Fourier Spectrum Characterizations of Clifford $H^{p}$ Spaces on $\mathbf{R}^{n+1}_+$ for $1\leq p \leq \infty$

TL;DR

This paper characterizes Clifford algebra-valued Hardy spaces on  with Fourier spectrum conditions, extending classical harmonic analysis results and connecting to the work of Stein, Weiss, and McIntosh.

Contribution

It provides a Fourier spectrum characterization of Clifford Hardy spaces for all p in [1, ], generalizing previous results and linking to conjugate harmonic systems.

Findings

Fourier spectrum condition  characterizes boundary functions in Clifford Hardy spaces.

Results extend McIntosh's context to all p in [1, ].

Vector-valued Clifford Hardy functions coincide with conjugate harmonic systems.

Abstract

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further the non-tangential boundary limit of some function in $H^p(\mathbf R^{n+1}_+),$ $1\leq p\leq \infty,$ if and only if $\hat{f}=\chi_+\hat{f},$ where $\chi_+(\underline{\xi})=\frac{1}{2}(1+i\frac{\underline \xi}{|\underline \xi|}),$ 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 $1\leq p<\infty.$