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 L p ( T ) (b) SL ∞ ( T ), Interpolation, Approximation. 2. Stochastic Proofs (a) SL ∞ (Ω) (b) Conditioned Brownian Motion (c) Permanence theorem

Fouriermultipliers. To u ∈ L p ([0 , 2 π ]) with Fourier series ∞ � u ( θ ) = a k cos kθ + b k sin kθ. k =1 form the dyadic blocks 2 n +1 − 1 � ∆ n ( u )( θ ) = a k cos kθ + b k sin kθ k =2 n and define the transform ∞ � v ( θ ) = ǫ n ∆ n ( u )( θ ) , ǫ n ∈ {− 1 , 1 } . n =0 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 .

Littlewood-Paley Function. Let u ( z ) , z ∈ D denote the harmonic exten- sion of u ∈ L p ([0 , 2 π ]) obtained by integration against the Poisson kernel P θ ( z ) = 1 − | z | 2 | e iθ − z | 2 . The Littlewood Paley Funktion D |∇ u ( z ) | 2 log 1 � g 2 D ( u )( θ ) = | 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 � v � L p ≤ C p � g D ( v ) � L p ≤ C p � v � L p . p

Uniformly bounded Littlewood Pa- ley Functions SL ∞ ( T ) denotes the space of all functions u with uniformly bounded Littlewood Paley Func- tion. � 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 � 2 π exp( cu 2 ( θ )) dθ < ∞ . 0 On the other hand there exist E ⊆ [0 , 2 π [ so that � g 2 D (1 E ) � ∞ = ∞ .

Multipliers into SL ∞ ( T ) and Marcinkiewicz-decomosition We get two different endpoints of the L p scale.  L ∞ L 2 ⊃ · · · ⊃ L p ⊃ · · · ⊃ BMO ⊃  SL ∞ ( T )  The relation of the endpoint SL ∞ to the L p scale is clarified by a Marcinkiewicz decompo- sition and by pointwise multipliers with values in SL ∞ ( T ) .

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 � f − g � 1 ≤ λ 1 − p � f � p � g � SL ∞ + � g � ∞ ≤ C 0 λ, p Non trivial pointwise multipliers. Theorem 3 (P. W. Jones & P.F.X.M.) To each E ⊆ [0 , 2 π [ there exists ∗ 0 ≤ m ( θ ) ≤ 1 so that � 2 π � m 1 E � SL ∞ < C 0 and m 1 E dθ ≥ | E | / 2 . 0

Conditioned Brownian Motion and Littlewood-Paley Let (Ω , P ) be Wiener Space. 2D-Brownian mo- tion 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 0 |∇ u ( B s ) | 2 ds. � u ( B τ ) � = Form the expectation � u ( B τ ) � under the con- dition { B τ = e iθ } , to obtain g D ( u )( θ ) . Thus g 2 D ( u ) = E ( � u ( B τ ) �| B τ = e iθ ) .

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 representa- � H s dB s has quadratic variation, tion X = E X + � ∞ | H s | 2 ds. � X � = 0 1 < p < ∞ . The Permanence-theorem due to Zygmund, Burkholder, Doob (combined) as- serts that � g 2 ( N ( X )) � L p ( T ) ≤ C p � N � X �� L p ( T ) , where C p → ∞ if p → ∞ or p → 1 .

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 calcu- lus to obtain random variables with bounded quadratic variation, then transfer and get functions in SL ∞ ( T ) .

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 esti- mates g 2 D ( N ( X ))( · ) ≤ C 0 N ( � 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 ∈ H p ( T ) induces holomorphic mar- tingal � t z t = B (1) + iB (2) 0 f ′ ( z s ) dz s , f ( z t ) = t t With the stopping time � r 0 | f ′ ( z s ) | 2 ds > λ, | f ( z r ) | > λ } ρ ( ω ) = inf { r : f ( z ρ ) on Wiener space is bounded with uni- formly bounded quadratic variation.

Part 2. First, g = N ( f ( z ρ )) is in H ∞ ( T ) , with � g � L ∞ ≤ � f ( z ρ ) � ∞ ≤ λ. And by the SL ∞ − permance theorem it satis- fies � 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 .

Solution of Multiplier Problems. Part 1. E ⊆ [0 , 2 π [ . Let 1 E ( z ) be the harmonic extension of the indicator function 1 E . Define the bounded and non-negative multi- plier in Wiener space � t 0 |∇ 1 E ( B s ) | 2 (1 1 µ t = exp(1 E ( B t ) − 2+ 1 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.

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

Holomorphic Random Variables X ∈ L p (Ω) , is holomorphic RV if � X = E X + F s dz s . � ρ 0 f ′ ( z s ) dz s . Example: f ( z ρ ) = f (0) + If X ∈ L p (Ω) is holomorphic RV then the har- monic 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 � τ r E ( XP θ ( z τ r )) = E X + E F s ∂ z P θ ( z s ) ds. 0 Use Power series ∞ a n ( z s ) e inθ � ∂ z P θ ( z s ) = n =1 � τ r and put b n = E 0 a n ( z s ) ds to get ∞ b n e inθ . � E ( XP θ ( z τ r )) = E X + n =1

The harmonic Extension of N ( X ) . X ∈ L 2 (Ω) , � X = E X + F s dz s + G s d ¯ z s . The sum of � τ �� π � π ∂ z P θ ( z t ) ∇ w P θ ( w ) dθ E 0 F s dsd P . and � τ �� π � z P θ ( z t ) ∇ w P θ ( w ) dθ E 0 G s π ∂ ¯ dsd P . is ∇ w N ( X )( w ) . Start with � τ N ( X )( θ ) = E X + E 0 F s ∂ z P θ ( z s )+ G s ∂ ¯ z P θ ( z t ) ds integrate against the Poisson kernel P θ ( w ) and form the gradient with respect to w.

The Whitney Decomposition of the Unit Disk. The Littlewood Paley Funktion of N ( X ) D |∇ w N ( X )( w ) | 2 log 1 � g 2 D ( N ( X ))( θ ) = | w | P θ ( w ) dA ( w ) satisfies the pointwise estimate � τ 0 ( | F s | 2 + | G s | 2 ) P θ ( z t ) ds. g 2 D ( N ( X )( θ ) ≤ C 0 E 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 ∇ w N ( X )( w ) lead to almost diagonal matrices: For Q 1 , Q 2 ∈ W , put �� π � k ( Q 1 , Q 2 ) = − π ∂ z P 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 encoun- tered in the proof of the David Journe T(1) theorem. Hence ℓ 2 estimates.