Weak type estimates for cone multipliers on $H^p$ spaces, $p < 1$

Abstract:

We consider operators $T^{\delta }$ associated with the Fourier multipliers \begin{equation*}{\bigg (}1- \frac {|{\xi }’|^{2}}{{\xi }_{n+1}^{2}} {\bigg )}_{+}^{\delta },\quad \quad ({\xi }’,\xi _{n+1}) \in {\mathbb R}^{n} \times {\mathbb R},\end{equation*} and show that $T^{\delta }$ is of weak type $(p,p)$ on $H^{p}({\mathbb R}^{n+1})$, $0 < p < 1$, for the critical value $\delta = n(\frac {1}{p}-\frac {1}{2}) - \frac {1}{2}$.