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.