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We establish necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality \(\displaystyle {} \left \| I_{\alpha } f \,\right \|_{ L^{p}_v( \mathbb {R}^n) } \leqslant C \left \| f\,\right \|_{ L^{p,1}(\mathbb {R}^n) }, \;\; f\in L^{p,1}(\mathbb {R}^n), \) for the Riesz potential \(I_{\alpha }\) , where \(L^{p,1}(\mathbb {R}^n)\) is the Lorentz space. The condition on \(\,v\) is of D. Adams type. The same problem is studied for potentials defined on spaces of homogeneous type. We also find necessary and sufficient conditions on a weight v for the boundedness of multilinear Riemann–Liouville operators from \(\prod _{j=1}^m L^{p_j}\) to \(L^q_v\) .

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Trace Inequalities for Fractional Integrals

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  • 发表日期：

    2025年

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  • Alexander Meskhi,
  • Humberto Rafeiro,
  • Stefan Samko

摘要

We establish necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality \(\displaystyle {} \left \| I_{\alpha } f \,\right \|_{ L^{p}_v( \mathbb {R}^n) } \leqslant C \left \| f\,\right \|_{ L^{p,1}(\mathbb {R}^n) }, \;\; f\in L^{p,1}(\mathbb {R}^n), \) for the Riesz potential \(I_{\alpha }\) , where \(L^{p,1}(\mathbb {R}^n)\) is the Lorentz space. The condition on \(\,v\) is of D. Adams type. The same problem is studied for potentials defined on spaces of homogeneous type. We also find necessary and sufficient conditions on a weight v for the boundedness of multilinear Riemann–Liouville operators from \(\prod _{j=1}^m L^{p_j}\) to \(L^q_v\) .