Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

TL;DR

This paper develops non-smooth atomic decompositions for Besov spaces, enabling the computation of traces on Lipschitz domain boundaries and exploring applications in pointwise multipliers, thus advancing the understanding of function space boundary behavior.

Contribution

It introduces non-smooth atomic decompositions for Besov spaces and applies them to compute traces on Lipschitz domain boundaries without restrictions on parameters.

Findings

Derived non-smooth atomic decompositions for Besov spaces

Computed traces of Besov spaces on Lipschitz domain boundaries

Explored applications to pointwise multipliers

Abstract

We provide non-smooth atomic decompositions for Besov spaces $\Bd(\rn)$, $s>0$, $0<p,q\leq \infty$, defined via differences. The results are used to compute the trace of Besov spaces on the boundary $\Gamma$ of bounded Lipschitz domains $\Omega$ with smoothness $s$ restricted to $0<s<1$ and no further restrictions on the parameters $p,q$. We conclude with some more applications in terms of pointwise multipliers.