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

TL;DR

This paper establishes bounds for variation and jump inequalities of convolution operators on Hardy spaces, extending understanding of their behavior in harmonic analysis.

Contribution

It introduces new bounds for variation and jump operators on $H^p$ spaces under specific Fourier decay conditions, advancing the theory of singular integrals.

Findings

Boundedness of variation operators on $H^p$ spaces for $p > n/(n+1)$.

Uniform bounds for $ ext{lambda}$-jump operators on $H^p$ spaces.

Extension of classical inequalities to a broader class of convolution operators.

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 $C_3>0$.