THE LINEAR BOUND FOR THE NATURAL WEIGHTED RESOLUTION OF THE HAAR SHIFT

Transcription

1 THE INEAR BOUND FOR THE NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT SANDRA POTT, MARIA CARMEN REGUERA, ERIC T SAWYER, AND BRETT D WIC 3 Abstract The Hilbert transform has a linear bound in the A characteristic on eighted, H () () [] A, and e extend this linear bound to the nine constituent operators in the natural eighted resolution of the conjugation M SM induced by the canonical decomposition of a multiplier into paraproducts: M f = P f + P 0 f + P + f The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function Contents Introduction 83 Notation and Preliminary Estimates 85 3 Proof of Theorem 89 3 Estimating Easy Terms 89 3 Estimating Hard Terms 9 33 An Alternate Approach to Estimating Term (30) 99 References 04 Introduction et (R) denote the space of square integrable functions over R For a eight, ie, a positive locally integrable function on R, e set () (R; ) In particular, e ill be interested in A eights, hich are defined by finiteness of their A characteristic, [] A sup I I, I here I denotes the average of over the interval I Research supported by grants 009SGR (Generalitat de Catalunya) and MTM (Spain) Research supported in part by a NSERC Grant 3 Research supported in part by National Science Foundation DMS grants # and # The authors ould like to thank the Banff International Research Station for the Banff PIMS Research in Teams support for the project: The Sarason Conjecture and the Composition of Paraproducts 83

2 84 S POTT, MC REGUERA, E T SAWYER, AND B D WIC An operator T is bounded on () if and only if M T M - the conjugation of T by the multiplication operator M ± - is bounded on Moreover, the operator norms are equal: T () () = M T M In the case that T is a dyadic operator adapted to a dyadic grid D, it is natural to study eighted norm properties of T by decomposing the multiplication operators M ± into their canonical paraproduct decomposition relative to the grid D and Haar basis { h 0 } I, ie M ± f = P(0,) + P (,0) ± f + P (0,0) ± ±, h 0 I f, h I ± f h 0 I + ±, h 0 I f, h 0 I h I + ±, h I f, h 0 I h 0 I, (above h I is the averaging function) and then decomposing M T M into the nine canonical individual paraproduct composition operators: ( ) ( ) M T M = P (0,) + P (,0) + P (0,0) T P (0,) + P (,0) + P (0,0) () Q (0,),(0,) T, + Q (0,),(,0) T, + Q (0,),(0,0) T, +Q (,0),(0,) T, + Q (,0),(,0) T, + Q (,0),(0,0) T, +Q (0,0),(0,) T, + Q (0,0),(,0) T, + Q (0,0),(0,0) T,, here Q (ε,ε ),(ε 3,ε 4 ) T, P (ε,ε ) T P (ε 3,ε 4 ) We refer to this decomposition of the conjugation M T M into nine operators as the natural eighted resolution of the operator T The operators e ill be interested in have the property that T () () [] A And e ill be interested in demonstrating that the operator norms of the Q (ε,ε ),(ε 3,ε 4 ) T, are linear in the A characteristic: Q (ε,ε ),(ε 3,ε 4 ) [] A T, We ish to apply this general idea to the Haar shift operator This is the folloing dyadic model operator M SM : here S is a shift operator defined on the Haar basis by h I h I h I+ Because of linearity, it suffices to consider just half of the shift operator S defined on the Haar basis by the operator Sh I h I Petermichl proved the folloing result:

3 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 85 Theorem (Petermichl, [6]) et A Then S () () [] A This result can be used ith an averaging technique, [7], to sho that the same estimate persists for the Hilbert transform on () The idea of decomposing a general Calderón Zygmund operator into shifts as further refined by Hytönen in [, ] (see also [3, 9, ]) and as used in his solution of the A -Conjecture for all Calderón Zygmund operators For the simplest proof of the A -Conjecture the interested reader should consult [4] In this paper e ill implement the strategy outlined above to extend this result to the natural eighted resolution of the Haar shift transform and obtain the folloing result Theorem et A and S the Haar shift on Then for the resolution of S into its canonical paraproducts as given in () e have that each term can be controlled by a linear poer of [] A In particular, Q (ε,ε ),(ε 3,ε 4 ) [] A S, If one could control each term appearing in Theorem independent of the Theorem, this ould imply Petermichl s Theorem But, for one of the terms, e unfortunately need to resort to the estimate in Theorem, and ill point out an interesting question that e are unable to resolve as of this riting Hoever, our point is to demonstrate that the paraproduct operators arising in the canonical resolution of the Haar shift S are also bounded, and linearly in the A characteristic Finally, e mention that the operators Q (ε,ε ),(ε 3,ε 4 ) I, in the resolution of the identity I = M IM in (), are all bounded on if and only if A, and if so, the operator norm is linear in the A characteristic The proof of this fact is easy using the techniques in the proof of Theorem, and amounts to little more than repeating the arguments ithout the shift involved ey to this proof orking is that the identity is a purely local operator, and this points to the difficulty of Haar shift and its non-local nature Throughout this paper means equal by definition, hile A B means that there exists an absolute constant C such that A CB Notation and Preliminary Estimates Before proceeding ith the proof of Theorem, e collect a fe elementary observations and necessary notation that ill be used frequently throughout the remainder of the paper To define our paraproducts, let D denote the usual dyadic grid of intervals on the real line Define the Haar function h 0 I and averaging function h I by h 0 I h I I ( I I+ ) and h I I I, I D The paraproduct operators considered in this paper are the folloing dyadic operators

4 86 S POTT, MC REGUERA, E T SAWYER, AND B D WIC Definition Given a symbol b = {b I } and a pair (α, β) {0, } {0, }, define the dyadic paraproduct acting on a function f by P (α,β) b f b I f, h β I hα I, here h 0 I is the Haar function associated ith I, and h I is the average function associated ith I The index (α, β) is referred to as the type of P (α,β) b For a function b and I D e let b(i) b, h 0 I b I b, h I denote the corresponding sequences indexed by the dyadic intervals I D With this notation, the canonical paraproduct decomposition of the pointise multiplier operator M b is given by M b = P (0,) b + P (,0) b, + P (0,0) b At points in the argument belo e ill have to resort to the use of disbalanced Haar functions To do so, e introduce some additional notation Given a eight σ on R e set σ + σ () C (σ) and D (σ) σ() σ σ Then e have () h = C (σ)h σ + D (σ)h here {h σ } is the (σ) normalized Haar basis For an interval D e let E σ (f) fσdx σ() here σ() = σdx When σ is ebesgue measure e simply rite f = E (f) fdx For to functions f and g e have the folloing product formula that appears in higher dimensions in [0] fg (I) = I f () ĝ () δ (, I) I + f (I) g I + ĝ (I) f I I D, here δ(, I) = if I and δ(, I) = if I + For a sequence a = {a I } define a l sup a I ; a CM sup I The folloing estimates are ell-knon a I

5 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 87 emma We have the characterizations: P (0,0) (3) a = a l ; (4) P (0,) = P (,0) a CM a The Carleson Embedding Theorem is a fundamental tool in this paper and ill be used frequently It is the folloing: Theorem (Carleson Embedding Theorem) et v 0, and let {α I } be positive constants The folloing to statements are equivalent: α I E v I (f) 4C f (v) f (v); sup v(i) α v C I As a simple application of the Carleson Embedding Theorem, one can prove: a emma 3 et a = {a I } be a sequence of non-negative numbers Then (5) P (,) a CM a Recall that the dyadic square function is given by Sφ(x) φ(i) h I (x), and that for any eight v 0 e have Sφ (v) = φ(i) v I Since {h I } is an orthonormal basis for (R), it is trivial that Sf = f Hoever, in [8] it as shon for A that Sf () [] A f () Applying this inequality to f = I for I D, along ith some obvious estimates, yields the folloing: (6) () [] A I I D I A trivial consequence of (6) is, (7) () I [] A I I D, since Because of the symmetry of the A condition, e also have these estimates ith the roles of and interchanged All of these estimates play a fundamental role at various times hen applying the Carleson Embedding Theorem

6 88 S POTT, MC REGUERA, E T SAWYER, AND B D WIC We also need a modified version of estimate (6), but hich incorporates a shift in the indices Define the modified square function S π by, S π φ(x) φ (I) I πi(x) here πi is the dyadic parent of the dyadic interval I Unfortunately, S π f is not pointise bounded by Sf (S π h on π but Sh vanishes outside ), yet e sho in the theorem belo that S π has a linear bound in the characteristic on eighted spaces Theorem 4 For any φ () e have S π φ () [] A φ () Proof The proof of this theorem uses the ideas in [8] Without loss of generality e may assume both and are bounded so long as these bounds do not enter into our estimates First note that S π φ () = φ (I) πi = D φ, φ, here D : is the discrete multiplier map that sends h I πi h I We also have φ () = M φ, φ, here M : is the operator of pointise multiplication by We first claim the operator inequality M [] A D, hich upon taking inverses, is equivalent to (8) D [] A (M ) = [] A M, here D : is the map that sends h I h I But, (8) is πi equivalent to φ (I) [] A φ ( ) πi At this point e simply observe that since πi = I and (I) (πi), e have φ (I) φ (I) πi I and then e invoke the folloing inequality in [8]: φ (I) [] A φ ( ), I and this proves (8) No e apply a duality argument to obtain (9) D [] A M,

7 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 89 hich is equivalent to the desired inequality in the statement of the theorem To see (9), e note that D [] A ( D ), follos from D φ, φ = φ (I) πi = φ (I) πi πi πi [] A φ (I) ( = [] A D ) φ, φ πi No e continue by applying (8) ith replaced by to get ( ) [ D [] A D []A ] A (M ) = [] A M Again, using this Theorem 4 ith the function φ = I for I D yields (0) () π [] A I I hich ill be used frequently in the proof belo I D 3 Proof of Theorem Our method of attack on Theorem ill use the language of paraproducts We expand the operator M SM in term of the canonical paraproducts ith symbols and to obtain, ( ) ( ) (3) P (0,) + P (,0) + P (0,0) S P (0,) + P (,0) + P (0,0) This results in nine terms and e ill study each of these separately to see that they are controlled by (no orse than) the linear characteristic of the eight Doing so ill provide a proof of Theorem In hat follos, any time there are internal zeros, the Haar shift S can be absorbed, and e are left ith simply studying a modified (shifted) paraproduct 3 Estimating Easy Terms There are four easy terms that arise from (3), and they are easy because the composition of the paraproducts reduce to classical paraproduct type operators To motivate our approach, e point out that simple computations (that absorb the Haar shift and are given explicitly belo) give (3) P (,0) SP (0,) (33) P (,0) SP (0,0) P (,) P(,0) ; ;

8 90 S POTT, MC REGUERA, E T SAWYER, AND B D WIC (34) P (0,0) SP(0,) P (0,) ; (35) P (0,0) SP(0,0) P(0,0) Here a b is the Schur product of the sequences a and b, and e are letting mean that there has been a shift in one of the basis elements appearing in the definition of the Haar basis (this presents no problem in the analysis of these operators since it simply results in a change in the absolute constants appearing; e ill make this rigorous shortly) Notice that (33), (34) and (35) are of the type considered in emma and so have a complete characterization, hile (3) can be handled by emma 3 In each of these characterizations e are of course left ith shoing that the norm on the symbol for these classical operators can be controlled by the linear poer of the characteristic of the eight We no make these ideas precise 3 Estimating Term (3) Note that a simple computation shos that P (,0) SP (0,) = (I )ŵ (I)h I h I By the Carleson Embedding Theorem e see that P (,0) SP (0,) φ, ψ = (I )ŵ (I) φ, h I ψ, h I ŵ (I )ŵ (I) φ, h I ψ, h I Hoever, by Cauchy-Scharz e then have ŵ (I )ŵ (I) hich gives I P(,0) SP (0,) ( ) ŵ (I )ŵ (I) φ ( I ŵ (I )ŵ (I) ψ I φ ψ sup ŵ (I )ŵ (I) I = ( [] A [] A, yielding the desired linear bound for term (3) ) [] A, )

9 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 9 3 Estimating Terms (33) and (34) These terms have are symmetric and so e focus only on the first term In this case term (33) reduces to a classical paraproduct operator ith a symbol given by the product of to sequences as indicated by the notation Indeed, e have the identity, P (,0) SP (0,0) = (I ) I h I h I No by the Carleson Embedding Theorem applied to ψ e have that But P (,0) SP (0,0) φ, ψ sup I = (I ) I φ, h I ψ, h I ( φ ŵ (I ) I ŵ (I ) I sup ψ I ) φ ψ sup ŵ (I ) I I (I ) I [] A, here e have used the linear bound for the square function in (0) in the last inequality Thus, P(,0) SP (0,0) [] A hich gives the desired linear bound in terms of the A characteristic 33 Estimating Term (35) Term (35) reduces to a standard Haar multiplier Indeed, e have the identity I P (0,0) SP(0,0) = I I h I h I, here (h h ) f f, h h, and then P(0,0) SP(0,0) = sup I I sup I I = [] A [] A This gives the desired linear estimate for (35) 3 Estimating Hard Terms There are five remaining terms to be controlled These include the four difficult terms, (36) (37) (38) (39) P (0,) SP (0,) P (0,) SP (0,0) P (,0) P (0,0) SP (,0) SP(,0) P (0,) P (0,) ; P (0,) P (0,0) P (,0) P (,0) ; P (0,0) P(,0)

10 9 S POTT, MC REGUERA, E T SAWYER, AND B D WIC To estimate terms (36) and (38) e ill rely on disbalanced Haar functions adapted to the eight and For these terms e ill also proceed by computing the norm of the operators in question by using duality ey to this ill be the application of the Carleson Embedding Theorem The proof of the necessary estimates for these terms are carried out in subsection 3 Terms (37) and (39) ill be handled via a similar method; their analysis is handled in subsection 3 The remaining very difficult term is the one for hich the Haar shift can not be absorbed into one of the paraproducts Namely, e need to control the folloing expression, (30) P (0,) SP (,0) In this situation, e ill explicitly compute Sh, h and see that the shift presents no problem in ho the operator behaves, nor in estimating its norm in terms of the A characteristic [] A This is carried out in subsection 33 3 Estimating Terms (36) and (38) Once again these to terms are symmetric and so it suffices to prove the desired estimate only for (36) We need to sho that (3) P (0,) P(0,) SP (0,) [] A Proceeding by duality, e fix φ, ψ and consider SP (0,) φ, ψ SP (0,) φ, P (,0) ψ = =, () φ ŵ () ψ() h, h = ( () φ () ψ() ) h, h = () φ () ψ()δ (, ) = T + T + T 3 ŵ () φ (ψ ( ) ŵ ( ) ψ ψ( ) The fifth equality follos by an application of the product formula for Haar coefficients We need to sho that each of these terms has the desired estimate T j [] A φ ψ since this ill imply that P(0,) SP (0,) = sup φ,ψ P (0,) SP (0,) φ, ψ [] A hich implies the desired estimate on the norm of the operator )

11 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 93 Consider term T 3 This term can be controlled by T 3 () φ ψ( ) ψ ( [] A φ ψ, () φ ith the last estimate folloing by an application of the Carleson Embedding Theorem using (7) For the term T, e have T ()ŵ ( ) ψ φ ) ( ) ( ŵ ()ŵ ( ) ψ ŵ ()ŵ ( ) φ [] A φ ψ Again e have used here to applications of the Carleson Embedding Theorem, one for φ and one for ψ, since e have that ŵ ()ŵ ( ) ( ) ( ) () ( ) ()() = [] A ()() Finally, e prove the estimate for term T, hich requires the use of disbalanced Haar functions We expand the terms in T using Haar functions ith respect to to different disbalanced bases In particular, using () e have ψ, h = C () ψ, h + D () ψ ( ) = C () ψ, h () + D () E ψ ) and, h = C ( ), h + D ( ) = C ( ), h + D ( ) ( ) Then e can rite the term T as a sum of four terms, namely T = S + S + S 3 + S 4

12 94 S POTT, MC REGUERA, E T SAWYER, AND B D WIC ith S = φ C ()C ( ), h ( ) ψ, h S = φ C ()D ( ) ψ, h () S 3 = ) φ D ()C ( ) E (ψ, h S 4 = ( ) φ D ()D ( ) E ψ () ( ) We need to sho that each term can be estimated by a constant times [] A φ ψ, hich ould then imply T [] A φ ψ as required We no proceed to prove the necessary estimates Consider the term S We have S ψ φ C ()C ( ) (), h ( ) ψ, h ( φ C () C ( ), h () ( = ψ φ + ( [] A ψ φ, h [] A φ ψ + ( ) ) () ), h ( ) ) Here the last inequality follos by the Carleson Embedding Theorem since, h ( ) D

13 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 95 Turning to term S e have S = ( [] ψ A [] A ψ φ C ()D ( ) ψ, h () ) ψ, h + ( () () ( [] A φ ψ ( ŵ () φ ŵ () φ ) ) ŵ () φ ) ith the last estimate folloing from the Carleson Embedding Theorem, once e prove the folloing estimate: (3) ŵ () [] A D To prove (3) recall the folloing to estimates in [6, emma 5 and emma 53], translated to the notation of this paper: ŵ () [] A () D ŵ () 3 () D These to estimates coupled ith a simple application of Cauchy-Scharz then proves (3) Indeed, ŵ () = ŵ () ŵ () ŵ () [] A () () = [] A () () [] A

14 96 S POTT, MC REGUERA, E T SAWYER, AND B D WIC For the term S 3 e have S 3 = = = ) φ D ()C ( ) E (ψ, h ( ) ŵ( ) ( ) φ E ψ +, h ) φ ŵ( )E (ψ +, h ( ) ( φ, h φ ( [] A ψ φ ( ) ) ( + ) ( ) ŵ( ) E ψ ( ) + ) ( ) ŵ( ) E ψ For the last inequality e used the Carleson Embedding Theorem tice, first applied to φ as above, and then to ψ upon noting that + ŵ( ) ŵ( ) [] A () D via the linear bound for the modified square function (0) Finally consider the term S 4 We have S 4 = = = ( ) φ D ()D ( ) E ψ ŵ( ) ŵ φ () ( ) E ψ ŵ φ ()ŵ( ) ( ) E ψ ( φ ) ŵ ()ŵ( ) ( ( φ ŵ ()ŵ( ) ) ( ) ŵ ()ŵ( ) ( ) E ψ ŵ ()ŵ( ) E ( ψ ) )

15 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 97 No note that in [6] the folloing estimates are proved ŵ ()ŵ( ) [] A D ŵ ()ŵ( ) [] A () D The Carleson Embedding Theorem applied to φ and ψ then gives that as desired S 4 [] A ψ φ 3 Estimating Terms (37) and (39) Once again there is symmetry beteen these terms, so e focus only on (37) Fix φ, ψ We compute P (0,) SP (0,0) φ, ψ = SP (0,0) φ, P (,0) ψ = φ() ψ()ŵ () h, h, φ() ψ() ŵ ()δ(, ) = φ() = T + T + T 3 (ψ ( ) ψ( ) ) ŵ ( ) ψ We sho that each term satisfies T j [] A φ ψ, hich ould imply that P(0,) SP (0,0) = sup SP (0,0) φ, ψ [] A φ,ψ P (0,) proving the desired norm on the operator in question The term T is easiest since e have T = φ() ψ( ) [] A φ() ψ( ) [] A φ ψ

16 98 S POTT, MC REGUERA, E T SAWYER, AND B D WIC using the definition of A, Cauchy-Scharz and Parseval For term T 3, e find T 3 = φ φ() ( [] A φ ψ ( ) ψ ) ( ) ψ ith the last estimate folloing from the Carleson Embedding Theorem and (0) (hich shos that the Carleson Embedding Theorem applies) For term T, e require disbalanced Haar functions again Note that it is possible to rite T = = = + S + S φ() ψ ( ) φ() ψ, C ()h + D ()h φ() φ() φ() C () ψ, h + φ() C () ψ, h () D () E ( ψ ) D () ψ, h For term S observe that C () [] A, and so S [] A φ ψ = [] A φ ψ ()

17 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 99 For term S e have ( ) S = φ() D () E ψ φ = φ = φ ( ( ( [] A φ ψ ) ( ) D () E ψ ) ŵ( ) ( ) E ψ ) ( ) ŵ( ) E ψ ith the last estimate folloing from the Carleson Embedding Theorem and the linear bound for the modified square function (0) 33 Estimating Term (30) We are left ith analyzing P (0,) SP (,0) an operator on But, by the analysis above e have from (3) that as P (0,) SP (,0) = M SM (3) (33) (34) (35) (36) (37) (38) (39) And, using all the estimates above, e find that P(0,) SP (,0) [] A + M SM No using Petermichl s result, Theorem, (see also [ 3]) e have that M SM [] A This combines to give: P(0,) SP (,0) [] A as required We pose a question that e are unable to resolve as of this riting Resolution of this question ould provide an alternate proof of the linear bound of the Hilbert transform Question 3 Can one give a direct proof that SP (,0) [] A P(0,) ithout resorting to the use of Theorem? A direct proof of the above estimate ould provide an alternate proof of Theorem 33 An Alternate Approach to Estimating Term (30) In this final section e sho that the tools of this paper are almost sufficient to provide a direct proof of (30)

18 00 S POTT, MC REGUERA, E T SAWYER, AND B D WIC Fix φ, ψ No, observe that e have P (0,) SP (,0) φ, ψ = SP (,0) φ, P (,0) ψ = ()ŵ ()φ() ψ() Sh, h, D We then further specialize to the case hen = and hen The case hen can be handled via the techniques of this paper The case hen = requires a ne idea 33 Estimating Term (30): We can then split this into three sums, hen =, hen and hen Doing so, e see that P (0,) SP (,0) φ, ψ ()ŵ ()φ() ψ() Sh, h = + ()φ() + () ψ() D = T + T + T 3 Consider no term T and observe T = ()ŵ ()φ() ψ() Sh, h ŵ ()ŵ ()φ() ψ() h ()ŵ () φ() ψ() [] A φ ψ () ψ() Sh, h ()φ() Sh, h Here e make an obvious estimate using the definition of A, Cauchy- Scharz and that S : has norm at most one There is a symmetry beteen terms T and T 3, so e only handle term T We first make a computation of Sh, h hen Set s() : etting h () h (c()) herein c() is the center of the dyadic interval and using the definition of S it is a straightforard computation to sho

19 that NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 0 Sh, h = = = = = h () h, h h ()h () δ(, ) δ(, ) : δ(, )δ(, ) s() We have used that =, and that δ(, ) = δ(, ) since Note that e have s() = ( ) No define the function v + and observe () : v () = ŵ () : + Then using the product formula e have () ψ() Sh, h = s() () ψ() = s() () ψ() Hoever, simple computations sho that ŵ + ( ŵ ()) ψ() = s() ψ()v ()δ(, ) ( = s() v ψ() ψ() ) v v () ψ v = ŵ () v () = ψv () = ψ Thus, e have () ψ() Sh, h = s() ψ s()ŵ () ψ() s() ψ

20 0 S POTT, MC REGUERA, E T SAWYER, AND B D WIC This then yields that T = ()φ() = S + S + S 3 () ψ() Sh, h ( ()φ() s() ψ s()ŵ () ψ() s() Using the estimate that s(), it is immediate to see that S [] A φ ψ It is also an easy application of the Carleson Embedding Theorem, again using the linear bound for the square function and that s(), to see that S 3 φ ( () ψ ) [] A φ ψ Finally, note that for term S, by an application of Cauchy-Scharz and again that s(), e have S = s() ŵ ()φ() ψ ( ) = s() ŵ ()φ() E ψ φ ( ) ( ) () E ψ We claim that the folloing Carleson estimate holds: (33) () [] A () D Assuming (33) holds, e can apply the Carleson Embedding Theorem to conclude that S [] A φ ψ = [] A () φ ψ Then combining the estimates on S j ith j =,, 3 gives that T [] A φ ψ Further, combining the estimates on T j for j =,, 3 gives that sup P (0,) φ,ψ SP (,0) φ, ψ [] A φ ψ, hich is the desired estimate Thus it only remains to demonstrate (33) Fix and start ith a standard Calderón Zygmund stopping time argument on the function ) ψ

21 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 03 et Q 0 Fix γ > large and let G 0 = { Q 0 } be the ground zero ( ) } generation Then let G = G Q 0 = {Q j be the first generation of maximal dyadic subintervals Q j j Γ 0 of such that Q j > γ Q 0, j Γ 0 Then for each i Γ 0 define G ( ) { } Q i = Q j j Γ i maximal dyadic subintervals Q j of Q i such that to be the collection of Q j > γ Q i, j Γ i Set G = Q i G G k Q k i G k G ( Q i ) to be the second generation of subintervals of Continue by recursion to define the k th generation G k of subintervals of ( ) ( ) { } by G k here G k Q k = Q k j is the collection Q k i of maximal dyadic subintervals Q k j of Qk i i such that j Q k j > γ Q k i, j Γ k i Then define the corona C Q k j = { Q k j : Q k+ i for any Q k+ i G k+ }, and denote by G = k=0 each corona C G for G G, e get G k the collection of all stopping intervals in Within ŵ () γ ŵ () G C G C G ( γ G γ G G ) G G

22 04 S POTT, MC REGUERA, E T SAWYER, AND B D WIC Summing up over all the coronas C G for G G ith γ = e get () G G G G = Q k j k,j Q k j Q k j = R k,j Q k Q k (x) (x) dx j Q k j j sup R kj: x Q k j Q k (x) dx j Q k j M [] A = [] A hich gives (33) Note that e used that the sequence of numbers Q k j Q k j k,j: x Q k j is super-geometric in the sense that consecutive terms have ratio exceeding γ =, and so the sum of their squares is essentially the square of the largest one 33 Estimating Term (30): = We must consider this case since the Haar shift is not a completely local operator It instead sees some long range interaction, and e need to account for this since there can be terms for hich = and Sh, h 0 A simple example occurs hen and are dyadic brothers Fortunately, in the case of the identity operator, this term does not appear and the proof ould terminate The tools used in this paper currently appear to be unable to resolve this term We thus pose a refined version of Question 3 Question 3 Is the folloing estimate ()ŵ ()φ() ψ() Sh, h true?, D: = [] A We again kno that this estimate is true by using Theorem, but are interested in a direct resolution of Question 3 A direct proof of the estimate above ould provide a ne proof of Theorem References [] Tuomas P Hytönen, The sharp eighted bound for general Calderón-Zygmund operators, Ann of Math () 75 (0), no 3, , 99,

23 NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT 05 [] Tuomas Hytönen, On Petermichl s dyadic shift and the Hilbert transform, C R Math Acad Sci Paris 346 (008), no -, (English, ith English and French summaries) 85, 99 [3] Michael T acey, Stefanie Petermichl, and Maria Carmen Reguera, Sharp A inequality for Haar shift operators, Math Ann 348 (00), no, , 99 [4] A erner, A Simple Proof of the A Conjecture, Internat Math Res Notices (0),, to appear, available at 85 [5] Stefanie Petermichl, The sharp eighted bound for the Riesz transforms, Proc Amer Math Soc 36 (008), no 4, [6] S Petermichl, The sharp bound for the Hilbert transform on eighted ebesgue spaces in terms of the classical A p characteristic, Amer Math 9 (007), no 5, , 95, 97 [7] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators ith matrix symbol, C R Acad Sci Paris Sér I Math 330 (000), no 6, (English, ith English and French summaries) 85 [8] S Petermichl and S Pott, An estimate for eighted Hilbert transform via square functions, Trans Amer Math Soc 354 (00), no 4, (electronic) 87, 88 [9] S Petermichl, S Treil, and A Volberg, Why the Riesz transforms are averages of the dyadic shifts?, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 000), 00, pp [0] Eric T Sayer, Chun-Yen Shen, and Ignacio Uriarte-Tuero, The To eight theorem for the vector of Riesz transforms: an expanded version, available at org/abs/305093v3 86 [] Armen Vagharshakyan, Recovering singular integrals from Haar shifts, Proc Amer Math Soc 38 (00), no, S Pott, Centre for Mathematical Sciences, University of und, und, Seden address: sandra@mathslthse M C Reguera, Department of Mathematics, Universitat Autònoma de Barcelona, Barcelona, Spain address: mreguera@matuabcat E T Sayer, Department of Mathematics, McMaster University, Hamilton, Canada address: sayer@mcmasterca Brett D Wick, Department of Mathematics, Washington University, St ouis, MO USA address: ick@mathustledu