$$L^p$$ -Sobolev spaces and coupled potential operators associated with coupled fractional Fourier transform

Abstract

This paper is devoted in investigations concerning the study of the coupled potential operator \(J_{s}^{\alpha , \beta }\) and corresponding \(L^p\)-Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space \(\mathcal {S}_{\alpha ,\beta }\) is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The \(L^p\) norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator \(J_{s}^{\alpha , \beta }\) is defined as a pseudo-differential operator related with a precise symbol. The operator \(J_{s}^{\alpha , \beta }\) is extended to a space of distributions. An \(L^p\)-Sobolev boundedness result for the operator \(J_{s}^{\alpha , \beta }\) is shown. The spaces \(H^{m,\alpha ,\beta }_{p}\) and \(\mathcal {H}^{m,\alpha ,\beta }_{p}\) introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces.