Tops of dyadic grids

TL;DR

This paper introduces the concept of 'tops' of dyadic grids, extending traditional dyadic cubes to include infinite ones, and explores their role in tiling Euclidean space and in two weight norm inequalities.

Contribution

It extends dyadic grid theory by defining 'tops' that include infinite cubes, linking to tiling and two weight inequalities in harmonic analysis.

Findings

Defined 'tops' as infinite dyadic cubes.

Connected 'tops' to tiling Euclidean space.

Applied 'tops' in two weight norm inequalities.

Abstract

We extend the notion of a dyadic grid of cubes in Euclidean space to include infinite dyadic cubes. These `tops' of a dyadic grid form a tiling of Euclidean space which is subject to the constraints similar to those arising in tiling Euclidean space by (finite) unit cubes. These tops arise in the theory of two weight norm inequalities through weighted Haar and Alpert wavelets.