Lower bound of the parabolic Hilbert commutator

TL;DR

This paper establishes a lower bound for the BMO norm of functions in relation to the operator norm of their commutator with the parabolic Hilbert transform, advancing understanding in harmonic analysis.

Contribution

It provides the first explicit lower bound linking BMO space norms to commutator operator norms for the parabolic Hilbert transform, addressing an open problem.

Findings

Established a lower bound for BMO norm in terms of commutator norm

Connected BMO space properties with parabolic Hilbert transform behavior

Resolved an open question from recent harmonic analysis research

Abstract

Answering a key point left open in a recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ \|b\|_{\text{BMO}_{\gamma}(\mathbb{R}^2)}\lesssim \|[b,H_{\gamma}]\|_{L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)}, $$ where $H_{\gamma}$ is the parabolic Hilbert transform.