The Kontorovich–Lebedev Transform and Sobolev Type Space

In this paper boundedness of convolution operator in $$L^p(\mathbb {R}_+; ~xdx)$$ is given and continuity of pseudo-differential operator $$\mathcal {P}_a$$ associated with the Kontorovich–Lebedev transform (KL-transform) from $$H(\mathbb {R}_+)$$ into itself is discussed. The Kontorovich–Lebedev potential (KL-potential) $$\mathcal {P}_a^s$$ is defined on $$H(\mathbb {R}_+)$$ and then it is extended to the space of distributions also some of its properties are studied. A Sobolev type space $$W^{s,p}(\mathbb {R}_+)$$ 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 $$W^{s,p}$$ . An $$L^p$$ -boundedness for the KL-potential is obtained and at the end applications of the KL-potential in pseudo-differential equation are discussed.