Estimates near \(L^ 1\) for Fourier multipliers and maximal functions (Q1109270)

The following endpoint estimate of the Hörmander multiplier theorem is proved: Suppose that the bounded function \(m\) belongs to the localized Besov-space defined by the norm \(\|\varphi m(t\cdot)\| _{B^2_{n| 2,1}}\), where \(\varphi\) is a suitable bump-function supported away from 0. Then the operator \(T_ m\), defined by \([T_ mf]^\wedge =mf^\wedge\), is bounded from the Hardy-space \(H^1\) into the Lorentz-space \(L^{12}\). Furthermore, a weak-type-(1-1)-inequality for maximal-functions \(\sup | F^{-1}[m(t\cdot)f^\wedge)| (x)\) is obtained.