Endpoint multiplier theorems of Marcinkiewicz type

Abstract

We establish sharp $(H^1,L^{1,q})$ and local ($L$ log${}^{r}L,L^{1,q}$) mapping properties for rough one-dimensional multipliers. In particular we show that the multipliers in the Marcinkiewicz multiplier theorem map $H^1$ to $L^{1,\infty }$ and $L$ log${}^{1/2}L$ to $L^{1,\infty }$, and that these estimates are sharp.