Variation and $\lambda$-jump inequalities on $H^p$ spaces

Let $\phi\in \mathscr{S}$ with $\displaystyle\int\phi (x) dx=1$, and define
$$\phi_t(x)=\frac{1}{t^n}\phi \left(\frac{x}{t}\right),$$
and denote the function family $\{\phi_t\ast f(x)\}_{t>0}$ by $\Phi\ast f(x)$. Let $\mathcal{J}$ be a subset of $\mathbb{R}$ (or more generally an ordered index set), and suppose that there exists a constant $C_1$ such that
$$\sum_{t\in\mathcal{J}} |\hat{\phi}_t(x)|^2<C_1$$
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 $f\in H^p(\mathbb{R}^n)$, $\dfrac{n}{n+1}<p\leq 1$.
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 $C_3>0$.