THS This is to certify that the dissertation entitled Some Sharp Estimates Involving Hilbert Transform presented by Stefanie Petermichl has been accepted towards fulfillment of the requirements for Ph.D. degree inMthfimaiLCjL Mm Major protéssor Date April 26, 2000 MS U L1 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 FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATRflyE DATE DUE DATE DUE Mews 2003 MROIS 40200 CO 1100 W.“ Some Sharp Estimates Involving Hilbert Transform By Stefanie Petermichl A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2000 ABSTRACT Some Sharp Estimates Involving Hilbert Transform By Stefanie Petermichl We construct the Hilbert transform on R as an average of dyadic operators, al- lowing us to translate norm estimates to estimates of dyadic type. We are going to apply this representation to give the sharp dimensional growth of the commutator of the Hilbert transform with matrix multiplication by a BM 0 matrix of size n x n. The bound is a multiple of logn times the BM O-norm of the matrix. Furthermore we will apply this representation to give an elegant proof of the fact that H, as an operator in scalar but weighted L2, is bounded by the cube of the classical A2 -norm of the weight. By the different method of Bellman functions, we show the quadratic bound for H in terms of invariant A2 and this estimate is sharp. ACKNOWLEDGMENTS I would like to express my deep gratitude to my advisor Prof. Alexander Volberg for introducing me to the world of research mathematics in the best possible way, for sharing his knowledge and insight and for his belief in me. I would like to express my thanks to Prof. Fedor Nazarov for many conversations, his outstanding lectures and his office. I want to thank Prof. Michael Frazier for his help, time and orange juice. I thank Prof. Sergei Treil for helpful discussions. My school teacher Dr. Karl Weill has my gratitude for supporting my interests in mathematics. I also want to thank Dr. Sandra Pott for our collaboration and the good effects that her presence had on me. I thank Dr. Teddi Draghici for his valuable advice, and the members of my committee Prof. Sledd, Prof. Weil and Prof. Zeidan. I also want to thank Max Mindel and Ralf Kristel for listening and their support. Last, I would like to thank Dr. Alberto Corso for his professional advice. iii TABLE OF CONTENTS 1 Introduction 2 The Hilbert Transform and Dyadic Shift Operators 2.1 Definitions ................................... 2.2 The Representation of the Kernel via Limits of Averages ......... 2.2.1 Proof that K(1) = co > 0 ......................... 3 Application to Hankel Operators with Matrix Symbol 3.1 Definitions and Statement .......................... 3.2 The Proof ................................... 3.3 Sharpness of Result .............................. 4 The Cubic Bound for the Hilbert Transform 4.1 Definitions and Statement .......................... 4.2 The Proof ................................... 5 A Sharp Bound for Weighted Hilbert 'IH'ansform 5.1 Definitions and Statement .......................... 5.2 Proof ...................................... 5.2.1 The First Integral ............................. 5.2.2 The Second and the Third Integral .................... 5.2.3 The Fourth Integral ............................ 5.3 Shortcut .................................... 5.4 Sharpness of Result .............................. 6 Harmonic Bilinear Imbedding Theorem 6.1 Statement and Proof ............................. BIBLIOGRAPHY iv CHAPTER 1 Introduction It has been of interest for a long time to give sharp estimates for the norm of the Hilbert transform and related operators in BUR). In the present work we look at the estimates of the Hilbert transform in weighted and / or vector spaces. We restrict ouselves to the case p = 2. It will be of great help to reduce various estimates for H to estimates of dyadic type. We will prove that there is a very nice representation of the Hilbert transform via averaging operators that we will refer to as dyadic shifts. Let us start with vector problems. Recently some activity has been focused on the area of non-commutative weighted estimates. By non-commutative weighted es- timates we understand, for example, the estimate of the singular integral operators T, say, the Hilbert transform H, in the space of vector functions with matrix or operator weights. Let us refer the reader to [16], [17], [8], [18]. The problem of estimating ||T|| L2(W)_,L2(w) is equivalent to estimating ||W1/2TW‘1/2||L2_,L2 which is non-linear in W. But there is a linearized counter- part, which comes down to estimates of commutators of the form TB -— BT. This problem is linear in B. It is well known that estimates of commutators of matrix multiplication with the Hilbert transform yield estimates of Hankel operators. In the present work we are going to give the estimate from above in terms of the 1 dimension n, ”HB — BHHL2(C")—)L2(C") S Clog nllBllBMO (1.1) We refer to [11]. In [7] it has been proven that there exists an n x n matrix function B such that ”HB - BH]]L2(C")—->L2(C") 2 C, logn||B||BM0 (1.2) (the symbol ||B|| 3M0 is defined below), which proves sharpness. The same idea of averaging can be tried for weighted estimates of the Hilbert transform and other CZ operators. Now we are in the scalar but weighted situation. The space considered is L2(w), where w is in A2, the exact class of weights that allows the Hilbert transform to be bounded. The question for sharp estimates for the Hilbert transform, the square function and a uniform bound for martingales on weighted L2 spaces in terms of the A2 constant of the weight has attracted considerable interest in recent years. S. Buckley in [1] proved that the square function is bounded by ||w||§42 and that the Hilbert transform is bounded by ||w||f42. More recently, in [5] the quadratic bound for the square function has been proven. This bound is sharp. An alternative proof can be found in [19] and [12], where the estimate is found using the lower bound for the square function, which is linear. The linear lower bound for a harmonic version has been proven in [2] and independently for the dyadic version in [12]. We prove the cubic bound for all dyadic shifts and hence the Hilbert transform. By the different method of Bellman functions we obtained the sharp estimate in terms of invariant A2, which is the version of A2 using Poisson averages instead of box averages. The proof uses a nice ‘duality’ between sharp uniform estimates for dyadic martingales and its continuous analog, the Hilbert transform. The proof includes an alternative to what is known as bilinear imbedding theorem in [10] involving two 2 weights. We establish slightly simpler conditions on the two weights 1) and w for the imbedding. The function to prove the theorem is B(X)Y)x3ylr)S?G)H7M’N’K) : I172 y2 $2 y2 $2 y2 $23 — 2333/52. + yzr 7‘ 8 T+-qu 8+6: T+a7 S+'Q-.{ T's-a CHAPTER 2 The Hilbert Transform and Dyadic Shift Operators 2.1 Definitions We will be using a variety of dyadic grids in IR. The standard dyadic grid is the collection of the following intervals with disjoint interiors : {[2nk, 2"(k+ 1)], n, k E Z}. The point 0 has the significance to be the only point that does not lie in the interior of any of the intervals. We call the standard dyadic grid Do'1 and we denote by hg’l the Haar function for J 6 D04, namely ho’1 = 1/ m (X J_ —- X J+) where J. is the left half of J and J+ the right half of J. We obtain a variation of 190'1 by first shifting the starting point 0 to a 6 IR and secondly choosing intervals of length r - 2" for positive 1' The resulting grid is called D0". The corresponding Haar functions ha" are chosen so that they are still normalized in L2. We often omit the indices a, r in our notations for the Haar functions. The following is an illustration of D0". For f E L2(R) we have f0?) = Z (f,hr)h1($)- [Evam We define a dyadic shift operator III”: (HI“"f)(x) = 2 (f, h1)(h1_ (17) — ’11,. (53))- lava-r The symbol ‘III’ is a cyrillic letter that reads ‘sha’. Its L2 operator norm is \/2 and its representing kernel is Ka’r(t,$) : Z h1(t)(h1— (x) _ h1+(.’13)), (2'1) 1613“" by which we mean that (,ma Tf)($ )=‘/l; K0, ,T‘ (t, xv for smooth compactly supported f and 1: outside the support of f. 2.2 The Representation of the Kernel via Limits of Averages 1 The kernel of the Hilbert transform is K (t,a:) = ——Z. Constant multiples of this function are characterized by the four properties that K only depends on the distance :r — t, K 5.5 0, the correct degree of homogeneity and antisymmetry. We will pick the correct averaging process to form such properties from K a” defined in equation (2.1). Lemma 2.1 The convergence of sum (2.1) is uniform for lax—t] Z 6 for every 6 > 0. For 3:7ét let 1 L 1 R d K =' —— l' — a” 4. (t,:r) Ilfi2logL/1/LR1—13602RZRK (t,:r)da7. The limits exist pointwise and the convergence is bounded for [2: - t| 2 6 for every 6 > 0. K (t, :r) is a nonzero constant multiple of the kernel of the Hilbert transform: K(t,:r) = if“; for some co > 0. PROOF. First note that Va 6 R and Vr > 0: Z lh1(t)(h1_(x) — hm)» s If—fi- Ievam — ml In fact, if [I] < It — as] then h;(t)(h1_ (:13) — h1+(:c)) = 0, so picking no minimal such that r2"° 2 It —- a:| we get 2 |h1(t)(h1_($) — hI+($))| 16m”. x _<_ Z Z Ih,(t)(h._(x)—m.l n=no I E’D‘” |I|=r2" °°\/§ Z— r2“ n=no 2J5 lt-évl' In particular, the sum converges absolutely and uniformly for [re - t| _>_ 6 for every 6>0. So 1 R AfifiIRK ’ (t,a:)da co . 1 R = 7;. $3051}. [.3 16;? h1(t)(h1_(a:) —- h1+(:c))da |I|=r2" by dominated convergence. The limit R , 1 ,lggo m _R I; h1(t)(hl-($) — hm» da |I|=r2fl exists for all fixed r because shifting a certain grid by r2" will result in the same intervals of that particular length: 2 h1(t)(h,_ (513) —hI+($)) = Z h1(t)(hr-($) —h1+($))- IE’DQ" [Ego-Hann- '1 [”2" |I|=r2" The outer limit in L exists for similar reasons. Here the entire grid repeats itself, i.e. D0” = Da'zk' for any integer It, so K a" = K “’2" The main point is to show that K (t, x) = 259;. For this purpose it is enough to prove the following properties of K (t, 3:): 1. Translation invariance, i.e. K(t,:r) = K(t + c,:l: + c) \7’c E IR, so K(t,;c) = K(t — :r) 2. Antisymmetry, i.e. K(t,:r) = —K(—t, —:r) , so K(a: — t) = —K(t — :r) 3. Dilation invariance, i.e. K(t,:r) = AK(/\t, Ax) VA > 0 4. K(l) = co > 0 In order to check the first three properties we observe the following simple relation- ships between the Haar functions of different dyadic grids for translations, reflections and dilations. For any interval I E ’D°‘" there exists an interval of the same length in Da‘c” so that h?"(t + c) = h?”°"(t). In a similar sense h?"(—t) = —h,’°‘"(t) when changing grids from D” to 13—0" and h?"()\t) = A‘l/L’hCIVA’r/AU) when changing from 'D‘” to ’Da/A‘r/A. In more detail: 2: h‘,’”(t + c)(h‘;f(a: + c) — h‘;f(x + c)) (2.2) IeDOfi' |I|=r2n = Z h‘}'c”'(t)(h‘;_"c’r(:r) — h‘,’:c"(a:)) VC 6 n. Jena-cw |I|=r2" Z h?"‘<-t>(h?:'<—x) — m-..» (2,3) IG‘DOn‘ |I|=r2" = Z —h;°"< ><—h;:” 0. leva/AJ/A |I|=r/»\ 2" Now we are ready to prove the first three properties. Proof of translation invariance: K(t+c,:r+c) . 1 L , 1 R a, d, —Ill—>Hol<>210gL//LRh-£I002? lim— 1 [Llim —/: —K"°’ '(t, 3:) da— dr L—mo 2 log L 1/L R—Enoo 2R —K(t, x) by applying (2.3) and a substitution in a . Proof of dilation invariance: K(At, Ar) 1 L 1 R d = l ' — a’r l LflflogL‘A/th—ngo 2R/_R K (At,A:r)da r 1 L - a/A, r/A d?“ 35202logL‘/1/LRi—+oozll2/: K (t :13)da .1. A 1 1 L R/A _ - c,r/A d__7_' AIll—roo 210gL//LRi—)oo— 2R _R/AK (t$)d01 l A .1. A 1 “A adr 1. K0, ,1' (t, [41-1330 210gLf/(LA)R->OO 1112112]: x) K(t, x) by first using (2.4), then a substitution in a, then a substitution in r and the fact that the integrand lS bounded by M II- t| These three properties prove that the kernel we obtain must be K (t, :r) = 7:95. For our purposes it is essential to know that co # 0. 2.2.1 Proof that K(l) = co > 0 Let us first illustrate h1(t)(h1_(a:) —- h1+ (1.)) h1(t)(h1_(a:) — h1+(x)) 75 0 if and only if the point (t, 2:) lies in this square. Its value is ifi, where the correct sign is indicated inside the smaller rectangles. 10 Let us first compute 7' ' 1 R max) := 11320 a [_R g): h1 — h1+ 0 and n E Z and assuming t > :13. The picture is the following: The exact location of the squares along the diagonal is influenced by the starting point a. The picture will repeat for two values of a that differ by an integer multiple of II | We compute (2.5) in (t,a:) by considering the probability that (t, :2) lies in any of the squares. Due to the averaging process in a, this is only going to depend on t — x. We only need to compute for It — xl on the dotted lines: IL‘ If t — a: = 0 then K;(t,:1:) = 3g + %(— II?) , and similarly E t— a: = fiIII then K;(t,:c) = %£ t— a: = %|I| then K;(t,a:) = 0 t— a: = §|I| then K;(t,a:) = — t-xZ |I| then K;(t,:c)=0. 11 Since K;(t,a:) = K;(t — as) is piecewise linear in t — as, we obtain the following graph, depending on n and r: (l 3%? #- Next we compute 7' 1%. _ a,(r a. . K (t,:1:):= 1m 0022 :'K (,tx)da (26) First note that 1mm) zzgfl .2? 1/: Z h1(t) —,h1+(:c))da (2.7) 1162 1617'" |I|=r2" so we compute K '(t, 1:) using K ;(t, 11:) defined in equation (2.5) for different values of n and summing over n e Z. It suffices to compute K '(t, ax) for values t — :1; = %r2" and t—a: = r2" since the graph is piecewise linear on intervals [2"‘1, %2"] and E2", 2"] for all n. For t — :1: = g-TZ" we obtain that 3 If 3f 9f \f2' K (2’2 )= ‘15 + 163: + 6—4-1~2_"(1 + 4 + E + ~=> grew (28) and for t - a: = r2" we get . .. 3 f \/§ K (7‘2 )16'r_22n( +4+'11_6+” '=) 47-21" (2'9) Equations (2.8) and (2.9) imply that 3‘5 ‘5 V7" > 0. (2.10) 32(t-$)— K’(t— x)‘<'4(t—x) The expression in Lemma 2.1 IS obtained from K '(t — :13) by a limit of averages in r, so it is clear from equation (2.10) that co > O. I 12 CHAPTER 3 Application to Hankel Operators with Matrix Symbol 3.1 Definitions and Statement We consider vector valued L2 = L§(C”) for n 2 2, i.e., measurable f : IR -+ C" such that fR( f, f)Cn =: N f ”flaw,” < 00. We also consider BMO - matrix functions B of size n x n with ‘B E BM 0’ in the following sense. If B is an n x n matrix function on IR, we say that B E BM 06 if for all e E C", ||e|| denoting its Hilbert space norm, sgp 0 such that for all B E BM 0 IIHB - BHIILg(cn)_.L,2,(cn) S 0 log nllB “BMO- 3.2 The Proof PROOF. We estimate ||HB — BH|| by relating it to “1118 — Bill”. The operators 111“” are norm bounded by J2, so the collection of operators Tfi, definedby 1 R Tfisz-Zfilfllll’fda are, as averages, also norm bounded by \/2. By compactness of the closed unit ball in 3(L2) in the weak operator topology and the fact that the unit ball is metrizable, there is a subsequence Tfik such that Tfik converges weakly, which means there exists ’1" with norm S \/2 such that Vf,g E L2 : (Tgkf,g) ——> (T'f,g) as k —> 00. Then consider T -fI—> 1 [LT’fd—T L. 2logL1/L 7". By the same argument there exists a subsequence TLk that converges weakly to a bounded operator T. 14 Let us show that T is represented by the kernel co / (t — 2:). Because of bounded convergence in Lemma 2.1, we have for f, g compactly supported with disjoint sup- ports, (T fig) : klino(Tkaig) 1 L" d = lim —/ Tr , l k—roo210g(Lk) 1/L,.( f9) 7' - 1 Lk 1 R1 ar d?” - ham/M; 312.2121,” 159”“? Li: 1 R1 01' d?" (/f( t) lim—— lim— K ’ (t, :13) da Tdt , g) k—>oo log1(Lk)121/Lk l—+oo 2R-_l com]: 9.) , By [15] (p. 33) there exists a bounded measurable function a such that T f (1:) = con(a:) + a.(x)f(:1:) (i.e., hence co(HB — BH)f = (TB — BT)f. By convexity ||TB — ET” 3 sup” ||IH°"B — Bmwu. So we are left to estimate the commutators with 1.11“” uniformly. We show that for all 01 E R and for all r > 0 “1110MB - Bma’ |IL§(C")—>L§ (on) < 0108 "HBIIBMO (3-3) In the following a, 1' will be omitted because all estimates do not depend on the dyadic grid. It suffices to consider f E ’D(R) only, because the estimates do not depend on the support of f. First let us decompose the product Bf, at least for ‘almost compactly supported’ B in the sense that B differs from a compactly sup- ported function only by an additive constant. Note that for any such BM 0 matrix function, the sum Ba) = [(3, h;)h1(:c) (3.4) 1619 is meaningful, in the sense that it converges unconditionally to an L2 function. The reason is that coefficients of constant functions vanish, so (3.4) ignores constants and 15 treats our B like a compactly supported one, whose entries are in L2. For our vector function f we have f(w) = EU. haw). (3.5) IED By multiplying the sums (3.4) and (3.5) formally one gets Bf = AB(f) + HB(f) + RBU), where Ago) = 209, hm, hm, 16D HBU) = 2(B,hi)1 In. 160 RB(f) = 2(B)1(f,h1)h1. IE’D The formal multiplication is meaningful due to our assumptions on f and B. Hence SB — BS = SAB — ABS + $113 — HES + SR3 — RES and we can estimate the terms separately for our special class of B. By [9] and [6] llHBl|L§(Cn)—>L§(cn) S Clog nHBHBMOa where C does not depend on n. The proof in [9] is by a Bellman function construction, whereas the independent proof in [6] is by a stopping time procedure. For the part involving AB, note that A}; = HB-. So “AB“ S ClOSTIIIEIIBMo = ClognllBHBMo, so the parts involving I13 and A3 are bounded by fiC log n||B|| 3 Mo- 16 We estimate the last term as commutator: IHRBf — Remf = Z (f, huh, — 2(8 > (f,hz)h1_ + I 23B > (f hi) )-h1+ [(8)1415 ham. 2 -Z((B —_(B>1 )(f,hz)(h1_ — m. So it suffices to show that ||((B)1+ -<13>1_)€||2 S C||B||23Mo llell2 V6 6 C" with C independent of n. In fact, l|((B)z+ - (3)1.)ell2 = 4H((B>I-(B > -e>u2 = 4ll-I-—I_I/_1—t) ))edtll"’ I——,_I/_(I((B >BI— (newt (178/, ((((B>I —- Bowen? dt S 8||Bll2amolle||2- IA |/\ Now we pass from our ‘almost compactly supported’ B to general B E BM 0. For fixed B we consider the sequence of intervals 1,, = [—k, k]. We can construct Bk so that Bk = B on 1],, Bk = (B) [k outside the interval concentric with 1,, and three times its length and furthermore, “BkHBMO S cIIBIIBMo with c independent of k. A suggestion for such a construction can be found in [3] (p.269). So the family of operators IIIB,c — 81,111 is uniformly bounded in L2, by C log n||B|| 3M0. Hence by a weak compactness argument, the operator IIIB — BHI is bounded by the same norm, which gives us the estimate for general B, finishing the proof of inequality (3.3). I 17 3.3 Sharpness of Result In [7] it has been proven that there exists an n x n matrix function B such that HHB — BHHL2(C")->L2(C") 2 0' log nllBHBMO (3-6) which shows that the estimate in terms of logn is sharp. We would like to point out that the same averaging technique has been used to obtain the lower bounds for paraproduct operators and hence for Carleson imbedding theorem using the lower bounds for commutators. The proofs can be found in [7]. As a consequence, the log n upper bounds for paraproducts and Carleson imbedding theorem are sharp. 18 CHAPTER. 4 The Cubic Bound for the Hilbert Transform 4.1 Definitions and Statement 1 ICC, where We are now in scalar but weighted situation. We consider weights (.0 E L w 6 A2 with norm “LUNA; = Slip (w)1(w“1)1, where the supremum runs over all intervals. We also consider a dyadic version, Ag, where the supremum runs over dyadic intervals only. So the norm ||w|| Ag may depend on the choice of the dyadic grid. We are concerned with weighted L2 spaces, denoted by L§(w), containing functions so that [I f “E, := In | f |2w < 00. Our main theorem in this chapter is the following: Theorem 4.1 H : Lid”) —> Lfi(w) has operator norm ”H” S cllwlliz. We will reduce the problem to upper and lower bounds of certain square functions, using averaging technique from chapter 2. The square function 5' is defined by 19 Sf(t) = \//2 |(Tef)(t)l2d5 = l/ZI: |(f,hz)|22(IIT(It2 where 2 denotes the space {-1, 1}” provided with the natural measure (16 which assigns equal measure 2"“ to every cylindrical subset of {—1,1}D of length 2" and T5 is the martingale transform T. = f e Eamon ham I associated with the sequence 6(1) 6 {—1,1}D. The following has been proven in [2] (in a harmonic version) and independently in [12]: Theorem 4.2 There exists c > 0 such that for all f E L2(w) llfllw S cllwllAgHSwa- In [5] the sharp upper bound has been proven: Theorem 4.3 There exists c > 0 such that for all f E L2(w) “‘8wa S CIIWIliallfllw. 2 Both [19] and [12] contain a proof of the fact that the quadratic upper bound follows from the linear lower bound, which implies that the lower bound is sharp as well. 4.2 The Proof We will give a short and elegant proof of Theorem 4.1. It was found together with Sandra Pott. As seen in Chapter 2, H lies in the closed convex hull of dyadic shift operators. The square function does not ‘see’ the dyadic shift: 20 Proposition 4.4 (SIIIf)(x) = x/2(Sf)(x) for all x. PROOF. 3mm»? = / ((T.mf>(a:)(2de = / (2134001112 hawnzde A l Z(f,h1)(6(I—)h1_ — €(I+)h1+)|2d€ I (L) x 2 — 2 f2 |;8(I)(f,h1)hz( )lde = 2Sf(x)2 where (*) is an effect of the averaging over sequences of signs 5(1) and the fact that for each fixed x there exists a sequence of signs 5(1) so that we have for all 1: fihzfiv) = 5(1 )(€(1—)hz_ - 6(1+)ht+)(-B)- I Now it is easy to prove Theorem 4.1: PROOF. Dyadic shifts with respect to all translates and dilates of the standard dyadic grid have cubic bound, indeed a (1) r (2) (3) “HI ’ero < CllwllAgamHSIHa’ fllw = CllwllAgamHSfHo < Cllwlligwllfllo where (1) is by Theorem 4.2, the lower bound for the square function, (2) by Propo- sition 4.4, the fact that the square function does not see the dyadic shift and (3) by Corollary 4.3, the upper bound for the square function. By convexity, as before, we Obtain the desired bound for the Hilbert transform: IIHHL2(w)—)L2(w) S CSIIp Ilma’rllL2(w)—*L2(W) S CSIIp “w“:ga’r S Cllwuiz. c,r c,r I 21 CHAPTER 5 A Sharp Bound for Weighted Hilbert Transform 5.1 Definitions and Statement In this section, it is more convenient to work on the unit circle '1‘. We consider the space L?r(w) where, as before, w is a positive L1 function, called a weight. Let m be normalized Lebesgue measure on '11‘. The norm of f E L%(w) is ( f1. I f lzwdmr/ 2 and denoted by N f llw. We are, as before, concerned with a special class of weights, called A2. We say (.0 6 A2 if (MIA. == sgp (w>11 < oo where the supremum is taken over all dyadic subarcs I C '11‘. The notation (no); means the average of the function w over I . We also consider a version of A2 that is invariant under MObius transforms, called A2,,nv. A weight w 6 A2,,“ if “th... == sup mow-1(2) < oo 26!) where (0(2) denotes the harmonic extension of 4.), so w(z) = f w(t)Pz(t)dm(t), where Pz(t) = Ill—jg); Note that in general w'1(z) and (.4)(.z)‘1 have different meaning. 22 The first expression means taking the harmonic extension of the reciprocal of w, the second one means taking the reciprocal of the harmonic extension of w. Observe that w(z)w‘1(z) Z 1 by Jensen’s inequality, so "(all/42‘,“ > 1. In [4] it has been proven that the exact sharp relationship between the two different A2 norms is as follows: CllleAa _<. llwlle,.-... S 02IIWIlf42- In particular, to 6 A2 if and only if w 6 Ann”. In what follows, H stands for Hilbert transform on the circle '1‘. H acts on trigonometric polynomials as follows: H (2 akew") = —i Z akewk + i Z akeio". 1:20 k<0 Let Ho be the Operator H + iPo where Po : f v—-> f (0) Theorem 5.1 H : L%~(w) —> L%(w) has operator norm “H“ S cllwllfag’m. Note again, that in our notation “WW/12,4“ = supzen w(z)w"1(z). 5.2 Proof ||P0|| L2(w)_, 1.20.1) g ||w||A2Im. Indeed, applying Jensen’s inequality we obtain ||Po(f)||.2o = If (0)I2w(0) S (|fl2w)(0) w'1(0)W(0) S llwllig,....||f||3- As ”H” S ”Hall + ”Poll and “WM/12...... 2 1 it suffices to show that “Ho“ 3 cllwllizm. We estimate ||H0||L%(w)—+L%(w) by duality. Since (Ho t f, g/t) = (Ho f, 9), it is enough to Show that |(Hof,9)| S cllwlliah...(llfllfi + Halli-1) for all f 6 Lila!) and 9 E L-iW") (just use t = \/||g||w-1/||f||w if f 76 0). It suffices to consider real valued and non- negative functions f and 9. By polarizing [3] (p. 236) we have [f(Hof — Hof(0))(9 — g<0>>dm = % f(VHof)(Vg) log iii/1(2). lzl 23 Due to Ho f (0) = O the left hand side equals (Ho f, 9). Since IVHO f | = [V f l we get |(Hof,g)| _<. 511; [D IVfIIVgllog 1 lzl dA(z). Note that for f real valued, |V f I = 2|(9 f /az|. We will write f (z)’ for the holo- morphic function (9 f / Bz. We split the integral into four parts. One can see that [D lf(z)’l|g(z)’llog / |f(z)||g(2)| D + flf(2)ng(2)l + /|f(z)llg(z)| + (mange) I—iIdA(2)s f(Z)’_w‘1(z)’ g(2)’_w(2)' 0 i z 1(2) 24(2) 4(2) 2(2) lglzldA(’ 24(2) g(2)’_w(2)' 0 _1_ z 24(2) 4(2) 2(2) lngldA() 2(2) f(Z)’_w“(z)’ 0 i z 2(2) 1(2) w-1(2)’gl2("A(’ w‘1(z)’ w(z)’ i z 24(2) 2(2) longldA( " The first integral can be controlled in the same way as done in [9], in fact, the proofs are identical. For the second and third integral we need to proceed in two steps. Again, we want tO use the same proof as in [9], but in order to do that we need to have an estimate for a certain Green’s potential function involving the weight. But there is a dyadic analog for this estimate, which showed up in the proof for sharp bounds for the dyadic square function in L2(w) and was proven by Bellman function technique ( see [5]). We will use the same Bellman function to obtain estimates for the Green’s potential. The fourth integral requires what is known as bilinear harmonic imbedding theorem. The appropriate Bellman function was constructed with help from [9]. We give an explicit expression for the function. The imbedding conditions are again certain Green’s potentials. We find the appropriate bounds using Bellman functions found in [9] and [19]. 24 Before we start to estimate the four integrals we need the following lemma to relate Laplacians to second differentials: Lemma 5.2 If b(z) = B(h(z)) where h = (fi), : C ——) IR” and B : 1R" ——) IR with B and h sufficiently smooth, then Ab(z) = 4 (423(h(2)) (63?), (if) ) + 4(VB)(h(z)) ( :3; )1. (5.1) 1 In particular, if all f,- are harmonic, then 0 i 0 ,- A.(.)=.(2B(h(.»(6:),(39) (5.2) PROOF. By elementary computation. I We will be using the appropriate Bellman functions to bound all integrals. Each variable carries meaning, usually harmonic extensions of functions or Green’s poten- tials for some fixed .2. The following variables show up frequently: X = f2w(z) Y = 92w‘1(2) If we assume f, g to be real and nonnegative, all variables will be nonnegative. Furthermore we have the following natural estimates: 1 3 rs S Q2 if we write Q for ”(dummy (5.3) x2 S X r and y2 3 Y3 because of Jensen’s inequality. (5.4) These restrictions give a natural domain of our Bellman functions. 25 5.2.1 The First Integral Consider the following function of six real variables $2 312 B(X,£L',T,Y,y,3) : X — T +Y _ —S- then we get the following size estimate within the natural domain of B: OSBSX+Y. and by direct computation of the second differential we get $2 —d2B=—$— 7‘ dy ds2 y 8 dx dr2 LE 7' +2312 3 (5.5) Also consider the function b : (C —+ R 5(2) = 3 (11(2)) = B(f2w(2),f(z),w'1(2),92w’1(2),9(2),w(2)), then we obtain the following estimate for —Ab(z) using (5.2) and (5.5) —Ab(z) 8f|f(z)|’ 1(2) _24(2)'2 (4(2))? g(2)’_2(2)'2 824(2) (2) 24(2) 2(2) 9(2) 2(2) If(2)g(2)( f(z)’_w“1(z)’ 4(2) _2(_2_)_' 2 16 w‘1(z)w(z) f(z) 224(2) 9(2) 2(2) If(2)g(2)| 1(2) _ 24(2) 9(2) _ 2(2)’ 2 16 c2 f(2) 24(2) 4(2) 2(2)‘ integral: 1 log Ian/1(2) We use the above estimate for —Ab(z) and Green’s formula to estimate the first f((2__)_' _w”(2)’ 9(Z)’ _ w(2)’ 1W) f(z) 24(2) 4(2) 2(2) _ OSB(r,s) ScQ4r, 1 _<_ rs S Q2 => —dZB Z Cs(dr)2. Let us also consider the function b : C —-) R b(Z) = B(11(2)) = B(w’1(2),w(z)), 0 S b(z) S cQ4w"(z) and — Ab(z) Z cw(z)|w—1(z)’|2. This function will help us to estimate the following Green’s potential: GUM1 'lzw)(2) _ _1__ —1 I 2 — / log I s.(4)I"" (o I 2(e)dA(<) 1 s c [D —Ab(4) ammo (:2 _ .1. — 2]” Ab(S—z(£))10gI€IdA(€) = b - bd .( <4 f. m) s cQ4w'1(Z), 27 where 33(5) = £15? In (*) we just did a change of variables 5 1—) S_z(£) ( note that the symbol A carries the variable as well). Hence we proved that C(lw‘1’lzw)(z) S Cllwll‘342,,,,w"(2) and analogously G(lw'l2w"1)(z) S Cllwlliz.,,.w(z)- The reader should note the similarity between the estimate for the Green’s poten- tial and its dyadic analog found in [5] I—j—IZ I(2)z. — z_|’z|1| s 224(2)» ICJ Functions of similar form as discussed in the proposition below will appear fre- quently. We take care of their concavity. Proposition 5.3 Functions of the form $2 y+z f(waxi 31,2) z w _ (56) with y > 0 and 2 Z 0 are concave. PROOF. The matrix {00 0 0) 0 _2_ —2x —2x _ d2 f = y+z (14+sz (y+z)"’ 0 -2x 2::2 2x2 (y+z)’ (y+2)3 (2+2)3 -2 2x2 222 K 0 (2+5)? (IMP (y+2)3 / is positive semidefinite. I 28 Let us introduce a new variable 0 = G(lw’1 ’lzw)(2)- Now we are ready to steal from [9] the Bellman function used to prove weighted dyadic imbedding theorem. We let B(X,x,r,G,Y,y,s)=X— G +Y—— r B is, as a sum of two functions of the form discussed in Proposition 5.3, concave. Consider W) = B(11(2)) = B(f2w(2),f(2),w"(2), GUM1 'lgw)(2),92w"(z),w(2)) We will have to estimate —Ab(z) from below. We use equation (5.1) to estimate the part involving X, x, r, G, where the concavity of B allows us to drop the part involving the second differential. We only need to consider partial derivative in the ‘non—harmonic variable’ G. Note that —AG’(|w‘1 ’|2w) = [of1 ’l2w. We use (5.2) and (5.5) for the part involving Y, 3;, s. —Ab(z) 2 ‘24 (2f1(8::3Fg(It-ITZE)(»))” 835(2)) 9412))" 2:) 2 Z CQM’(24($(:)2343(I:|—23(I:32)(2))2+0272): 21—3-38? 2 CQ_4f(z)2|:::E:);l’w(2) +c%_ %_ ago 2 co 2|f(z)g(Z) “5:18 2):) it) Now we use Green’s formula, the fact that b _>_ O on T and b(0) S Ilfllf, + Ilgllffl 29 to estimate the second integral: :‘1(z)’9(2)’ w(2)’ flf(2)IIg(2)I () 9(2) 2(2) —1-dA(z) s cllwll’ / —Ab(2)log n lzl _<_ Cllwll2(llf||3 + Halli—1). 1 log mdA(z) 5.2.3 The Fourth Integral We will apply Lemma 6.1, the harmonic bilinear imbedding theorem whose statement and proof can be found in chapter 6. We apply it for the weights w and v = w“ with ||w|| Ann” = Q and, up to a normalization constant not depending on Q, |w(Z)'||w'l(Z)| w(z)w‘1(z) ° We need to prove the imbedding condition inequalities (6.1), (6.2) and (6.3). We (1(2) = first turn to (6.1): Consider the function from [19] 2 B(s, r)— - s(—% - 2% + 4Q2 + 1), this function has the following properties: lsrsSQ2:0£B(r,s)Scst dsdr sr ° lsrsSQ2=>—d2BZC's Let us also consider the function b : C —> R 11(2) = B(h(Z)) = B(w(2),w‘1(2)), then 0 S b(z) S cQ2w(z) and 30 —Ab(z) > 40w (7. )Iw W< )’||w‘1 (2)" = 4Cw(z)a(z). w(2)w“(z) This function will help us to estimate the following integral: / 124— I S (Ola ae( e)2(4)dA(4) 1, g / —Ab()£ logI—gzITIIdAc) = /_ —Ab(S'_z(€))logI— 421(4) =(2 I2) _<_CQw 6| similarly we obtain 1 lo ——a 5 (fl 5 dAE S csz"1z, fogISzIIIIu () () () which give the desired estimates (6.1) and (6.2). We are left to show the inequality (6.3), namely that / log I S —I—III2(4)2(4)2 (024(2) 3 co . Consider the function from [9] B(s, r) = 4Q\/s7 — sr, this function has the following properties: lgrssQ2=>OSB(r,s)S4Q2 1 3 rs S Q2 => —dQB Z c|dsdr|. Let us also consider the function b : C —) IR 5(2) = 3072(2)) = B(W(Z),w"(2)), 31 then 0 s 13(2) 3 CQ2 and —Ab(z) 2 wimp-1(2) 'WYHWWI = 4Cw(z)w_l(z)oz(z). w(z)w‘1(2) This function will take care of the following integral: / log— IS I“), 2( )2(:)2- (924(5) 1 g 0/0 —Ab(£) )log Mama) 2 CA—Ab(S—z(€))10g = c (be) — [T bdm) Q2 proving (6.3). 1 “lad/1K) 5.3 Shortcut There is a faster, but less instructive way to obtain the desired result. We use the dyadic analog and deduce the existence of the corresponding Bellman function. In [19] it has been proven that Z |(f:h1)||(9,h1)| s Cllwllizllfllwllgllw—x. I By restricting f and g to J and discarding some positive terms on the left, we can deduce the following for all dyadic J .7. Z IIH —(g)1_( s cllwlli,x/(f22)1(g22-1>J. ICJ Again, the reader should note the similarity with our integral: I I 1 If (Z) “9(2) llog -dA(Z)- n lzl 32 Since the above dyadic estimate holds, there exists a function B : D -—> IR where D: {v= (X,Y,s,r,:c,y) ERgo: 1 S STS Q2} so that vED=>OSB_<_cQ2\/)TI7 (5.7) and 8(2) — $092.) + 8(2)) 2 212+ — any. — y-) (5.8) whenever 22, 22+, (1. E D and v = 1/2(v+ + 2)-). Such a function is B(X.Y,s,.:=2,y) —sup— n: |I||gz_||< ) —z-l, lch where the supremum runs over functions f E L2(w), g E L2 (w‘l) and weights (0 6 A2 with norm ”cull/12 = Q so that (f)J = 517, (9).! = y, (00h = 3, (w’lh = 7‘, (fzwh = X, (gzw'llJ = Y- The lower bound in (5.7) is clear by definition of B and the upper bound is just the fact that the dyadic estimate holds. Inequality (5.8) follows by investigating the relationship between the contributions to the supremum that are made by the right and left hand sides of the interval J. In fact, for v E D and v+, v- E D so that 1/2(v+ + v-) = v we have that zl—sup WEN”) mm > —Ji : xi) Ji = yia Ji = 3i: (w—llJi = Ti, 5, consider dilates (1)6(33) = 1/66(:r/6) of a smooth bellshaped function supported in the unit ball of R6. Then the convolution B5 = B * (be is smooth and satisfies the same estimates than B in the set K with different constants not depending on 6. So we have the following size condition on Bc(v) for all 1) E K. o s 36(2) 3 csz. The condition (5.8) implies the following estimate for the second differential of Be: —d23£(v) Z cldxlldyl. (5.9) NOW let 2(2) = (f2w(z),g2w‘l(z),w(2),224(2),f(Z),9(Z)). The set {11(3) = IZI S r} is a compact subset of D. Choose 6 accordingly and consider b.(2) = Be(f2w(2),92w’1(2),w(2),w"1(2), f (2), 9(2)). Then applying (5.2) together with (5.9) gives -Abe(2) Z C|f(2)'||9(2)'|- 34 Now we are ready to estimate the fourth integral: [D |f(z)’||g(z)’llog int/1(2) IZI - I I 1 — hm f... (f(z) llg(z) (log -dA(z) r—>l '2' _<_ limc/ —-Ab£(z)log-—1-dA(z) rD r-)1 '2' = 6(1),,(0) — clim / bcdm) r—+l rT CQ2(|lf||2||9||2-1)- |/\ 5.4 Sharpness of Result Sharpness can be seen using power weights. We refer the reader to [5] and [13]. 35 CHAPTER 6 Harmonic Bilinear Imbedding Theorem 6.1 Statement and Proof Lemma 6.1 Let a(z) Z O and w,v be two weights so that 1 S w(z)v(z) S Q2 for all z E D and / a(5)2(5) 122— I S (5)) 24(5) s 222(2) (21) [D 2(5)2(5 )1og I 1(5)) 24(5) s 2222(2) (22) and [D a(n)w(n)v(n) 125 @222) s 222. (6.3) Then for f, g 2 0 E L2('Il‘) we have foawe )g (z z)1ogI— I_2A(z) < cQ2||f||v1||gHw-1. 36 PROOF. As before, it is more convenient to switch to Young’s inequality. It suffices to show that A2(z)f(z)g(z)1ogI:IdA(z) s cQ2(|lf||3-1 + Ilglli-1). Let us consider the following variables: X = f2v“1(z) a: = f(z) 'r = v(z) Y=92w“1(2) y =9(z) s =w(2) and the following non-harmonic variables 1 M: f‘w" 10g(£)I/IS )log)ldA(n)|S5( 24(5) N— [2(02 )1|__°g1()/(€)la(n)w()1m51°gl—_ ln)|d A(n)21(5) 1 K: /a(17)v(77)w(77)510gI'—— InIIdAOI) We have the natural estimates: 13733622 2:2 S X r and y2 S Y3 by Jensen’s inequality M S Q47“ and N S Q43 by (6.1), (6.2), (6.3) K S Q2 by (6.3) . 37 Let us consider the following function of nine (l) real variables: B(X,x,r,Y,y,s,M,N,K) = Bl(X, 33,1‘, M) + Bg(Y,y,s, N) + B3(X, x, r, Y, y, s, K) where $2 B X, ,,M =X— 1( :cr ) r+$§ 2 B2(YayaSaN)=Y_ yN 8+6? 9:23 — 2x313; + y2r 2 Bg(X,x,r,Y,y,s,K) =X+Y— and, as before, b(z), b1(z), b2(z), b3(z) the corresponding functions on ID. We discuss the properties of B. Derivative estimates: 2 63—le 2 47124:; since M S Q47" 632 > 1 y2 EV— _ 4724-55—2 since N S Q43 1‘8 033 3‘35}! if both K S 622% and K S Q23 0 else 38 By exchanging a: and y we only need to consider the case {— 2 y;- for the derivative estimate of Ba. Let us point out where 3;; came from. It was taken from an early version of [9], where it was written (up to normalization) in the following form: B3(X,x3r,Y’y’s,K) =X+Y—Supfl(a,X’x)r3Ky,S’K) aZO where 2:2 y2a aaxaxarax ,8,K = _— m y ) r+agg as+g§ Let us write R for K / Q2. It has been shown in an early version of [9] that under the restrictions above 2 18(a’X)$)T)Y,y)s)K) Z x 2 + E’— for 00:91; (6.4) 3 CBS [\DIr-a ’l‘ we will include the proof for the sake of completeness. Let us first observe that 2 2 ~ 2 2 ~ 2 a: .. Zi—aKE-z— and ——-y——1—~—>g-—a’1K&2- r+aK 1' 7' s+a-1K s s andhence x2 yz ~$2 _ ~y2 fi(a,X,x,r,Y,y,s,K)ZT+:-(aK-fi+a le—i)‘ (6.5) The part in parentheses for a = a0 = g is where (2:) uses the assumption R _<_ 1%. Now we obtain the required estimate from below for 6 at a0: V to [O (* a(ao,X. 33,73 Y. y. s, K) 2 3:— + m 313 S NIH y_2 _ s r where (2:) uses (6.5) for a0 together with (6.6). So taking supremum in the first variable yields $2 supfl(a,,,,X:crY,y,s, K)>—+1y2. a>0 28 39 Note that 6 is continuously differentiable in a for a > 0 and that [3 is close to 5r: for a near 0 and close to ”:2 for very large a. 80 6 as a function of a attains its maximum in (0,00). Testing for critical points yields 2" 2" _ ~ Qéz— 3K. + yK~ and a—B=O<=>a=yr—£I—:. 30 (r-l-aK)2 (as+K)2 00 xs—yK By the above it is already clear that am := 3333; > O and that 6 attains its maximum at this point. We found our B3 by letting a = am — — m1; We consider zs—yK the one parameter family of functions B§(X,a:,r,Y,y,s,K) :2 X + Y -— fi(a,X,a:,r,Y,y,s,K). In an early version of [9] the following derivative estimate has been proven: aBg .. Z cay- where am = M , (6.7) 6K a=am 7‘3 $3 — yK but B3(X,:c,r,Y,y, s, K) = B§m(X,:c,r,Y,y, s, K) so 923. _ BB? . 22m. + 03:? 6}? 0a 0:0,“ 01? 61? 0:0," . Note that 6755- = -— gg— a=am = 0 since 6 attains its maximum in am. We have the derivative estimate 999. :2 6K _ers' Let us include the proof of (6.7). First observe that according to (6.4) we have that $2 212 2 a x Z 3 Nli—I + 2 _ y_ 7' 8 ——.. + ———7 r + amK s + ,‘an . . . 2 2 _1 ~ . 2 Wthh implies Eff—I? 2 32’; and hence s 2 am K. But smce $7 _>_ 3’; inequality (6.4) implies also ~ and similarly we obtain 1' Z amK . Now we are ready to show the estimate in (6.7). .. — .. + .. .. .. 6K (7‘+aK)2 (3+a‘1K)2_ (r+aK)(s+a—1K) 68; (1:1:2 a‘ly2 > my now using 1' Z amff and s 2 (1,7111? we obtain the desired estimate. Size: We have the following obvious size estimates for Bi: OSBISX OSB2_<_Y OSBgSX+Y where 0 _<_ 83 follows from the fact that X — mix 2 O and Y — Elli—11? Z 0 for positive a. Concavity: BI and 82 are of the form (5.3) so —d2B1 2 0 — 01ng _>_ 0. Functions B; are concave for all parameters a, so B3 is, as infimum of a family of concave functions, concave, so —d2Bg 2 0. So B is concave. We turn to the main estimate. The functions B1, 82 and B3 will play their main roles in different parts of the unit disk. We divide ID into three parts A1={z€D;K(Z)2Q2%} 41 A2 = {z E 11);K() szf+:()2)} A3=1D>\(A1LJA2) If 2 6 A1, then flav229fifj§> and similarly, if z 6 A2, then —Ab2(z)> 16162201(2)f(2)gz 921 ) If 2 E A3 then 633 —Ab3(Z)_ >-——- ——K—-—-( AK) 0 f(2)9(z) 2 Q? T—wz)vIz)a(2)w(2)v(Z) = @361 a»2'(2)f(Z)g( )- Since —Ab1,2,3 2 O on all of II) we have all together -CQ2Ab(2) Z a(Z)f(Z)9(Z) and we are ready to run the Green’s formula trick: [0 a