On derivatives, Riesz transforms and Sobolev spaces for Fourier-Bessel expansions

We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are $L^p$-boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fourier-Bessel situations that were considered earlier in the literature. We believe that our observations shed some new light on analysis of Fourier-Bessel expansions.