Maximal functions and Fourier transforms (Q1088101)

The aim of the paper is to obtain sharp \(L^ p\) bounds for maximal operators \(T^*f(x)=\sup_{t>0}| T_ tf(x)|,\) \((T_ tf){\hat{\;}}(\xi)=\hat f(\xi)m(t\xi)\) (where \(m\in L^{\infty}({\mathbb{R}}^ n)\) is a given multiplier) in terms only of the decay at infinity of m(\(\xi)\). In general, if \(| m(\xi)| \leq C| \xi |^{-a}\) and \(a>1/2\), it is proved that \(T^*\) is bounded in \(L^ p({\mathbb{R}}^ n)\) for \(2n/(n+2a-1)<p<(2n-2)/(n-2a).\) Moreover, if m is the Fourier transform of a compactly supported measure, then the range is \(p>(2a+1)/2a\). This covers \textit{E. M. Stein's} theorem on maximal spherical means [Proc. Nat. Acad. Sci. USA 73, 2174-2175 (1976; Zbl 0332.42018)] and extensions of it due to Greenleaf and Sogge- Stein. The proofs are rather elementary and based on \(L^ 2\) methods (g- functions) and vector valued Calderón-Zygmund theory.