Paraproducts and the bilinear Calderón-Zygmund theory

Transcription

1 Paraproducts and the bilinear Calderón-Zygmund theory Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS th New Mexico Analysis Seminar April 23-25, 2009

2 Outline of the talk The paraproduct paradigm Variations and applications Bilinear Calderón-Zygmund singular integral operators

3 All the paraproducts, the paraproduct The name paraproduct denotes an idea rather than a unique definition; several definitions exist and can be used for the same purposes. S. Janson and J. Peetre, TAMS 1988.

4 Leibnitz rule Hence For f, g C 1 0 (R) we have (fg) = f g + fg. Define f (x)g(x) = x Π 1 (f, g)(x) = x f (t)g(t) dt + f (t)g (t) dt x f (t)g(t) dt Properties Π 1 satisfies 1 fg = Π 1 (f, g) + Π 1 (g, f ) (Product reconstruction) 2 Π 1 (f, g) = f g (Leib rule = half of Leibnitz rule) 3 If h(0) = 0, then Π 1 (f, h (f )) = h(f ) (Linearization) Π 1 : X Y Z not well-suited for L p -spaces (no Hölder inequality).

5 A. Calderón 1 (1965) For F, G analytic on the upper-half plane {s + it : s R, t > 0}, Calderón defined Π 2 (F, G)(s) = i Properties Π 2 verifies 0 F (s + it)g(s + it) dt. 1 FG = Π 2 (F, G) + Π 2 (G, F ) (Product reconstruction) 2 Π 2 (F, G) = Π 2 (F, G) + Π 2 (F, G ) (Leibnitz rule) 3 If H(0) = 0, then Π 2 (F, H (F )) = H(F ) (Linearization) 4 For 1 < p, q, r < with 1 r = 1 p + 1 q Π 2 (F, G) L r c F H p G H q (Hölder s inequality) 1 Proc. Nat. Acad. Sci., 53,

6 Simple complex manipulations F (s + it) = F 1 (s, t) + if 2 (s, t) G(s + it) = G 1 (s, t) + ig 2 (s, t) Write where and F 1 (s, t) = (f 1 P t )(s), P(x) = 1 π(1 + x 2 ), x R, P t (x) = 1 t P ( x t ), t > 0. Set Q(x) = P (x) so that Q(x) dx = P( ) P( ) = 0 and (P t ) (x) = 1 ( x ) t 2 P = 1 t t Q t(x).

7 Back to reality Rewrite F (s + it) = F 1 s (s, t) + i F 2 (s, t) s = (f 1 P t ) (s) + i (f 2 P t ) (s) s s = f 1 (P t ) (s) + if 2 (P t ) (s) = 1 t (f 1 Q t )(s) + i 1 t (f 2 Q t )(s) and since G(s + it) = (g 1 P t )(s) + i(g 2 P t )(s), we only care for Π 3 (f, g)(s) := 0 (Q t f )(s)(p t g)(s) dt t No more complex variables!

8 Prehistoric paraproducts Π 3 (f, g)(s) = has a discrete version (think t = 2 j ) 0 (Q t f )(s)(p t g)(s) dt t Π 4 (f, g)(s) = j Z(Q j f )(s)(p j g)(s) Question What properties do Π 3 and Π 4 have?

9 Elements of Littlewood-Paley theory φ S(R n ), real, radially symmetric, and χ { ξ 1/2} ˆφ χ { ξ 1} Why should any one be frightened by a bump function?

10 Elements of the Littlewood-Paley theory Define ψ S(R n ) by ˆψ(ξ) := ˆφ(ξ/2) ˆφ(ξ), ξ R n. For j Z, set ψ j (ξ) := ˆψ(ξ/2 j ) and φ j (ξ) := ˆφ(ξ/2 j ) Hence, supp( ψ j ) {ξ R n : 2 j 1 ξ 2 j+1 }, for j Z, and ψ j (ξ) = 1, ξ R n \ {0}. j Z

11 Low-pass and band-pass filters Define the operators S j and j by S j (f ) = φ j f, and In particular, j (f ) = ψ j f. S j (f ) = k j k (f ) and f = j Z j (f ),

12 Product vs. Paraproduct Given f, g S(R n ), fg = ( ) j (f ) k (g) = j (f ) k (g) j Z j Z k Z k Z Introduce the (idea of a) paraproduct between f and g as Π 5 (f, g) := j (f ) k (g) = j (f )S j (g) j Z k j j Z That is, Π 5 (f, g) = j Z(ψ j f )(φ j g) Compare with Π 4 (f, g) = j Z(Q j f )(P j g)

13 Allowing for errors Although Π 5 gives perfect product reconstruction, we consider Π 6 (f, g) = j Z(ψ j f )(φ j 2 g) so that where fg = Π 6 (f, g) + Π 6 (g, f ) + R(f, g), R(f, g) = j,k Z j k <2 Why the 2? Why allowing error at all? To make room for bigger elephants. (ψ j f )(ψ k g).

14 The paraproduct paradigm Π 6 verifies Product reconstruction fg = Π 6 (f, g) + Π 6 (g, f ) + R(f, g) Hölder s inequality (or other bilinear estimates) For 1 < p, q, r < with 1 r = 1 p + 1 q Π 6 (f, g) L r c f L p g L q Para-Leib rule (or other commutativity properties) α Π 6 (f, g) = Π 6 ( α f, g) Here Π 6 is another paraproduct(!). Linearization formula (modulo smooth errors)

15 Example of the paradigm in action Proving the Kato-Ponce inequality In 1988, T. Kato and G. Ponce 2 proved that, if α > 0 and 1 < p 1, p 2, q 1, q 2, r <, with and D α h(ξ) = ξ α ĥ(ξ), then 1/r = 1/p 1 + 1/q 1 = 1/p 2 + 1/q 2, D α (fg) L r C ( D α f L p 1 g L q 1 + f L p 2 D α g L q 2 ) Short proof due to C. Muscalu, J. Pipher, T. Tao, and C. Thiele 3 D α (fg) L r = D α (Π 5 (f, g)) + D α (Π 5 (g, f )) L r = Π 5 (D α f, g) + Π 5 (D α g, f ) L r C ( D α f L p 1 g L q 1 + f L p 2 D α g L q 2 ) 2 Comm. Pure Appl. Math. 41, no. 7, Acta Math. 193 (2004), no. 2,

16 J.-M. Bony 4 (1981) Paradifferential operators - Paraproducts - Paramultiplication Suppose that f C α, g C β (both bounded), with 0 < α < β. Then C {}}{ α C β C {}}{ α C {}}{{}} α+β { fg = Π 6 (f, g) + Π 6 (g, f ) + R(f, g) Suppose F C (R) with F (0) = 0. Then where e F (u) is smoother than u. F (u) = Π 6 (u, F (u)) + e F (u), 4 Ann. Sci. École Norm. Sup. 14,

17 R. R. Coifman and Y. Meyer (1978) Au-delà des Opérateurs Pseudo-différentiels au-delà = beyond beyond = para Paradifferential operators

18 The dyadic or generalized Littlewood-Paley decomposition J. Peetre 5 ( ), H. Triebel 6 (1977) fg = (ψ j f )(φ j g) + j g)(φ j f ) j Z j Z(ψ In the context of pointwise multiplication of function spaces. 5 New thoughts on Besov spaces. Chapter 7 6 Ann. Mat. Pura Appl. (4) 114 (1977)

19 Some famous applications. Apologies for any omissions 1 Bony s linearization formula 2 Boundedness of Calderón commutators (Coifman-Meyer) 3 T (1) and T (b) theorems (David-Journé-Semmes) 4 Pointwise multipliers of function spaces (Peetre, Sickel, Triebel,...) 5 Fluid dynamics (Cannone, Chemin, Lemarié-Rieusset, Meyer,...) 6 Decomposition of operators (Coifman-Meyer, Gilbert-Nahmod, Grafakos-Kalton, Lacey-Thiele...) 7 Bilinear pseudo-differential operators, reduced symbols (Coifman-Meyer, Grafakos-Torres, Bényi-Torres) 8 Paracommutators (Janson-Peetre) 9 Compensated compactness (Coifman, P.L. Lions, Meyer, Semmes, Peng, Wong,...) 10 Dyadic versions (P. Auscher, O. Beznosova, O. Dragicevic, L. Grafakos, N. Katz, M. Lacey, X. Li, T. Mei, C. Muscalu, F. Nazarov, M. C. Pereyra, S. Petermichl, T. Tao, C. Thiele, S. Treil, A. Volberg, L. Ward, B. Wick,...)

20 As bilinear multipliers We have Π 6 (f, g) = j Z(ψ j f )(φ j 2 g) Using the Fourier transform Π 6 (f, g)(x) = ψ j (ξ) φ j 2 (η) f (ξ)ĝ(η)e ix(ξ+η) dξdη j Z A closer look at the bilinear symbols ψ j (ξ) φ j 2 (η) For ξ and η such that 2 j 1 ξ 2 j+1 and η 2 j 2 it holds 2 j 2 ξ + η 2 j+2

21 Bigger elephants Let Ψ S(R n ), radial, such that Ψ(ω) = 1, ω [2 2, 2 2 ] and Therefore, supp( Ψ) {ω R n : ω [2 2 1/8, /8]}. Ψ j (ξ + η) ψ j (ξ) φ j 2 (η) = ψ j (ξ) φ j 2 (η).

22 One way to use Ψ j Π 6 (f, g)(x) = j Z = j Z ψ j (ξ) φ j 2 (η) f (ξ)ĝ(η)e ix(ξ+η) dξdη Ψ j (ξ + η) ψ j (ξ) φ j 2 (η) f (ξ)ĝ(η)e ix(ξ+η) dξdη = j Z Ψ j ((ψ j f )(φ j 2 g))(x). By duality, Π 6 (f, g), h = j Z (ψ j f )(φ j 2 g), Ψ j h. Advantage: immediate access to the Littlewood-Paley pieces of h.

23 Typical example Let f BMO and g, h L 2 Π 6 (f, g), h = j Z (ψ j f )(φ j 2 g), Ψ j h j Z (ψ j f )(φ j 2 g) L 2 Ψ j h L 2 j Z ψ j f 2 φ j 2 g 2 j Z 1/2 Ψ j h 2 1/2 C f BMO Mg L 2 Sh L 2 C f BMO g L 2 h L 2

24 Freezing a variable: linear singular integrals That is, Π 6 : BMO L 2 L 2. Moreover, for fixed f BMO, the mapping g Π 6 (f, g) has a Calderón-Zygmund kernel. Thus, g Π 6 (f, g) is a linear Calderón-Zygmund singular integral operator. This explains Π 6 : L C α C α, and many other mapping properties of the form Π 6 : L X X.

25 A convenient third wheel The bilinear operator Π 6 (f, g) = j Z (ψ j f )(φ j 2 g) = j Z Ψ j ((ψ j f )(φ j 2 g)) leads to considering bilinear operators of the form Π 7 (f, g)(x) := j Z φ 1 j ((φ 2 j f )(φ 3 j g))(x) = =: j Z φ 1 j (x w)φ 2 j (w y)φ 3 j (w z) dw f (y)g(z) dydz K Π7 (x, y, z)f (y)g(z) dydz, where φ 1, φ 2, and φ 3 are smooth functions, some of them with appropriate cancelation.

26 Dyadic cubes We write Q D if Q is a dyadic cube. That is, for some k Z n, ν Z Q = {x R n : k i 2 ν x i k i + 1; i = 1,..., n} Set x Q = 2 ν k and also write Q = Q νk. Notice that Q = 2 νn.

27 Towards a molecular representation I Think j = ν, so that K Π7 (x, y, z) = φ 1 ν(x w)φ 2 ν(w y)φ 3 ν(w z) dw ν Z = 2 νn φ 1 (2 ν (x w))2 νn φ 2 (2 ν (w y))2 νn φ 3 (2 ν (w z)) dw ν Z k Z n Q νk = ν Z k Z n Q νk 1 2 Q νk Q νk 2 3νn 2 φ 1 (2 ν (x w))φ 2 (2 ν (w y))φ 3 (2 ν (w z)) dw = Q νk 1 2 φ 1 Q (x)φ 2 Q (y)φ3 Q (z) + E(x, y, z). ν Z k Z n Here φ j νn Q (x) = 2 2 φ j (2 ν (x 2 ν k)), for Q = Q νk and j = 1, 2, 3, and E(x, y, z) is the kernel of a smoothing operator (see M.-Naibo, 2009)

28 Towards a molecular representation II Define the bilinear kernel K Π8 (x, y, z) = Q νk 1 2 φ 1 Q (x)φ 2 Q (y)φ3 Q (z) ν Z k Z n and its associated bilinear operator Π 8 (f, g)(x) = Q νk 1 2 φ 2 Q, f φ 3 Q, g φ1 Q (x). ν Z k Z n Recall that φ j νn Q (x) = 2 2 φ j (2 ν (x 2 ν k)), Q = Q νk, j = 1, 2, 3.

29 Smooth molecules Let Q D and M, N N. A smooth molecule of regularity M and decay N associated to Q is a function denoted by φ Q = φ Qνk = φ νk that satisfies for all γ M. γ φ νk (x) C γ,n 2 νn/2 2 γ ν (1 + 2 ν x 2 ν k ) N, x Rn, Yes, just like φ Q (x) = 2 νn 2 φ(2 ν (x 2 ν k)) for a smooth φ. M and N are specified during the applications.

30 Molecular paraproducts Let {φ 1 Q } Q D, {φ 2 Q } Q D, {φ 3 Q } Q D be three families of molecules. The associated molecular paraproduct is given by Π 9 (f, g)(x) = Q 1/2 f, φ 1 Q g, φ2 Q φ3 Q (x) Q D Heuristics for the para-leib rule. Duality yields Π 9 (f, g), h = Q 1/2 f, φ 1 Q g, φ2 Q φ3 Q, h. Q D Excellent set up for molecular representations of functions.

31 Π 9 as a bilinear CZ SIO. Bényi-M.-Nahmod-Torres (2008) Motto: Don t freeze, bilinearize! 1 Bilinear almost diagonal estimates 2 Molecular paraproducts have bilinear Calderón-Zygmund kernels 3 Molecular paraproducts with cancelation are bilinear Calderón-Zygmund operators 4 Mapping properties of the form X Y Z (as opposed to X Y Y )

32 Bilinear almost orthogonality estimates For every N > n + 1 there is a constant C, depending only on N and n, such that for any γ, ν, µ, λ Z and any x, y, z R n the following inequality holds k Z n 2 γn2νn/22µn/22λn/2 [(1 + 2 ν x 2 γ k )(1 + 2 µ y 2 γ k )(1 + 2 λ z 2 γ k )] 5N C 2 max(µ,ν,λ)n/2 2 med(µ,ν,λ)n/2 2 min(µ,ν,λ)n/2 ((1 + 2 min(ν,µ) x y )(1 + 2 min(µ,λ) y z )(1 + 2 min(ν,λ) x z )) N

33 Paraproducts have bilinear C-Z kernels Estimating the size of K Π9 (x, y, z). Assume N > 10n K Π9 (x, y, z) 2 nν/2 φ 1 νk (y) φ2 νk (z) φ3 νk (x) ν Z k Z n C2 2nν [(1 + 2 ν x y ) (1 + 2 ν z y ) (1 + 2 ν x z )] N ν Z C ( x y + z y + x z ) 2n. To estimate K Π9 (x, y, z) K Π9 (x, y, z), ask for M = 1. No cancelation needed.

34 The L 2 L 2 L 1 bound (cancelation required) Suppose N > 10n + 10, M = 1, and {φ 1 Q } Q D, {φ 2 Q } Q D with cancelation. Π 9 (f, g), h Q D f, φ 1 Q 2 Q D C f L 2 g L 2 h L. f, φ 1 Q g, φ2 Q 2nν/2 h, φ 3 Q 1/2 g, φ 2 Q 2 Q D 1/2 sup 2 nν/2 h, φ 3 Q Q D

35 Further extensions 1 Paraproducts with Dini continuous molecules, M.-Naibo, Paraproducts and bilinear C-Z operators in spaces of homogeneous type, M. (PhD thesis, KU 2005) and Grafakos-Liu-M.-Yang (work in progress)

36 Thank you!