Conditioned Brownian Motion, Hardy spaces, Square Functions Paul - - PDF document

Conditioned Brownian Motion, Hardy spaces, Square Functions

Paul F.X. M¨ uller Johannes Kepler Universit¨ at Linz

Topics

• 1. Problems in Harmonic Analysis

(a) Fourier Multipliers in Lp(T) (b) SL∞(T), Interpolation, Approximation.

• 2. Stochastic Proofs

(a) SL∞(Ω) (b) Conditioned Brownian Motion (c) Permanence theorem

Fouriermultipliers.

To u ∈ Lp([0, 2π]) with Fourier series u(θ) =

∞

• k=1

ak cos kθ + bk sin kθ. form the dyadic blocks ∆n(u)(θ) =

2n+1−1

• k=2n

ak cos kθ + bk sin kθ and define the transform v(θ) =

∞

• n=0

ǫn∆n(u)(θ), ǫn ∈ {−1, 1}. Theorem 1 (Littlewood-Paley, Marcinkiewicz) There exists Cp > 0 so that for all ǫn ∈ {−1, 1}, vLp ≤ CpuLp, and Cp → ∞ for p → ∞ or p → 1.

Littlewood-Paley Function.

Let u(z), z ∈ D denote the harmonic exten- sion of u ∈ Lp([0, 2π]) obtained by integration against the Poisson kernel Pθ(z) = 1 − |z|2 |eiθ − z|2. The Littlewood Paley Funktion g2

D(u)(θ) =

• D |∇u(z)|2 log 1

|z|Pθ(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 gD(v)(θ) ≤ CgD(u)(θ), and Lp integral estimates C−1

p

vLp ≤ CpgD(v)Lp ≤ CpvLp.

Uniformly bounded Littlewood Pa- ley Functions

SL∞(T) denotes the space of all functions u with uniformly bounded Littlewood Paley Func- tion. uSL∞(T) = gD(u)∞. The conditions gD(u)∞ < ∞ contolls the growth

• f u and also its oscillationen. Chang-Wilson-

Wolff proved that there existists c > 0 so that

2π

exp(cu2(θ))dθ < ∞. On the other hand there exist E ⊆ [0, 2π[ so that g2

D(1E)∞ = ∞.

Multipliers into SL∞(T) and Marcinkiewicz-decomosition

We get two different endpoints of the Lp scale. L2 ⊃ · · · ⊃ Lp ⊃ · · · ⊃ BMO ⊃

  

L∞ SL∞(T) The relation of the endpoint SL∞ to the Lp scale is clarified by a Marcinkiewicz decompo- sition and by pointwise multipliers with values in SL∞(T).

A function f ∈ Lp is in the Hardy space Hp when its harmonic extension to the unit disc is analytic. Theorem 2 (P. W. Jones & P.F.X.M.) To f ∈ Hp and λ > 0 there exists∗ g ∈ SL∞∩H∞ so that gSL∞ + g∞ ≤ C0λ, f − g1 ≤ λ1−pfp

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 m1ESL∞ < C0 and

2π

m1Edθ ≥ |E|/2.

Conditioned Brownian Motion and Littlewood-Paley

Let (Ω, P) be Wiener Space. 2D-Brownian mo- tion Bt : Ω → R2 starting at B0 = 0 leaves the unit disk for the first time at τ = inf{t > 0 : |Bt| > 1}. The harmonic extension of u ∈ Lp defines the martingal u(Bt), t ≤ τ, with quadratic variation u(Bτ) =

τ

0 |∇u(Bs)|2ds.

Form the expectation u(Bτ) under the con- dition {Bτ = eiθ}, to obtain gD(u)(θ). Thus g2

D(u) = E(u(Bτ)|Bτ = eiθ).

Classical Lp Permanence.

Let 1 ≤ p ≤ ∞, X ∈ Lp(Ω) and N(X)(eiθ) = E(X|Bτ = eiθ). N : Lp(Ω) → Lp(T) contracts, u = Nu(Bτ). X ∈ Lp(Ω) with stochastic integral representa- tion X = EX+

HsdBs has quadratic variation,

X =

∞

|Hs|2ds. 1 < p < ∞. The Permanence-theorem due to Zygmund, Burkholder, Doob (combined) as- serts that g2(N(X))Lp(T) ≤ CpNXLp(T), where Cp → ∞ if p → ∞ or p → 1.

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

The SL∞−Permanence theorem. For X ∈ Lp(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|Bτ = eiθ) is bounded, since we have the pointwise esti- mates g2

D(N(X))(·) ≤ C0N(X)(·).

Using random variables with uniformly bounded quadratic variation as input the SL∞− perma- nence theorem generates functions with uni- formly bounded Littlewood Paley function.

Applications of SL∞−Permanence.

Solution of the Interpolation Problem. Part 1. f ∈ Hp(T) induces holomorphic mar- tingal f(zt) =

t

0 f′(zs)dzs,

zt = B(1)

t

+ iB(2)

t

With the stopping time ρ(ω) = inf{r :

r

0 |f′(zs)|2ds > λ, |f(zr)| > λ}

f(zρ) on Wiener space is bounded with uni- formly bounded quadratic variation.

Part 2. First, g = N(f(zρ)) is in H∞(T), with gL∞ ≤ f(zρ)∞ ≤ λ. And by the SL∞− permance theorem it satis- fies gSL∞ ≤ CNf(zρ)∞ ≤ Cλ. With Theorems of Doob and Burkholder we get for h = f − g the error estimates h1 ≤ Cpλ1−pfp

p.

Solution of Multiplier Problems. Part 1. E ⊆ [0, 2π[. Let 1E(z) be the harmonic extension of the indicator function 1E. Define the bounded and non-negative multi- plier in Wiener space µt = exp(1E(Bt)−

t

0 |∇1E(Bs)|2(1

2+ 1 1E(Bs))ds). By Feynman-Kac-Stochastic Calculus: µt1E(Bt) t < τ, is a martingale and has uniformly bounded quadratic variation µτ1E(Bτ) ≤ C.

Part 2. Define mulitplier on the disk m = N(µτ). Then by SL∞−Permanence theorem m1E = N(µτ1E(Bτ)) has bounded Littlewood Paley function and mean

• m1Edt = Eµτ1E(Bτ) = µ01E(B0).

Holomorphic Random Variables X ∈ Lp(Ω), is holomorphic RV if X = EX +

• Fsdzs.

Example: f(zρ) = f(0) +

ρ

0 f′(zs)dzs.

If X ∈ Lp(Ω) is holomorphic RV then the har- monic Extension of E(X|zτ = eiθ) is analytic and

E(X|zτ = eiθ) ∈ Hp.

Covariance formula for holomorphic RV gives

E(XPθ(zτr)) = EX + E

τr

Fs∂zPθ(zs)ds. Use Power series ∂zPθ(zs) =

∞

• n=1

an(zs)einθ and put bn = E

τr

0 an(zs)ds to get

E(XPθ(zτr)) = EX +

∞

• n=1

bneinθ.

The harmonic Extension of N(X). X ∈ L2(Ω), X = EX +

• Fsdzs + Gsd¯

zs. The sum of

E

τ

0 Fs

π

π ∂zPθ(zt)∇wPθ(w)dθ

• dsdP.

and

E

τ

0 Gs

π

π ∂¯ zPθ(zt)∇wPθ(w)dθ

• dsdP.

is ∇wN(X)(w). Start with N(X)(θ) = EX+E

τ

0 Fs∂zPθ(zs)+Gs∂¯ zPθ(zt)ds

integrate against the Poisson kernel Pθ(w) and form the gradient with respect to w.

The Whitney Decomposition of the Unit Disk. The Littlewood Paley Funktion of N(X) g2

D(N(X))(θ) =

• D |∇wN(X)(w)|2 log 1

|w|Pθ(w)dA(w) satisfies the pointwise estimate g2

D(N(X)(θ) ≤ C0E τ

0 (|Fs|2 + |Gs|2)Pθ(zt)ds.

W = {Q : Q is Whitney Cube in D}. Whitney cubes are pairewise disjoint and sat- isfy 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 in- variant.

The integral kernels defining ∇wN(X)(w) lead to almost diagonal matrices: For Q1, Q2 ∈ W, put k(Q1, Q2) =

π

−π ∂zPθ(Q1)∇wPθ(Q2)dθ

• .

Then, k(Q1, Q2) ∼ 0 except when Q1 is close to Q2, (in the sense of the Poincare metric). This gives rise to almost diagonal matrices encoun- tered in the proof of the David Journe T(1)

• theorem. Hence ℓ2 estimates.