We prove that the Riesz potential operator \(I_\alpha \) of order \(\alpha \) embeds from Musielak–Orlicz–Morrey spaces \(L^{\Phi ,\nu }(\mathbf{R}^N)\) of the double phase functionals \(\Phi (x,t)= t^{p} + (b(x) t)^{q}\) to Campanato–Morrey spaces, where \(1<p<q\) and \(b(\cdot )\) is non-negative, bounded and Hölder continuous of order \(\theta \in (0,1]\). We also study the continuity of Riesz potentials \(I_\alpha f\) of functions in \(L^{\Phi ,\nu }(\mathbf{R}^N)\) and show that \(I_\alpha \) embeds from \(L^{\Phi ,\nu }(\mathbf{R}^N)\) to vanishing Campanato–Morrey spaces.

中文翻译：

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双重功能的Campanato-Morrey空间

我们证明了阶\（\ alpha \）的Riesz势算子\（I_ \ alpha \）从Musielak–Orlicz–Morrey空间\（L ^ {\ Phi，\ nu}（\ mathbf {R} ^ N）嵌入\）双相函数\（\ Phi（x，t）= t ^ {p} +（b（x）t）^ {q} \）到Campanato-Morrey空间，其中\（1 <p <q \）和\（b（\ cdot）\）是非负的，有界且Hölder连续为\（\ theta \ in（0,1] \）。我们还研究了Riesz势\（I_ \ alpha \（L ^ {\ Phi，\ nu}（\ mathbf {R} ^ N）\）中的函数f \）并显示\（I_ \ alpha \）从\（L ^ {\ Phi，\ nu }（\ mathbf {R} ^ N）\） 消失的坎帕纳托-莫里空间。