A version of the Kontorovich-Lebedev transformation with the Hankel function of second kind in the kernel is investigated in a space of distributions of doubly exponential descent. The inversion theorem is rigorously established making use in some steps of the proof of a relation of this transform with the Laplace one. Finally, the theory developed is illustrated in solving certain type of partial differential equations.

The analysis of a kernel involving the product of three Bessel functions motivates the introduction of the translation operator and the convolution associated to the Hankel-Kontorovich-Lebedev tranformation, first in a classical framework, and then in certain spaces of generalized functions. The main properties of this convolution are investigated, the more important operational rules are obtained and some applications are shown.

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