A generalization of parabolic potentials associated to Laplace-Bessel differential operator and its behavior in the weighted Lebesque spaces

Authors: ÇAĞLA SEKİN

Abstract: In this work we introduce some generalizations of the singular parabolic Riesz and parabolic Bessel potentials. Namely, $\Delta _{\nu }$ being the Laplace-Bessel singular differential operator, we define the families of operators \begin{equation*} H_{\beta ,\nu }^{\alpha }=\left( \frac{\partial }{\partial t}+(-\Delta _{\nu })^{\beta /2}\right) ^{-\alpha /\beta }\text{ and }\mathcal{H}_{\beta ,\nu }^{\alpha }=\left( I+\frac{\partial }{\partial t}+(-\Delta _{\nu })^{\beta /2}\right) ^{-\alpha /\beta }\text{ , (}\alpha ,\beta >0\text{),} \end{equation*} and investigate their properties in the special weighted $L_{p,\nu }$-spaces.

Keywords: Laplace-Bessel differential operator, Fourier-Bessel transform, singular parabolic potentials, generalized translation operator, Hardy-Littlewood-Sobolev type inequality

Full Text: PDF