Contents. 1. The simplest operator whose average is the Hilbert. transform WHY THE RIESZ TRANSFORMS ARE AVERAGES OF THE DYADIC SHIFTS?

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1 Publ. Ma. (22), Proceedings of he 6 h Inernaional Conference on Harmonic Analysis and Parial Differenial Equaions. El Escorial, 2. WHY THE RIESZ TRANSFORMS ARE AVERAGES OF THE DYADIC SHIFTS? S. Peermichl, S. Treil and A. Volberg Absrac The firs auhor showed in [8] ha he Hilber ransform lies in he closed convex hull of dyadic singular operaors so-called dyadic shifs. We show here ha he same is rue in any R n he Riesz ransforms can be obained as he resuls of averaging of dyadic shifs. The goal of his paper is almos enirely mehodological: we simplify he previous approach, raher han presening he new one. Conens. The simples operaor whose average is he Hilber ransform Wha is he dyadic shif? Planar case Dimension uning Geomeric applicaion 225 References 226. The simples operaor whose average is he Hilber ransform Le L denoe a dyadic laice in R. By L(k) we undersand he dyadic grid of inervals from L having lengh 2 k, k Z. For he convenience we would like o use he noaions D =: L(). We consider firs such a dyadic laice ha he grid D has he poin as one of he end-poins of is inervals. To emphasize ha we wrie D. Laer we will have D he poin plays he role of. 2 Mahemaics Subjec Classificaion. Primary: 47B37; Secondary: 47B4. Key words. Riesz ransforms, Haar funcions, dyadic shif, dyadic laice, recifiable measures, Menger s curvaure. Auhors are parially suppored by NSF gran DMS

2 2 S. Peermichl, S. Treil, A. Volberg Le us consider he following linear operaion f ϕ(x) := Σ I D (f, h I )χ I (x). Here h I denoes he Haar funcion of he inerval I, ha is I, for x I h I (x) = /2 I, for x I +, /2 and I, I + are lef and righ halves of he inerval I correspondingly. Symbol χ I as usual sands for he characerisic funcion of he inerval I. This linear operaion is no even a bounded operaor in L 2 (R), bu i will be our main building block, so i deserves a name P. Acually, we will call i P, hus ϕ (x) := P f := Σ I D (f, h I )χ I (x). Index indicaes he end-poin of one of he inervals from D. So similarly we consider ϕ (x) := P f defined exacly as before, bu wih respec o he grid D of uni inervals such ha he end-poin of one of hem is in R. Noice ha he family of grids D, R, can be naurally provided wih he srucure of probabiliy space. This space is (R/Z, d) = ((, ], d). As usual we can use he leer ω for a poin from (, ], and dp (ω) denoes he probabiliy in his case jus Lebesgue measure on he inerval (, ]. We wan o fix x R and o wrie a nice formula for E (ϕ ω (x) dp (ω)). So we wan o average operaors P ω. I can be noiced immediaely ha EP ω is a convoluion operaor. In fac, le us denoe by L a he shif operaor: L a (f)(x) = f(x + a). Then obviously P a L a = L a P. Applying averaging (and he fac ha our dp (ω) is invarian wih respec o he naural shif on R/Z induced by he shif on R) we immediaely ge (.) EP ω L a = L a EP ω.

3 Riesz Transforms as Averages of Dyadic Shifs 2 So he average operaor EP ω is a convoluion operaor, we will wrie his as follows (.2) E (ϕ ω (x) dp (ω)) = EP ω f(x) = F f(x). I is easy o compue F. By he definiion of ϕ (x) one can wrie (see Figure ) (.3) ϕ (x) = f(s)h 2 (s) ds, x 2 < 2 < x + 2, where h (s) = {, s ( /2, ) +, s (, + /2). Bu h (s) = k ( s), where { +, s ( /2, ) k (s) =, s (, /2). So (.3) can be rewrien as follows (.4) ϕ + (x) = f(s)k 2 ( s) ds, x 2 < < x + 2. Thus comparing his wih (.2) (and using again he shif invariance of dp(ω)) we ge F f(x) = E (ϕ ω (x) dp (ω)) ( ) = E ϕ ω+ (x) dp (ω) = 2 x+ 2 x 2 ( ) f(s)k ( s) ds d. From which we ge he formula for F : (.5) F (x) = x+ 2 x 2 k () d = k χ (x), where χ is he characerisic funcion of he uni inerval ( /2, /2). On Figure 2 one can see he graph of F.

4 22 S. Peermichl, S. Treil, A. Volberg x s Figure. Funcion h (s) /2 x Figure 2. Funcion F Le us sar over he beginning of his secion wih one sligh difference we rescale all our operaors, and now P ρ, ϕ ρ, F ρ, kρ are precisely as above, bu when he uni lengh inervals are replaced by inervals of lengh ρ >. We jus change he scale nohing else. In paricular, ϕ ρ (x) := Pρ f := Σ I D ρ (f, h I)χ I (x)/ ρ

5 Riesz Transforms as Averages of Dyadic Shifs 23 where D ρ is he grid of inervals of lengh ρ such ha is he end-poin of wo inervals from his grid. We wan o remind ha h I here is always normalized in L 2. Again we have a naural ) probabiliy space of all grids of inervals of size ρ: (R/ρZ; ρ d ( ρ, ]. ϕ ρ (x) := P ρ f := Σ I D ρ (f, h I)χ I (x)/ ρ. Averaging over all grids of inervals of size ρ makes P ρ a convoluion operaor here is no difference wih our reasoning above. I is easy o see ha his is he convoluion operaor wih he kernel F ρ (x) := x+ ρ ( ) 2 ρ ρ k d = ( ) x (.6) ρ ρ F. ρ x ρ 2 The firs ρ is because of he form our probabiliy has. The second ρ because we should average a funcion normalized in L. Le us now consider all convoluion operaors wih kernels F ρ. Le us fix r [, 2) and le us ake a look a he convoluion operaor wih kernel (.7) F r = Σ n= F 2n r. The grids D 2n r ( is fixed) can be unied ino a dyadic laice L r. Here means he reference poin one of he end-poin of inervals from our laice, and r means he lengh of one of he inervals of he laice le us call r he calibre of he laice. Obviously he convoluion operaor wih he kernel F r is he averaging over all dyadic laices (no grids!) L r of fixed calibre r of he operaors given by P L r f = Σ I L r (f, h I )χ I (x)/ I F r f = EP L r f. This is jus because he kernel F r is he sum of kernels, each of which appeared as averaging of he grid opearaors assigned o grids of size 2 n r, n =, ±, ±2,..., where we summed up over he grids, and he laice of calibre r is he union of such grids. Now le us finally average over r [, 2): F (x) := 2 F r (x) dr r.

6 24 S. Peermichl, S. Treil, A. Volberg (.8) Now we have from one side F f = (Average P L ) f, where averaging is performed over all laices L r. (.9) On he oher hand i is easy o compue Φ. F (x) = = 2 2 F r (x) dr r Σ n= F 2n r dr r = F ρ dρ ρ = We used (.6) here. Finally we have (see Figure 2) (.) F (x) = x F () d = 4 x. F ( x ρ ) dρ ρ 2. Theorem.. Averaging of operaors P L r over boh parameers and r is equal o one quarer of he Hilber ransform. Proof: Jus compare (.8) and (.). We have a good hing: The Hilber ransform is he averaging over he family of laices and a bad hing: of very simple operaors These simple operaors are no bounded in L 2. We wan he bes of he boh worlds: a) he Hilber ransform is he averaging over he family of laices of very simple operaors; b) hese simple operaors have o be bounded in L Wha is he dyadic shif? The funcion ha generaed everyhing in he firs secion was funcion F he kernel of he convoluion operaor which is he averaging of grid operaors P. I is easy o see ha F (x±) are also kernels of he convoluion operaors which are he averagings of some grid operaors. Given f, le us consider ϕ (x) as above and also ϕ (x + ) = Σ I D (f, h I )χ I (x) = Σ I D (f, h I )χ I+ (x) =: P + (f) ϕ (x ) = Σ I D (f, h I+ )χ I (x) = Σ I D (f, h I )χ I (x) =: P (f). So we es f on h I and pu he resul on I ±.

7 Riesz Transforms as Averages of Dyadic Shifs 25 Wha if we average hese operaors? Repeaing (.2) we ge ( ) (2.) d f = F (x ) f. Consider P ± (2.2) S(x) := F (x) 2 [F (x + ) + F (x )]. Supposedly S is a kernel of a convoluion operaor corresponding o averaging over grids of a cerain grid operaor (we will show which one). If we build S ρ as before for all calibres, we can consider again S r := Σ n= S 2nr. Operaors S r are averagings over all laices of calibre r of he operaors which are sums of our hypoheical grid operaors. Averaging over r [, 2) wih respec o he measure dr/r, we will ge he operaor wih kernel (see Figure 3 and (.9)) 2 S r (x) dr ( ) x dρ r = S ρ ρ 2 = (2.3) S() d = x 4 x. So we are lef o inven a simple grid operaor, whose average will give us S(x). Theorem 2.. Le D (2) be a grid of inervals of lengh 2 such ha is he end-poin. Consider operaors f Σ (2) J D (f, h J )χ J+ f Σ (2) J D (f, h J+ )χ J f Σ (2) J D (f, h J )χ J f Σ (2) J D (f, h J+ )χ J+. The averaging over of he firs operaor gives a convoluion wih kernel 2 F (x ), he averaging over of he second operaor gives a convoluion wih kernel 2 F (x + ), and he averaging over of he hird and he fourh operaor gives a convoluion wih kernel 2 F (x) each. Proof: Le us call he firs operaor H, and le us average EH i over is probabiliy space ( R/2Z; 2 d ( 2, ]). Insead of considering he grid of inervals of lengh 2 le us consider he grid of inervals of lengh we call i D. Consider operaors A : f Σ I is odd, I D () B : f Σ I is even, I D () (f, h I )χ I+, (f, h I )χ I+. Clearly A + = B. Also i is clear

8 26 S. Peermichl, S. Treil, A. Volberg ha A + B = P +, where he las operaor is our grid operaor from he beginning of Secion 2. EH = 2 From (2.) we ge ha (A + A + ) = 2 EH = 2 F (x ). (A + B ) = 2 P + Similarly, if we call he second operaor G we ge from (2.) EG = 2 F (x + ). Using (.2) and (.5) we show ha averagings of he hird and he fourh operaors give us convoluion operaor wih kernel 2 F. Theorem 2. is proved. Theorem 2.2. Le us consider he following grid operaor f Σ (2) J D (f, h J+ h J )h J, d. (R/2Z; 2 d ( 2, ] ). Then is averaging is he convoluion operaor wih kernel 2 S(x). Proof: We wrie h J as 2 ( χ J + χ J+ ). Then i is an obvious algebraic remark ha 2 our operaor = hird operaor of Theorem fourh operaor of Theorem 2.2 firs operaor of Theorem 2.2 second operaor of Theorem 2.2. Averaging his and using Theorem 2. finishes he proof. As in he previous secion, given he laice L = L r, we can consider he laice operaor L f := Σ J L (f, h J+ h J )h J amalgamaed from he grid operaors of Theorem 2.2. This operaor is called he dyadic shif. I has been proved in [8] ha averaging of dyadic shifs over all laices gives us operaor which is proporional o he Hilber ransform (we cerainly mean ha coefficien of proporionaliy is no zero).

9 Riesz Transforms as Averages of Dyadic Shifs 27 Le us reproduce his resul. Fixing r and averaging over laices wih fixed calibre r (we leave for he reader o inven he naural probabiliy space of all laices wih fixed calibre r) we ge he convoluion operaor wih he kernel 2 n= ( x ) 2 n r S 2 n =: r 2 S r (x). Averaging convoluion operaors wih kernels 2 S r over ( ) [, 2); dr, we ge (see (2.3)) he operaor wih he kernel 4 2 x. So we ge (2.4) Averaging of he shif operaors over all laices of all calibres = 4 2 he Hilber ransform. r 4 2 x Figure 3. Funcion S 3. Planar case We can and will reason by analogy. We have laices L ρ of squares, where now is in Ω ρ := R 2 /ρz 2 wih normalized Lebesgue measure (Lebesgue measure on he orus Ω ρ divided by ρ 2 ). We have he main grid operaor P f := Σ Q D (f, h Q )χ Q

10 28 S. Peermichl, S. Treil, A. Volberg where D is a grid of uni squares such ha R 2 is a verex for 4 of hem, where, for x Q l Q h Q (x) :=, for x Q r Q, oherwise. Here Q l, Q r are lef and righ halves of Q, funcion h Q is normalized in L 2. We consider he same ype of grid operaors for grids D ρ of squares of side ρ he only change is ha we divide χ Q by ρ o make i normalized in L 2. Le us denoe by k he funcion h Q, where Q is he uni square cenered a. Also χ denoes he characerisic funcion of his square. Consider Φ := χ k, Φ ρ (x) := ρ 2 ρ 2 χ ( ) ( ) k = ρ ρ ρ 2 Φ ( ) x. ρ Exacly as before (in one dimensional case) funcion Φ is he kernel of he convoluion operaor, which appears as averaging of P over Ω. Funcion Φ ρ is he kernel of he convoluion operaor, which appears as averaging of P ρ over Ω ρ. Again, we can consider kernel ( ) x ω k(x) := Φ ρ (x)dρ ρ = x x 2. And i is very easy o see ha ω is an odd non-zero funcion on he uni circle. Lierally as before we can see ha k is he convoluion operaor which is he average wih respec o measure dr r [, 2) of he convoluion operaors wih kernels k r (x) := n= Φ r 2n (x). In is urn, k r is he average of he laice operaors which are sums of corresponding grid operaors, here are hose laice operaors: P L r := Σ Q L r(f, h Q )χ Q / Q.

11 Riesz Transforms as Averages of Dyadic Shifs 29 Here r is fixed and denoes he calibre of he laice. The averaging over he laices of his fixed calibre gives us he convoluion operaor wih kernel k r. So he averaging over he calibres (= 2... dr r ) gives us he averaging over all laices, over all ( calibres. As a resul we ge he convoluion operaor wih kernel k = ω x ) x x. 2 Again we would like o repea all his bu wih slighly differen laice operaors jus because here are nicer ones and because P L r are no L 2 bounded. Anoher problem we face now is ha k is no necessarily a kernel of a Riesz ransform. So we will need o work a bi more han in he one-dimensional case o obain he Riesz ransform kernel. For a square Q consider is pariion o 4 equal squares and le us call hem Q nw, Q ne, Q sw, Q se according o norhwes, norheas,.... Le us consider he following grid operaor f Σ Q D (2) (f, h Q ne + h Q se h Q nw h Q sw)h Q, Ω (2) := Consider also he funcion (x = (x, x 2 )) (R 2 /2Z 2 ; 4 Lebesgue measure ). S(x, x 2 ) = Φ (x, x 2 ) 2 Φ (x +, x 2 ) 2 Φ (x, x 2 ) (3.) + 2 Φ (x, x 2 + ) 4 Φ (x +, x 2 + ) 4 Φ (x, x 2 + ) + 2 Φ (x, x 2 ) 4 Φ (x +, x 2 ) 4 Φ (x, x 2 ). Theorem 3.. The averaging of he grid operaor above over Ω (2) gives he convoluion operaor wih kernel 2 S(x). The proof is lierally he same as he proof of Theorem 2.2. Le us sar wih one observaion abou (3.). Funcion Φ is he convoluion χ k. Bu boh funcions χ and k are producs of funcions of one variable -Φ (x, x 2 ) = f (x 2 ) F (x ). Moreover, funcion f is nonnegaive. Acually f (x 2 ) is a convoluion square of he characerisic funcion of he uni inerval cenered a. Formula (3.) now looks like ( S(x, x 2 ) = f (x 2 ) + 2 f (x 2 + ) + ) 2 f (x 2 ) ( F (x ) 2 F (x + ) ) 2 F (x ).

12 22 S. Peermichl, S. Treil, A. Volberg For he fuure purposes we can say wha happens in n > 2 case easily. We ge S n (x) = S n (x, x 2,..., x n ) and ( (3.2) S n (x) = F (x ) 2 F (x + ) ) 2 F (x ) n i=2 ( f (x i ) + 2 f (x i + ) + ) 2 f (x i ). As in he previous secion his S generaes kernel s by formula ( ) ( ) x x dρ ξ n s(x) = ρ n S ρ ρ = x x n. And i is very easy o see ha ξ n is an odd non-zero funcion on he uni sphere. We will show i below. Lierally as before we can see ha s is he convoluion operaor which is he average wih respec o measure dr r [, 2) of he convoluion operaors wih kernels s r (x) := n= S r 2n (x). In is urn, s r is he average of he laice operaors which are sums of corresponding grid operaors, here are hose laice operaors: (3.3) S L r := Σ Q L r(f, h Q ne + h Q se h Q nw h Q sw)h Q. Here r is fixed and denoes he calibre of he laice. The averaging over he laices of his fixed calibre gives us he convoluion operaor wih kernel s r. So he averaging over he calibres (= 2... dr r ) gives us he averaging over all laices, over all calibres. ( As a resul we ge he x ) convoluion operaor wih kernel s = ξn x x. n Le S n denoe as always he boundary sphere of he n-dimensional uni ball. Denoe by S+ n he righ half sphere he half ha lies in {x R n : x > }. Le e be a uni vecor in he direcion of coordinae axis x. Le denoe he scalar produc in R n. Le σ denoe Lebesgue measure of S n. I would be imporan o prove (3.4) ξ n (ω) ω, e dσ(ω) <. S n + For n = 2 we can jus prove ha ξ 2 (ω) < for any ω S+. Then (3.4) follows immediaely. To do his we use formula (3.2) and noice ha f (x) + 2 f (x + ) + 2 f (x ) = ( 2 x) +. Then he fac ha ξ 2 (ω) < follows from he following lemma.

13 Riesz Transforms as Averages of Dyadic Shifs 22 Lemma 3.2. For any k [, ) we have 2 ( 2 ) ( kx F (x) ) 2 F (x ) x dx <. + Proof: If k 2 hen he firs facor vanishes everywhere where he second facor is posiive. So we are done for such k. For k we have ( 2 kx) + = ( 2kx) on [, 2], and we can make an easy calculaion of he inegral. For he range < k < 2 he calculaion becomes unpleasan, bu sill sraighforward, we skip i jus o avoid direc and simple calculaions. For n = 2, ω can be idenified wih a poin of [ π, π). Under his idenificaion he kernel ξ 2 becomes an even funcion skew symmeric on [, π] wih respec o he poin π/2. Roaion of he kernel ξ 2 (ω) means jus he new kernel ξ 2 (ω φ). Then ( π ) (ξ 2 cos)(φ) = cos φ ξ 2 (s) cos s ds (3.5) π ( ) = cos φ ξ n (ω) ω, e dσ(ω) S = c 2 cos φ, and consan c 2 = S ξ n (ω) ω, e dσ(ω) because of (3.4). π cos s ds π Consider A 2 := c 2. Noice ha roaion of kernel ξ 2 corresponds o roaion of dyadic laices on he plane. We have jus proved he following heorem. Theorem 3.3. The Riesz ransform ) x ξ 2 (U ψ cos ψ x c 2 x 3 x x 3 x x is he operaor inegral 3 dψ. In paricular, his means ha operaor wih he kernel A 2 lies in he closed convex hull (in he weak operaor opology) of he planar dyadic shifs. Thus, uniform boundedness of dyadic shif operaors in any Banach space implies he boundedness of he Riesz ransform in he same space. For he case n > 2 we again sar wih (3.4). Le us average ξ n wih respec o all roaions ha leave e fixed. We ge a new funcion η n (ω) = f( ω, e ). Obviously, (3.6) f( ω, e ) ω, e dσ(ω) <. S n + Le SO is he group of orhogonal roaions of S n.

14 222 S. Peermichl, S. Treil, A. Volberg Le us calculae c n = SO f( Ue, e ) Ue, e du. Obviously, c n = f( ω, e ) ω, e dσ(ω), S n because of (3.6). Now le us consider he roaed funcions f( Uω, e ). Consider g(ω) = f( Uω, e ) Ue, e du. SO Then i is clear ha g(rω) = g(ω) for every R SO ha fixes e. In fac, g(rω) = f( URω, e ) Ue, e du = = SO SO SO f( V ω, e ) V R e, e dv f( V ω, e ) V e, e dv = g(ω). On he oher hand, i easy o see ha (3.7) g(ω) = f( ω, ξ ) ξ, e dσ(ξ). S n Such a funcion (as we saw) depends only on ω, e. Bu moreover, i can be wrien as f( e S n, ξ ) ξ, ω dσ(ξ). This is a resricion of a linear polynomial ono he sphere. This linear polynomial depends on ω, e only, and, hus, is c ω, e. The consan c is jus our c n. One can see ha by plugging ω = e ino our formula (3.7) for g(ω). Ue,e du SO Consider A n := c n. Noice ha roaion of kernel ξ n corresponds o roaion of dyadic laices on he plane. We have jus proved he following heorem. Theorem 3.4. The Riesz ransform η n (U x c x n Ue, e SO x n+ x x is he operaor inegral n+ ) du. In paricular, his means ha operaor wih he kernel A n x lies n+ in he closed convex hull (in he weak operaor opology) of he planar dyadic shifs. Thus, uniform boundedness of dyadic shif operaors in any Banach space implies he boundedness of he Riesz ransform in he same space. x

15 Riesz Transforms as Averages of Dyadic Shifs 223 Remind ha we have proved (3.4) inequaliy only for he case n = 2 so far (see Lemma 3.2). We do no know how o compue he inegral in (3.4) for he n > 2 dimensional analog of he operaor in (3.3). However, we are going o inroduce he operaor similar o he one in (3.3) ha will give us (3.4) immediaely. Here is he descripion of his operaor. For every cube Q of a dyadic laice L in R n we denoed by h Q he funcion suppored by Q and such ha i is equal o on Q /2 he lef half Q l of Q and is equal o on he righ half Q Q /2 r of Q. This jus one of he Haar funcions. We are going o choose where his funcion should be mapped by our dyadic shif by using he following consideraions. The image mus be he combinaion of Haar funcions of he previous generaion. Tha means ha i should be suppored by he faher Q of Q, should be consan on each son of Q (including Q), and he sum of hese cosans mus be zero. So he only choice is he choice of consans c B, where B is eiher Q or one of 2 n of is brohers. When n = we made a correc choice by using he rule c Q =, c B = for he only broher B of Q. One of he naural choices now would be c Q = 2n Q, c /2 B =, B is he broher of Q. Q /2 The operaor which sends each h Q, Q L, o he corresponding funcion on Q is called L (we should say ha i maps all oher Haar funcions o zero). And he operaor, which does his for a dyadic grid G will be called P G. Le us consider all dyadic grids of cubes Q wih sidelengh 2. And le us consider he averaging P = EP G over a probabiliy space of all such dyadic grids. Operaor P is of course a convoluion operaor. Le us call is kernel p. Remind ha S n + = {ω R n : ω =, ω, e > } is he righ half sphere. Obviously he nex Lemma 3.5 proves (3.4), and we finish he proof ha he Riesz ransform R can be decomposed ino he dyadic shifs. Lemma 3.5. For any ω S n +, p( ω ρ ) ρ n dρ ρ >. Proof: For any given G of cubes Q of sidelengh 2 le us spli P G ino wo operaors. The firs will be called V G and i maps h Ql ino 2 n χ Ql, h Qr ino 2 n χ Qr. In oher words, if Q is a uni cube ha happens a son of a cube from G, hen V G (h Q ) = 2 n χ Q. And i maps all oher Haar funcion o zero. The res will be called W G. In oher words, if Q is a uni cube ha happens a son of a cube Q from G, hen W G (h Q ) = B is he son of Q χ Q = χ Q. As always, le us consider all dyadic grids of cubes Q wih sidelengh 2. And le us consider he averagings V = EV G,

16 224 S. Peermichl, S. Treil, A. Volberg W = EW G over a probabiliy space of all such dyadic grids. Operaors V, W are of course convoluion operaors. Le us call heir kernels v, w. Denoe w 2 (x) = 2 n w(2x). Noice ha w 2 ( ω ρ ) dρ ρ n ρ = w( ω ρ ) dρ ρ n ρ. Remind ha p = v w. Now we can see ha o prove he lemma we obviously jus need o show ha for any ω S+ n ( ) ω dρ ( ) ω (3.8) v ρ ρ n ρ dρ w 2 ρ ρ n ρ <. (3.9) We will jus see ha v(x) w 2 (x). I is very easy o see ha inequaliy (3.9) gives a sric inequaliy in (3.8). So le us see why (3.9) holds by jus compuing he kernels. We will use he noaion F from he firs secion and f is he convoluion of he characerisic funcion χ [ /2,/2] wih iself. Then i is easy o see ha v(x) = 2 n F (x )f (x 2 )... f (x n ) w(x) = 2 n (2F (x ) + F (x ) + F (x + ))(2f (x 2 ) + f (x 2 ) + f (x 2 + ))... (2f (x n ) + f (x n ) + f (x n + )). Le a() := 2F ()+F ( )+F (+), b() := 2f ()+f ( )+f (+). I is easy o check ha 4F () 2a(2), 4f () = 2b(2). Thus, 2 n F (x )f (x 2 )... f (x n ) 2 n w(2x). Inequaliy (3.9) is compleely proved, and his proves he lemma. 3.. Dimension uning. Le us consider a varian of dyadic shif, bu slighly rescaled. Namely, le L be any dyadic (acually r imes dyadic) laice dropped on he plane. We have inroduced dyadic planar shifs L = Σ Q L sh Q, here sh Q is a rank one operaor described in he previous secion (or, for ha maer, any oher dyadic shif operaor, for example, he one from (3.3)). Le us call hem dyadic planar shifs of order 2. The dyadic planar shif of order d is d L := Σ Q L Q 2 d 2 shq. As before by averaging over laices we ge kernels p d (x), ζ d α(x). Wih no changes we ge

17 Riesz Transforms as Averages of Dyadic Shifs 225 Theorem 3.6. The planar Riesz ransform wih kernel x x +d is he operaor inegral g(α)ζα, d in paricular, he A muliple of his operaor is equal o a cerain averaging of dyadic planar shifs of order d. This means ha operaor wih he kernel A x lies in he closed convex x +d hull (in he weak operaor opology) of he planar dyadic shifs of order d. Thus, uniform boundedness of dyadic shif operaors of order d in any Banach space implies he boundedness of he Riesz ransform in he same space. 4. Geomeric applicaion The mos ineresing case is d =. Then we know wha are he measure µ on he plane such ha operaors x are bounded in x 2, x 2 x 2 L 2 (µ). See [3], [2]. Descripion of such measures was he soluion of he famous problem. For arclengh measure on curves i has been found by Guy David [4]. They are he same as hose for which he Cauchy ransform is bounded. We will call hem here recifiable measures. One can find he explanaion for his name in he groundbreaking aricle []. We have he following heorem in which l(q) denoes he side lengh of he square Q. Theorem 4.. If dyadic shif operaors L := Σ Q Ll(Q)sh Q of order are all bounded as operaors from L 2 (µ) o iself, hen µ is recifiable. One can prove easily Theorem 4.2. The dyadic shif operaors L := Σ Q L Q 2 sh Q of order are all bounded as operaors from L 2 (µ) o iself if and only if (4.) µ(q) C l(q), and for any laice L and any square R L he following oscillaion crierion for he measure µ is saisfied (µ(q l ) µ(q r )) 2 (4.2) C 2 µ(r). l(q) Q L, Q R Unforunaely, he condiion (4.2) is oo srong i is no saisfied even for Lebesgue measure on a sraigh segmen! We can propose a much weaker condiion on µ which corresponds o boundedness of dyadic operaors in average versus heir uniform boundedness. Here is his condiion: for any dyadic laice L µ(q)(µ(q l ) µ(q r )) 2 (4.3) l(q) 2 C 2 µ(r). Q L, Q R Q

18 226 S. Peermichl, S. Treil, A. Volberg One can compare i wih curvaure condiion in [2] and may wonder wheher hey are equivalen. If yes, i is geing ineresing because (4.3) can be obviously exended o R n, n > 2 and i is a very ineresing problem wha replaces he curvaure condiion for he case n > 2. Our (4.3) or is modificaion can be a curious candidae. References [] J. Bourgain, Some remarks on Banach spaces in which maringale difference sequences are uncondiional, Ark. Ma. 2(2) (983), [2] J. Bourgain, Vecor-valued singular inegrals and he H -BMO dualiy, in: Probabiliy heory and harmonic analysis (Cleveland, Ohio, 983), Monogr. Texbooks Pure Appl. Mah. 98, Dekker, New York, 986, pp. 9. [3] D. L. Burkholder, A geomeric condiion ha implies he exisence of cerain singular inegrals of Banach-space-valued funcions, in: Conference on harmonic analysis in honor of Anoni Zygmund, Vol. I, II (Chicago, Ill., 98), Wadsworh Mah. Ser., Wadsworh, Belmon, CA, 983, pp [4] G. David, Opéraeurs inégraux singuliers sur ceraines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 7() (984), [5] J. B. Garne, Bounded analyic funcions, Pure and Applied Mahemaics 96, Academic Press, Inc., New York-London, 98. [6] T. A. Gillespie, S. Po, S. Treil and A. Volberg, Logarihmic growh for marix maringale ransforms, J. London Mah. Soc. (2) 64(3) (2), [7] T. A. Gillespie, S. Po, S. Treil and A. Volberg, The ransfer mehod in esimaes for vecor Hankel operaors, (Russian), Algebra i Analiz 2(6) (2), 78 93; ranslaion in S. Peersburg Mah. J. 2(6) (2), [8] N. H. Kaz, Marix valued paraproducs, in: Proceedings of he conference dedicaed o Professor Miguel de Guzmán (El Escorial, 996), J. Fourier Anal. Appl. 3, Special Issue (997), [9] M. T. Lacey and C. M. Thiele, L p esimaes on he bilinear Hilber ransform for 2 < p <, Ann. of Mah. (2) 46(3) (997), [] F. Lus-Piquard, Opéraeurs de Hankel -somman de l (N) dans l (N) e muliplicaeurs de H (T), C. R. Acad. Sci. Paris Sér. I Mah. 299(8) (984),

19 Riesz Transforms as Averages of Dyadic Shifs 227 [] P. Maila, M. S. Melnikov and J. Verdera, The Cauchy inegral, analyic capaciy, and uniform recifiabiliy, Ann. of Mah. (2) 44() (996), [2] M. S. Melnikov and J. Verdera, A geomeric proof of he L 2 boundedness of he Cauchy inegral on Lipschiz graphs, Inerna. Mah. Res. Noices 995(7) (995), [3] F. Nazarov, S. Treil and A. Volberg, Cauchy inegral and Calderón-Zygmund operaors on nonhomogeneous spaces, Inerna. Mah. Res. Noices 997(5) (997), [4] F. Nazarov, S. Treil and A. Volberg, Weak ype esimaes and Colar inequaliies for Calderón-Zygmund operaors on nonhomogeneous spaces, Inerna. Mah. Res. Noices 998(9) (998), [5] F. Nazarov, S. Treil and A. Volberg, Accreive sysem T b heorem of M. Chris for non-homogeneous spaces, Duke Mah. J. (o appear). [6] F. Nazarov, S. Treil and A. Volberg, Counerexample o he infinie-dimensional Carleson embedding heorem, C. R. Acad. Sci. Paris Sér. I Mah. 325(4) (997), [7] F. Nazarov, S. Treil and A. Volberg, The Bellman funcions and wo-weigh inequaliies for Haar mulipliers, J. Amer. Mah. Soc. 2(4) (999), [8] S. Peermichl, Dyadic shifs and a logarihmic esimae for Hankel operaors wih marix symbol, C. R. Acad. Sci. Paris Sér. I Mah. 33(6) (2), [9] X. Tolsa, Colar s inequaliy wihou he doubling condiion and exisence of principal values for he Cauchy inegral of measures, J. Reine angew. Mah. 52 (998), [2] X. Tolsa, L 2 -boundedness of he Cauchy inegral operaor for coninuous measures, Duke Mah. J. 98(2) (999), [2] X. Tolsa, Curvaure of measures, Cauchy singular inegral, and analyic capaciy, Thesis, Deparmen of Mahemaics, Universia Auònoma de Barcelona (998). [22] S. Treil and A. Volberg, Waveles and he angle beween pas and fuure, J. Func. Anal. 43(2) (997), [23] S. Treil and A. Volberg, Compleely regular mulivariae saionary processes and he Muckenhoup condiion, Pacific J. Mah. 9(2) (999), [24] A. Volberg, A p weighs via S-funcions, J. Amer. Mah. Soc. (2) (997),

20 228 S. Peermichl, S. Treil, A. Volberg S. Peermichl: Insiue for Advanced Sudies Princeon, NJ 854 U.S.A. address: S. Treil: Deparmen of Mahemaics Brown Universiy Providence, RI U.S.A. address: A. Volberg: Deparmen of Mahemaics Michigan Sae Universiy Eas Lansing, Michigan U.S.A. and Equipe d Analyse Universié Paris VI 4, place Jussieu Paris Cedex 5 France address: volberg@mah.msu.edu