On $H^p(\mathbf {R}^n)$-multipliers of mixed-norm type

Abstract:

For a function m in ${L^\infty }({{\mathbf {R}}^n})$, an appropriately chosen function $\eta$ in ${C^\infty }({{\mathbf {R}}^n})$ and $\delta > 0$ we define ${m_\delta }$ by ${m_\delta }(\xi ) = m(\delta \xi )\eta (\xi )$. We show that if $0 < p \leq 1$ and if the sequence $((m_{2^n})\hat \emptyset )$ belongs to a certain mixed-norm space, depending on p, then m is a Fourier multiplier for the corresponding Hardy space ${H^p}({{\mathbf {R}}^n})$. Moreover, we prove the sharpness of our multiplier theorem. Comparable results had been proved earlier for multipliers for Hardy spaces defined on a locally compact Vilenkin group.