The Fractional Maximal and Riesz Potential Operators Involving the Lebedev–Skalskaya Transform

Abstract

In this paper, the boundedness of the fractional maximal function and Riesz potential for the LS transform from $L^p(\mathbb {R}_+;\frac{\exp ({-x \cos (\rho )})}{\sqrt{x}}\mathrm{{dx}})$ to $L^p(\mathbb {R}_+;x^{\frac{p}{2}}\mathrm{{dx}})$ and from $L^1(\mathbb {R}_+;\frac{\exp ({-x \cos (\rho )})}{\sqrt{x}}\mathrm{{dx}})$ to the weak space $\mathrm{{WL}}^1(\mathbb {R}_+;x^{\frac{1}{2}}\mathrm{{dx}})$ are studied. Relevance of the work In this work, we define the fractional integral and the fractional maximal operators using the translation operator associated with LS transform. The boundedness of these integral operators is investigated in the framework of Lebesgue spaces. These fractional integral operators are applied to the study of partial differential equations and Sobolev spaces.