Weak type (1,1) jump inequalities in a nonsymmetric Gaussian setting

TL;DR

This paper establishes a weak type (1,1) jump inequality for a nonsymmetric Ornstein--Uhlenbeck semigroup in a nondoubling measure space, refining previous variation seminorm results.

Contribution

It provides the first weak type (1,1) jump inequality for a nonsymmetric semigroup in a nondoubling setting, extending endpoint analysis.

Findings

Proves weak type (1,1) inequality for jump quasi-seminorm of order 2.

Demonstrates the inequality in a nondoubling measure space.

Refines previous results on variation seminorms for semigroups.

Abstract

We prove that the jump quasi-seminorm of order $\varrho= 2$ for a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$ defines an operator of weak type $(1,1)$ with respect to the invariant measure. This provides an example of a weak-type jump inequality for a nonsymmetric semigroup in a nondoubling measure space. Our result may be seen as an endpoint refinement of the weak type $(1,1)$ inequality for the $\varrho$-th order variation seminorm of $\left(\mathcal H_t\right)_{t>0}$, recently proved by the authors when $\varrho>2$, and disproved for $\varrho=2$.