Sparse Brudnyi and John-Nirenberg Spaces

TL;DR

This paper introduces a new class of sparse Brudnyi and John-Nirenberg spaces, connecting local polynomial approximation with sparse domination, and establishes key properties including an analogue of the maximal theorem and a characterization of John-Nirenberg spaces.

Contribution

It generalizes existing theories by linking Brudnyi's local approximation approach with sparse domination techniques, providing new insights into John-Nirenberg spaces and their structure.

Findings

SJN_{p} coincides with L^{p} for 1<p<.

Established an analogue of the maximal theorem for fractional maximal functions.

Provided a new characterization of John-Nirenberg spaces using sparse domination.

Abstract

A generalization of the theory of Y. Brudnyi \cite{yuri}, and A. and Y. Brudnyi \cite{BB20a}, \cite{BB20b}, is presented. Our construction connects Brudnyi's theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak--Kuznetsov. Our spaces shed light on the structure of the John--Nirenberg spaces. We show that $SJN_{p}$ (sparse John--Nirenberg space) coincides with $L^{p},1<p<\infty.$ This characterization yields the John--Nirenberg inequality by extrapolation and is useful in the theory of commutators.