Parabolic Extrapolation and Its Applications to Characterizing Parabolic BMO Spaces via Parabolic Fractional Commutators

TL;DR

This work develops a parabolic analogue of Rubio de Francia extrapolation and uses it to characterize parabolic BMO-type spaces via time-lagged commutators of parabolic fractional operators. It introduces a parabolic iteration framework, a Cauchy integral technique, and a Fourier-series argument tailored to parabolic geometry, establishing extrapolation for commutators and an extrapolation-at-infinity principle in the weighted setting. The authors prove that PBMO and related Campanato spaces are precisely captured by the boundedness of various commutators of parabolic maximal and fractional operators with time lag, across high dimensions and under parabolic Muckenhoupt weights $A_{r,q}^+(\gamma)$. These results provide robust tools for weighted harmonic analysis in the parabolic setting and have potential applications to regularity theory in parabolic PDEs.

Abstract

In this article, we establish the parabolic version of the celebrated Rubio de Francia extrapolation theorem. As applications, we obtain new characterizations of parabolic BMO-type spaces in terms of various commutators of parabolic fractional operators with time lag. The key tools to achieve these include to establish the appropriate form in the parabolic setting of the parabolic Rubio de Francia iteration algorithm, the Cauchy integral trick, and a modified Fourier series expansion argument adapted to the parabolic geometry. The novelty of these results lies in the fact that, for the first time, we not only introduce a new class of commutators associated with parabolic fractional integral operators with time lag, but also utilize them to provide a characterization of the parabolic BMO-type space in the high-dimensional case.

Theorems & Definitions (45)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Definition 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10