Singular stochastic integral operators Emiel Lorist Delft - PowerPoint PPT Presentation

Singular stochastic integral operators Emiel Lorist Delft University of Technology, The Netherlands B¸ edlewo, Poland May 21, 2019 Joint work with Mark Veraar

Overview 1 Motivation: SPDEs 2 Stochastic singular integral operators 3 Stochastic Calder´ on–Zygmund theory 4 Sparse domination and weighted results 1 / 12

Stochastic abstract Cauchy problem � d u + Au d t = G d W on R + u (0) = 0 • − A the generator of an analytic semigroup e − tA on X • For example A = − ∆ • X a UMD Banach space • For example X = L q for q ∈ (1 , ∞ ) • W a standard Brownian motion • G ∈ L p F ( R + × Ω; X ) for some p ∈ (1 , ∞ ) The mild solution u is given by the variation of constants formula � t e − ( t − s ) A G ( s ) d W ( s ) , u ( t ) = t ∈ R + 0 1 / 12

Stochastic abstract Cauchy problem � d u + Au d t = G d W on R + u (0) = 0 • − A the generator of an analytic semigroup e − tA on X • For example A = − ∆ • X a UMD Banach space • For example X = L q for q ∈ (1 , ∞ ) • W a standard Brownian motion • G ∈ L p F ( R + × Ω; X ) for some p ∈ (1 , ∞ ) The mild solution u is given by the variation of constants formula � t e − ( t − s ) A G ( s ) d W ( s ) , u ( t ) = t ∈ R + 0 1 / 12

Stochastic maximal regularity Definition We say that A has stochastic maximal L p -regularity if 1 2 u ∈ L p ( R + × Ω; X ) A or in other words, if S : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) given by � t 1 2 e − ( t − s ) A G ( s ) d W ( s ) , SG ( t ) := A t ∈ R + 0 is a bounded operator. Previously studied by: • Da Prato et al. for p = 2 and X Hilbert • Krylov et al. for A = − ∆ and X = L q with 2 ≤ q ≤ p < ∞ • van Neerven–Veraar–Weis for A with bounded H ∞ -calculus for p ∈ (2 , ∞ ) and X = L q with q ∈ [2 , ∞ ) 2 / 12

Stochastic maximal regularity Definition We say that A has stochastic maximal L p -regularity if 1 2 u ∈ L p ( R + × Ω; X ) A or in other words, if S : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) given by � t 1 2 e − ( t − s ) A G ( s ) d W ( s ) , SG ( t ) := A t ∈ R + 0 is a bounded operator. Previously studied by: • Da Prato et al. for p = 2 and X Hilbert • Krylov et al. for A = − ∆ and X = L q with 2 ≤ q ≤ p < ∞ • van Neerven–Veraar–Weis for A with bounded H ∞ -calculus for p ∈ (2 , ∞ ) and X = L q with q ∈ [2 , ∞ ) 2 / 12

Stochastic singular integral operators Goal Study the p -independence of the L p -boundedness of � ∞ S K G ( t ) := K ( t , s ) G ( s ) d W ( s ) , t ∈ R + 0 where K : R + × R + → L ( X ) is a singular kernel. Of particular interest is 2 e ( t − s ) A 1 t > s 1 K ( t , s ) = A • We call S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) a singular stochastic integral operator • Deterministic case studied through Calder´ on–Zygmund theory • No Calder´ on–Zygmund theory yet in the stochastic setting Topic of this talk! 3 / 12

Stochastic singular integral operators Goal Study the p -independence of the L p -boundedness of � ∞ S K G ( t ) := K ( t , s ) G ( s ) d W ( s ) , t ∈ R + 0 where K : R + × R + → L ( X ) is a singular kernel. Of particular interest is 2 e ( t − s ) A 1 t > s 1 K ( t , s ) = A • We call S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) a singular stochastic integral operator • Deterministic case studied through Calder´ on–Zygmund theory • No Calder´ on–Zygmund theory yet in the stochastic setting Topic of this talk! 3 / 12

Stochastic singular integral operators Goal Study the p -independence of the L p -boundedness of � ∞ S K G ( t ) := K ( t , s ) G ( s ) d W ( s ) , t ∈ R + 0 where K : R + × R + → L ( X ) is a singular kernel. Of particular interest is 2 e ( t − s ) A 1 t > s 1 K ( t , s ) = A • We call S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) a singular stochastic integral operator • Deterministic case studied through Calder´ on–Zygmund theory • No Calder´ on–Zygmund theory yet in the stochastic setting Topic of this talk! 3 / 12

Stochastic Calder´ on–Zygmund theory Stochastic Calder´ on–Zygmund theorem (L., Veraar ’19) Let X be a UMD Banach space with type 2, p 0 ∈ [2 , ∞ ) and let K : R + × R + → L ( X ) satisfy the 2-H¨ ormander condition. If S K : L p 0 F ( R + × Ω; X ) → L p 0 ( R + × Ω; X ) , is bounded, then S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) is bounded for all p ∈ (2 , ∞ ). 4 / 12

Stochastic Calder´ on–Zygmund theory Stochastic Calder´ on–Zygmund theorem (L., Veraar ’19) Let X be a UMD Banach space with type 2, p 0 ∈ [2 , ∞ ) and let K : R + × R + → L ( X ) satisfy the 2-H¨ ormander condition. If S K : L p 0 F ( R + × Ω; X ) → L p 0 ( R + × Ω; X ) , is bounded, then S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) is bounded for all p ∈ (2 , ∞ ). • X has type 2 if for any x 1 , · · · , x n ∈ X n � n � � � x k � 2 � 1 / 2 � � � � ε k x k L 2 (Ω; X ) � � � k =1 k =1 where ( ε k ) n k =1 is a Rademacher sequence. • L q for q ∈ [2 , ∞ ) is a UMD Banach space with type 2 4 / 12

Stochastic Calder´ on–Zygmund theory Stochastic Calder´ on–Zygmund theorem (L., Veraar ’19) Let X be a UMD Banach space with type 2, p 0 ∈ [2 , ∞ ) and let K : R + × R + → L ( X ) satisfy the 2-H¨ ormander condition. If S K : L p 0 F ( R + × Ω; X ) → L p 0 ( R + × Ω; X ) , is bounded, then S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) is bounded for all p ∈ (2 , ∞ ). • X has type 2 if for any x 1 , · · · , x n ∈ X n � n � � � x k � 2 � 1 / 2 � � � � ε k x k L 2 (Ω; X ) � � � k =1 k =1 where ( ε k ) n k =1 is a Rademacher sequence. • L q for q ∈ [2 , ∞ ) is a UMD Banach space with type 2 4 / 12

Stochastic Calder´ on–Zygmund theory Stochastic Calder´ on–Zygmund theorem (L., Veraar ’19) Let X be a UMD Banach space with type 2, p 0 ∈ [2 , ∞ ) and let K : R + × R + → L ( X ) satisfy the 2-H¨ ormander condition. If S K : L p 0 F ( R + × Ω; X ) → L p 0 ( R + × Ω; X ) , is bounded, then S K : L p F ( R + × Ω; X ) → L p ( R + × Ω; X ) is bounded for all p ∈ (2 , ∞ ). • If K : R + × R + → L ( X ) satisfies �� � ∂ � ∂ � C � � � max ∂ t K ( t , s ) � , ∂ s K ( t , s ) ≤ | t − s | 3 / 2 , t � = s � then it satisfies the 2-H¨ ormander condition. 4 / 12

Comments on the proof • Extrapolation upwards from p 0 to p ∈ ( p 0 , ∞ ) • Prove BMO-endpoint and interpolate • Extrapolation downwards from p 0 to p ∈ (2 , p 0 ) • Use L 2 -Calder´ on–Zygmund decomposition • Cannot exploit mean zero of “bad” part of decomposition • Using ideas from Duong–McIntosh ’99 Some differences with classical Calder´ on–Zygmund theory: • For X = L q by the Itˆ o isomorphism it is equivalent to consider the operators �� t � 1 / 2 2 d s T K f ( t ) := | K ( t , s ) f ( s ) | , t ∈ R + 0 � �� � ∈ L q for f ∈ L p ( R + ; L q ) • The integrals converge absolutely, cancellation takes the form �� � � 1 / 2 � | K ( t , s ) x | 2 d s t ∈ R + , x ∈ L q � � � s �→ L q � � x � L q , � R + • If X = C and K ( t , s ) = k ( t − s ), then S K is bounded if and only if k ∈ L 2 ( R + ) (analog of k ∈ L 1 ( R + ) in deterministic setting) 5 / 12

Comments on the proof • Extrapolation upwards from p 0 to p ∈ ( p 0 , ∞ ) • Prove BMO-endpoint and interpolate • Extrapolation downwards from p 0 to p ∈ (2 , p 0 ) • Use L 2 -Calder´ on–Zygmund decomposition • Cannot exploit mean zero of “bad” part of decomposition • Using ideas from Duong–McIntosh ’99 Some differences with classical Calder´ on–Zygmund theory: • For X = L q by the Itˆ o isomorphism it is equivalent to consider the operators �� t � 1 / 2 2 d s T K f ( t ) := | K ( t , s ) f ( s ) | , t ∈ R + 0 � �� � ∈ L q for f ∈ L p ( R + ; L q ) • The integrals converge absolutely, cancellation takes the form �� � � 1 / 2 � | K ( t , s ) x | 2 d s t ∈ R + , x ∈ L q � � � s �→ L q � � x � L q , � R + • If X = C and K ( t , s ) = k ( t − s ), then S K is bounded if and only if k ∈ L 2 ( R + ) (analog of k ∈ L 1 ( R + ) in deterministic setting) 5 / 12

Comments on the proof • Extrapolation upwards from p 0 to p ∈ ( p 0 , ∞ ) • Prove BMO-endpoint and interpolate • Extrapolation downwards from p 0 to p ∈ (2 , p 0 ) • Use L 2 -Calder´ on–Zygmund decomposition • Cannot exploit mean zero of “bad” part of decomposition • Using ideas from Duong–McIntosh ’99 Some differences with classical Calder´ on–Zygmund theory: • For X = L q by the Itˆ o isomorphism it is equivalent to consider the operators �� t � 1 / 2 2 d s T K f ( t ) := | K ( t , s ) f ( s ) | , t ∈ R + 0 � �� � ∈ L q for f ∈ L p ( R + ; L q ) • The integrals converge absolutely, cancellation takes the form �� � � 1 / 2 � | K ( t , s ) x | 2 d s t ∈ R + , x ∈ L q � � � s �→ L q � � x � L q , � R + • If X = C and K ( t , s ) = k ( t − s ), then S K is bounded if and only if k ∈ L 2 ( R + ) (analog of k ∈ L 1 ( R + ) in deterministic setting) 5 / 12