On Some Transformations Associated to a Certain Cone

TL;DR

The paper studies a model elliptic pseudo-differential equation on a 4-faced cone in R^3, deriving an explicit Bochner kernel and solving the equation via wave factorization and transmutation operators. It provides a concrete formula for the Bochner kernel of the cone, an explicit operator representation for V_φ, and solvability results for a boundary value problem with an integral condition under symbol restrictions. The results yield exact solution representations and a priori estimates, contributing concrete tools for conical elliptic problems in Sobolev–Slobodetskii spaces. These constructions enable explicit computations on conical domains and advance understanding of model problems in higher dimensions.

Abstract

A model elliptic pseudo-differential equation in $4$-faced cone is studied in Sobolev--Slobodetskii space. The Bochner kernel for such a cone is evaluated and explicit formula for unique solution to the considered equation is presented under certain restrictions on the symbol. Boundary value problem with additional integral condition is considered and unique solvability to the boundary value problem is proved.