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

中文翻译：

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域上 Besov 型空间的复插值

令 $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) 是一个有界 Lipschitz 域。在本文中，作者主要研究域$\Omega$上Besov型空间的复杂插值，即我们研究插值$$ [B_{p_0,q_0}^{s_0,\tau_0}(\Omega), B_{p_1,q_1}^{s_1,\tau_1}(\Omega)]_\Theta = B_{p,q}^{\diamond s,\tau}(\Omega) $$ 在一定条件下对参数，其中 $B_{p,q}^{\diamond s,\tau}(\Omega)$ 表示与 Besov 型空间相关的所谓菱形空间。为此，我们首先根据 Littlewood–Paley 分解和差异建立菱形空间 $B_{p,q}^{\diamond s,\tau}(\mathbb{R}^d)$ 的等效表征。通过一些例子，我们还表明这个插值结果在参数的一些其他假设下或当 $\Omega=\mathbb{R}^d$ 时不成立。