Proceedings of the Centre for Mathematics and its Applications

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 $f \in L^p([0,T] \times \mathbb{R}^N)$.

Article information

Dates
First available in Project Euclid: 3 December 2014