Maximal $L^1$-regularity for parabolic initial-boundary value problems with inhomogeneous data

TL;DR

This paper establishes sharp endpoint maximal $L^1$-regularity results for parabolic initial-boundary value problems with inhomogeneous data, using harmonic analysis techniques in Besov and Lizorkin-Triebel spaces.

Contribution

It provides the first sharp endpoint maximal $L^1$-regularity results for such problems in the non-UMD Banach space setting, employing harmonic analysis methods.

Findings

Sharp trace estimates achieved

Maximal $L^1$-regularity established for inhomogeneous data

Method applicable to boundary potential analysis

Abstract

End-point maximal $L^1$-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal $L^1$-regularity for initial-boundary value problems is established in time end-point case upon the homogeneous Besov space $\dot B_{p,1}^s({\mathbb R}^n_+)$ with $1< p< \infty$ and $-1+1/p<s\le 0$ as well as optimal trace estimates. The main estimates obtained here are sharp in the sense of trace estimates and it is not available by known theory on the class of UMD Banach spaces. We utilize a method of harmonic analysis, in particular, the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity in the Besov and the Lizorkin-Triebel spaces.