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.