  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.
