An inequality for analytic functions (Q1104438)

``Theorem 2 enables us to answer positively the question of V. I. Arnol'd posed in the paper by \textit{O. A. Oleĭnik} and \textit{M. A. Shubin} [Usp. Mat. Nauk 37, No.6, 261-285 (1982)], Problem 15] in connection with the work of A. I. Neĭshtadt. Theorem 1: Let \(\phi(I,q)\) be an analytic function defined in a neighborhood of the compact set \(T\times Q\), where \(T\subset {\mathbb{R}}^ n,\) \(Q\subset {\mathbb{R}}^ m.\) Assume that for all \(q\in Q\) the function \(\phi(\cdot,q)\) is not identically 0. Then there exists an \(r>0\) such that the estimate \(mes\{I: | \phi(I,q)| \leq \alpha \}\leq M| \alpha |^{2/r}\) holds uniformly with respect to q. Let f be an analytic mapping given in the neighborhood of T with values in \({\mathbb{R}}^ m,\) where for all \(q\in {\mathbb{R}}^ m,\) \(q\neq 0\), the function \(\phi (\cdot,q)=<f(\cdot),q>_{{\mathbb{R}}^ m}\) is not identically equal to zero. Theorem 2: There exists an \(r>0\) such that the measure of the set \(I\subset T\) for which \(| <f(I),q>_{{\mathbb{R}}^ m}| \geq c(I)\| g\|^{-mr}\) for all \(q>{\mathbb{Z}}^ m \)is complete. Here \(c(I)>0.\)''