Continuous embeddings of Besov-Morrey function spaces
Let \(\mathcal S(\mathbb R^d)\) and \(\mathcal S'(\mathbb R^d)\) be the space of Schwartz functions and its dual space, respectively. Let \(\varphi_0:=\varphi\in\mathcal S(\mathbb R^d)\) satisfy that \(\text{supp}\, \varphi\subset\{y\in\mathbb R^d:|y|<2\}\) and \(\varphi(x)=1\) if \(|x|\leq 1\), and, for each \(j\in\mathbb N\), let \(\varphi_j(x):=\varphi(2^{-j}x)-\varphi(2^{-j+1}x)\). Given any \(f\in\mathcal S'(\mathbb R^d)\), denote by \(\mathcal Ff\) and \(\mathcal F^{-1}f\) its Fourier transform and its inverse Fourier transform, respectively. Let \(0<q\leq p<\infty\), \(r\in(0,\infty]\), \(s\in\mathbb R\). The Besov-Morrey space \(MB_{p,q}^{s,r}(\mathbb R^d)\) is defined to be the set of all \(f\in\mathcal S'(\mathbb R^d)\) such that \[ \|f|MB_{p,q}^{s,r}(\mathbb R^d)\| :=\biggl(\sum_{j=0}^\infty 2^{jsr} \|\mathcal F^{-1}(\varphi_j\mathcal Ff)|M_{p,q}(\mathbb R^d)\|^r \biggl)^\frac1r \] is finite with the usual modification when \(r=\infty\), where the norm \(\|f|M_{p,q}(\mathbb R^d)\|\) for a locally \(q\)-integrable function \(f\) is defined by \[ \|f|M_{p,q}(\mathbb R^d)\|:=\sup_{x\in\mathbb R^d,\,R>0} R^{\frac dp-\frac dq}\biggl(\int_{\{y\in\mathbb R^d:\;|y-x|<R\}}|f(y)|^q\,dy \biggl)^\frac 1q \] In this paper, the authors first show the following. Let \(s_1,s_2\in\mathbb R\), \(r_1,r_2\in(0,\infty]\) and \(0<q_i\leq p_i<\infty\) for \(i\in\{1,\;2\}\). Then there is a continuous embedding \(MB_{p_1,q_1}^{s_1,r_1}(\mathbb R^d)\hookrightarrow MB_{p_2,q_2}^{s_2,r_2}(\mathbb R^d)\) if and only if {\parindent=1.0cm\begin{itemize}\item[(i)] \(p_1\leq p_2\), \(\frac{q_2}{p_2}\leq \frac{q_1}{p_1}\) and \item[(ii)] \(s_1-\frac d{p_1}>s_2-\frac d{p_2}\), or \(s_1-\frac d{p_1}=s_2-\frac d{p_2}\) and \(r_1\leq r_2\). \end{itemize}} Given \(r\in(0,\infty]\), \(p\in(0,\infty)\), \(s\in\mathbb R\) and a Muckenhoupt \(\mathcal A_\infty\) weight \(\omega\), let \(B_{p,q}^s(\mathbb R^d,\omega)\) be the weighted Besov space. In this paper, the authors also obtain necessary and sufficient conditions for the embedding \(B_{p_1,r_1}^{s_1}(\mathbb R^d,\omega)\hookrightarrow MB_{p_2,q_2}^{s_2,r_2}(\mathbb R^d)\), where \(p_1\in(0,\infty)\), \(r_1,r_2\in(0,\infty]\), \(s_1>s_2\), and \(0<q_2\leq p_2<\infty\).
scientific article; zbMATH DE number 6102791
scientific article