Complex interpolation of Besov-type spaces on domains

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ ($d\ge 2$) be a bounded Lipschitz domain. In this article, the author mainly studies complex interpolation of Besov-type spaces on the domain $\Omega$, namely, we investigate the interpolation

under certain conditions on the parameters, where $B_{p,q}^{\diamond s,\tau }(\Omega )$ denotes the so-called diamond space associated with the Besov-type space. To this end, we first establish the equivalent characterization of the diamond space $B_{p,q}^{\diamond s,\tau }({\mathbb{R}}^d)$ in terms of Littlewood–Paley decomposition and differences. Via some examples, we also show that this interpolation result does not hold under some other assumptions on the parameters or when $\Omega ={\mathbb{R}}^d$.