Davis-Garsia Inequalities for Hardy Martingales Paul F.X. M uller - PowerPoint PPT Presentation

Davis-Garsia Inequalities for Hardy Martingales Paul F.X. M¨ uller Johannes Kepler Universit¨ at Linz

Topics 1. Basic Examples 2. Maximal Functions 3. Davis Decomposition 4. Martingale Transforms and Consequences 5. Davis Garsia Inequalities

The main sources A. Pelczynski, Banach Spaces of analytic functions and absolutely summing operators, (1977) Embedding L 1 to L 1 /H 1 , TAMS 278 J. Bourgain. (1983). PFXM. A decomposition for Hardy Martingales, Indiana Univ. Math. J. (2012) PFXM. A decomposition for Hardy Martingales II, Math. Proc. Cambr. Philos. Soc. (2014)

Complex analytic Hardy Spaces f ∈ L p ( T , X ) , T = { e iθ : | θ | ≤ π } , D = { z ∈ C : | z | < 1 } . The harmonic extension of f to the unit disk � π 1 − | z | 2 f ( z ) = 1 | z − e iα | 2 f ( e iα ) dα, z ∈ D . 2 π − π Define f ∈ H p ( T , X ) if f ∈ L p ( T , X ) and the harmonic extension of f is analytic in D .

Hardy Martingales H 1 ( T N , X ) T N the infinite torus-product with Haar measure d P . F k : T N → C is F k measurable iff x = ( x i ) ∞ F k ( x ) = F k ( x 1 , . . . , x k ) , i =1 An ( F k ) martingale F = ( F k ) with differences ∆ F k = F k − F k − 1 is a Hardy martingale if ∈ H 1 y → ∆ F k ( x 1 , . . . , x k − 1 , y ) 0 ( T , X ) . Conditional expectation E k F is integration � E k F ( x ) = T N F ( x 1 , . . . , x k , w ) d P ( w ) .

Example: Maurey’s embedding. Fix ǫ > 0 , w = ( w k ) ∈ T N . Put ϕ 1 ( w ) = ǫw 1 , and ϕ n ( w ) = ϕ n − 1 ( w ) + ǫ (1 − | ϕ n − 1 ( w ) | ) 2 w n . Then lim | ϕ n | = 1 and ϕ = lim ϕ n is uniformly dis- tributed over T . For any f ∈ H 1 ( T , X ) w ∈ T N F n ( w ) = f ( ϕ n ( w )) , is an integrable Hardy martingale with uniformly small increments � � sup E ( � F n � X ) = T � f � X dm and � ∆ F n � X ≤ 2 ǫ T � f � X dm. n ∈ N

Pointwise estimates for ∆ F n . Fix w ∈ T N , n ∈ N , z = ϕ n ( w ) , u = ϕ n − 1 ( w ) ∆ F n ( w ) = f ( ϕ n ( w )) − f ( ϕ n − 1 ( w )) . Cauchy integral formula � � ζ ζ � f ( z ) − f ( u ) = ζ − z − f ( ζ ) dm ( ζ ) . ζ − u T Triangle inequality | z − u | � � f ( z ) − f ( u ) � X ≤ T � f � X dm (1 − | u | )(1 − | z |

Example: Rudin Shapiro Martingales k =1 | c k | 2 ≤ 1 . Fix a complex sequence ( c n ) with � ∞ Define recursively: F 1 = G 1 = 1 and for w = ( w n ) ∈ T N F m +1 ( w ) = F m ( w ) + G m ( w ) c m +1 w m +1 , G m +1 ( w ) = G m ( w ) − F m ( w ) c m +1 w m +1 . Pythagoras for ( F m , G m ) and ( G m , − F m ) gives | F m +1 ( w ) | 2 + | G m +1 ( w ) | 2 = (1+ | c m +1 | 2 )( | F m ( w ) | 2 + | G m ( w ) | 2 ) . and repeat m +1 | F m +1 ( w ) | 2 + | G m +1 ( w ) | 2 = (1 + | c k | 2 )2 . � k =1

Rudin Shapiro Martingales II F = ( F n ) a uniformly bounded Hardy martingale n � F n ( w ) = G m ( w ) c m +1 w m +1 m =1 for which the martingale differences reproduce the ( c m ) . E w ( w m ( F n ( w ) − F n − 1 ( w )) = c m +1 E w G m ( w ) = c m +1 . Rudin Shapiro martingales gives the cotype 2 estimate for L 1 /H 1 n � 1 / 2 �� � x m � 2 � E w � w m x m � L 1 /H 1 ≥ c . L 1 /H 1 m =1 when the x m have well separated Fourier spectrum.

The Origins I A. Pelczynski posed famous problems in “Banach Spaces of analytic functions and absolutely summing operators, (1977).” Does H 1 have an unconditional basis? Does there exist a subspace of L 1 /H 1 isomorphic to L 1 ? Does L 1 /H 1 have cotype 2? Are the spaces A ( D n ) and A ( D m ) not isomorphic when n � = m ?

The Origins II Hardy martingales gave rise to the operators by which Maurey proved that H 1 has an unconditional basis; and to the isomorphic invariants by which Bourgain proved the dimension conjecture, that L 1 /H 1 has co- type 2 and that L 1 embeds into L 1 /H 1 . Pisier’s L 1 /H 1 valued Riesz products form a Hardy martingale that is strongly intertwined with Bourgain’s solutions and played an important role for the work of Garling, Tomczak-Jaegermann, W. Davis on Hardy martingale cotype and complex uniformly convex renorm- ings of Banach spaces.

Garling’s Maximal Functions estimate I . For any X valued Hardy martingale F = ( F k ) E (sup � F k � ) ≤ e sup E ( � F k � ) . k ∈ N k ∈ N For any 0 < α ≤ 1 , ( � F k − 1 � α X ) is a non- negative sub- martingale � F k − 1 � α X ≤ E k − 1 ( � F k � α X ) .

Brownian Motion Let Ω denote the Wiener space { z t : t > 0 } denotes complex Brownian Motion started at 0 ∈ D , and define τ = inf { t > 0 : | z t | > 1 } . For f ∈ H 1 ( T , X ) , 0 < α < 1 and 0 < t < τ, � f ( z t ) � α X ≤ E ( � f ( z τ ) � α X |F t ) , and E (sup t<τ � f ( z t ) � X ) ≤ e sup t<τ E ( � f ( z t ) � X ) , where the integration is over the Wiener space Ω .

Garling’s Maximal Functions estimate II . Σ = T k − 1 × Ω , x ∈ T k − 1 , ω ∈ Ω . For any X valued Hardy martingale F = ( F k ) , the max- imal function � � F ∗ k ( x, ω ) = max m ≤ k − 1 � F m ( x ) � X , sup max t<τ � F k ( x, z t ( ω )) � X satisfies E Σ ( F ∗ k ) ≤ e 2 E ( � F k � X ) .

Davies Decomposition I. Let F = ( F k ) n k =1 be an X valued Hardy martingale. With the maximal function estimates, the standard B. Davies decomposition and Doob’s projection we obtain a splitting of F into Hardy martingales F = G + B satisfying � ∆ G k � X ≤ m ≤ k − 1 � F m � X , max and n � E ( � ∆ B k � X ) ≤ C E ( � F � X ) . k =1

Sketch of Proof. Fix x ∈ T k − 1 , v ∈ T . Define f ( v ) = ∆ F k ( x, v ) , λ = m ≤ k − 1 � F m ( x ) � X . max and ρ = inf { t < τ : � f ( z t ) � X > 2 λ } , R k = f ( z ρ ) , S k = f ( z ρ ) − f ( z τ ) . • F ∗ k ( x, ω ) ≤ 4( F ∗ k ( x, ω ) − F ∗ k − 1 ( x, ω )) , ω ∈ A = { ρ < τ } . •� S k � X ≤ 2 F ∗ k ≤ 8( F ∗ k − F ∗ � n k =1 � S k � X ≤ 8 F ∗ k − 1 ) , n . • By choice of the stopping time ρ, � R k � ≤ 2 λ. Doob’s projection generates the analytic functions ∆ B k = E ( S k | z τ = z ) , ∆ G k = E ( R k | z τ = z ) , z ∈ T .

Improved Davies Decomposition (PFXM) A Hardy martingale F = ( F k ) can be decomposed into Hardy martingales as F = G + B such that � ∆ G k � X ≤ C � F k − 1 � X , and ∞ � E ( � ∆ B k � X ) ≤ C E ( � F � X ) . k =1 Lemma If h ∈ H 1 0 ( T , X ) , z ∈ X there exists g ∈ H ∞ 0 ( T , X ) with � g ( ζ ) � X ≤ C 0 � z � X , ζ ∈ T and � z � X + 1 � � T � z + h � X dm. T � h − g � X dm ≤ 8

Sketch of Proof. Fix x ∈ T k − 1 . Put h ( y ) = ∆ F k ( x, y ) and z = F k − 1 ( x ) . Lemma yields a bounded analytic g with � � � z � X +1 / 8 T � h − g � X dm ≤ T � z + h � X dm ; � g ( ζ ) � X ≤ C 0 � z � X . Define ∆ G k ( x, y ) = g ( y ) , ∆ B k ( x, y ) = h ( y ) − g ( y ) . Then � F k − 1 � X + 1 / 8 E k − 1 ( � ∆ B k � X ) ≤ E k − 1 ( � F k � X ) . Integrate and take the sum, � E ( � ∆ B k � X ) ≤ 4 sup E ( � F k � X ) .

Iterating Maurey’s embedding: An Alternative to Decomposing. Given η > 0 and a X valued Hardy martingale ( g k ) there exists a vector valued Hardy martingale ( G k ) , and an increasing sequence of integers m (0) < m (1) < · · · < m ( n ) < . . . so that: ( G k ) , has small previsible increments , � ∆ G k � X ≤ ηβ k − 1 , E (sup k ∈ N β k ) ≤ sup k ∈ N E ( � g k � X ) , and on the subsequence m ( n ) it has almost identitcal L p norms E � g k � 1 ± η E � ∆ g k � p 1 ± η ∼ E � G m ( k ) − G m ( k − 1) � p . ∼ E � G m ( k ) � ,

We Continue with scalar valued Martingales Martingale Norms and Spaces Let G = ( G k ) be an integrable ( F n ) martingale. n E k − 1 | ∆ G k | 2 ) 1 / 2 , � � G � P = E ( k =1 n n | ∆ G k | 2 ) 1 / 2 � � � G � H 1 = E ( and � G � A = E ( | ∆ G k | ) . k =1 k =1 The resulting spaces are related as follows H 1 ⊆ P + A . A ⊆ H 1 . P ⊆ H 1 , Triangle Inequality, Burkholder-Gundy, Davis-Garsia.

Martingale Transforms Let (Ω , ( F n ) , P ) be a filtered probability space. Let w k is complex valued, adapted, and | w k | ≤ 1 . The martingale transforms   n  , � T ( G ) = ℑ w k − 1 · ∆ G k  k =1 is a contraction on H 1 , as well as on P . In general T is unbounded on L 1 .

The Transform Estimate: (PFXM) Let F = ( F k ) be a martingale. Define the transform �� � T ( G ) = ℑ w k − 1 · ∆ G k w k − 1 = F k − 1 / | F k − 1 | . , If G satisfies | ∆ G k | ≤ A | F k − 1 | , then � T ( G ) � P ≤ C � F � 1 / 2 L 1 � F � 1 / 2 H 1 + C � F − G � A , where C = C ( A ) . Proof exploits non-linear telescoping.

Davis-Garsia inequalities for Hardy Martingales PFXM. Every scalar valued Hardy martingale F = ( F k ) has a decomposition into Hardy martingales as F = G + B so that � G � P + � B � A ≤ C � F � L 1 , | ∆ G k | ≤ C | F k − 1 | . Compare with the classical Davis-Garsia inequality. A general martingale F = ( F k ) has a decomposition F = G + B so that � G � P + � B � A ≤ C � F � H 1 , | ∆ G k | ≤ C max m ≤ k − 1 | F m | .