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Zeitschrift für Analysis und ihre Anwendungen

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Volume 40, Issue 3, 2021, pp. 313–347
DOI: 10.4171/ZAA/1687

Published online: 2021-06-24

Complex interpolation of Besov-type spaces on domains

Ciqiang Zhuo[1]

(1) Hunan Normal University, Changsha, China

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 $\Omega=\mathbb{R}^d$.

Keywords: Besov-type space, diamond space, extension operator, Lipschitz domains, complex interpolation

Zhuo Ciqiang: Complex interpolation of Besov-type spaces on domains. Z. Anal. Anwend. 40 (2021), 313-347. doi: 10.4171/ZAA/1687