Weak type (1, 1) of some operators for the Laplacian with drift

Abstract

Let $v = (v_1, \ldots , v_n)$ be a vector in $\mathbb {R}^n {\setminus } \{ 0 \}$. Consider the Laplacian on $\mathbb {R}^n$ with drift $\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )$ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$, with respect to which $\Delta _{v}$ is self-adjoint. Let d and $\nabla$ denote the Euclidean distance and the gradient operator on $\mathbb {R}^n$. Consider the space $(\mathbb {R}^n, d, d\mu )$, which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform $\nabla (- \Delta _{v} )^{-\frac{1}{2}}$ and for the heat maximal operator, with respect to $d\mu$. Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on $L^p$ for $1 < p \le +\infty$ but not of weak type (1, 1) if $n \ge 2$.