Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

TL;DR

This paper proves two-weight boundedness for local fractional maximal operators and integrals on Gaussian measure spaces, expanding understanding of weighted inequalities in this setting.

Contribution

It introduces new two-weight weak and strong type estimates for local fractional operators on Gaussian spaces, using radialization and dyadic methods.

Findings

Established two-weight weak-type estimates for local fractional maximal operators.

Proved two-weight strong-type boundedness for local fractional integrals.

Extended weighted inequality theory to Gaussian measure spaces.

Abstract

In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a} A_{p,q,\alpha}^{b'}$, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.