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

Abstract

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