Riesz transforms of some parabolic operators

Abstract

We study boundedness on $L^p([0,T]) \times \mathbb{R}^N$ of Riesz transforms $\nabla(\mathcal{A})^{-1/2}$ for class of parabolic operators such as $A = \frac{\partial}{\partial t} - \Delta + V(t,x)$. Here $V(t,x)$ is a non-negative potential depending on time ${t}$ and space variable ${x}$. As a consequence, we obtain $W_{x}^{1,p}$-solutions for the non-homogeneous problem \[ \partial_{t}u - \Delta u + V(t,.)u = f(t,i), u(0) = 0 \] for initial data $f \in L^p([0,T] \times \mathbb{R}^N)$.