Riesz transforms related to Bessel operators

Abstract

Riesz transforms $R_\mu$ related to the Bessel operators $$ \Delta_\mu=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2} $$ are studied in this work. We develop for $R_\mu$ a theory that runs parallel to that for the Euclidean Hilbert transform. It is proved that $R_\mu$ is actually a Calderon–Zygmund singular integral operator. Also, $R_\mu$ is seen to be the boundary value of the appropriate harmonic extension for this context. Finally, we analyse weighted inequalities involving $R_\mu$ .