Vector-Valued Extensions of Operators on Martingales

Tozoni, Sérgio Antonio

doi:10.1006/jmaa.1996.0246

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…Proposition 3.6 is an extension of [24,Theorem 4.7] to the weighted setting. The result remains true for UMD Banach function spaces X and can be proved using the same techniques of [24] where one needs to apply the weighted extension of [29,Theorem 3] which is obtained in [32]. The endpoint case s = 1 of Proposition 3.6 plays a crucial rôle in the proof of Theorems 1.1 and 3.10.…”

Section: It Follows From Proposition A1 That In This Waymentioning

confidence: 87%

“…Using deep connections between harmonic analysis with weights and martingale theory, Theorem 2.6 was obtained in [2] and [29,Theorem 3] for UMD Banach function spaces in the case w = 1. It has been extended to the weighted setting in [32]. As our main result Theorem 3.10 is formulated for iterated L q (Ω)-spaces we prefer the above more elementary treatment.…”

Section: 2mentioning

confidence: 99%

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On the $\ell^s$-boundedness of a family of integral operators

Gallarati¹,

Lorist²,

Veraar³

2016

Rev. Mat. Iberoam.

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Abstract. In this paper we prove an ℓ s -boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal L p -regularity for parabolic problems with time-dependent generator is developed. IntroductionIn the influential work [34,35], Weis has found a characterization of maximal L p -regularity in terms of R-sectoriality, which stands for R-boundedness of a family of resolvents on a sector. The definition of R-boundedness is given in Definition 3.15. It is a random boundedness condition on a family of operators which is a strengthening of uniform boundedness. Maximal regularity of solution to PDEs is important to know as it provides a tool to solve nonlinear PDEs using linearization techniques (see [4,23,25]). An overview on recent developments on maximal L pregularity can be found in [7,21]. Maximal L p -regularity means that for all f ∈ L p (0, T ; X), where X is a Banach space, the solution u of the evolution problemhas the "maximal" regularity in the sense that u ′ , Au are both in L p (0, T ; X). Using a mild formulation one sees that to prove maximal L p -regularity one needs to bound a singular integral with operator-valued kernel Ae (t−s)A . In [11] the first and third author have developed a new approach to maximal L pregularity for the case that the operator A in (1.1) depends on time in a measurable way. In this new approach R-boundedness plays a central rôle again. Namely, the R-boundedness of the family of integral operators

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Section: It Follows From Proposition A1 That In This Waymentioning

confidence: 87%

Section: 2mentioning

confidence: 99%

On the $\ell^s$-boundedness of a family of integral operators

Gallarati¹,

Lorist²,

Veraar³

2016

Rev. Mat. Iberoam.

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“…Proof. This can be easily derived from [29,Theorem 2.6], which is a weighted version of the special case of the L p -boundedness of the Banach lattice version of the Hardy-Littlewood maximal function [8,30,58,64] for mixed-norm spaces (also see [29, Remark 2.7]).…”

mentioning

confidence: 99%

Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

Lindemulder

2019

J. Evol. Equ.

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In this paper we establish weighted L q -L p -maximal regularity for linear vectorvalued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin and Besov type, whose trace theory is also subject of study.

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Section: Convolution Operatorsmentioning

confidence: 89%