Embedding theorems for Besov–Morrey spaces

The purpose of this paper is to investigate the embedding theorems for Besov–Morrey spaces using the equivalence theorem for the K-functional and the modulus of continuity on Morrey spaces. First, we obtain some theorems in ball Banach function space and then focus on Morrey spaces. The Marchaud’s inequality on Morrey spaces and a specific case of embedding theorems for Sobolev–Morrey spaces are crucial tools. We show that the Besov–Morrey space \(B_{\alpha , a}^{p,\lambda }(\mathbb {R}^{n})\) $B_{\alpha ,a}^{p,\lambda }({\mathbb{R}}^n)$ is continuously embedded in the Morrey-Lorentz space \(\mathcal {M}_{q,p}^{\lambda }(\mathbb {R}^{n})\) ${\mathcal{M}}_{q,p}^{\lambda }({\mathbb{R}}^n)$ , and also, for any \(\alpha , \beta > 0\) $\alpha ,\beta >0$ and \(1< a\le p < q \le \infty \) $1<a\le p<q\le \infty$ , the Besov–Morrey space \(B_{\alpha + \beta , a}^{p,\lambda }(\mathbb {R}^{n})\) $B_{\alpha +\beta ,a}^{p,\lambda }({\mathbb{R}}^n)$ is continuously embedded in the Besov–Morrey space \(B_{\beta , a}^{q,\lambda }(\mathbb {R}^{n})\) $B_{\beta ,a}^{q,\lambda }({\mathbb{R}}^n)$ .