Composition of Haar Paraproducts

Transcription

1 Composition of Haar Paraproucts Brett D. Wick Georgia Institute of Technology School of Mathematics Chongqing Analysis Meeting IV Chongqing University Chongqing, China June 1, 2013 B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 1 / 34

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 CAM IV 2 / 34

3 Motivations 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 CAM IV 3 / 34

4 Motivations 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 CAM IV 4 / 34 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 Motivations 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 CAM IV 5 / 34

6 Motivations 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 CAM IV 6 / 34

7 Motivations 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 CAM IV 7 / 34

8 Motivations Classical Characterizations Internal Cancellations an Simple Characterizations When there are internal zeros the ehavior of P (α,0) the ehavior of P a (α,β) 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 CAM IV 8 / 34

9 Motivations 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 CAM IV 9 / 34

10 Motivations 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 CAM IV 10 / 34

11 Motivations 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 CAM IV 11 / 34

12 Motivations 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 CAM IV 12 / 34 J I

13 Motivations Classical Characterizations Decomposing (1, 1) into Pieces Expan the averaging functions in a Haar series: hi 1 = hi 1, h J h L J I 2 J, P (1,1) a f = a I f, hi 1 L h1 2 I = a I f, hi 1, h J J I = + + J K K J J=K h L 2 J I J K P (1,0) Ŝ(a) f + P(0,1) Ŝ(a) f + P(0,0) E(a) f. L 2 K I a I h 1 I, h J L 2 h 1 I, h K h L 2 K h 1 I, h K L 2 f, h J L 2 h B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 13 / 34

14 Motivations 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 CAM IV 14 / 34

15 Motivations 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; u) 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 CAM IV 15 / 34

16 Motivations 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 CAM IV 16 / 34

17 Motivations 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 CAM IV 17 / 34

18 Motivations 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 CAM IV 18 / 34

19 Motivations 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 CAM IV 19 / 34

20 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 CAM IV 20 / 34

21 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 CAM IV 21 / 34

22 Proofs of Main Results The Characterization of Type (0, 1, 1, 0) The Gram Matrix of P (0,1) P (1,0) Let G (0,1) P P (1,0) = [G I,J ] I,J D e the Gram matrix of the operator P (0,1) P (1,0) relative to the Haar asis {h I }. A simple computation show that it has entries: G I,J = = P (0,1) P (1,0) h J, h I = L 2 J hj 1, I hi 1 = I J I J I J L 2 = P (1,0) h J, P (1,0) I J 1 I if J I I J 1 J if I J 0 if I J =. h I L 2 Iea: Construct T (0,1,1,0), : Lc(H) 2 Lc(H) 2 that has the same Gram matrix as P (0,1) P (1,0) }, ut with respect to the asis { 1 T(I ). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 22 / 34

23 The Operator T (0,1,1,0), Proofs of Main Results The Characterization of Type (0, 1, 1, 0) an it Gram Matrix For λ R an a = {a I } the multiplication operator M λ a is efine on asis elements 1 T(K) y Define an operator T (0,1,1,0), T (0,1,1,0), M λ a 1 T(K) a K K λ 1 T(K). Then the Gram matrix G (0,1,1,0) T }, the asis { 1 T(I ) has entries G I,J = 1 T(J), 1 T(I ) T (0,1,1,0), on Lc 2 (H) y M 0 1 T(K) 1 Q(K) K D M 1. = [G I,J ] I,J D of T (0,1,1,0), L 2 (H) relative to Q (I ) T (J) = I J 2 I J 2 = 1 { 1 I J I if J I 2 0 if J I. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 23 / 34

24 Proofs of Main Results The Characterization of Type (0, 1, 1, 0) Connecting the Prolem to a Two Weight Inequality Up to an asolute constant, G (0,1,1,0) T, triangle where J I. So, P (0,1) P (1,0) T (0,1,1,0) L 2 L 2, matches G (0,1) P P (1,0) in the lower T + (0,1,1,0) L 2 (H) L 2 (H),. L 2 (H) L 2 (H) The inequality we wish to characterize is M 0 UM 1 f L T = (0,1,1,0) 2 c (H), f f L 2 c (H) Lc 2 (H). Define U on L 2 c (H), where U K D 1 T(K) 1 Q(K). For appropriate choice of weights σ an w on H the esire estimate is simply: U (σk) L 2 c (H;w) k L 2 c (H;σ). B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 24 / 34

25 Proofs of Main Results The Characterization of Type (0, 1, 1, 0) A Two Weight Theorem for Positive Operators Theorem (S. Pott, E. Sawyer, M. Reguera-Roriguez, BDW) Let w an σ e non-negative weights on H. Then U (σ ) : L 2 (H; σ) L 2 (H; w) is oune if an only if the following testing conition hols: ) 2 1 Q(I ) U (σ1 Q(I ) C 2 2 L Q(I ). (H;w) L 2 (H;σ) The proof of this Theorem is a translation of Sawyer s proof strategy for two weight inequalities for positive operators. Choosing w I 2 1 T(I ) an σ I 2 1 I 2 T(I ) (an unraveling the efinitions) gives the forwar testing conition. Appropriate choice of w an σ will then provie the ackwar testing conition when stuying T (0,1,1,0),. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 25 / 34

26 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 CAM IV 26 / 34

27 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 CAM IV 27 / 34

28 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 CAM IV 28 / 34

29 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 CAM IV 29 / 34

30 Proofs of Main Results The Characterization of Type (0, 1, 0, 0) Theorem (S. Pott, E. Sawyer, M. Reguera-Roriguez, BDW) 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, we have that U L 2 c (H;µ) L 2 c (H;ν) C 1 + C 2 where C 1 an C 2 are the est constants appearing aove. B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 30 / 34

31 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 CAM IV 31 / 34

32 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 CAM IV 32 / 34

33 Conclusion Thanks (Moifie from the Original Dr. Fun Comic) Thanks to Dechao for Organizing the Meeting! B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 33 / 34

34 Conclusion Thanks Thank You! B. D. Wick (Georgia Tech) Composition of Haar Paraproucts CAM IV 34 / 34