On the generalized Lebedev index transform

TL;DR

This paper generalizes the Lebedev index transform using the square of the Macdonald function, analyzing its properties and providing inversion formulas, with applications to differential difference equations involving the Laplacian.

Contribution

It introduces a new family of integral operators with the Macdonald function, studying their properties and solving related differential difference equations.

Findings

Operators are bounded, compact, and invertible in weighted Lebesgue spaces.

Inversion formulas for the operators are established.

Application to solving initial value problems for differential difference equations.

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 $|K_{(i\tau+\alpha)/2}(x)|^2, \alpha \ge 0,\ x >0, \ \tau \in \mathbb{R},$ 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 the Lebesgue weighted 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.