Variational characterization of H^p

TL;DR

This paper provides a variational characterization of Hardy spaces $H^p$ for certain $p$ values, along with estimates for related operators, and discusses limitations of these operators in characterizing $H^p$.

Contribution

It introduces a variational characterization of $H^p$ spaces and analyzes the effectiveness of oscillation and jump operators in this context.

Findings

Established variational characterization of $H^p$ for $p\in(\frac{n}{n+1},1]$.

Derived estimates for oscillation and $\ ext{lambda}$-jump operators on $H^p$.

Provided counterexamples showing limitations of these operators in characterizing $H^p$.

Abstract

In this paper we obtain the variational characterization of Hardy space $H^p$ for $p\in(\frac n{n+1},1]$ and get estimates for the oscillation operator and the $\lambda$-jump operator associated with approximate identities acting on $H^p$ for $p\in(\frac n{n+1},1]$. Moreover, we give counterexamples to show that the oscillation and $\lambda$-jump associated with some approximate identity can not be used to characterize $H^p$ for $p\in(\frac n{n+1},1]$.