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

TL;DR

The paper establishes a weak-type jump inequality at the endpoint $\varrho=2$ for the non-symmetric Ornstein–Uhlenbeck semigroup in a nondoubling Gaussian setting, namely $J_2^{1,\infty}(\mathcal H_t f:\, t>0) \lesssim \|f\|_{L^1(\gamma_\infty)}$ for $f\in L^1(\gamma_\infty)$. The authors develop a localization strategy, decompose the semigroup into a local small-time part and a global part, and further split the local part into three difference operators plus a main convolution-type operator $M_t$. They obtain strong-type and weak-type bounds for the difference pieces and, crucially, a weak-type jump bound for $M_t$ via a reduction to a convolution kernel and an application of Liu’s jump inequality result. Together, these components yield an endpoint refinement linking variational bounds (valid for $\varrho>2$) to jump inequalities at $\varrho=2$, and they discuss the sharpness by noting that the corresponding $L^{1}$-type bounds fail for $\varrho<2$.

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 $\varrho=2$.

Theorems & Definitions (17)

• Proposition 1.1
• Theorem 1.2
• Proposition 5.1
• proof
• Proposition 5.2
• Proposition 6.1
• proof
• Lemma 8.1
• proof
• Lemma 8.2