On maximal functions for Mikhlin-Hoermander multipliers

Abstract

Given Mikhlin-Hörmander multipliers $m_i$, $i=1,...,N$, with uniform estimates we prove an optimal $\sqrt{\log (N+1)}$ bound in $L^p$ for the maximal function $\sup_i|\cF^{-1}[m_i\hat f]|$ and related bounds for maximal functions generated by dilations.