Variaiton and $λ$-jump inequalities on $H^p$ spaces

Abstract: 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 $C_3>0$.