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

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.