## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- AMSI International Conference on Harmonic Analysis and Applications. Xuan Duong, Jeff Hogan, Chris Meaney, and Adam Sikora, eds. Proceedings of the Centre for Mathematics and its Applications, v. 45. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2013), 115 - 123

### 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

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1417630508

**Mathematical Reviews number (MathSciNet)**

MR3424870

**Zentralblatt MATH identifier**

1338.42028

#### Citation

Ouhabaz, E. M.; Spina, C. Riesz transforms of some parabolic operators. AMSI International Conference on Harmonic Analysis and Applications, 115--123, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2013. https://projecteuclid.org/euclid.pcma/1417630508