  New index transforms of the Lebedev type are investigated. It involves the
real part of the product of the modified Bessel functions as the kernel. The
boundedness and invertibility are examined for these operators in the Lebesgue
weighted spaces. Inversion theorems are proved. Important particular cases are
exhibited. The results are applied to solve an initial value problem for the
fourth order PDE, involving the Laplacian. Finally, it is shown that the same
PDE has another fundamental solution, which is associated with the generalized
Lebedev index transform, involving the square of the modulus of Macdonald's
function, recently considered by the author.
