Representation of the generalized Bessel potential by means of a Poisson kernel of the general form

The article considers a generalization of the Poisson kernel, which in potential theory is an integral kernel used to simply represent the generalized Bessel potential as a one-dimensional integral and studies its properties. The concept of a common Poisson kernel is introduced. Further, it is shown that the generalized Bessel potential of a function integrable in the p-th degree with a power weight can be represented by an integral of a very simple form, using a Poisson kernel. Also in this paper, Young's inequality for B-convolutional operators in spaces $ \mathbf{B}_{\gamma}^{\alpha} $ is proved and some applications using the Laplace-Bessel differential operator are given. Generalized Poisson kernels find application in control theory and electrostatics problems.