The Kontorovich-Lebedev Transform and Sobolev Type Space

In this paper boundedness of convolution operator in is given and continuity of pseudo-differential operator associated with the Kontorovich-Lebedev transform (KL-transform) from into itself is discussed. The Kontorovich-Lebedev potential (KL-potential) is defined on and then it is extended to the space of distributions also some of its properties are studied. A Sobolev type space associated with the KL-transform is discussed and proved as a Banach space. Moreover, it is shown that the KL-potential is an isometry of . An -boundedness for the KL-potential is obtained and at the end applications of the KL-potential in pseudo-differential equation are discussed.