Fourier spectrum of Clifford Hp spaces on [formula omitted] for 1 ≤ p ≤ ∞

Abstract

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces Hp(R+n+1),1≤p≤∞. Namely, for f∈Lp(Rn), Clifford algebra-valued, f is further the non-tangential boundary limit of some function in Hp(R+n+1), 1≤p≤∞, if and only if fˆ=χ+fˆ, where χ+(ξ_)=12(1+iξ_|ξ_|), 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 singular integral version of the vector-valued cases for 1≤p<∞. We also obtain the generalized conjugate harmonic systems for the whole Clifford algebra-valued Hardy spaces rather than only the vector-valued cases in the Stein-Weiss setting.