T (1) and T (b) Theorems on Product Spaces

Transcription

1 T (1) and T (b) Theorems on Product Spaces Yumeng Ou Department of Mathematics Brown University Providence RI yumeng ou@brown.edu June 2, 2014 Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

2 Classical SIO with standard kernel Definition K : R n R n \ C is locally integrable away from the diagonal. It is called a standard kernel, if for some 0 < δ 1, C > 0 we have K (x, t) C x t n K (x, t) K (x, t) C( x x ) δ x t n δ K (t, x) K (t, x ) C( x x ) δ x t n δ whenever x x x t /2. K denotes the best constant in both inequalities. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

3 Space of δ-czo Definition A continuous linear operator T : C0 (C 0 ) is called a Calderón-Zygmund operator, if it is bounded on L 2 (R n ), and there is a standard kernel K such that for f, g C0 (Rn ) with disjoint supports, Tf, g = K (x, t)f (t)g(x) dtdx. The set of all the δ-czo is a Banach space (denoted by CZ) with norm T CZ := T L 2 L2 + K. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

4 Classical one-parameter T (1) and T (b) Theorems Theorem (T (1)) An operator T : C 0 (C 0 ), associated with a standard kernel K, extends to a bounded operator on L 2 (R n ) if and only if T 1, T 1 BMO and T has the weak boundedness property (WBP). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

5 Classical one-parameter T (1) and T (b) Theorems Theorem (T (1)) An operator T : C 0 (C 0 ), associated with a standard kernel K, extends to a bounded operator on L 2 (R n ) if and only if T 1, T 1 BMO and T has the weak boundedness property (WBP). Theorem (T (b)) Let b, b be two accretive functions. An operator T : bc 0 (b C 0 ), associated with a standard kernel K, extends to a bounded operator on L 2 (R n ) if and only if Tb, T b BMO and M b TM b has the WBP. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

6 Example: Cauchy integral along Lipschitz curves Let y = A(x) be a Lipschitz function that defines a Lipschitz curve Γ in the complex plane. A L. Define the Cauchy integral operator associated to Γ: C Γ f (x) = p.v. Then, C Γ is bounded on L 2. R f (y) (x y) + i(a(x) A(y)) dy. (Check C Γ (1 + ia ) = 0 by Cauchy integral formula.) Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

7 Multi-parameter operators Example 1 (Tensor product type) For f C0 (R) C 0 (R), define the double Hilbert transform as f (y 1, y 2 ) [(H 1 H 2 )f ] (x 1, x 2 ) = lim ɛ1 0 x 1 y 1 >ɛ 1 (x 1 y 1 )(x 2 y 2 ) dy 1dy 2 ɛ 2 0 x 2 y 2 >ɛ 2 Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

8 Multi-parameter operators Example 1 (Tensor product type) For f C0 (R) C 0 (R), define the double Hilbert transform as f (y 1, y 2 ) [(H 1 H 2 )f ] (x 1, x 2 ) = lim ɛ1 0 x 1 y 1 >ɛ 1 (x 1 y 1 )(x 2 y 2 ) dy 1dy 2 ɛ 2 0 x 2 y 2 >ɛ 2 Example 2 (Non tensor product type) For complex valued function a L (R 2 ) with a < 1, define the double Cauchy operator associated to the kernel K a as K a (x, y) = 1 2 i=1 (x i y i ) + y 1 x 1 y2 x 2 a(u 1, u 2 ) du Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

9 Journé s class of bi-parameter SIO Definition (Vector-valued Kernels) Let B be a Banach space. A continuous function K : R 2 \ B is called a B-δ-standard kernel, if for some 0 < δ 1, C > 0 we have K (x, t) B C x t 1 K (x, t) K (x, t) B C( x x ) δ x t 1 δ K (t, x) K (t, x ) B C( x x ) δ x t 1 δ whenever x x x t /2. K denotes the best constant in both inequalities. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

10 Journé s class of bi-parameter SIO Definition (Vector-valued Kernels) Let B be a Banach space. A continuous function K : R 2 \ B is called a B-δ-standard kernel, if for some 0 < δ 1, C > 0 we have K (x, t) B C x t 1 K (x, t) K (x, t) B C( x x ) δ x t 1 δ K (t, x) K (t, x ) B C( x x ) δ x t 1 δ whenever x x x t /2. K denotes the best constant in both inequalities. We will use the case B = CZ in the following. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

11 Definition (Bi-parameter SIO) Let T : C0 (R) C 0 (R) [ C0 (R) C 0 (R)] be a continuous linear mapping. It is a δ-sio on R R if there exists a pair (K 1, K 2 ) of CZ-δ-standard kernels so that, f 1, f 2, g 1, g 2 C0 (R), with suppf 1 suppg 1 =, g 1 g 2, T (f 1 f 2 ) = g 1 (x 1 ) g 2, K 1 (x 1, t 1 )f 2 f 1 (t 1 ) dt 1 dx 1, g 2 g 1, T (f 2 f 1 ) = g 1 (x 2 ) g 2, K 2 (x 2, t 2 )f 2 f 1 (t 2 ) dt 2 dx 2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

12 Definition (Bi-parameter SIO) Let T : C0 (R) C 0 (R) [ C0 (R) C 0 (R)] be a continuous linear mapping. It is a δ-sio on R R if there exists a pair (K 1, K 2 ) of CZ-δ-standard kernels so that, f 1, f 2, g 1, g 2 C0 (R), with suppf 1 suppg 1 =, g 1 g 2, T (f 1 f 2 ) = g 1 (x 1 ) g 2, K 1 (x 1, t 1 )f 2 f 1 (t 1 ) dt 1 dx 1, g 2 g 1, T (f 2 f 1 ) = g 1 (x 2 ) g 2, K 2 (x 2, t 2 )f 2 f 1 (t 2 ) dt 2 dx 2. Partial adjoint operators: T 1 (f 1 f 2 ), g 1 g 2 = T (g 1 f 2 ), f 1 g 2, analogously for T 2. Note that T 2 = T 1. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

13 We can check the double Hilbert transform is indeed in Journé s class. For fixed x 1, y 1, define K 1 (x 1, y 1 ) to be a constant multiple of the usual Hilbert transform: 1 h(y 2 ) K 1 (x 1, y 1 )h(x 2 ) = lim dy 2. x 1 y 1 ɛ 0 x 2 y 2 >ɛ x 2 y 2 It is CZ valued, with norm C x 1 y 1. Similarly for K 2(x 2, y 2 ). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

14 We can check the double Hilbert transform is indeed in Journé s class. For fixed x 1, y 1, define K 1 (x 1, y 1 ) to be a constant multiple of the usual Hilbert transform: 1 h(y 2 ) K 1 (x 1, y 1 )h(x 2 ) = lim dy 2. x 1 y 1 ɛ 0 x 2 y 2 >ɛ x 2 y 2 It is CZ valued, with norm C x 1 y 1. Similarly for K 2(x 2, y 2 ). To check K 1 is associated to a standard kernel. It is easily seen that we have the size condition: K 1 (x 1, y 1 ) CZ C x 1 y 1. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

15 We can check the double Hilbert transform is indeed in Journé s class. For fixed x 1, y 1, define K 1 (x 1, y 1 ) to be a constant multiple of the usual Hilbert transform: 1 h(y 2 ) K 1 (x 1, y 1 )h(x 2 ) = lim dy 2. x 1 y 1 ɛ 0 x 2 y 2 >ɛ x 2 y 2 It is CZ valued, with norm C x 1 y 1. Similarly for K 2(x 2, y 2 ). To check K 1 is associated to a standard kernel. It is easily seen that we have the size condition: K 1 (x 1, y 1 ) CZ C x 1 y 1. For Hölder conditions, note that K (x 1, y 1 ) CZ C x 1 y 1 2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

16 BMO in the product spaces Standard dyadic grid: D 0 := {2 k ([0, 1) d + m) : k Z, m Z d } Shifted dyadic grid: Let ω = (ω j ) j Z ({0, 1} d ) Z and I ω := I + 2 j ω j. j:2 j <l(i) Then D ω := {I ω : I D 0 }. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

17 Dyadic double square function For any given dyadic grids D n, D m in R n, R m, respectively, define S 2 D nd m (f ) = K D n f, h K u V 2 χ K χ V K V V D m. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

18 Dyadic double square function For any given dyadic grids D n, D m in R n, R m, respectively, define S 2 D nd m (f ) = K D n f, h K u V 2 χ K χ V K V V D m We say a function f HD 1 nd m (R n R m ) if S DnDm f L 1 (R n R m ). Let f H 1 = Sf D L 1. ndm BMO DnD m is defined as the dual of H 1 D nd m.. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

19 Dyadic double square function For any given dyadic grids D n, D m in R n, R m, respectively, define S 2 D nd m (f ) = K D n f, h K u V 2 χ K χ V K V V D m We say a function f HD 1 nd m (R n R m ) if S DnDm f L 1 (R n R m ). Let f H 1 = Sf D L 1. ndm BMO DnD m is defined as the dual of H 1 D nd m. Definition f is in product BMO iff f is in product dyadic BMO uniformly with respect to every shifted dyadic grid.. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

20 Bi-parameter T (1) Theorem 1 Theorem (J. Journé) Let T be a δ-sio on R R satisfying the WBP and T (1), T (1), T 1 (1), T 1 (1) BMO prod(r R). Then T extends boundedly on L 2 (R R). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

21 Bi-parameter T (1) Theorem 1 Theorem (J. Journé) Let T be a δ-sio on R R satisfying the WBP and T (1), T (1), T 1 (1), T 1 (1) BMO prod(r R). Then T extends boundedly on L 2 (R R). Simplest case: T (1) = T (1) = T 1 (1) = T 1 (1) = 0 View T as a classical vector valued SIO acting on C 0 (R) L2 (R, dx 2 ), and use the hilbertian version of the classical T(1) argument. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

22 Pott-Villarroya s class of bi-parameter SIO Novelty: Drop the a priori partial boundedness assumption. Replace the vector-valued assumptions by an enlarged set of mixed conditions combining more size and Hölder conditions together. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

23 P-V s class of bi-parameter SIO (reformulated by Martikainen) For f = f 1 f 2, g = g 1 g 2 with f 1, g 1 : R n C, f 2, g 2 : R m C, and sptf 1 sptg 1 = sptf 2 sptg 2 =, we have Tf, g = K (x, y)f (y)g(x) dxdy. R n+m R n+m Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

24 P-V s class of bi-parameter SIO (reformulated by Martikainen) For f = f 1 f 2, g = g 1 g 2 with f 1, g 1 : R n C, f 2, g 2 : R m C, and sptf 1 sptg 1 = sptf 2 sptg 2 =, we have Tf, g = K (x, y)f (y)g(x) dxdy. R n+m R n+m (1) Size condition: K (x, y) C 1 x 1 y 1 n 1 x 2 y 2 m. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

25 (2) Hölder conditions: 1 K (x, y) K (x, (y 1, y 2 )) K (x, (y 1, y 2)) + K (x, y ) C y 1 y 1 δ x 1 y 1 n+δ y 2 y 2 δ x 2 y 2 m+δ whenever y 1 y 1 x 1 y 1 /2 and y 2 y 2 x 2 y 2 /2, 2 K (x, y) K ((x 1, x 2 ), y) K ((x 1, x 2), y) + K (x, y) C x 1 x 1 δ x 1 y 1 n+δ x 2 x 2 δ x 2 y 2 m+δ whenever x 1 x 1 x 1 y 1 /2 and x 2 x 2 x 2 y 2 /2, 3 K (x, y) K ((x 1, x 2 ), y) K (x, (y 1, y 2)) + K ((x 1, x 2 ), (y 1, y 2)) C y 1 y 1 δ x 1 y 1 n+δ x 2 x 2 δ x 2 y 2 m+δ whenever y 1 y 1 x 1 y 1 /2 and x 2 x 2 x 2 y 2 /2, 4 K (x, y) K (x, (y 1, y 2 )) K ((x 1, x 2), y) + K ((x 1, x 2), (y 1, y 2 )) C x 1 x 1 δ x 1 y 1 n+δ y 2 y 2 δ x 2 y 2 m+δ whenever x 1 x 1 x 1 y 1 /2 and y 2 y 2 x 2 y 2 /2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

26 (3) Mixed Hölder and size conditions: 1 K (x, y) K ((x 1, x 2), y) C x 1 x 1 δ x 1 y 1 n+δ 1 x 2 y 2 m whenever x 1 x 1 x 1 y 1 /2, 2 K (x, y) K (x, (y 1, y 2)) C y 1 y 1 δ x 1 y 1 n+δ 1 x 2 y 2 m whenever y 1 y 1 x 1 y 1 /2, 3 K (x, y) K ((x 1, x 2 1 x ), y) C 2 x x 1 y 1 n 2 δ x 2 y 2 m+δ whenever x 2 x 2 x 2 y 2 /2, 4 K (x, y) K (x, (y 1, y 2 )) C 1 x 1 y 1 n y 2 y 2 δ x 2 y 2 m+δ whenever y 2 y 2 x 2 y 2 /2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

27 (4) Partial CZ structure: If f = f 1 f 2 and g = g 1 g 2 with sptf 1 sptg 1 =. Then assume Tf, g = K f2,g 2 (x 1, y 1 )f 1 (y 1 )g 1 (x 1 ) dx 1 dy 1. R n R n The kernel K f2,g 2 : (R n R n ) \ {x 1 = y 1 } satisfies: K f2,g 2 (x 1, y 1 ) C(f 2, g 2 ) 1 x 1 y 1 n K f2,g 2 (x 1, y 1 ) K f2,g 2 (x 1, y 1) C(f 2, g 2 ) x 1 x 1 δ x 1 y 1 n+δ whenever x 1 x 1 x 1 y 1 /2, K f2,g 2 (x 1, y 1 ) K f2,g 2 (x 1, y 1 ) C(f 2, g 2 ) y 1 y 1 δ x 1 y 1 n+δ whenever y 1 y 1 x 1 y 1 /2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

28 Control for C(f 2, g 2 ) in the diagonal: For any cube V R m, C(χ V, χ V ) + C(χ V, u V ) + C(u V, χ V ) C V, whenever u V is V -adapted with zero-mean (i.e. sptu V V, u V 1, u V = 0). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

29 Control for C(f 2, g 2 ) in the diagonal: For any cube V R m, C(χ V, χ V ) + C(χ V, u V ) + C(u V, χ V ) C V, whenever u V is V -adapted with zero-mean (i.e. sptu V V, u V 1, u V = 0). And we also assume the analogous representation and properties with a kernel K f1,g 1 in the case sptf 2 sptg 2 =. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

30 Equivalence of the two classes A very recent result of A. Grau De La Herrán shows that under the additional condition that T is bounded on L 2, the classes of bi-parameter SIO defined by Journé and P-V are equivalent. This implies in particular that for a SIO T in P-V s class to be bounded on L 2, conditions T (1), T (1) BMO are necessary. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

31 Boundedness and cancellation assumptions T 1, T 1, T 1 (1), T 1 (1) BMO(Rn R m ). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

32 Boundedness and cancellation assumptions T 1, T 1, T 1 (1), T 1 (1) BMO(Rn R m ). (Dyadic) WBP: T (χ K χ V ), χ K χ V C K V for every cube K R n, V R m. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

33 Boundedness and cancellation assumptions T 1, T 1, T 1 (1), T 1 (1) BMO(Rn R m ). (Dyadic) WBP: T (χ K χ V ), χ K χ V C K V for every cube K R n, V R m. Diagonal BMO conditions: for any cube K R n, V R m, and every zero-mean functions a K and b V which are K and V adapted respectively: 1 T (a K χ V ), χ K χ V C K V 2 T (χ K χ V ), a K χ V C K V 3 T (χ K b V ), χ K χ V C K V 4 T (χ K χ V ), χ K b V C K V Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

34 Bi-parameter T (1) Theorem 2 Theorem (H. Martikainen) For a bi-parameter SIO T of P-V s class satisfying the boundedness and cancellation assumptions, there holds for some bi-parameter shifts S i 1i 2 j 1 j 2 D nd m that Tf, g = CE ωn E ωm Corollary i 1,i 2,j 1,j max(i 1,i 2 ) δ 2 2 max(j 1,j 2 ) δ 2 S i 1 i 2 j 1 j 2 D nd m f, g. A bi-parameter singular integral T as defined above is L 2 bounded. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

35 Bi-parameter T (b) Theorem Definition A function b L (R n R m ) is called pseudo-accretive if there is a constant C such that for any rectangle R in R n R m with sides parallel 1 to axes, R R b > C. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

36 Bi-parameter T (b) Theorem Definition A function b L (R n R m ) is called pseudo-accretive if there is a constant C such that for any rectangle R in R n R m with sides parallel 1 to axes, R R b > C. Theorem (Y. Ou) Let b = b 1 b 2, b = b 1 b 2 be pseudo-accretive. For a bi-parameter SIO T of a class similar to P-V s, satisfying Tb, T b, T 1 (b 1 b 2), T1 (b 1 b 2 ) BMO(Rn R m ), and M b TM b has the WBP, then T is bounded on L 2 (R n R m ). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

37 Assumptions on the SIO for T (b) Theorem (1) Full C-Z structure: If f = f 1 f 2 and g = g 1 g 2 with f 1, g 1 C0 (Rn ), f 2, g 2 C0 (Rm ), sptf 1 sptg 1 = and sptf 2 sptg 2 =, then M b TM b f, g = K (x, y)f (y)g(x)b(y)b (x) dxdy. R n+m R n+m The kernel K satisfies all the size, Hölder and mixed size-hölder conditions. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

38 (2) Partial C-Z structure: If f = f 1 f 2, g = g 1 g 2 and sptf 1 sptg 1 =, then M b TM b f, g = K f2,g 2 (x 1, y 1 )f 1 (y 1 )g 1 (x 1 )b 1 (y 1 )b 1 (x 1) dx 1 dy 1. R n R n The partial kernel K f2,g 2 satisfies all the partial size and Hölder conditions with constant C(f 2, g 2 ) satisfying C(χ V, χ V ) + C(χ V, u V b 1 2 ) + C(u V b 1 2, χ V ) C V. We also assume the symmetric partial kernel representation and conditions on kernel K f1,g 1 for the other variable. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

39 (3) Boundedness and cancellation assumptions: WBP: M b TM b (χ K χ V ), χ K χ V C K V. BMO: Let d = b 1 b 2, d = b 1 b 2, assume Tb, T b, T 1 d, T1 d BMO(Rn R m ). Diagonal BMO: Let a K, b V be any K, V adapted zero-mean functions, M b TM b (a K b 1 1 χ V ), χ K χ V C K V M b TM b (χ K χ V ), a K b 1 1 χ V C K V M b TM b (χ K b V b 1 2 ), χ K χ V C K V M b TM b (χ K χ V ), χ K b V b 1 2 C K V Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

40 Difficulties from T (1) to T (b) Define T (b). (In the case b is a tensor product) Develop estimates of bi-parameter b-adapted paraproducts. Generalize the argument for dyadic shifts to deal with b-adapted martingale differences directly instead of Haar basis. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

41 Bi-parameter b-adapted martingale differences For each p Z, let Dp n be the collection of cubes of side length 2 p in D n, define E b 1 p f = I fb 1 χ I, E b 1 I f = χ I E b 1 p f. I D n p I b 1 Similarly for the other variable. Then E b p,q = E b 1 p E b 2 q = E b 2 q E b 1 p. Let b 1 p = E b 1 p+1 E b 1 p, b 1 I = χ I b 1 p for each I Dp, n similarly for the other variable. The b-adapted double martingale difference is defined as b p,q = b 1 p b 2 q = b 2 q b 1 p. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

42 Properties 1 b I Jf is supported on the dyadic rectangle I J, and is a constant on each of its children; 2 b1 b p,qf dx 1 = b 2 b p,qf dx 2 = 0; 3 b p,q b k,l = 0 unless p = k, q = l, and in this case it equals b p,q; 4 If f L 2 (R n R m ), then f = p,q b p,qf with convergence in L 2, and f 2 L 2 p,q b p,qf 2 L 2 f 2 L 2. 5 M b b p,q = b p,qm b Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

43 Definition of T (b) Let A be the subspace of H 1 d (Rn R m ) containing all the finite combinations of b b 1 I b 2 J f. Then, Tb BMO means Tb is a bounded functional on A. To define b b 1 I b 2 J f, Tb, split into b b 1 I b 2 J f, T (bχ 3I χ 3J ) + b b 1 I b 2 J f, T (bχ 3I χ (3J) c) + b b 1 I b 2 J f, T (bχ (3I) c χ 3J) + b b 1 I b 2 J f, T (bχ (3I) c χ (3J) c). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

44 Tools: b-adapted bi-parameter paraproduct Definition For a BMO(R n R m ), operator π b,b a defined as π b,b a (f ) = is called full paraproduct, K D n,v D m f b K V M b b 1 K b 2 V a. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

45 Tools: b-adapted bi-parameter paraproduct Definition For a BMO(R n R m ), operator π b,b a defined as Proposition π b,b a (f ) = is called full paraproduct, K D n,v D m f b K V M b b 1 K b 2 V a. Full paraproducts are bounded operators on L 2 (R n R m ). Specifically, π b,b a (f ) L 2 (R n R m ) a BMO(R n R m ) f L 2 (R n R m ). Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

46 Tools: b-adapted bi-parameter paraproduct Definition For a BMO(R n R m ), operator π b,b a defined as Proposition π b,b a (f ) = is called full paraproduct, K D n,v D m f b K V M b b 1 K b 2 V a. Full paraproducts are bounded operators on L 2 (R n R m ). Specifically, π b,b a (f ) L 2 (R n R m ) a BMO(R n R m ) f L 2 (R n R m ). In the definition, if f is averaged only on one variable but has a martingale difference attached to the other one, what obtained is called a mixed paraproduct, which satisfies a similar L 2 estimate. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

47 Key steps in the proof of the proposition: 1 Define b-adapted square function and maximal function: S b f (x) = ( p,q Z b 1 p b 2 q f (x) 2 ) 1/2 = ( b1 I b 2 I D n,j D m J f (x) 2 ) 1/2, fb (x) = sup E b 1 p E b 2 q f (x) = sup E b 1 p,q Z I D n,j D m I E b 2 J f (x). 2 Observe f H 1 b f b L 1, where f H1 b iff fb H1. 3 Prove f b L 1 S bf L 1. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

48 Demonstrate Step 3 Define K 1 b (Rn R m ) = {f : S b f L 1 (R n R m )}. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

49 Demonstrate Step 3 Define K 1 b (Rn R m ) = {f : S b f L 1 (R n R m )}. Theorem (Atomic Decomposition) Given f K 1 b, there exists a sequence of atoms an and a sequence of scalars λ n such that (1) f = n λ na n, a.e. (2) n λ n f K 1 b. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

50 Demonstrate Step 3 Define K 1 b (Rn R m ) = {f : S b f L 1 (R n R m )}. Theorem (Atomic Decomposition) Given f K 1 b, there exists a sequence of atoms an and a sequence of scalars λ n such that (1) f = n λ na n, a.e. (2) n λ n f K 1 b. Proposition If a is an atom, then a C B, where B is the unit ball in H 1 b or K 1 b, and C is a universal constant independent of a. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

51 Tools: Control lemma imitating dyadic shifts Lemma For fixed i 1, i 2, j 1, j 2 N and any f L 2 (R n R m ), g L 2 (R n R m ), (i 1,i 2 ) (j 1,j 2 ) K D n I 1,I 2 K J 1,J 2 V V D m f L 2 g L 2, I 1 1/2 I 2 1/2 J 1 1/2 J 2 1/2 K V b 1 I 1 b 2 J 1 f L 2 b 1 I 2 b 2 J 2 g L 2 where the constant doesn t depend on i 1, i 2, j 1, j 2. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

52 Sketch of the Proof of T (b) Theorem 1 Averaging formula M b TM b f, g = 1 E ω ne ω m π n good πm good I 1,I 2 D n J 1,J 2 D m M b TM b b 1 I 1 b 2 J 1 f, b 1 I 2 b 2 J 2 g. χ good (sm(i 1, I 2 ))χ good (sm(j 1, J 2 )) 2 Case by case manipulation: relative positions of cubes Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

53 Good and bad cubes Definition A cube I is called bad if there exists Ĩ D n so that l(ĩ) 2r l(i) and d(i, Ĩ) 2l(I)γn l(ĩ)1 γn, where γ n = δ/(2n + 2δ). For I D 0, the position and badness of I ω are independent random variables. The probability of a particular cube I ω being bad is equal for all I D 0 : P ω (I ωbad) = π bad = π bad (r, n, δ) Key lemma: π bad < 1 if r = r(n, δ) is chosen large enough. Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36

54 Thank you! Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces June 2, / 36