On maximal functions generated by Fourier multipliers (Q1070472)

We consider maximal operators of type \(M_ mf(x)=\sup_{t>0}| F^{-1}[m(A_{1/t}\xi)]*f(x)|\) with \(F^{-1}\) being the inverse Fourier transformation, m a bounded function and \(A_ t\) a dilation matrix. Conditions are given on the Fourier multiplier m such that \(M_ m\) is of strong type \((p,p),1<p\leq \infty\), and of weak type (1,1). These conditions (in dependence of p) become particularly simple if m is quasi- radial, i.e., \(m-m_ 0\circ \rho\), where \(m_ 0:(0,\infty)\to {\mathbb{C}}\) and \(\rho\) is an \(A_ t\)-homogeneous distance function. The Hardy- Littlewood maximal function and Stein's surface spherical maximal function fall under the latter scope.