On a progress in the Kontorovich–Lebedev transform theory and related integral operators

Abstract

This survey represents mainly recent results of the author, which were obtained during the last decade and concern the theory of the Kontorovich–Lebedev transformation and some related integral operators of the convolution and non-convolution type. In particular, we discuss their boundedness in the Lebesgue, Hardy and Sobolev type spaces and in some spaces of distributions. Special attention is given to the L 2-case and Plancherel's theory for these operators and to the harmonic analysis of the so-called Lebedev–Stieltjes integrals. Moreover, we concentrate on asymptotic, summation formulas and uncertainty principles for the Kontorovich–Lebedev transform.