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The special atom space and Haar wavelets in higher dimensions

Kwessi Eddy, Souza de Geraldo, Djitte Ngalla, Ndiaye Mariama

Demonstratio Mathematica

2020

Vol. 53, nr 1

131--151

In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spacessuch as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higherdimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.