Hardy-Stein identity and Square functions

Transcription

1 Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

2 Motivation: Littlewood-Paley theory Question If u = f and f L p for 1 < p <, then is 2 u x i x j L p? Connection to Riesz transform We define Riesz transform R i by R i f(ξ) = iξ i ξ f(ξ), then 2 u x i x j = R i R j u = R i R j f. So, we want to have R i R j f p f p. For f L 2 L p and h L 2 L q with h q = 1, it follows from Plancherel s theorem that 1 ξ i ξ j R i R j f(x)h(x)dx = R n (2π) n R n ξ f(ξ)ĥ(ξ)dξ 2 1 = (2π) n (ξ i e t 2 ξ 2 f)(ξi e t 2 ξ 2 ĥ)dtdξ R n Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

3 Motivation: Littlewood-Paley theory Littlewood-Paley square functions Let B t be Brownian motion and P t f(x) = E x [f(b t )]. Note that P t f(ξ) = e t 2 ξ 2 f(ξ) and Pt f = p t f where p t (x) = (2πt) n 2 e x 2 2t. Then, we have R i R j f(x)h(x)dx = i P t f(x) j P t h(x)dtdx R n R n P t f(x) P t h(x) dtdx R n R n ( G(f) p G(h) q ( where G(f)(x) = P t f(x) 2 dt ) 1 ( P t f 2 2 dt ) 1 2 We call G(f) is a Littlewood-Paley square function.. P t h 2 dt ) 1 2 dx Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

4 Motivation: Littlewood-Paley theory So, the L p -boundedness of second order Riesz transform R i R j can be obtained by showing G(f) p f p. Indeed, we have better than this:. c p f p G(f) p C p f p. How can we show this two sides L p -bounds for G(f)? 1. Analytic way: Calderon-Zygmund theory (singular integral) 2. Probabilistic way: Itô s formula and Burkholder-Davis-Gundy inequality Goal 1. L p -bounds for Lévy-Fourier multiplier operators (like Riesz transforms) 2. Via Littlewood-Paley square functions, which is defined by a pure jump Lévy process (like Brownian motion) Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

5 Preliminaries: Levy processes Definition A stochastic process X on (Ω, F, P) is called a Lévy process if (i) independent increments: for t < t 1 < < t n <, {X tk X tk 1 } k 1 are independent. (ii) stationary increments: for < s < t < and a Borel set A R n, P(X t X s A) = P(X t s A). (iii) stochastically continuous: for all ε > and s, lim P( X t X s > ε) =. t s Characterization Lévy-Khintchine formula says: for ξ R n, E[e iξxt ] = e tψ(ξ) and ψ(ξ) = ib ξ ξ Aξ + R n (1 e iξy + iξ y1 B1 ()(y))ν(dy) where b R n, A a n n positive semi-definite symmetric matrix, and min{1, y 2 }ν(dy) <. R n \{} Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

6 Preliminaries: Levy processes Conversely, given (b, A, ν), there exists a Lévy process X t such that ψ(ξ) = ib ξ ξ Aξ + R n (1 e iξy + iξ y1 B1 ()(y))ν(dy). Example 1: Brownian motion For (b, A, ν) = (, I n, ), X t is the standard Brownian motion. Example 2: Gaussian process with drift For (b, A, ) with A >, X t = bt + G t where G t is a stochastic process such that mean zero, covariance E[G i tg j s] = A ij (t s), and for each t >, G t has the normal distribution ( 1 (2πt) n/2 det(a) exp 1 ) 2t x A 1 x. The process G t is called a Gaussian process. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

7 Preliminaries: Levy processes We say X t is a pure jump Lévy process if b = and A =. Example 3: Poisson process For (,, δ 1 ), X t = N t is an integer-valued process such that P(N t = k) = e t t k, k =, 1, 2,. k! Example 4: Compound Poisson process Suppose {Y i } are i.i.d. random variables with density ν and N t is an independent Poisson process with rate 1, then X t = Y 1 + Y Y Nt a Lévy process with ψ(ξ) = R (1 e iξ y )ν(dy). In this case, n (b, A, ν) = ( y <1 yν(dy),, ν). is Example 5: Symmetric α-stable process If E[e iξxt ] = e t ξ α and < α < 2, X t is a symmetric α-stable process. In this case, the triplet is (,, ν α ) with ν α (dx) = c n,α x n α dx. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

8 Preliminaries: Levy processes Jump processes The jump of X t at time s is denoted by X s = X s X s. For t, a Borel subset A R n, we define the jump measure of for X t by N(t, A) = the number of jumps during time [, t] of size in A = {s [, t] : X s A}. Note that this is a Poisson random measure with intensity dt ν. One can decompose X t into continuous part and jump part: X t = Xt c + X s s: s t = bt + G(t) + x 1 xn(t, dx) + x <1 = (Gaussian with drift)+(pure jump Lévy) xñ(t, dx) where Ñ(t, A) := N(t, A) tν(a) is the compensated jump measure. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

9 Preliminaries: Levy processes Semigroup and generator Let X t be a Lévy process with (b, A, ν). We define a semigroup P t f(x) := E x [f(x t )] and the infinitesimal generator L by Lf(x) = lim t P t f(x) f(x) t whenever the limit exists. It is well-known that Lf(x) = b f(x) + n A ij ij f(x) i,j=1 + (f(x + y) f(y) y f(x) 1 B1 ()(y))ν(dy). R n Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

10 Preliminaries: Itô s formula Setting Let (Z t ) t be a stochastic process defined by Z t = Z + M t + A t + where t R n G(s, x)n(ds, dx) + t H(s, x)ñ(ds, dx), R n (i) N(t, A) is a Poisson random measure with intensity ν (which is the jump measure of X t ), (ii) M t is a n-dimensional continuous square integrable local martingale, (iii) A t is a continuous adapted process of bounded variation with A =, (iv) G(t, x), H(t, x) are predictable with G i (t, x) H j (t, x) = for all 1 i, j n, and (v) Uniform boundedness of H: sup t T sup H(t, x) < for all T >. x R n Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

11 Preliminaries: Itô s formula Itô s formula for processes with jumps If ϕ is a C 2 function on R n and Z t is a process defined as above, we have ϕ(z t ) = ϕ(z ) t t t t t ϕ(z s ) dm s ϕ(z s ) da s Z t = Z + M t + A t + n i,j=1 t ij ϕ(z s )d M i, M j s (ϕ(z s + G) ϕ(z s )) N(ds, dx) R n (ϕ(z s + H) ϕ(z s )) Ñ(ds, dx) R n R n (ϕ(z s + H) ϕ(z s ) H ϕ(z s )) ν(dx)ds. t R n G(s, x)n(ds, dx) + t H(s, x)ñ(ds, dx) R n Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

12 Preliminaries: Itô s formula Example Let ϕ C 2 and X t be a pure jump Lévy process, that is, t t X t = xn(ds, dx) + xñ(ds, dx). x 1 x <1 Then, Itô s formula yields that t ϕ(x t ) = ϕ(x ) + (ϕ(x s + x) ϕ(x s ))N(ds, dx) x 1 t + (ϕ(x s + x) ϕ(x s ))Ñ(ds, dx) x <1 t + (ϕ(x s + x) ϕ(x s ) x ϕ(x s ))ν(dx)ds x <1 t t = ϕ(x ) + Lϕ(X s )ds + (ϕ(x s + x) ϕ(x s ))dñ. R n Lf(x) = (f(x + y) f(y) y f(x) 1 B1 ()(y))ν(dy) R n Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

13 Preliminaries: Fourier multipliers Definition Let m : R n C be a function in L and 1 p. For f L 2 L p, we define an operator T m by T m f(ξ) = m(ξ) f(ξ). If T m f p f p for all f L 2 L p, then T m can be extended to L p uniquely. We say T m is a L p -Fourier multiplier operator with symbol m. In this case, we denote by m M p. Examples (i) Riesz transforms R j is a Fourier multiplier operator with symbol m = iξ j ξ. (ii) Beurling-Ahlfors operator is a L p (C)-Fourier multiplier with symbol m(ξ) = ξ ξ. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

14 Preliminaries: Fourier multipliers Hörmander multipliers theorem If k > n 2 and m : Rn C is of class C k on R n \ {} with α m(ξ) ξ α, ξ R n for α = (α 1, α 2,, α n ) with α k. Then, m M p for all 1 < p <, that is, T m f p f p. Question 1. Does a given function m define a L p -Fourier multiplier operator T m? 2. Can we find the best constant C p such that T m p C p f p? Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

15 Previous works R. Banuelos, P. J. Mendez-Hernández (23) Using the space-time Brownian motion and Burkholder s inequality, found a better constant for Beurling-Ahlfors operator. R. Banuelos, K. Bogdan (27) L p boundedness of T m where R (cos(ξ z) 1)φ(z)ν(dz) m(ξ) = n. R (cos(ξ z) 1)ν(dz) n R. Banuelos, F. Baudoin (213), R. Banuelos, D. Applebaum (214), R. Banuelos, A. Osekowski(215) L p boundedness of T m on manifolds, Lie groups and Gauss spaces Note: These works rely on martingale transform and Burkholder s inequality. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

16 Previous works Question: can we get L p -Fourier multipliers via Littlewood-Paley square function? R. Banuelos, K. Bogdan and T. Luks (216) (a) G(f) p f p where G(f) is a square function for a symmetric pure jump Lévy process X t, defined by ( ) 1 G(f) = (P t f(x + y) P t f(x)) 2 2 ν(dy)dt. R d (b) To obtain this, the authors prove the Hardy-Stein identity using the properties of the semigroup P t. (c) As a corollary, they derive L p bounds for certain Lévy-Fourier multiplier. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

17 Main result Our contribution is... (a) to give a simple, direct proof for the Hardy-Stein identity, using Itô s formula, and (b) to extend L p bounds for certain Lévy-Fourier multipliers to the non-symmetric case. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

18 Main result Theorem (K., R. Bañuelos, 217) Let 1 < p < and F (a, b; p) = b p a p pa a p 2 (b a). If f L p (R d ), then we have f(x) p dx = F (P t f(x), P t f(x + y); p)ν(dy)dtdx R d R d R d and f p p R d with constants depending only on p. (P t f(x + y) P t f(x)) 2 R d ( P t f(x + y) P t f(x) ) p 2 ν(dy)dtdx Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

19 Remark on the main result 1. The function F (a, b; p) is the second-order Taylor reminder of the function x x p. It is well-known that F (a, b; p) b a 2 (a b) p There is an assumption on the Lévy measure ν, the Hartman-Wintner condition: lim ξ Re(ψ(ξ)) log(1 + ξ ) =. This makes the semigroup P t regular enough. 3. This result holds for non-symmetric ν. But, to obtain L p bounds for Littlewood-Paley square functions from this, we need to assume that ν is symmetric, that is, ν(b) = ν( B). Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

20 Main result We define Λ φ (f, g) for f L 2 L p and g L 2 L q, 1 p + 1 q = 1, by Λ φ (f, g)(x) = (P t f(x + y) P t f(x))(p t g(x + y) P t g(x))φ(t, y)ν(dy)dtdx. Theorem (K., R. Bañuelos (217)) (i) The bilinear operator Λ φ (f, g) is absolutely convergent for f L 2 L p and g L 2 L q. (ii) Furthermore, there is a unique linear operator S φ on L p such that Λ φ (f, g) = S φ (f), g and S φ = T mφ with the Fourier multiplier m φ given by m φ (ξ) = e iξ y 1 2 e 2t Re(ψ(ξ)) φ(t, y)ν(dy)dt. R d Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

21 Sketch of proof Step 1 Let T > be fixed, f L p (R d ) for 1 p <, and P t f(x) = E x [f(x t )]. If we define Y t := P T t f(x t ), then by Itô s formula, we have t Y t = Y + (P T s f(x s + y) P T s f(x s ))Ñ(ds, dy). R d P T t f(x) C 2? By Hartman-Wintner condition! Step 2 Let p 2. Apply Itô s formula to ϕ(x) = x p and Y t := P T t f(x t ) with H(s, y) = P T s f(x s + y) P T s f(x s ) for fixed T >. Y t p = Y p + (a Martingale) t + (ϕ(y s + H) ϕ(y s ) H ϕ(y s )) ν(dy)ds, R d for all t T. Is H(s, y) uniformly bounded? By Hartman-Wintner condition! Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

22 Sketch of proof Step 3 Taking expectation and integral, we have E x [ Y T p ]dx E x [ Y p ]dx R d R d T = F (P T s f(x s + y), P T s f(x s ); p)ν(dy)dsdx R d R d Here, we have used the fact that ϕ(y s +H) ϕ(y s ) H ϕ(y s ) = F (P T s f(x s +y), P T s f(x s ); p). After careful manipulation, we send T to obtain the desired identity. Step 4 For 1 < p < 2, the map x x p is not C 2. So, we used the map x x 2 + ε 2 p 2 instead. Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23

23 Thank you! Daesung Kim (Purdue) Hardy-Stein identity UIUC / 23