Let $\phi\in \mathscr{ S}$ with $\int\phi (x)\, dx=1$, and define $$\phi_t(x)=\frac{ 1}{ t^n}\phi (\frac{ x}{ t}),$$ and denote the function family $\{ \phi_t\ast f(x)\}_{ t>0}$ by $\Phi\ast f(x)$. Suppose that there exists a constant $C_1$ such that $$\sum_{ t>0} |\hat{ \phi}_t(x)|^2<C_1$$ for all $x\in \mathbb{ R}^n$. Then (i) There exists a constant $C_2>0$ such that $$\|\mathscr{ V}_2(\Phi\ast f)\|_{ L^p}\leq C_2\|f\|_{ H^p},\;\;\frac{ n}{ n+1}<p\leq 1$$ for all $f\in H^p(\mathbb{ R}^n)$, $\frac{ n}{ n+1}<p\leq 1$. (ii) The $\lambda$-jump operator $N_{ \lambda}(\Phi\ast f)$ satisfies $$\|\lambda [N_{ \lambda}(\Phi\ast f)]^{ 1/2}\|_{ L^p}\leq C_3\|f\|_{ H^p},\;\;\frac{ n}{ n+1}<p\leq 1,$$ uniformly in $\lambda >0$ for some constant $C_3>0$.
Categories: math.CA