Conditioned Brownian Motion, Hardy spaces, Square Functions

Transcription

1 Conditioned Brownian Motion, Hardy spaces, Square Functions Paul F.X. Müller Johannes Kepler Universität Linz

2 Topics 1. Problems in Harmonic Analysis (a) Fourier Multipliers in L p (T) (b) SL (T), Interpolation, Approximation. 2. Stochastic Proofs (a) SL (Ω) (b) Conditioned Brownian Motion (c) Permanence theorem

3 Fouriermultipliers. To u L p ([0,2π]) with Fourier series u(θ) = k=1 form the dyadic blocks a k cos kθ + b k sin kθ. 2 n+1 1 n (u)(θ) = a k cos kθ + b k sin kθ k=2 n and define the transform v(θ) = n=0 ǫ n n (u)(θ), ǫ n { 1,1}. Theorem 1 (Littlewood-Paley, Marcinkiewicz) There exists C p > 0 so that for all ǫ n { 1,1}, v L p C p u L p, and C p for p or p 1.

4 Littlewood-Paley Function. Let u(z), z D denote the harmonic extension of u L p ([0,2π]) obtained by integration against the Poisson kernel P θ (z) = 1 z 2 e iθ z 2. The Littlewood Paley Funktion g 2 D (u)(θ) = D u(z) 2 log 1 z P θ(z)da(z) plays a central role in proving the multiplier theorem: Its proof consists of basically two independent components Pointwise estimates between the g functions g D (v)(θ) Cg D (u)(θ), and L p integral estimates C 1 p v L p C p g D (v) L p C p v L p.

5 Uniformly bounded Littlewood Paley Functions SL (T) denotes the space of all functions u with uniformly bounded Littlewood Paley Function. u SL (T) = g D (u). The conditions g D (u) < contolls the growth of u and also its oscillationen. Chang-Wilson- Wolff proved that there existists c > 0 so that 2π exp(cu 2 (θ))dθ <. 0 On the other hand there exist E [0,2π[ so that g 2 D (1 E) =.

6 Multipliers into SL (T) and Marcinkiewicz-decomosition We get two different endpoints of the L p scale. L 2 L p BMO L SL (T) The relation of the endpoint SL to the L p scale is clarified by a Marcinkiewicz decomposition and by pointwise multipliers with values in SL (T).

7 A function f L p is in the Hardy space H p when its harmonic extension to the unit disc is analytic. Theorem 2 (P. W. Jones & P.F.X.M.) To f H p and λ > 0 there exists g SL H so that g SL + g C 0 λ, f g 1 λ 1 p f p p Non trivial pointwise multipliers. Theorem 3 (P. W. Jones & P.F.X.M.) To each E [0,2π[ there exists 0 m(θ) 1 so that m1 E SL < C 0 and 2π 0 m1 E dθ E /2.

8 Conditioned Brownian Motion and Littlewood-Paley Let (Ω, P) be Wiener Space. 2D-Brownian motion B t : Ω R 2 starting at B 0 = 0 leaves the unit disk for the first time at τ = inf{t > 0 : B t > 1}. The harmonic extension of u L p defines the martingal u(b t ), t τ, with quadratic variation τ u(b τ ) = u(b s) 2 ds. 0 Form the expectation u(b τ ) under the condition {B τ = e iθ }, to obtain g D (u)(θ). Thus g 2 D (u) = E( u(b τ) B τ = e iθ ).

9 Classical L p Permanence. Let 1 p, X L p (Ω) and N(X)(e iθ ) = E(X B τ = e iθ ). N : L p (Ω) L p (T) contracts, u = Nu(B τ ). X L p (Ω) with stochastic integral representation X = EX+ H s db s has quadratic variation, X = 0 H s 2 ds. 1 < p <. The Permanence-theorem due to Zygmund, Burkholder, Doob (combined) asserts that where g 2 (N(X)) L p (T) C p N X L p (T), if p or p 1. C p

10 Due to the behaviour of the constants C p the classical theorem is limited to 1 < p <. With a permanence theorem valid for p =!! we could use stopping times and Ito calculus to obtain random variables with bounded quadratic variation, then transfer and get functions in SL (T).

11 The SL Permanence theorem. For X L p (Ω) define X SL (Ω) if X L (Ω). Theorem 4 (P.W. Jones, P.F.X. M. ) N : SL (Ω) SL (T), X E(X B τ = e iθ ) is bounded, since we have the pointwise estimates g 2 D (N(X))( ) C 0N( X )( ). Using random variables with uniformly bounded quadratic variation as input the SL permanence theorem generates functions with uniformly bounded Littlewood Paley function.

12 Applications of SL Permanence. Solution of the Interpolation Problem. Part 1. f H p (T) induces holomorphic martingal f(z t ) = t f (z s )dz s, z t = B t (1) + ib t (2) 0 With the stopping time r ρ(ω) = inf{r : f (z s ) 2 ds > λ, f(z r ) > λ} 0 f(z ρ ) on Wiener space is bounded with uniformly bounded quadratic variation.

13 Part 2. First, is in H (T), with g = N(f(z ρ )) g L f(z ρ ) λ. And by the SL permance theorem it satisfies g SL C N f(z ρ ) Cλ. With Theorems of Doob and Burkholder we get for h = f g the error estimates h 1 C p λ 1 p f p p.

14 Solution of Multiplier Problems. Part 1. E [0,2π[. Let 1 E (z) be the harmonic extension of the indicator function 1 E. Define the bounded and non-negative multiplier in Wiener space µ t = exp(1 E (B t ) t 0 1 E(B s ) 2 ( E (B s ) )ds). By Feynman-Kac-Stochastic Calculus: µ t 1 E (B t ) t < τ, is a martingale and has uniformly bounded quadratic variation µ τ 1 E (B τ ) C.

15 Part 2. Define mulitplier on the disk m = N(µ τ ). Then by SL Permanence theorem m1 E = N(µ τ 1 E (B τ )) has bounded Littlewood Paley function and mean m1 E dt = Eµ τ 1 E (B τ ) = µ 0 1 E (B 0 ).

16 Holomorphic Random Variables X L p (Ω), is holomorphic RV if X = EX + F s dz s. Example: f(z ρ ) = f(0) + ρ 0 f (z s )dz s. If X L p (Ω) is holomorphic RV then the harmonic Extension of E(X z τ = e iθ ) is analytic and E(X z τ = e iθ ) H p. Covariance formula for holomorphic RV gives E(XP θ (z τr )) = EX + E Use Power series z P θ (z s ) = n=1 τr 0 F s z P θ (z s )ds. a n (z s )e inθ and put b n = E τ r 0 a n(z s )ds to get E(XP θ (z τr )) = EX + n=1 b n e inθ.

17 The harmonic Extension of N(X). X L 2 (Ω), The sum of τ X = EX + { π F s dz s + G s d z s. E F s zp θ (z t ) w P θ (w)dθ 0 π and τ { π E G s zp θ (z t ) w P θ (w)dθ 0 π is w N(X)(w). } } dsdp. dsdp. Start with τ N(X)(θ) = EX+E F s z P θ (z s )+G s z P θ (z t )ds 0 integrate against the Poisson kernel P θ (w) and form the gradient with respect to w.

18 The Whitney Decomposition of the Unit Disk. The Littlewood Paley Funktion of N(X) g 2 D (N(X))(θ) = D wn(x)(w) 2 log 1 w P θ(w)da(w) satisfies the pointwise estimate g 2 D (N(X)(θ) C 0E τ 0 ( F s 2 + G s 2 )P θ (z t )ds. W = {Q : Q is Whitney Cube in D}. Whitney cubes are pairewise disjoint and satisfy dist(q, D) diamq. Lokalization: On Whitney Cubes we obtain stabilization of Green-funktion, Poisson-kern, Poincare-metrik studied in Potential theory. Die Whitney decomposition defines a conformal invariant.

19 The integral kernels defining w N(X)(w) lead to almost diagonal matrices: For Q 1, Q 2 W, put k(q 1, Q 2 ) = { π π zp θ (Q 1 ) w P θ (Q 2 )dθ }. Then, k(q 1, Q 2 ) 0 except when Q 1 is close to Q 2, (in the sense of the Poincare metric). This gives rise to almost diagonal matrices encountered in the proof of the David Journe T(1) theorem. Hence l 2 estimates.