A Convolution Related to the Inverse Kontorovich-Lebedev Transform

We establish the mapping properties in the space L₂(₊;dt/t ν t), 0<ν≤π for a convolution related to the transformation\[F(x)=∫_0^∞ f(t) Kᵢₜ(x)\,dt, x∈{ R}₊\]involving the modified Bessel function Kᵢₜ(x) as a kernel. It is shown that the convolution operator of two L₂-functions exists as a Lebesgue integral and represents a continuous function on R₊. As a consequence, we get the multiplication theorem for two modified Bessel functions of different subscripts. Further applications to the corresponding class of convolution integral equations are obtained. 2000 Mathematics Subject Classification. 44A15, 44A05, 44A35, 33C10, 45A05