Boundedness properties of the operators of best approximation by analytic and meromorphic functions (Q1803529)

For a function \(f\in\text{BMO}\) there exists a function \(f_ 0\) analytic in \(| z|<1\) such that \(\| f-f_ 0\|_{L^ \infty(T)}=\inf\| f-g\|_{L^ \infty(T)}<\infty\), the inf being taken over all functions analytic in \(| z|<1\). In the case \(f\in\text{VMO}\) the function \(f_ 0\) is unique and is called the best approximation of \(f\) by analytic functions and is denoted by \(f_ 0=Af\). The operator \(A\) is not linear and not bounded in general. This paper is devoted to the question of boundedness of \(A\). The following theorem is proved Theorem: The operator \(A\) is not bounded in the Hölder classes \(\Lambda_ s\), \(s>0\), \(s>0\), and in the Bessov classes \(B^ s_ p\), \(p>0\), \(s>1/p\).