Extending BMO functions in parabolic setting

TL;DR

This paper demonstrates that functions in the parabolic BMO space defined on a cube can be extended to the entire space with minimal change to their BMO seminorm, advancing understanding of function extension in parabolic analysis.

Contribution

It introduces a method to extend parabolic BMO functions from a cube to the whole space while nearly preserving their BMO seminorm, a novel result in parabolic function theory.

Findings

Extension preserves BMO seminorm approximately

Extension applies to functions in parabolic setting

Advances in parabolic BMO function analysis

Abstract

We prove that one can extend any $BMO^{x}$ function $a$ given in a cube in $\mathbb{R}^{d+1}$ to become a $BMO^{x}$ functions $\hat a$ in $\mathbb{R}^{d+1}$ almost preserving its $[a]^{\sharp}$ seminorm, which is, loosely speaking, $L_{\infty}$-norm of the maximal function in $t$ and $BMO$-norm in $x$.