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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]

The pseudo-differential operator involving the Kontorovich–Lebedev transform is defined. The boundedness of pseudo differential operator in a certain Sobolev-type space is shown. Composition of pseudo-differential operatorsis defined in terms of the two symbols, and also its boundedness in certain Sobolev-type space is given. Some special cases are also discussed. Finally, the Kontorovich–Lebedev transform is used in the solution of some partial differential equations. [ABSTRACT FROM PUBLISHER]