Sharp Hardy space estimates for multipliers

Abstract

We provide an improvement of Calderón and Torchinsky's version of the Hörmander 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 Slavíková 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 $L_s^{r,q}(A_0)$ with $r<{\tau }^{(s,p)}$ or $q>\min (1,p)$.

To appear in Int. Math. Res. Not