Weighted norms inequalities for a maximal operator in some subspace of amalgams

TL;DR

This paper establishes weighted norm inequalities for a maximal fractional operator and fractional integral within specific amalgam and Lebesgue spaces on spaces of homogeneous type, using Orlicz norm conditions.

Contribution

It introduces new weighted inequalities for maximal and fractional integral operators in amalgam spaces on spaces of homogeneous type, expanding the theoretical framework.

Findings

Weighted inequalities are proved for the maximal fractional operator.

Inequalities are established between amalgam and Lebesgue spaces.

Conditions on weights are expressed via Orlicz norms.

Abstract

We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta}$ of Hardy-Littlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha}(X,d,\mu)$ spaces (which are super spaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$, and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.