Composition of Haar Paraproducts

Transcription

1 Composition of Haar Paraproucts Brett D. Wick Georgia Institute of Technology School of Mathematics Analytic Spaces an Their Operators Memorial University of Newfounlan July 9-12, 2013 B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 1 / 29

2 This talk is ase on joint work with: Eric T. Sawyer McMaster University Sanra Pott Lun University Maria Reguera Roriguez Universia Autónoma e Barcelona B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 2 / 29

3 Sarason s Conjecture H 2 (D), the L 2 (T) closure of the analytic polynomials on D. P : L 2 (T) H 2 (D) e the orthogonal projection. A Toeplitz operator with symol ϕ is the following map from H 2 (D) H 2 (D): T ϕ (f ) P (ϕf ). An important question raise y Sarason is the following: Conjecture (Sarason Conjecture) The composition of T ϕ T ψ is oune on H 2 (D) if an only if sup z D T 1 z 2 1 zξ 2 ϕ(ξ) 2 m(ξ) T 1 z 2 1 zξ 2 ψ(ξ) 2 m(ξ) < Unfortunately, this is not true! A counterexample was constructe y Nazarov. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 3 / 29

4 Deep work y Nazarov, Treil, Volerg, an then susequent work y Lacey, Sawyer, Shen, Uriarte-Tuero allow for an answer in terms of the Hilert transform. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 4 / 29 The Sarason Conjecture & Hilert Transform Question (Sarason Question (Revise Version)) Otain necessary an sufficient (testale (?)) conitions so that one can tell if T ϕ T ψ is oune on H 2 (D) y evaluating these conitions. Possile to rephrase this question as one aout the two-weight ouneness of the Hilert transform. Let M φ enote multiplication y φ: M φ f φf ; H 2 ( φ 2 ) is the L 2 (T) closure of pφ where p is an analytic polynomial; H 2 T ϕt ψ H 2 ( M ψ M ϕ L 2 T; ψ 2) H ( L 2 T; ϕ 2)

5 Notation Haar Paraproucts L 2 L 2 (R); D is the stanar gri of yaic intervals on R; Define the Haar function h 0 I an averaging function h1 I y h 0 I h I 1 I ( 1I + 1 I+ ) h 1 I 1 I 1 I I D. I D h 1 [0,1] (x) {h I } is an orthonormal asis of L 2. h 0 [0,1] (x) B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 5 / 29

6 Notation Haar Paraproucts from Multiplication Operators Given a function an f it is possile to stuy their pointwise prouct y expaning in their Haar series: f =, h I L 2 h I f, h J L 2 h J J D = = I,J D, h I L 2 f, h J L 2 h I h J + + I =J I J J I, h I L 2 f, h J L 2 h I h J =, h I L 2 f, h I L 2 hi 1 +, h I L 2 +, hi 1 f, h L 2 I L 2 h I. f, hi 1 h L 2 I B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 6 / 29

7 Notation Haar Paraproucts Definition (Haar Paraproucts) Given a symol sequence = { I } an a pair (α, β) {0, 1} 2, efine the yaic paraprouct acting on a function f y P (α,β) f I f, h β I L hα 2 I. The inex (α, β) is referre to as the type of P (α,β). Question (Discrete Sarason Question) For each choice of pairs (α, β), (ɛ, δ) {0, 1} 2, otain necessary an sufficient conitions on symols an so that P (α,β) P (ɛ,δ) <. L 2 L 2 B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 7 / 29

8 Classical Characterizations Internal Cancellations an Simple Characterizations When there are internal zeros the ehavior of P (α,0) the ehavior of P (α,β) a f g : L 2 L 2 e the map given y P (0,β) reuces to for a special symol a. For f, g L 2, let f g(h) f g, h L 2. Then: P (α,0) P (0,β) = I hi α h I J h J h β J J D = I I h α I h β I = P (α,β). Here is the Schur prouct of the symols, i.e., ( ) I = I I. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 8 / 29

9 Classical Characterizations Norms an Inuce Sequences For a sequence a = {a I } efine the following quantities: a l sup a I ; a CM sup 1 a J 2. I J I Associate to {a I } two aitional sequences inexe y D: E(a) Ŝ (a) 1 a J ; I J I a J hj 1, h I J D L 2 = J I a J ĥj 1 (I ). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 9 / 29

10 Classical Characterizations Classical Characterizations Theorem (Characterizations of Type (0, 0), (0, 1), an (1, 0)) The following characterizations are true: P a (0,0) = a L 2 L 2 l ; = a L 2 L 2 L 2 L 2 CM. P a (0,1) P (1,0) a P (1,1) a = P (1,0) Ŝ(a) + P(0,1) Ŝ(a) + P(0,0) E(a). Theorem (Characterization of Type (1, 1)) The operator norm P a (1,1) of P(1,1) a on L 2 satisfies L 2 L 2 P a (1,1) Ŝ(a) CM + E(a) L 2 L 2 l. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 10 / 29

11 Classical Characterizations Proof of the Easy Characterizations A simple computation gives P a (0,0) f 2 = L 2 I,I D a I a I f, h I L 2 f, h I L 2 h I, h I L 2 = a I 2 f, h I L 2 2, f 2 L 2 = f, h I L 2 2. Similarly, P a (0,1) f 2 = L 2 I,I D a I a I = a I 2 f, hi 1 f, hi 1 f, h 1 L 2 I L h 2 I, h I L 2 L 2 2 B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 11 / 29

12 Classical Characterizations Proof of the Easy Characterizations Theorem (Carleson Emeing Theorem) Let {α I } e positive constants. The following two statements are equivalent: 2 α I f, hi 1 L C f 2 L 2 f L2 ; 1 sup α J I J I C. Then the Carleson Emeing Theorem gives a L 2 L 2 CM. P (0,1) a Let Î enote the parent of I, an then we have P (0,1) a 2 P (0,1) L 2 L 2 a hî 2 1 a J 2. L 2 I B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 12 / 29 J I

13 Classical Characterizations Alternate Interpretations: Testing Conitions It is easy to see for paraproucts of type (0, 0) that: P a (0,0) = a L 2 L 2 l = sup P (0,0) L a h I. 2 Moreover, P a (1,0) = L 2 L 2 P (0,1) a L 2 L 2 a CM sup P (0,1) L a h I. 2 These oservations suggest seeking a characterization for the other compositions in terms of testing conitions on classes of functions. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 13 / 29

14 Classical Characterizations Two Weight Inequalities in Harmonic Analysis Given weights u an v on R an an operator T a prolem one frequently encounters in harmonic analysis is the following: Question Determine necessary an sufficient conitions on T, u, an v so that is oune. T : L 2 (R; u) L 2 (R; v) Meta-Theorem (Characterization of Bouneness via Testing) The operator T : L 2 (R; u) L 2 (R; v) is oune if an only if T(u1 Q ) L 2 (v) 1 Q L 2 (u) T (v1 Q ) L 2 (u) 1 Q L 2 (v). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 14 / 29

15 Main Results Characterization of Type (0, 1, 1, 0) For a sequence a, an interval I D let Q I a J I a J h J. Theorem (E. Sawyer, S. Pott, M. Reguera-Roriguez, BDW) The composition P (0,1) P (1,0) is oune on L 2 if an only if oth Q I P (0,1) ( ) 2 Q I 2 I D; P (1,0) Q I P (0,1) P (1,0) ( Q I C 2 L 2 1 Q I 2 L ) 2 C 2 L 2 2 Q I 2 L2 I D. Moreover, the norm of P (0,1) P (1,0) on L 2 satisfies P (0,1) P (1,0) C L 2 L C 2 where C 1 an C 2 are the est constants appearing aove. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 15 / 29

16 Main Results Rephrasing the Testing Conitions We want to rephrase the testing conitions on Q I an Q I : Q I P (0,1) P (1,0) ( Q I P (0,1) P (1,0) ) 2 Q I C 2 L 2 1 Q I 2 L2 I D; ( ) 2 Q I C 2 L 2 2 Q I 2 L2 I D. It isn t har to see that these are equivalent to the following inequalities on the sequences: J 2 1 J I J 2 L 2 L J J 2 1 J I J 2 L 2 L J 2 2 C 2 1 C 2 2 L 2 I D; L I L 2 I D. L I B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 16 / 29

17 Main Results Characterization of Type (0, 1, 0, 0) Theorem (E. Sawyer, S. Pott, M. Reguera-Roriguez, BDW) The composition P (0,1) P (0,0) is oune on L 2 if an only if oth I 2 (0,1) 2 P h I C 2 L 2 1 I D; Q I P (0,0) P (1,0) Q I 2 C 2 L 2 2 Q I 2 L2 I D. Moreover, the norm of P (0,1) P (0,0) on L 2 satisfies P (0,1) P (0,0) C L 2 L C 2 where C 1 an C 2 are the est constants appearing aove. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 17 / 29

18 Main Results Rephrasing Testing Conitions Again, it is possile to recast the conitions: I 2 (0,1) 2 P h I C 2 L 2 1 I D; Q I P (0,0) P (1,0) Q I 2 C 2 L 2 2 Q I 2 L 2 I D as expressions epening only on the sequences. In particular, these are equivalent to the following inequalities: J I J 2 J I 2 I L 2 C1 2 I D; L I K 2 K 2 K J + K J 2 C 2 2 L 2 I D. L I B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 18 / 29

19 Proofs of Main Results Preliminaries For I D set [ ] I T (I ) I 2, I Q (I ) I [0, I ] = T (J) J I (Carleson Tile); (Carleson Square). The yaic lattice D is in corresponence with the Carleson Tiles. Let H enote the upper half plane C + : H = T (I ). For a non-negative function σ let L 2 (H; σ) enote the functions that are square integrale with respect to σ A, i.e, f 2 L 2 (H;σ) f (z) 2 σ(z) A(z) <. H When σ 1, L 2 (H; 1) L 2 (H). For f L 2 (H), let f enote the normalize function. f f L 2 (H) B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 19 / 29

20 Proofs of Main Results Functions Constant on Tiles Let L 2 c (H) L 2 (H) e the suspace of functions which are constant on tiles. Namely, f : D C Then L 2 c (H) f = f I 1 T(I ). f : D C : f (I ) 2 I 2 < ; f 2 Lc 2 (H) 1 f (I ) 2 I 2. 2 Easy to show: } { 1T(I ) is an orthonormal asis of L2 c (H) ; } { 1 Q(I ) is an Riesz asis of L2 c (H). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 20 / 29

21 Proofs of Main Results The Characterization of Type (0, 1, 0, 0) The Gram Matrix of P (0,1) P (0,0) Let G (0,1) P P (0,0) = [G I,J ] I,J D e the Gram matrix of the operator P (0,1) P (0,0) relative to the Haar asis {h I }. A simple computation shows its entries are: G I,J = = P (0,1) P (0,0) h J, h I = L 2 J h J, I hi 1 L 2 = I J ĥi 1 (J) = I J P (0,0) h J, P (1,0) h I L 2 1 if I J J 1 I J if I J + J 0 if J I or I J =. Iea: Construct T (0,1,0,0), : Lc(H) 2 Lc(H) 2 that has the same Gram matrix as P (0,1) P (0,0) }, ut with respect to the asis { 1 T(I ). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 21 / 29

22 The Operator T (0,1,0,0), Now consier the operator T (0,1,0,0), Here T (0,1,0,0), Proofs of Main Results The Characterization of Type (0, 1, 0, 0) M 1 K D efine y 1 Q± (K) 1 T(K) 1 Q± (K) 1 T(L) + 1 T(L). L K L K + A straightforwar computation shows M Q± (K) = K L 2 (H) 2 ; M λ a1 Q± (K) = a L L λ 1 T(L) + a L L λ 1 T(L). L K L K + B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 22 / 29

23 Proofs of Main Results The Characterization of Type (0, 1, 0, 0) The Gram Matrix for the Operator T (0,1,0,0), The Gram matrix G (0,1,0,0) T = [G I,J ] I,J D of T (0,1,0,0), relative to the }, asis { 1 T(I ) then has entries given y G I,J = T (0,1,0,0), = 2 1 T(J), 1 T(I ) L 2 (H) I J J 1 2 if I J I J J 1 2 if I J + 0 if J I or I J =. Thus, up to an asolute constant, G (0,1,0,0) T = G (0,1), P P (0,0), an so P (0,1) P (0,0) T (0,1,0,0) L 2 L 2,. L 2 (H) L 2 (H) B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 23 / 29

24 Proofs of Main Results The Characterization of Type (0, 1, 0, 0) Connecting to a Two Weight Inequality The inequality we wish to characterize is: M 1UM 1 2 f = T (0,1,0,0), f f L 2 c (H) L 2 c (H) Lc 2 (H). Where the operator U on L 2 (H) is efine y U K D 1 Q± (K) 1 T(K). One sees that the inequality to e characterize is equivalent to: U (µg) L 2 c (H;ν) g L 2 c (H;µ), where the weights µ an ν are given y ν I 2 I 2 1 T(I ) µ I 2 I 1 1 T(I ). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 24 / 29

25 Proofs of Main Results The Characterization of Type (0, 1, 0, 0) Theorem (S. Pott, E. Sawyer, M. Reguera-Roriguez, BDW) Let U K D 1 Q± (K) 1 T(K) an suppose that µ an ν are positive measures on H that are constant on tiles, i.e., µ µ I 1 T(I ), ν ν I 1 T(I ). Then U (µ ) : L 2 c (H; µ) L 2 c (H; ν) if an only if oth ( ) L U µ1 T(I ) 2 c (H;ν) ( ) 1 Q(I ) U L ν1 Q(I ) 2 c (H;µ) C 1 1T(I ) L 2 c (H;µ) = µ (T (I )), C 2 1Q(I ) L 2 c (H;ν) = ν (Q (I )), hol for all I D. Moreover, U L 2 c (H;µ) L 2 c (H;ν) C 1 + C 2. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 25 / 29

26 Conclusion Application An Application: Linear Boun for Hilert Transform For a weight w, i.e., a positive locally integrale function on R, let L 2 (w) L 2 (R; w). A weight elongs to A 2 if: [w] A2 sup I w I w 1 I < +. The Hilert transform is the operator: H (f )(x) p.v. R Theorem (Petermichl) f (y) y x y. Let w A 2. Then H L 2 (w) L 2 (w) [w] A 2, an the linear growth is optimal. T L 2 (w) L 2 (w) = Mw 1 2 TM w 1 2 L 2 L 2; H is the average of yaic shifts X; M 1 w 2 XM w 1 can e written as a sum of nine compositions of 2 paraproucts; Some of which are amenale to the Theorems aove. However, each term can e shown to have norm no worse than [w] A2. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 26 / 29

27 Conclusion Open Question An Open Question Unfortunately, the methos escrie o not appear to work to hanle type (0, 1, 0, 1) compositions. However, the following question is of interest: Question For each I D etermine function F I, B I L 2 of norm 1 such that P (0,1) P (0,1) is oune on L 2 if an only if P (0,1) P (0,1) L F I C 2 1 I D; P (1,0) P (1,0) L B I C 2 2 I D. Moreover, we will have P (0,1) P (0,1) C L 2 L C 2. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 27 / 29

28 Conclusion Thanks (Moifie from the Original Dr. Fun Comic) Thanks to Jie an Kehe for Organizing the Meeting! B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 28 / 29

29 Conclusion Thanks Thank You! B. D. Wick (Georgia Tech) Composition of Haar Paraproucts AS&O 29 / 29