Fourier Spectrum Characterizations of Clifford H^p Spaces on Rⁿ⁺¹₊ for 1≤ p ≤ ∞

2017/11/07 by Pei Dang, Dang, Pei, Weixiong Mai +3
Mathematics · #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1711.02610

openalex publication_date 2017/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces H^p(\mathbf Rⁿ⁺¹₊), 1≤ p≤ ∞. Namely, for f∈ L^p(\mathbf Rⁿ), Clifford algebra-valued, f is further the non-tangential boundary limit of some function in H^p(\mathbf Rⁿ⁺¹₊), 1≤ p≤ ∞, if and only if f=χ₊f, where χ₊(\underlineξ)=(1)/(2)(1+i(\underline ξ)/(|\underline ξ|)), where the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIntosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the corresponding results in terms of the single integral form for the cases 1≤ p

Citations

• Hp spaces of several variables

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