Functional Calculus on BMO-type Spaces of Bourgain, Brezis and Mironescu

TL;DR

This paper studies the behavior of nonlinear superposition operators on a new BMO-type space and its subspaces, establishing conditions for their boundedness and continuity, thus advancing understanding of function space mappings.

Contribution

It provides new necessary and sufficient conditions for the boundedness and continuity of superposition operators on BMO-type spaces introduced by Bourgain, Brezis, and Mironescu.

Findings

Characterized when $T_g$ is bounded on BMO-type spaces.

Established criteria for the continuity of $T_g$ on these spaces.

Extended results to VMO and CMO subspaces.

Abstract

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.