Laplace transform-type multipliers for Kontorovich-Lebedev transform

In this paper, we investigate Poison kernel associated with Kontorovich-Lebedev transform. We significantly simplify the integral yielding a tractable form which is more useful for explicit calculation. We establish also that Kontorovich-Lebedev multipliers of Laplace transform type are bounded from L-p(w) into it self when 1 < p < infinity, and from L-1(w) into L-1,L-infinity(w), provided that w is in the Muckenhoupt class Ap on ((0,infinity),dx). At the end, an integral representation of the operator Delta(-beta/2)f is obtained and its boundedness has been discussed in Lebesgue space.