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

TL;DR

The paper addresses boundedness properties of the maximal operator for the inverse Ornstein--Uhlenbeck semigroup in a nonsymmetric inverse Gaussian setting, where the underlying measure $d\gamma_{-\infty}$ grows superexponentially. It develops explicit kernel representations for the inverse Mehler semigroup, derives a Kolmogorov-type formula, and proves $L^p$-boundedness for $1<p\le\infty$ along with weak type $(1,1)$, distinguishing small-time (forbidden-zones) and large-time (new geometric tools) analyses. The main contributions include a complete weak-type and strong-type theory for the maximal operator in this setting, extended from the symmetric inverse Gaussian case, and a sharp logarithmic refinement for the global large-time part. This work advances harmonic-analysis techniques for nonsymmetric semigroups on spaces with non-doubling, superexponential volume growth, laying groundwork for a broader inverse-Gaussian theory.

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.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Theorem 3.1
• Theorem 3.2