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

TL;DR

This paper establishes $\, ext{ extlbrack}s extrbrackright$-boundedness of integral operators with operator-valued kernels using extrapolation techniques, enabling advances in maximal regularity for parabolic PDEs with time-dependent generators.

Contribution

It introduces a novel $ ext{ extlbrack}s extrbrackright$-boundedness result for integral operators with operator-valued kernels utilizing Rubio de Francia's extrapolation methods.

Findings

Proves $ ext{ extlbrack}s extrbrackright$-boundedness for a class of integral operators.

Employs extrapolation techniques with weights for the proof.

Lays groundwork for future applications in parabolic PDE regularity.

Abstract

In this paper we prove an $\ell^s$-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.