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Keywords:
fractional integral operator; Hausdorff capacity; two-weight inequality; trace inequality
fractional integral operator; Hausdorff capacity; two-weight inequality; trace inequality
Summary:
The purpose of this paper is to establish the trace inequality for fractional operators involving Hausdorff capacity and to characterize the Borel measures $\mu $ for which these inequalities hold. We further investigate two-weight norm inequalities for fractional integrals and fractional maximal operators on Choquet spaces with respect to Hausdorff capacities. As an application of the trace inequality, we derive an embedding inequality for Sobolev spaces with respect to the Hausdorff capacity into $L^{q}(\mu )$. This result, in turn, leads to a study of the Hardy inequality. In addition, we establish a generalized Stein-Weiss inequality through the two-weight inequalities.
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