$\mathcal {R}_{s}$ -Sectorial Operators and Generalized Triebel-Lizorkin Spaces

Kunstmann, Peer Christian; Ullmann, Alexander

doi:10.1007/s00041-013-9307-0

Cited by 27 publications

(44 citation statements)

References 29 publications

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“…Although ℓ s -boundedness is used implicitly in the literature for operators on L p -spaces, on Banach functions spaces it was introduced in [34] under the name R s -boundedness. An extensive study can be found in [20,33]. …”

Section: Resultsmentioning

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: Resultsmentioning

confidence: 99%

“…(1) follows from Calderón's theory of complex interpolation of vector-valued function spaces (see [3] and [20,Proposition 2.14] …”

Section: Definition 31 Let X and Y Be Banach Lattices And Let S ∈ [mentioning

confidence: 99%

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

Gallarati¹,

Lorist²,

Veraar³

2016

Rev. Mat. Iberoam.

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show abstract

“…We can now formulate the first main result of this article, which reduces the proof of

R_{s}

‐boundedness of the

H^{\infty}

‐functional calculus to

R_{s}

‐boundedness of single operators. A version for more general function spaces X will be contained in . For convenience of the reader, we present a detailed proof in Section .…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 99%

“…Moreover and more importantly,

R_{s}

‐boundedness of the

H^{\infty}

‐functional calculus has turned out to be a key assumption in the study of generalized Triebel‐Lizorkin spaces associated to a sectorial operator, cf. , . Actually, these spaces are defined by norms that can be viewed as counterparts to certain square function expressions which are essential in the theory of the