Fefferman-Stein decomposition for $Q$-spaces and micro-local quantities

TL;DR

This paper proves a decomposition theorem for $Q$-spaces, revealing how their predual spaces' norms depend on micro-local structures, thus solving a long-standing open problem.

Contribution

It provides an affirmative solution to an open problem about $Q$-spaces and introduces a micro-local analysis approach to understand their predual spaces.

Findings

Established Fefferman-Stein decomposition for $Q$-spaces.

Showed the predual space norm depends on micro-local structure.

Solved an open problem from 2000 regarding $Q$-spaces.

Abstract

In this paper, we consider the Fefferman-Stein decomposition of $Q_{\alpha}(\mathbb{R}^{n})$ and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ by the micro-local quantities. This result indicates that the norm of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ depends on the micro-local structure in a self-correlation way.