On the generalized Lebedev index transform

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.