Boundedness properties of the maximal operator in a nonsymmetric inverse Gaussian setting

TL;DR

This paper studies the boundedness of a maximal operator linked to a nonsymmetric inverse Gaussian Ornstein-Uhlenbeck semigroup, establishing its boundedness on L^p spaces and weak type (1,1) properties with novel proof techniques.

Contribution

It introduces a generalized inverse Gaussian framework and proves boundedness and weak type estimates for the associated maximal operator, extending Gaussian methods to a new setting.

Findings

Bounded on L^p for p in (1,∞]

Weak type (1,1) with respect to the relevant measure

New tools required for large time analysis

Abstract

We introduce a generalized inverse Gaussian setting and consider the maximal operator associated with the natural analogue of a nonsymmetric Ornstein--Uhlenbeck semigroup. We prove that it is bounded on $L^{p}$ when $p\in (1,\infty]$ and that it is of weak type $(1,1)$, with respect to the relevant measure. For small values of the time parameter $t$, the proof hinges on the "forbidden zones" method previously introduced in the Gaussian context. But for large times the proof requires new tools.