Endpoint multiplier theorems of Marcinkiewicz type

We establish sharp (H-1, L-1,L-q) and local (L log(r) L,L-1,L-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,L-infinity and L log (1/2) L to L-1,L-infinity, and that these estimates are sharp.