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

We consider operators T δ T^{\delta } associated with the Fourier multipliers ( 1 − | ξ ′ | 2 ξ n + 1 2 ) + δ , ( ξ ′ , ξ n + 1 ) ∈ R n × R , \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 δ T^{\delta } is of weak type ( p , p ) (p,p) on H p ( R n + 1 ) H^{p}({\mathbb R}^{n+1}) , 0 > p > 1 0 > p > 1 , for the critical value δ = n ( 1 p − 1 2 ) − 1 2 \delta = n(\frac {1}{p}-\frac {1}{2}) - \frac {1}{2} .