Sharp Hardy space estimates for multipliers

Abstract: We provide an improvement of Calder\'on and Torchinsky's version of the H\"ormander multiplier theorem on Hardy spaces $H^p$ ($0<p<\infty$), by replacing the Sobolev space $L_s^2(A_0)$ by the Lorentz-Sobolev space $L_s^{\tau^{(s,p)} ,\min(1,p) }(A_0)$, where $\tau^{(s,p)} =\frac{n}{s-(n/\min{(1,p)}-n)}$ and $A_0$ is the annulus $\{\xi \in \mathbb{R}^n: 1/2<|\xi|\<2\}$. Our theorem also extends that of Grafakos and Slav\'ikov\'a to the range $0<p\le 1$. Our result is sharp in the sense that the preceding Lorentz-Sobolev space cannot be replaced by a smaller Lorentz-Sobolev space $L^{r,q}_s(A_0)$ with $r< \tau^{(s,p)} $ or $q>\min(1,p)$.