On a testing-function space for distributions associated with the Kontorovich-Lebedev transform

We construct a testing-function space, which is equipped with the topology that is generated by L-nu,L-p - multinorm of the differential operator A(x) = x(2) - x d/dx [x d/dx], and its k-th iterates A(x)(k), where k = 0, 1, and A(x)(0) phi = phi. Comparing with other testing-function spaces, we introduce in its dual the Kontorovich-Lebedev transformation for distributions with respect to a complex index. The existence, uniqueness, imbedding and inversion properties are investigated. As an application we find a solution of the Dirichlet problem for a wedge for the harmonic type equation in terms of the Kontorovich-Lebedev integral.