Liouville property for functions having zero integrals on balls of fixed radius (Q2901625)

The admissible decrease rate of non-zero functions having zero integral on balls of fixed radius are studied. Suppose that \(f\in L_{\text{loc}}^1\left({\mathbb R}^n\right)\) and \(\int\limits_{| x| \leq r}f(x+y)dx=0\) for some fixed \(r\in A\) and all \(y\in {\mathbb R}^n\). Suppose that there exists a positive increasing function \(\varkappa\in C^1[0,+\infty)\) such that the following conditions are satisfied: (1) \(\lim\limits_{t\rightarrow\infty}\frac{\varkappa(t)}{\varkappa\left( t/\varkappa(t)\right)}=1\) and (2) \(| f(x)| \leq C_1 \) \(\exp{ \left( -\frac {| x_1| +\dots+| x_{n-1}| } {\varkappa\left(| x_1| +\dots+| x_{n-1}| \right)}+C_2| x_n| \right)}\) for a. e\(.x\in{\mathbb R}^n\) and some constants \(C_1, C_2\). Then \(f\equiv 0\). The conditions (1) and (2) cannot be omitted.