We study embeddings of Besov–Morrey spaces \({{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\) and of Triebel–Lizorkin–Morrey spaces \({{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\) in the limiting cases when the smoothness s equals \(s_o=d\max (1/u-p/u,0)\) or \(s_{\infty }=d/u\), which is related to the embeddings in \(L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})\) or in \(L_{\infty }({{{\mathbb {R}}}^d})\), respectively. When \(s=s_o\) we characterise the embeddings in \(L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})\) and when \(s=s_{\infty }\) we obtain embeddings into Orlicz–Morrey spaces of exponential type and into generalised Morrey spaces.

中文翻译：

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具有临界光滑度的Morrey空间的一些嵌入

我们研究Besov–Morrey空间\（{{\ mathcal {N}}} ^ {s} _ {u，p，q}（{{{\ mathbb {R}}} ^ d}）\）和TRIEBEL-Lizorkin空间-Morrey空间\（{{\ mathcal {E}}} ^ {S} _ {U，p，q}（{{{\ mathbb {R}}} ^ d}）\）在极限情况当平滑度s等于\（s_o = d \ max（1 / up / u，0）\）或\（s _ {\ infty} = d / u \）时，这与\（L_1 ^ { \ mathrm {loc}}（{{{\ mathbb {R}}} ^ d}）\）或在\（L _ {\ infty}（{{{\ mathbb {R}}} ^ d}）\）中，分别。当\（s = s_o \）表征\（L_1 ^ {\ mathrm {loc}}（{{{\ mathbb {R}}} ^ d}）\）中的嵌入特征时，以及\（s = s _ {\狡猾} \） 我们获得嵌入到指数类型的Orlicz-Morrey空间和广义Morrey空间中的嵌入。