Weak type \((1,1)\) of some operators for the Laplacian with drift (Q269860)

This paper is devoted to the boundedness of some classical operators with respect to the Laplacian with drift. Let \(v=(v_1,\ldots,v_n)\) be a non-zero vector in \(\mathbb{R}^n\). Define the Laplacian with drift \(\Delta_v=\sum_{i=1}^n(\frac{\partial^2}{\partial x_i^2}+2v_i\frac{\partial}{\partial x_i})\) and the measure \(d\mu(x)=e^{2\langle v,x\rangle}dx.\) Then it is well known that \(\Delta_v\) is self-adjoint with respect to \(\mu\). Let \(d\) be the Euclidean distance and \(\nabla\) the gradient operator on \(\mathbb{R}^n\). Then \((\mathbb{R}^n,d,\mu)\) is a metric space with exponential volume growth. In this paper, the authors obtain the weak type \((1,1)\) boundedness of the Riesz transform \(\nabla(-\Delta_v)^{-1/2}\) and the heat maximal operator with respect to \(\Delta_v\), as well as the \(L^p\) boundedness of the uncertered Hardy-Littlewood maximal operator for \(p\in(1,\infty]\). One interesting phenomenon observed in this paper is that, different from the usual case, the uncertered Hardy-Littlewood maximal operator is not of weak type \((1,1)\) if \(n\geq 2\) in the setting of the Laplacian with drift.