Geometric maximal operators and BMO on product bases

TL;DR

This paper investigates the boundedness of maximal operators on BMO spaces over various shape bases in ^n, extending known results to product structures and defining new boundedness spaces.

Contribution

It generalizes the boundedness of maximal operators from cubes to shapes with an engulfing property and introduces rectangular BLO for product shape bases.

Findings

Maximal operators are bounded from BMO to BLO for shape bases with an engulfing property.

Extension of boundedness results to product shape bases lacking an engulfing property.

Introduction of rectangular BLO space for product shape bases.

Abstract

We consider the problem of the boundedness of maximal operators on BMO on shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO.