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

Abstract

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

Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

“…The above proposition and its corollary remain valid for q = ∞. In this case the norm estimate corresponding to (29) can be obtained in a similar way, from which the unique extendability to a bounded linear operator (29) can be derived via the Fatou property, (10) and the case q = 1. The remaining statements can be established in the same way as for the case q < ∞.…”

“…These two inequalities imply (29) and (30), respectively. Let us finally show that the extension operator ext from Lemma 4.5 (withd = d [i] and a = a [i] , modified in the obvious way to the i-th multidimensional coordinate) restricts to a coretraction for tr [d ;i] .…”

“…Proof of Proposition 4.10. Let us first establish (29) and (30). Thanks to the Sobolev embedding Proposition 5.1 we may restrict ourselves to the case γ ∈ (−1,…”

“…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]).…”

The ℓ s-Boundedness of a Family of Integral Operators on UMD Banach Function Spaces

“…However none of these are applicable to the operator family of {I k : k ∈ K}. Therefore in [12] Gallarati, Veraar and the author show a sufficient condition for the R-boundedness of {I k : k ∈ K} on L p (R; X) in the special case where X = L q . This is done through the notion of ℓ s -boundedness, which states that for all finite sequences…”