Fourier Spectrum Characterizations of $H^{p}$ Spaces on Tubes Over Cones for $1\le p \le \infty$

Abstract

We give Fourier spectrum characterizations of functions in the Hardy $H^p$ spaces on tubes for $1\le p \le \infty .$ For $F\in L^p(\mathbb {R}^n),$ we show that F is the non-tangential boundary limit of a function in a Hardy space, $H^{p}(T_\Gamma ),$ where $\Gamma$ is an open cone of $\mathbb {R}^n$ and $T_\Gamma$ is the related tube in $\mathbb {C}^n,$ if and only if the classical or the distributional Fourier transform of F is supported in $\Gamma ^*,$ where $\Gamma ^*$ is the dual cone of $\Gamma .$ This generalizes the results of Stein and Weiss for $p=2$ in the same context, as well as those of Qian et al. in one complex variable for $1\le p\le \infty .$ Furthermore, we extend the Poisson and Cauchy integral representation formulas to the $H^p$ spaces on tubes for $p\in [1, \infty ]$ and $p\in [1,\infty ),$ with, respectively, the two types of representations.