Complex interpolation of Besov-type spaces on domains
Summary: Let \(\Omega\subset\mathbb{R}^d\) \((d\geq 2)\) 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 \[ [B_{p_0,q_0}^{s_0,\tau_0}(\Omega),B_{p_1,q_1}^{s_1,\tau_1}(\Omega)]_\Theta=B_{p,q}^{\diamond s,\tau}(\Omega) \] 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 \(\Omega=\mathbb{R}^d\).
scientific article; zbMATH DE number 7372684
scientific article