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This article studies the Fourier spectrum characterization of functions inthe 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, $f$ isfurther the non-tangential boundary limit of some function in $H^p(\mathbfR^{n+1}_+),$ $1\leq p\leq \infty,$ 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 suitablyinterpreted (for some cases in the distribution sense). These results furtherdevelop the relevant context of Alan McIntosh. As a particular case of ourresults, the vector-valued Clifford Hardy space functions are identical withthe conjugate harmonic systems in the work of Stein and Weiss. The latterproved the corresponding results in terms of the single integral form for thecases $1\leq p<\infty.$

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