On the generalized Lebedev index transform

Abstract

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel vertical bar K(ir+alpha)/2(x)vertical bar(2), alpha is an element of R, x > 0, T E R, where K-mu(z) is the Macdonald function and i is the imaginary unit. Mapping properties such as the boundedness, compactness, invertibility are investigated for these operators and their adjoints in weighted L-p spaces. Inversion theorems are proved. Important particular cases are exhibited. As an interesting application, a solution of the initial value problem for the second order differential difference equation, involving the Laplacian, is obtained.