Bilinear decompositions for the product space $H^1_L\times BMO_L$

TL;DR

This paper refines the understanding of products of functions in Hardy and BMO spaces associated with Schrödinger operators, decomposing these products into sums of bilinear operators with specific target spaces.

Contribution

It introduces a new bilinear decomposition for products in $H_L^1$ and $BMO_L$ spaces, extending previous results to include a sum into $L^1$ and a novel space $H^{ ext{log}}$.

Findings

Products can be expressed as sums of two bilinear operators.

One operator maps into $L^1(R^d)$.

The other maps into the space $H^{ ext{log}}(R^d)$.

Abstract

In this paper, we improve a recent result by Li and Peng on products of functions in $H_L^1(\bR^d)$ and $BMO_L(\bR^d)$, where $L=-\Delta+V$ is a Schr\"odinger operator with $V$ satisfying an appropriate reverse H\"older inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from $H_L^1(\bR^d)\times BMO_L(\bR^d) $ into $L^1(\bR^d)$, the other one from $H^1_L(\bR^d)\times BMO_L(\bR^d) $ into $H^{\log}(\bR^d)$, where the space $H^{\log}(\bR^d)$ is the set of distributions $f$ whose grand maximal function $\mathfrak Mf$ satisfies $$\int_{\mathbb R^d} \frac {|\mathfrak M f(x)|}{\log (e+ |\mathfrak Mf(x)|)+ \log(e+|x|)}dx <\infty.$$