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We establish the multilinear inequality \(\displaystyle {} \big \| g \, {\mathcal {I}}_{\alpha } (\vec {f}) \big \|_{ L^{q}_r } \leqslant C \big \| g \big \|_{ L^q_{\ell }} \prod _{j=1}^m \big \| f_j\big \|_{L^{p_j}_{s_j}}, \) where \(\displaystyle {\mathcal {I}}_{\alpha }(\vec {f}\, ) (x) = \int \limits _{({\mathbb {R}}^n)^m}\frac {f_1(y_1)\cdots f_m(y_m)}{(|x-y_1|+ \cdots + |x-y_m|)^{mn-\alpha }} d\vec {y}, \;\; x\in {\mathbb {R}}^n, \) with \(0<\alpha < mn\) , where \(L^{q}_r\) , \(L^{q}_{\ell }\) , \(L^{p_j}_{s_j}\) , \(j=1,\ldots , m\) , are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp in the sense that the Morrey norm in \(\| g \|_{ L^q_{\ell }}\) can not be replaced by the smaller one.

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Olsen Inequality

  • 章节

  • 发表日期：

    2025年

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  • Alexander Meskhi,
  • Humberto Rafeiro,
  • Stefan Samko

摘要

We establish the multilinear inequality \(\displaystyle {} \big \| g \, {\mathcal {I}}_{\alpha } (\vec {f}) \big \|_{ L^{q}_r } \leqslant C \big \| g \big \|_{ L^q_{\ell }} \prod _{j=1}^m \big \| f_j\big \|_{L^{p_j}_{s_j}}, \) where \(\displaystyle {\mathcal {I}}_{\alpha }(\vec {f}\, ) (x) = \int \limits _{({\mathbb {R}}^n)^m}\frac {f_1(y_1)\cdots f_m(y_m)}{(|x-y_1|+ \cdots + |x-y_m|)^{mn-\alpha }} d\vec {y}, \;\; x\in {\mathbb {R}}^n, \) with \(0<\alpha < mn\) , where \(L^{q}_r\) , \(L^{q}_{\ell }\) , \(L^{p_j}_{s_j}\) , \(j=1,\ldots , m\) , are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp in the sense that the Morrey norm in \(\| g \|_{ L^q_{\ell }}\) can not be replaced by the smaller one.