Riesz transforms for the Weinstein operator

ABSTRACT

In this paper we study the Riesz transforms ${\mathcal{R}}_w$ related to the Weinstein operators ${\mathrm{\Delta }}_w=\sum _{i=1}^d{x}_d^{-2\alpha -1}\left(\mathrm{\partial }/\mathrm{\partial }{x}_i\right)\left(x_d^{2\alpha +1}\right(\mathrm{\partial }/\mathrm{\partial }{x}_i\left)\right)$. We develop for ${\mathcal{R}}_w$ a theory that runs parallel to the one for the Euclidean Riesz Transform. It is proved that the Riesz–Weinstein transform in coordinates $i=1,\dots ,d$, ${\mathcal{R}}_w^i$ is actually a Calderón–Zygmund singular integral operator in the sense of the associated space of homogeneous type. Moreover, our Riesz–Weinstein transform can be written as a principal value.

KEYWORDS:

• Weinstein's operator
• Riesz transforms
• spaces of homogeneous type
• Calderón–Zygmund operators
• $A_p$ weights

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 26A33
• 42B10
• 42B20

Acknowledgements

I am grateful to Professor Néjib Ben Salem for suggesting the topic and for continuous encouragement to the study.

Disclosure statement

No potential conflict of interest was reported by the author.