Narrow Results

We provide a vertex operator realization for a two-parameter generalization of MacMahon’s formula introduced by M. Vuletić [Trans. Amer. Math. Soc.361, 2789 (2009)]. Since the generalized MacMahon function is the kernel function of some Macdonald symmetric function, we consider the action of two vertex operators on a state corresponding to a Macdonald symmetric function. It becomes evident that the vertex operators appear to be the creation and annihilation operators, respectively on the state.

In this paper, we prove the boundedness for the maximal and fractional maximal operators and Riesz potential-type operator associated with the Kontorovich–Lebedev transform (KL transform)in the $L^p({ℝ}_{+},x^{-\beta }dx)$ spaces.

The properties of the modified KONTOROVITCH-LEBEDEV transforms and their kernels are considered. The solutions of some inhomogeneous integral equations are derived by means of the use of these transforms. The applications for boundary value problems of mathematical physics in the wedge domains are given. The numerical solution is conducted.