Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

TL;DR

This paper establishes sharp endpoint estimates for Riesz transforms and Littlewood-Paley-Stein functions related to the Laplacian with drift in Euclidean space, considering the exponential growth measure and self-adjointness.

Contribution

It provides the first sharp endpoint estimates for these operators associated with the Laplacian with drift under exponential growth measures.

Findings

Proved weak type (1,1) bounds for Riesz transforms.

Established sharp endpoint estimates for Littlewood-Paley-Stein functions.

Analyzed operators in the context of exponential growth measures.

Abstract

Let $v \ne 0$ be a vector in $\R^n$. Consider the Laplacian on $\R^n$ with drift $\Delta_{v} = \Delta + 2v\cdot \nabla$ and the measure $d\mu(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $\Delta_{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions associated with the heat and the Poisson semigroups.