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

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 $\varrho =2$.

20 pages. Final version, to appear in Transactions of the American Mathematical Society