On one uniqueness theorem for M. Rietz potentials

TL;DR

This paper demonstrates the existence of nonzero Holderian functions with vanishing M. Rietz potentials on sets of positive length, extending previous results and applying to multidimensional cases.

Contribution

It improves prior theorems by constructing nonzero functions with vanishing potentials on positive-length sets and extends these results to higher dimensions.

Findings

Existence of nonzero Holderian functions with zero Rietz potential on positive-length sets

Extension of results to multidimensional Rietz potentials

Improvement over previous theorems by Beliaev and Havin

Abstract

We prove that there exists a nonzero holderian real-to-real function vanishing together with its M. Rietz potential in all points of some set of positive length. This result improves the one of D. Beliaev and V. Havin. We also extend the results to multidimensional M. Rietz potentials.