Uniqueness theorems for some classes of functions with zero spherical means (Q1277417)

Let \(p\in [1,\frac{2n}{n-1}] R>0 \), \(M_{\mu}(R)= R^{n-\frac{p(n-1)}{2}}\) for \(1\leq p< \frac{2n}{n-1}\) and \(M_{\mu}(R)=\ln R\) for \(p= \frac{2n}{n-1}\). Let \(M_{\mu}(f)_p= \int_{|x|\leq R}|f|^p dx\). Suppose that for some given \(r>0\) and for almost all \(x\in{\mathbb R}^n\) one has \[ \int_{|x|\leq r}f d{\sigma}=0\tag{1} \] where \(d\sigma\) is a normalised surface measure. Let \(E\) be a set of numbers \(\frac{\alpha}{\beta}\), where \(J_{n/2-1}(\alpha)= J_{n/2-1}(\beta)\), \(\alpha,\beta>0\). Typical results of the paper are of the following kind. Let \(f,g\in L^p_{\text{loc}}({\mathbb R}^n)\) satisfy (1) for given \(r= r_1,r_2\). If \(\frac{r_1}{r_2}\notin E\), and \(\underline {\lim}_{R\rightarrow\infty}\frac{M_R(f-g)_p}{\mu_p(R)}=0\), then \(f=g= 0\).