On functions in \(p\)-adic \(BMO\) and the distribution of prime integers (Q865360)

The author investigates the relationship between the classical BMO space on the real line, with a norm \[ \| f\| _*=\sup _{I}\frac{1}{| I|}\int_I | f(x)-f_I(x)|\,dx,\; f_I=\frac{1}{| I|}\int_I| f(x)|\,dx,\tag{*} \] where \(I\) is an interval, and the \(p\)-adic BMO space, denoted by \(\text{BMO}_p\), with the norm \(\| f\| _{*,p}\) given by \((*)\), but now the supremum is taken over all intervals of the form \((m+\frac{j-1}{p^k}, m+\frac{j}{p^k})\), where \(m\) is an integer, \(1\leq j\leq p^k\), and \(p\) a positive prime integer, \(k=1,2,\dots\). In particular, it is proved that \(\bigcap_{p}\text{BMO}_p\neq\text{BMO}\) and a relation to the distribution of primes is given.