Multilinear local Tb Theorem for Square functions
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1 Multilinear local Tb Theorem for Square functions (joint work with J. Hart and L. Oliveira) September 19, Sevilla.
2 Goal Tf(x) L p (R n ) C f L p (R n )
3 Goal Tf(x) L p (R n ) C f L p (R n ) Examples
4 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy
5 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy Riesz Transform xj y j R j f(x) = f(y)dy x y n+1
6 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy
7 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy Hilbert Transform Hf(x) = R f(y) x y dy
8 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy
9 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy Square functions ( Sf(x) = 0 Ψ t (x, y)f(y)dy 2 dt/t ) 1/2
10 Goal Tf(x) L p (R n ) C f L p (R n ) Examples Singular integral operators Tf(x) = K(x, y)f(y)dy Square functions ( Sf(x) = 0 Ψ t (x, y)f(y)dy 2 dt/t ) 1/2 where we note θ t f(x) = Ψ t (x, y)f(y)dy.
11 Examples Kato problem (AHLM c T) Lf 2 f 2
12 Examples Kato problem (AHLM c T) Lf 2 f 2 We reduce ourselves to prove the L 2 boundedness problem for the Square function θ t = (1 + t 2 L) 1 tdiv A
13 Examples Kato problem (AHLM c T) Lf 2 f 2 We reduce ourselves to prove the L 2 boundedness problem for the Square function θ t = (1 + t 2 L) 1 tdiv A Single layer potential (GH,R) θ t = t( t ) 2 S t where S t f(x) = E(x, t, y, 0)f(y)dy.
14 We have set a problem
15 We have set a problem We have some operators as motivation
16 We have set a problem We have some operators as motivation How do we solve the problem?
17 We have set a problem We have some operators as motivation How do we solve the problem? Method
18 We have set a problem We have some operators as motivation How do we solve the problem? Method 1 To solve the problem for p = 2.
19 We have set a problem We have some operators as motivation How do we solve the problem? Method 1 To solve the problem for p = 2. 2 Tool = To test how the operator behave locally.
20 We have set a problem We have some operators as motivation How do we solve the problem? Method 1 To solve the problem for p = 2. 2 Tool = To test how the operator behave locally. Constant function 1.
21 We have set a problem We have some operators as motivation How do we solve the problem? Method 1 To solve the problem for p = 2. 2 Tool = To test how the operator behave locally. Constant function 1. An accretive function b.
22 We have set a problem We have some operators as motivation How do we solve the problem? Method 1 To solve the problem for p = 2. 2 Tool = To test how the operator behave locally. Constant function 1. An accretive function b. Locally adapted functions.
23 T1 Theorem for Square functions [Christ-Journé] Let θ t f(x) R n ψ t (x, y)f(y)dy, and assume that ψ t is a LP family. Suppose that we have the Carleson measure estimate sup Q 1 Q l(q) 0 Q θ t 1(x) 2 dxdt t C Then we have that the square function is bounded in L 2. Remark.- The converse direction is essentially due to Fefferman and Stein
24 Tb Theorem for Square functions [Semmes] Let θ t f(x) R n ψ t (x, y)f(y)dy, assume that ψ t is a LP family and b an accretive function Suppose that we have the Carleson measure estimate sup Q 1 Q l(q) 0 Q θ t b(x) 2 dxdt t C Then we have the L 2 boundedness of the square function. Remark.- An accretive function b is an L function that satisfies Re(b) c > 0
25 Local Tb Theorem for Square functions [HMc, HLMc,AHLMCT] Assume that θ t is a Calderón Zygmund operator and 1 < p <. Suppose also that there exist δ > 0, C 0 < such that for any dyadic cube Q, there exists a function b Q satisfying: (i) R n b Q p C 0 Q (ii) Q ( l(q) 0 (iii)δ Q b Q ) p θ t b Q (x) 2 dt 2 t dx C0 Q Then we have L 2 boundedness of the square function.
26 Formal definition of the operators
27 Formal definition of the operators Definition (escalar-valued kernel) Given a function Ψ t : R n R n C we define the Square function associated to the function Ψ t : where f : R n C. ( S(f)(x) := Ψ t (x, 2 dt y)f(y)dy 0 R n t ) 1/2
28 Formal definition of the operators Definition (escalar-valued kernel) Given a function Ψ t : R n R n C we define the Square function associated to the function Ψ t : where f : R n C. ( S(f)(x) := Ψ t (x, 2 dt y)f(y)dy 0 R n t ) 1/2 Our goal is, ( ) 1/2 S(f)(x) 2 dx C f 2 R n
29 Definition (vector-valued kernel) Given a function Ψ t = (Ψ 1 t,..., Ψm t ) : R n R n C m we define the Square function associated to the function Ψ t : ( S(f)(x) := Ψ t (x, y) 2 dt f(y)dy 0 R n t m 2 = Ψ j 0 R n t (x, y)f j(y)dy dt t j=1 where f : R n C m (f = (f 1,..., f m )). ) 1/2 1/2
30 Definition (vector-valued kernel) Given a function Ψ t = (Ψ 1 t,..., Ψm t ) : R n R n C m we define the Square function associated to the function Ψ t : ( S(f)(x) := Ψ t (x, y) 2 dt f(y)dy 0 R n t m 2 = Ψ j 0 R n t (x, y)f j(y)dy dt t j=1 where f : R n C m (f = (f 1,..., f m )). ) 1/2 1/2 Our goal is, ( ) 1/2 S(f)(x) 2 dx) C f 2 R n
31 Definition (multilinear square function) Given a function Ψ t : R (m+1)n C we define the Square function associated to the function Ψ t : S t (f 1,..., f m )(x) := Ψ t (x, y 1,..., y m )f 1 (y 1 )dy 1...f m (y m )dy m R ( n m S(f 1,..., f m )(x) := S t (f 1,..., f m )(x) ) 1/2 2 dt t where f 1,..., f m : R n C. 0
32 Definition (multilinear square function) Given a function Ψ t : R (m+1)n C we define the Square function associated to the function Ψ t : S t (f 1,..., f m )(x) := Ψ t (x, y 1,..., y m )f 1 (y 1 )dy 1...f m (y m )dy m R ( n m S(f 1,..., f m )(x) := S t (f 1,..., f m )(x) ) 1/2 2 dt t where f 1,..., f m : R n C. Our goal is, ( ) 1/2 S(f 1,..., f m )(x) 2 dx C R n 0 m f i pi, 1/2 = i=1 m i=1 1 p i.
33 Definition (multi-parameter square function) Given a function Ψ t : R n n C we define the Square function associated to the function Ψ t : S t (f 1,..., f m )(x) := Ψ t (x, y 1,..., y m )f 1 (y 1 )dy 1...f m (y m )dy m R ( n S(f 1,..., f m )(x) := S t (f 1,..., f m )(x) ) 1/2 2 dt t 0 where f i : R n i C for i = 1,..., m and n = m i=1 n i.
34 Definition (multi-parameter square function) Given a function Ψ t : R n n C we define the Square function associated to the function Ψ t : S t (f 1,..., f m )(x) := Ψ t (x, y 1,..., y m )f 1 (y 1 )dy 1...f m (y m )dy m R ( n S(f 1,..., f m )(x) := S t (f 1,..., f m )(x) ) 1/2 2 dt t 0 where f i : R n i Our goal is, C for i = 1,..., m and n = m i=1 n i. ( ) 1/2 S(f 1,..., f m )(x) 2 dx C R n m f i 2. i=1
35 Multilinear standard Calderón-Zygmund kernel We say that Ψ t : R (m+1) n C is a multilinear standard Calderón-Zygmund kernel if for all x, y 1,..., y m, x, y 1,..., y m R n satisfies: 1 Ψ t (x, y 1,..., y m ) C t mn m i=1 (1+t 1 x y i ) n+α 2 Ψ t (x, y 1,..., y m ) Ψ t (x, y 1,..., y i,..., y m) C t mn (t 1 y i y i )α m i=1 (1+t 1 x y i ) n+α 3 Ψ t (x, y 1,..., y m ) Ψ t (x, y 1,..., y m ) C t mn (t 1 x x ) α m i=1 (1+t 1 x y i ) n+α
36 Comparison Standard C-Z kernel with Multilinear C-Z kernel 1 (Size condition) t α Ψ t (x, y) C (t + x y ) n+α vs t αm Ψ t (x, y 1,..., y m ) C m i=1 (t + x y i ) n+α
37 Comparison Standard C-Z kernel with Multilinear C-Z kernel 2 (Hölder condition on y) Ψ t (x, y) Ψ t (x, y y y α ) C (t + x y ) n+α vs Ψ t (x, y 1,..., y m ) Ψ t (x, y 1,..., y i,..., y y i y i m) C α m i=1 (t + x y i ) n+α
38 Comparison Standard C-Z kernel with Multilinear C-Z kernel 3 (Hölder condition on x) Ψ t (x, y) Ψ t (x x x α, y) C (t + x y ) n+α vs Ψ t (x, y 1,..., y m ) Ψ t (x x x α, y 1,..., y m ) C m i=1 (t + x y i ) n+α
39 (Multilinear Tb Theorem) Let S be a multilinear square function associated to Ψ t a multilinear standar C-Z kernel. Suppose that there exists q i, q > 1 for i = 1,..., m with 1 q = m 1 i=1 q i and functions b i indexed by Q dyadic cubes Q R n for i = 1,..., m such that for every dyadic cube there exists C > 0 such that 1 b i R n Q q i B 1 Q 2 1 B 2 1 m Q Q i=1 bi Q (x)dx 3 1 m R R i=1 bi Q (x)dx B m 3 i=1 1 R R bi Q (x)dx for all dyadic subcubes R Q ( ) q l(q) 4 S t (b 1 Q,..., bm 2 dt )(x) 2 Q t dx B4 Q Then ( Q 0 R n S(f 1,..., f m )(x) 2 dx ) 1/2 C m i=1 f i pi, 1/2 = m i=1 1 p i.
40 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions.
41 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system
42 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system 1 ɛ b j (x) ɛ 1 Q
43 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system 1 ɛ b j (x) ɛ 1 Q 2 Characteristic functions b Q (x) = χ Q (x)
44 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system 1 ɛ b j (x) ɛ 1 Q 2 Characteristic functions b Q (x) = χ Q (x) 3 Gaussian functions b Q (x) = e x x Q 2 l(q) 2
45 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system 1 ɛ b j (x) ɛ 1 Q 2 Characteristic functions b Q (x) = χ Q (x) 3 Gaussian functions b Q (x) = e x x Q 2 l(q) 2 4 Poisson kernels b Q (x) = l(q) n+1 (l(q) 2 + x x Q 2 ) (n+1)/2
46 Remark 1.- We say that {b Q } is a pseudo-accretive system if it satisfies the first two conditions. Remark 2.- We say that {b Q } is a m-compatible collection of pseudo-accretive system if it satisfies the first three conditions. Examples m-compatible system 1 ɛ b j (x) ɛ 1 Q 2 Characteristic functions b Q (x) = χ Q (x) 3 Gaussian functions b Q (x) = e x x Q 2 l(q) 2 4 Poisson kernels b Q (x) = non m-compatible system l(q) n+1 (l(q) 2 + x x Q 2 ) (n+1)/2 b Q (x) = b 1 Q (x) = b2 Q (x) = (x 1/2)χ [0,2](x)
47 Multilinear T1 Theorem (GLMY, GO and H) Let R be a Square function whose kernel satisfies the multilinear size condition and the multilinear Hölder condition on the y variables. If R t (1,..., 1) = 0 for t > 0 then R(f 1,..., f m ) L p m C f L p i i=1 Remark.- The condition R t (1,..., 1) = 0 is a sufficient but not necessary condition. We also have a version on the T1 theorem where the requirement is that we have the Carleson measure sup Q l(q) 0 Q S t(1,..., 1) 2 dxdt t C
48 Proposition For every dyadic cube Q there exists a family of subcubes {Q k }, C > 0 and η (0, 1) such that k Q k < (1 η) Q Q ( l(q) τ Q (x) S t(1,..., 1) 2 dt t ) q/2 C Q where E = Q \ k Q k and τ Q (x) = { l(qk ) x Q k 0 x E
49 Lemma There exists N > 0 and β (0, ) such that for every dyadic cube Q {x Q : g Q > N} (1 β) Q where g Q (x) = l(q) 0 S t (1,..., 1)(x) 2 dt t 1/2
50 Muchas gracias!!!
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