Sharp Hardy Space Estimates for Multipliers

Abstract We provide an improvement of Calderón and Torchinsky’s version [ 5] the Hörmander multiplier theorem on Hardy spaces $H^p$ ($0<p<\infty $), substituting Sobolev space $L_s^2(A_0)$ by Lorentz–Sobolev $L_s^{\tau ^{(s,p)},\min (1,p) }(A_0)$, where $\tau ^{(s,p)} =\frac{n}{s-(n/\min{(1,p)}-n)}$ $A_0$ is annulus $\{\xi \in{\mathbb{R}}^n:\,\, 1/2<|\xi |<2\}$. Our also extends that Grafakos Slavíková 10] to range $0 < p \leqslant 1$. result sharp in sense preceding cannot be replaced a larger $L^{r,q}_s(A_0)$ with $r< \tau $ or $q>\min (1,p)$.