# Uniqueness theorem for a functions with zero integrals over fourdimensional simpleces.

Throughout in this paper we assume that $A$ is a compact set in $\mathbb{R}^{n}$, $n>2$, of positive Lebesgue measure. As usual we denote by $M\left ( n \right )$ the group of Euclidean motions in $\mathbb{R}^{n}$. Under $\mathfrak{P}\left ( A,B \right )$ we mean the class of functions $L_{loc}\left ( B \right )$ such that the relation $\int _{\lambda A}f\left ( x \right )dx=0$, $\forall \lambda \in M\left ( n \right )$ is valid for any $\lambda \in M ot\left ( \overline{A} ,B\right )$.
We say that a set $A$ has the Pompeiu property if the only function $f\in L_{loc}\left ( \mathbb{R}^{n} \right )$ satisfying $\int _{\lambda A}f\left ( x \right )dx=0$ for any $\lambda \in M\left ( n \right )$ is $f = 0$. One says also that such a set $A$ is a Pompeiu set.
The Pompeiu problem in its original form is formulated as follows: "under what conditions does a set $A$ have the Pompeiu property?" This problem got its name from the Romanian mathematician D. Pompeiu who was the first to consider the relation $\int _{\lambda A}f\left ( x \right )dx=0$. A lot of researches were engaged in this problem but it still remains open.
Some sufficient conditions for $A\in Pomp\left ( \mathbb{R}^{n} \right )$ to meet were obtained by Pompeiu (1929), Nicolesco (1929), Christ (1943), Ilief (1946) and Chakalov (1949). Also, for many concrete cases there are a number of known results with the help of which one can determine whether $A$ is a Pompeiu’s set or not.
In the case where a set does not have the Pompeiu property, the presence of a non-zero function with the condition $\int _{\lambda A}f\left ( x \right )dx=0$ for any $\lambda \in M\left ( n \right )$ makes it possible to get nontrivial estimates of density of laying an arbitrary compact in $\mathbb{R}^{n}$ sets of the form $\lambda A$, $\lambda \in M\left ( n \right )$.Such estimates were obtained by Kotlyar. If A has the Pompeiu property then the Wiener theorem makes it possible to approximate indicators of sets of the type $\lambda A$, $\lambda \in M\left ( n \right )$ by linear combinations in L^{1}\left ( \mathbb{R}^{n} \right ).
A local version of Pompeiu’s problem was obtained by V.V. Volchkov in 1998: for a given set $A$, find the $\mathfrak{R}\left ( A \right )=\inf\left \{ R>0:A\in Pomp\left ( \mathbb{B}_{R} \right ) \right \}$ and investigate when the value $R\left ( A \right )$ is reached, i.e., $A\subset Pomp\left ( \mathbb{B}_{r} \right )$ whenever $r=\mathfrak{R}\left ( A \right )$.
In this paper we investigate the questions concerning the local version of Pompeiu’s problem. We consider the simplex $A=\left \{ x\in \mathbb{R}^{4}:x_{1}+x_{2}+x_{3}+x_{4}\leq 1, x_{j}\geq 0,j=1,2,3,4 \right \}$ with the vertices $z_{0}\left ( 0,0,0,0 \right )$, $z_{1}\left ( 1,0,0,0 \right )$, $z_{2}\left ( 0,1,0,0 \right )$, $z_{3}\left ( 0,0,1,0 \right )$, $z_{4}\left ( 0,0,0,1 \right )$. Let $r^{*}\left ( A \right )$ be the radius of the smallest closed ball containing the simplex closure, $r^{*}\left ( A \right )= \frac{\sqrt{3}}{2}$.
Our main result is the uniqueness theorem which implies that any locally integrable function with zero integrals over the simplexs and zero in the ball of radius $r$ $(r> \frac{1-\sqrt{4R^{2}-3}}{2})$ is equal to zero in the ball of radius $R> r^{*}$.
Theorem 1(The uniqueness theorem). Let $\frac{\sqrt{3}}{2}< R< 1$ and a function $f\in \mathfrak{P}\left ( A,\mathbb{B}_{R} \right )$. Let also $f=0$ in the ball $\mathbb{B}_{r}$ then $f=0$ in $\mathbb{B}_{r}$ for some $r> \frac{1-\sqrt{4R^{2}-3}}{2}$.
Earlier there were obtained results similar to Stokes’s formula for this simplex that allow to express the integral of some differential operator acting on a given function through its values over subsets of the simplex boundary of lower dimension. In particular, one can do it in the case when these subsets are vertexes and edges of the simplex considered before.
To prove the main result, we use given theorems as well as theorems obtained by V.V. Volchkov. Also a standard method of smoothing of functions is used.
The results can be used in the approximation theory and in the complex analysis (in particular, one can get a new version of Morera’s theorem for analytic functions).

Keywords: a local version of Pompeiu’s problem, Pompeiu’s radius, locally integrable functions, four-dimensional simplex, functions with zero integrals over sets

UDC:
517.988.28