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In the present paper, using a development of the technique of transmutation mappings, we obtain the first weighted analog of the well-known Brown–Schreiber–Taylor theorem on the eigenvalues of the Laplacian for corresponding spaces of continuous functio

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.

We obtain new sufficient conditions under which a set on the plane has the Pompeiu property. This result allows us to construct first examples of domains with the Pompeiu property with non-Lipschitz (and even fractal) boundary. In addition, we study the problem of determining the least radius of the ball on the sphere in which a given set is a Pompeiu set. We obtain the solution of this problem in the case of a biangle and a spherical half-disk. We also consider some applications to questions of complex analysis.

We obtain a description of the kernel of the local Pompeiu transform for some families of distributions on the sphere.

An exact characteristic of maximal smoothness of nontrivial functions with zero integrals over balls whose radii belong to a given two-element set is obtained..

New uniqueness theorems for periodic (in mean) functions on quaternion hyperbolic space are obtained.

The class of functions with zero integrals over all balls of one fixed radius on the complex hyperbolic space is described.