BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all x n, one can defne the W -weghted L space L (W = L ( n, W, as the space wth norm (ˆ f L (W = (ˆ W (xf(x C dx = d W (xf(x, f(x dx < In ths paper we consder the L (W space above wth matrx weghts that satsfy the A property, that s, [W ] A = sup W W <, W We clam where the supremum s taken over all cubes n n and W s defned as that f W belongs to the A class, then the esz transform s bounded L (W L (W and n fact L (W L (W [W ] 3 A log[w ] A Even more generally, under the same A assumpton,the Ahlfors-Beurlng transform T also satsfes T L (W L (W [W ] 3 A log[w ] A The proof of both results reles heavly on the technques and results descrbed n [] Outlne and Notaton Let D denote a dyadc lattce n n and let {h } D, represent the famly of Haar functons adapted to D Also defne, h = for all D Notce that h s not L normalzed Followng [], we wll consder the W -adapted dsbalanced Haar functons These functons were orgnally defned n the matrx settng by Trel and Volberg n [6] and are gven n terms of the set of egenvectors of W Let e, e,, ed be a set of orthonormal egenvectors of W and take w k = W e k W ek The dsbalanced Haar functons are defned as g W,k, = := w k h e k + h ẽ k,,
where ẽ k, = A(W,, ew and A(W,, = W (W, W +, W Here +, and, represent the subsets of the cube to whch the Haar functon h assgns postve and respectvely negatve value Note that, by calculatons n [6], ( g W,k,, g W,l,j = 0, k, l d,, j n One can also see that g W,k, L (W s unformly bounded for all k and Another mportant ngredent of the proof s a generalzed verson of the square functon Let ˆf( be the n dmensonal vector wth entres ˆf ( = f, h = L d f, h e j j= L e j, n, where {e j } d j= s an orthonormal bass n Gven ˆf(, denote by T σ the Haar multpler operator T σ f = σ, ˆf (h, D = where σ, s a d d matrx wth entres and Wth ths notaton t s possble to defne a generalzed verson of the square functon S W as S W : L ( n, L ( n,, S W f(x = E W (xt σ f(x, T σ f(x We then have S W L = W I ˆf (, ˆf ( D If W s a scalar A weght, the square functon S W s bounded and the constant depends lnearly on the A characterstc The followng theorem descrbes the relaton between the L norm of S W and the norm of f n the matrx settng Theorem Let W be a d d A matrx weght Then ( W I ˆf (, ˆf ( [W ] A log[w ] A f L (W, f L (W D To prove ths theorem, we proceed as n [] The result requres the estmate below, whch follows by the argument of Trel and Volberg n [6]: Theorem Let W be a d d A matrx weght and let h be Haar functons on the cubes n D Let and be two dsjont subsets of of equal measure such that each dyadc chld of s contaned n ether or Then for all f L, W (W W D W W W f [W ] A log[w ] A f L
Ths s just the n verson of Theorem 6 n [6] wth the constants stated explctly For the sake of completeness, we provde a sketch of the proof of ths result n Secton 3 We wll also requre the followng lower bound of the square functon: Theorem 3 Let W be a d d A matrx weght and let C W be defned as Then for any f L (W, C W = [W ] A log[w ] A (3 f L (W C W S W f L Ths bound wll arse as a consequence of Theorem under a change n notaton 3 Proof of Theorem In ths secton we provde the proof of the result of Trel and Volberg for n We wll track the dependence of the bound on [W ] A, but not on dmensonal constants We begn by statng some lemmas from [6] and provdng the proofs for those results that need to be adapted to hgher dmensons Lemma 4 Let W be a nonnegatve measurable d d matrx functon on a measure space (X, µ Then ˆ ˆ W (t dµ(t d W (tdµ(t X The proof s mmedate gven that the norm of any square matrx s equvalent to ts trace Usng Lemma and the smlar result for scalar weghts, one obtans: Lemma 5 For a d d matrx A weght W, the followng nequalty holds: ˆ ( ˆ W C W, where C s a dmensonal constant The followng theorem s the man result of [6] Theorem 6 Let W be a d d A weght and let h be a collecton of Haar functons on the dyadc cubes Let and be two subsets of of equal measure such that each dyadc chld of belongs only to ether or Then D, for all dyadc cubes W (W W X W Proof For any dyadc cube, we defne the followng µ( = det W ν( = det W m( = µ(ν( [W ] A log[w ] A,
Note that W = W +W, so by Lemma 4 n [6], Wth ths notaton, µ( µ( µ( = det W W (det W = det µ( µ( µ( and ν( ν( + ν( ( W W W ( det W W W and followng the argu- Lettng A = W W W and B = W W W ment n Proposton 4 n [6], we obtan that ( det A det B exp 4 trace((a B Therefore, µ( (µ( µ( exp ( 4 trace By the defnton of m, snce ν( ν( ν(, ( (4 m( (m( m( exp ( ( 4 trace W ( ( W (W W (W W W W It s now possble to apply (4 to and, takng and and and respectvely to be the subsets of and wth the propertes descrbed n (4 epeatng the process, we obtan ( ( n ( δ ( ( m( m( δ exp 8 trace W δ W δ W δ W δ, δ where δ runs over all possble n-tuples of and By the same reasonng as n [6], ( δ ( ( trace W δ W δ W δ W δ [W ] A log[w ] A We can now provde the proof of Theorem Proof Set µ( = W (W W W Let J be the embeddng operator J : L ( l ({µ(},, { } J f = W W f It can be checked that the operator J, the formal adjont of J has representaton J {α } D = χ W W α, {α } D l ({µ(}, D D
Notce that f =, then the block (J J, of the operator (J J s zero Furthermore, (J J, = ((J J,, and f, (J J, = W W Therefore, f, (J J, = (J J, = W W Set t, = t, and let them be equal to the expresson above f If =, set t, = t, = 0 We obtan a matrx {t, }, D To prove Theorem, t suffces to show that ths matrx generates a bounded operator on l ({µ(},, and n fact, snce the matrx {t, }, D s symmetrc, t s suffcent to show that the matrx {T, }, D defned as { t, T, = 0 otherwse generates a bounded operator on l ({µ(}, To do so, we must bound the sum T, T, µ( D If =, by defnton, T, T, = 0, so we can assume wthout loss of generalty that and In ths case, T, T, µ( = T, T, µ( D D, = W W W W µ( D, = W W W W W W µ( D, W W W W µ( D, By Lemma 35 n [6], the weght W W W satsfes the A condton wth A char- acterstc [W ] A Thus we can apply Lemma 5 to bound W W : W W = W W W ˆ W W W ( ˆ C W W ( ˆ = C W W χ
By Lemma 6, D, W (W W W [W ] A log[w ] A, so t s possble to apply the dyadc Carleson Embeddng Theorem (Theorem 6 n [6] One obtans: T W W, T, µ( D, C ( ˆ W W χ µ( D, W C[W ] A W χ L ˆ = C[W ] A W W W C[W ] A W W W (ˆ = C[W ] A log[w ] A, where the last nequalty was obtaned from Lemma 4 Thus, D so Theorem s proved T, T, µ( C[W ] A W W = C[W ] A (T, + T,, 4 Proof of theorem Proof Assume wthout loss of generalty that W and W are bounded (see emark 34 n [] and let DW be the dscrete multplcaton operator { DW h j e W k = h e k f = j 0 f j, where e k are the standard bass vectors n Defne the ( n ( n matrx operator D W by ( D W,j = D W Also let F be the n vector wth entres F = f and let M W be the ( n ( n dagonal matrx operator ( M W = M W, where M W s the multplcaton by W operator on L Note that S W f L = DW F, F L Therefore, nequalty (3 becomes MW F, F C n W DW F, F L L
Followng the argument n [] (whch tself follows [4], we proeed to nvert the nequalty Snce W and W, are bounded, the operators M W and DW are bounded and nvertble and ther nverses are M W = M W and { (DW h j e W k = h e k f = j 0 f j, respectvely Thus D W and M W are nvertble wth nverses D W and M W and the nequalty above can be nverted as: (5 D W F, F C W M L W L F, F In concluson, the proof of Theorem 3 amounts to showng that W ˆf (, ˆf ( C W f L (W f L D Frst defne, for D, a set of orthonormal egenvectors e, e,, ed of W Also defne w k = W e k W ek Then D W = ˆf (, ˆf ( = D d (w k f, h e k L Usng the dsbalanced Haar functons adapted to W, we expand h ek as: D d (w k f, h e k = L D D + D d (w k f, (wq k g W,k, L (w k A(W, h e k d f, g W,k, L + d f, A(W, h e k L = S + S + S 3 We now bound the three sums ndvdually By defnton, S = d f, g W,k, L D = d W f, g W,k, D f L (W, D L (W d f, g W,k, f, A(W, h e k L L where the last nequalty s a result of the fact that the functons g W,k, are unformly bounded n L (W and have dsjont support To bound S 3, one can rewrte t as
S 3 = D = D = D = D D D d d d d f, W (W, W +, W h e k L f, W Ŵ ( W h e k L f, W Ŵ ( W ek W f, W Ŵ ( W ek d W f W Ŵ ( W W f W Ŵ ( W ek Applyng Theorem to the functon W f, one obtans S 3 [W ] A log[w ] A f L (W Fnally, to bound S, notce that S S S 3, so the bounds on S and S 3 mply a bound on S Therefore, the quantty s bounded, that s whch completes the proof of Theorem S + S + S 3 C W f L (W, We can now apply Theorem to prove Theorem 3 Proof For ease of notaton, let B W = [W ] A log[w ] A, and as before assume wthout loss of generalty that W and W are bounded Usng the same notaton as above, we fnd that provng Theorem 3 s equvalent to showng that D W f, f L B W M W f, f L f L As before, t suffces to prove D W f, f L B W M W f, f L f L We clam that D W [W ] A ((D W By the A condton, W W e, W W e [W ] A e,
for all D and all vectors e In partcular, for a fxed g L and e = W ĝ (, W ĝ (, W W ĝ ( [W ] A ĝ (, W Then D W g, g = W L ĝ (, W ĝ ( D W [W ] A ĝ (, W ĝ ( D = [W ] A D W g, g L ĝ ( By equaton (5 n the proof of theorem 3, D W g, g L [W ] A (D W g, g L [W ] A C W M W g, g L = B W g L (W 5 The sez Transform In [5], Petermchl, Trel, and Volberg proved that the esz transform on L (W can be consdered essentally as an average of dyadc shfts Ths result s an extenson of the frst author s result n [3], whch stated that unform boundedness of the dyadc shfts mples the boundedness of the Hlbert transform The dyadc shft operator on a dyadc lattce D s defned as D f = f ((h + h D By the man results n [5], the esz transform on L ( n can be wrtten as = ct + M b, where c s a constant, T s the closure of the convex hull of the planar dyadc shfts n the weak operator topology, b s a bounded functon on n, and M b s the multplcaton by b operator Applyng ths result componentwse to the esz transform on L ( n,, we obtan the same expresson as before Gven ths equalty, we fnd that the bound proved n Theorem mples a smlar bound for the esz transform Theorem 7 If s the esz transform on L ( n, and W s a d d matrx A weght, f L (W [W ] 3 A log[w ] A f L (W Proof Note that the square functon S W does not depend on dyadc shfts More precsely, f s the parent of n the dyadc grd, W f(, f( D = = D D = S W L W f (, f ( W f (, f (
Now, by Theorems, and 3 f L (W [W ] A log[w ] A W f (, f ( D [W ] 3 A (log[w ] A f L (W The expresson for n terms of f then gves whch proves the theorem f L (W [W ]3 A (log[w ] A f L (W, A smlar argument mples the same bound for the Ahlfors-Beurlng transform Theorem 8 If T s the Ahlfors-Beurlng transform on L ( n, and W s a d d matrx A weght, T f L (W [W ] A log[w ] A f L (W Proof In [], t s shown that the Ahlfors-Beurlng transform T s n the weakly closed lnear span of two-dmensonal martngale transforms τ σ Consder the dyadc lattce D as the collecton of squares of the form = I J, where I and J are dyadc ntervals of the same length Then one can consder three Haar functons assocated to : h I (x, y = h I (x J (y J h J (x, y = h J (x I (y I h (x, y = h I (xh J (y Wth ths notaton, the two-dmensonal martngale transform s defned as τ σ f := σ I ( f, h I h I + σ J ( f, h J h J + σ( f, h h, D D D where σ I, σ J, σ : D {, } The proof of the theorem amounts to boundng the operators τ σ unformly by the quantty [W ] A log[w ] A We have τ σ f L (W σ I ( f, h I h I + σ J ( D f, h J h J + σ( f, h h L D (W L D (W = B + B + B 3 The bounds on B and B are establshed n [] For B 3, one can consder B 3 T σ L (W The bound on T σ L (W arses from a smlar argument to that n [] Note that D = W Tσ f (, T σ f ( = D = D = = σ, W σ, f (, σ, f ( L (W W σ, W W f (, σ, W W f ( D = W f (, f (
By Theorems and 3, T σ L (W [W ] A log[w ] A W Tσ f (, T σ f ( [W ] A log[w ] A σ, D = [W ] 3 A (log[w ] A σ, f L (W W f (, f ( Therefore, the bound on B 3 s proved Substtutng back nto the defnton of τ σ, we obtan τ σ f L (W B + B + B 3 the desred nequalty [W ] 3 A log[w ] A f L (W, eferences [] K Bckel, S Petermchl, and B Wck, Bounds for the Hlbert Transform wth matrx A weghts (04, arxv:403886v3 [] O Dragčevć and A Volberg, Sharp estmates of the Ahlfors-Beurlng transform va averagng martngale transforms, Mchgan Math J5 (003 [3] S Petermchl, Dyadc shfts and a logarthmc estmate for Hankel operators wth matrx symbol, C Acad Sc Pars Sér I Math 330 (000, no 6, 455 480 [4] S Petermchl and S Pott, An estmate for weghted Hlbert transform va square functons, Trans Amer Math Soc 354 (00, no 4, 699 703 [5] S Petermchl, S Trel, and A Volberg, Why the esz transforms are averages of the dyadc shfts, Publ Math46 (00, no, 09 8 [6] S Trel and A Volberg, Wavelets and the angle between past and future, J Funct Anal, 43 (997, no, 69 308