A remark on a maximal operator for Fourier multipliers
Given a set \(\Lambda\) of \(N\) distinct points, \(\Lambda\subset{\mathbf T},\) let \(\Lambda(m)=\cup_{\lambda\in\Lambda}(\lambda-2^{-m},\lambda+2^{-m}),\) and let \(T_m:f\to\{\check f\chi_{\Lambda(m)}\}^\wedge\) be the multiplier operator in \(l_2({\mathbf Z}),\) where \(\chi_E\) is the indicator of \(E.\) Let \(M_\Lambda:f\to\sup\limits_{m\in{\mathbf Z}}|(T_mf)(\cdot)|\) be the corresponding maximal operator. The main result reads as follows. Theorem. For any \(N\) there exist a set \(\Lambda\subset{\mathbf T},\) card\(\Lambda=N,\) and \(f\in l_2({\mathbf Z})\) such that \(||M_\Lambda f||\geq C \log^\alpha N ||f||,\) where \(C\) and \(\alpha\) are absolute constants, for example, one can take \(\alpha=1/4.\) It shows the sharpness of an earlier result of the first author.
scientific article; zbMATH DE number 1848888