1. Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting
- Author
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Agnieszka Hejna, Jacek Dziubański, and Jean-Philippe Anker
- Subjects
Pure mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,020206 networking & telecommunications ,02 engineering and technology ,Hardy space ,01 natural sciences ,Square (algebra) ,Riesz transform ,symbols.namesake ,Harmonic function ,Fourier analysis ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Maximal function ,0101 mathematics ,Laplace operator ,Analysis ,Mathematics - Abstract
In this work we extend the theory of the classical Hardy space $$H^1$$ to the rational Dunkl setting. Specifically, let $$\Delta $$ be the Dunkl Laplacian on a Euclidean space $$\mathbb {R}^N$$ . On the half-space $$\mathbb {R}_+\times \mathbb {R}^N$$ , we consider systems of conjugate $$(\partial _t^2+\Delta _{\mathbf {x}})$$ -harmonic functions satisfying an appropriate uniform $$L^1$$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $$H^1_{\Delta }$$ , can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.
- Published
- 2019