI S CIPI‘I/2IQI‘I/2 (2%) x ., (1+ 'x‘.:”.°)+‘)""“ (1+ ”2.3)“)"‘1 Since (x, t) E B, we have ILII—IIIPI"I"t)—n-‘E ( I$p—$QI)_n_e (II am 30 II IIQ) ’ 67 which together with Lemma 3.3 implies I/B TaP(x, tlmdpl S C (5(5)) n/2+e (1 + M) —n—¢' ((Q) ((Q) Again, we use the facts that a < e and £9?) S 1 to get the desired estimate for the C(Q) integral in I over the set B. If (x, t) E C, then Ix — xQI +t Z pr — xQI — (Ix — pr + t) Z %pr — xQI. This fact, inequality (3.4), applied to the first factor in the integral below, and Lemma 3.4, applied to the second factor in the integral below, yield —1/ —1/ {(32 .... ILTap(x.t)TaQ(cv.t)duISCIPI IIQI 2(,(Q) x pr—xQI ”HE Ix—xpl+t ”he XIII—IIQT) (C(I“ €(Q) I ‘II’ and Lemma 3.3 implies —1/2 —1/2 fl)n+£( IIP — 336.?I)_n—E S CIPI IQI (“Ql 1+ [(Q) IQI’ which, as in two previous cases, produces the desired estimate, which proves the lemma. I Proof of Theorem 3.2: Let f E L2(R",dx) and let f = 28pm» P be its atomic decomposition (The convergence Of the series is in a L2-sense.), where the summation goes over all dyadic cubes in R". Then, Lemma 3.6 implies: / ITfIIIdu = ((2 sPTap)IZ seTae)du P Q ((P)S3(Q) 68 [(P) a/2 I37? _ xQI —(n+e)/2} = 2 I3 I —) (1 + —— x I(P§(Q){ P (“QI ((Q) l x {ISQI (gE—gIHOW (1 + W) —(n+e)/2} , Applying the Cauchy-Schwarz inequality we Obtain 2 IIP) "‘ Imp-eel I we. _<_2{ I. I (_) (1+——— I x / II «PE... ” IIQ) IIQ) ) _“Pl. n+0 pr — mQI —n_€ % 1 1 X IS I2( ) (1+———) } :2Asz. {mega Q ((Q) ((Q) SO, let us estimate the factor A first. We can write: A2: 2 I3PI2 2 20-001 2: (1+M) . iEZ ((P)=2—1‘ jz—oo £(Q):2-i [(Q) Notice that the function ..): z (..I1,Q1;ol)‘"“ ((Q)=2‘j is periodic with the period 22 e, where e is any Of the canonical basis vectors (1, 0, 0, ...), (0, 1, 0, ...), in Z". Thus, without loss Of generality we can assume that x E Q“, = {x : 0 S x]- S 2‘2}. Then we have 2 (1.1—3334') so 2(1+Ikl)‘”“sa ((Q)=2—j (Q) IceZ" The last estimate implies A S C: 2 ISPI2 Z 2(j—IIa = C: ISPI2 = CIIfIIL2(IR",dx)° P iEZ ((P)=2‘i j=—00 In order to estimate the factor B, let us write B as 2 00 __( Imp _ IEQI —n-e B=ZI.QI 22....) 2 1+——— . Q 1:0 P:e(P)=2-1'e(Q) HQ) 69 Using the same argument as in the case Of the factor A (the estimate on the function g), and taking into account the fact that there are 2"" dyadic cubes P C Q with I(P) 2: 2“€(Q), we Obtain B < a; [Squz2’IIIIIa 211 1:0 = C: I362I2 2240 = C: ISQI2 = CIIfIIL2(R",dx)1 Q i=0 Q which completes the proof Of the theorem. I Notice that in each example in Chapter 2 the kernel K was Odd. Thus condition (3.1) is clearly satisfied. Condition (3.2) is a statement that connects the Carleson measure, ,u, and the kernel K. It is easy to see that when the kernel K is Odd in x, and the measure 71 is translation invariant in x, that condition (3.2) is satisfied. Let (p(t) = K (x’ , y, t), for some fixed x’, y E R". Then formally (assuming that all the integrals exist) /K( x, z, t)Ip )du( x, t) 2]“ K( (x z, t)Ip (t)d,u(x,t) +./n- K(x,z,t)go(t)d,u(x, t), where II+ = {x E R" : x1 > 0}, and H— = {x E R" : x, < 0}. Notice that the mapping w(x,t) = (-x,t) maps II+ into 11“, and that for every set E C R1“ we have u(E) = [.t(‘l()(E)). That together with oddness Of K in x implies that condition (3.2) is satisfied. One example of such a Carleson measure is the measure defined by u(E) = IE 0 {t = to}I for every set E C R1“, for any fixed to 2 0. Chapter 4 Applications to Tent Spaces In this chapter we are going to study the problem Of boundedness of the singular integral operator T : L”(R",dx) —> Tf(du), as defined in Chapter 3. The measure a is a Carleson measure on R1“, and the symbol T {(du) denotes the tent space defined as follows. Definition: Let u be a positive measure on R1“, and F (x) = {(y,t) E R1“ : Ix — yI < t} a cone with vertex at the point x E R" and aperture 1. For 1 S q < 00 we set 1/0 A,,(.) = If”, |f(v.t)|"d11(w.t)/t"} . e 11". A function f : R1“ —) C belongs to the tent space Tf(du) if and only if A, f E LP(R", dx), and the norm on Tf(du) is defined by |IfIIT:(d,.) = IIAquILP(IR",dx) for 1 g p,q < 00. In case q = 00, we define Aoof(x)= sup If(y,t)| xER". (y,t)EN-I‘) 70 71 If the measure 71 is defined by du = dxdt/ t, we Obtain the tent spaces defined in [6]. We denote such tent spaces by TI". A reader can find more about tent spaces in [6] or [19]. Lemma 1.5 implies that for any positive measure ,1 on R1“, and 1 < p < 00 there is a constant C > 0 so that IIHfIIrg’oIdh) S CIIfIILP(IR",dx) for every f E Lp(R",d$)- As a consequence Of this fact for any 1 < p < oo and any admissible kernel K there is a constant C > 0 so that we have IIK * fIITg’omp) S CIIfIILP(R",dx) for every f E LP(Rn,d$)- The following theorem, due to F. J. Ruiz and J. L. Torrea, (see [19]), is the essential technical tool when applying the vector-valued versions Of Theorem 2.1 and Theorem 2.4 to the tent spaces. Theorem 4.1 Let 71 be a Carleson measure and T a convolution operator, associated with the kernel K, that satisfies the following two conditions. There exist constants a > 0, and C > 0 so that l < Ct“ ‘ (I11 - yl + t)"+° IK(IL‘. y. t) (4-1) for every x,y E R”, with x 75 y, andt 2 0. There exists 6 > 0 so that whenever Ix — yI +t > 2Iy — y’I we have I < CIy — ylleta - (Ix — y’l +1)"+:+a’ IK(x, y. t) - K(x, y'. t)l + |K(y.$. t) - K(y’.$.t) (4-2) x,y,y’ E R", andt 2 0. Let S be the operator defined by Sf($)(ya t) = Tf(y: t)XI‘(x)(yi t)‘ 72 Then the following statements are equivalent. T : LP(R", dx) —> Tf(du) is bounded (4.3) S: L”(R",dx) -+ L1D Lq(R:+l,du/t")(Rn’ dx) is bounded. (4.4) Moreover, the operator 5 is a vector-valued singular integral operator whose (vector- valued) kernel K is given by ~ K(IL‘, Z)(y: t) : K(y, 2: ta )XI‘(x)(ya t), and satisfies the following two conditions. There exist positive constants C and 6 so that ~ C K n < —, 4.5 H (x,z)“LqMM/t ) — l1: _ zln ( ) for every x, z E R" with x aé z; and whenever 2|z — z’] < [x — 2| we have ~ ~ , ~ ~ , Clz — z’l‘ “K(m, Z) - K(x, Z )“Lq(du/t") + “K(ZHL‘) - K(Z ixllqu(dM/t" < _ (4-5) for every x, z, z’ E R", with x 75 2. Notice that the conditions (4.1) and (4.2) imposed on the kernel K are stronger than the conditions (2.2) and (2.3) imposed on kernel of singular integral Operator defined in Chapter 3. Proof: The equivalence is obtained by the following computations. llell’Z ..., = / ||Tf(y,t)xm)(y,t)lliqw,,,,.,dx Luna“ n") Z/(fm ITf( at t)lqdu/t")p/qd$ =A/( (Tf))”drv— — llellTp ..,.) 73 To prove that S is a vector-valued operator whose kernel K satisfy the conditions (4.5) and (4.6) we need the following lemma. Lemma 4.2 Let u be a Carleson measure, and n, b > 0. If we set I‘b(x) = I‘(x)fl{t S b}, and Pb(x) = F(x) H {t > b}, then we have / t_""’"du g Cb‘" I"’(13) and / t—"+"dp. g Cb". PHI) Proof of the lemma: For j 2 1, we set b __ '—1 ' 13a) _ F(x) 0 {21 b < t 3 23b}. The first statement of the lemma follows from 00 t’""’du = / t'"—"du Aha) 12:; P203) 00 1 the operator T : L”°(R", dx) —-> L”°(R", dm), is bounded for some p0 > O, which if we assume that K (x, y, 0) = k(x — y), where the function It is homogeneous with degree —n, would imply that the maximal singular integral operator Tf‘ , is LP-bounded for every p > 1. We also assume that condition (2.4) is satisfied, i. e. there exists a constant, B > 0, so that for every f E C§°(IR"), [Tf($,t) _ T0,tf($)l S BHf(IE,t), where the constant B > 0 does not depend on x E R" and t > 0. 76 Theorem 4.4 Let ,u be a Carleson measure on R1“, and let T be an operator that satisfies all the conditions above. Then T : L”(R", dx) —> Tf(dp) is bounded for every 1 < p, q < 00. Proof: Using Theorem 2.4 (See Remark 2 after the theorem.) we conclude that T : LP(IR", dx) —> U(R1+1,da) is a bounded operator for every 1 < p < 00. Lemma 4.3 implies that the operator T : L”(IR", dx) —) Tf(du) is bounded for every 1 < p < 00. Applying Theorem 4.1 we obtain that the operator S: L"(]R",dx) —> L" LQ(R1+1,du/t" )(Rn, dx), is bounded for each fixed q 6 (1,00). Let E = L" (1121“1 , du/t"). Then Theorem 2.2 applied to the vector valued convo- lution operator 8, on R", whose vector-valued kernel satisfies condition (4.6), yields S: L”(R",dx) —> LE(R",dx) is a bounded operator for every 1 < p < 00. By Theorem 4.1, the last statement is equivalent to the statement that the operator T: LP(R",dx) -—+ Tf(du) is bounded for every 1 < p, q < 00. Which proves the theorem. Bibliography [1] A. P. Calderon, Inequalities for the maximal function relative to the metric, Studia Math. 57 (1976) 297 — 306. [2] A. P. Calderon and A. Zygmund, 0n the existence on certain singular inte- grals, Acta Mathematica, 88 (1952), 85 — 139. [3] A. P. Calderon and A. Zygmund, On singular integrals, Amer. J. Math., 78 (1956), 289 — 309. [4] L. 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