Parabolic BMO and the forward-in-time maximal operator

TL;DR

This paper investigates the boundedness of the parabolic forward-in-time maximal operator on parabolic BMO spaces, revealing positive results for non-negative functions and more complex behavior for sign-changing functions, with implications for PDE regularity.

Contribution

It provides new insights into the boundedness of the maximal operator on parabolic BMO, especially distinguishing between non-negative and sign-changing functions, and extends analysis to one dimension.

Findings

Boundedness holds for non-negative functions.

Sign-changing functions exhibit more delicate behavior.

Results have implications for nonlinear parabolic PDE regularity.

Abstract

We study if the parabolic forward-in-time maximal operator is bounded on parabolic BMO. It turns out that for non-negative functions the answer is positive, but the behaviour of sign changing functions is more delicate. The class parabolic BMO and the forward-in-time maximal operator originate from the regularity theory of nonlinear parabolic partial differential equations. In addition to that context, we also study the question in dimension one.