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The main objective of this paper is to enrich the theoretical system of the linear canonical Fourier transform (LCFT) by introducing the canonical potential J m , A and corresponding L p -Sobolev space. Moreover, the Schwartz-type space S A is introduced. Further, pseudo-differential operator (PDO) is defined and obtained its another integral representation. The L p -boundedness result for the pseudo-differential operator associated with the LCFT is discussed. Some applications of Sobolev-type spaces H A m , p and H A m , p are given. [ABSTRACT FROM AUTHOR]

In this paper, some elementary and useful terminologies associated to Lebedev–Skalskaya transform (LS-transform) are notified and then estimates for R κ 1 2 + i τ (x) , F κ 1 2 + i τ (x) , translation and convolution operators associated to LS-transform are obtained. The continuity of LS-transforms, translation and convolution operators on functions spaces S α , k , H α , k and G α , k are discussed. The continuity of pseudo-differential operators in terms of LS-transforms is discussed on functions spaces S α , k , H α , k , G α , k and Lebesgue space. At the end, an application of LS-transform is shown by solving a differential equation of convolution type. [ABSTRACT FROM AUTHOR]