R-Boundedness of Smooth Operator-Valued Functions

Hytönen, Tuomas; Veraar, Mark

doi:10.1007/s00020-009-1663-4

Cited by 26 publications

(29 citation statements)

References 28 publications

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“…[6,Section 6], [7,Proposition 3.3 and Theorem 4.12], [13], [16,Section 3], [19,Section 4], [21,Chapter 2]). However, it seems they are of a different nature and cannot be used to prove Theorems 1.1, 3.10 and Corollary 3.14.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

Gallarati¹,

Lorist²,

Veraar³

2016

Rev. Mat. Iberoam.

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“…The case of R-boundedness has been considered in [16,Theorem 5.1]. The smoothness below is expressed in Besov and Hölder spaces.…”

Section: Smooth Operator-valued Functionsmentioning

confidence: 99%

“…We provide semi-R-bounded versions of results in [16] and prove sharp results for semigroups. Applications to stochastic equations are given in Section 5.…”

Section: Introductionmentioning

confidence: 97%

On semi-R-boundedness and its applications

Veraar

Weis

2010

Journal of Mathematical Analysis and Applications

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Type and cotype Multipliers Stochastic equations Schauder decompositions H ∞ -calculusR-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier's property (α) are required to guaranty the R-boundedness of a relevant set of operators.In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the H ∞ -calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions.

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“…By approximation, it suffices to consider finite sums 1 ≤ j ≤ n. This result can be found in [15], Lemma 3. Lemma 11.5.…”

Section: Bad Boundary Regionsmentioning

confidence: 99%

The vector-valued nonhomogeneous Tb Theorem

Hytönen

2012

International Mathematics Research Notices

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Abstract. The paper gives a Banach space -valued extension of the T b theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure µ, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces L p (µ; X) of functions with values in X-a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all p ∈ (1, ∞) and relies on delicate estimates for the non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.

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R-Boundedness of Smooth Operator-Valued Functions

Cited by 26 publications

References 28 publications

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

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

On semi-R-boundedness and its applications

The vector-valued nonhomogeneous Tb Theorem

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