Essential characterization of \(EH^ p\) space \((1 < p < +\infty)\) (Q1898434)

Let \(H^p\), \(1 < p < \infty\), be the usual space of harmonic functions on a half-space \(\mathbb{R}^n \times \mathbb{R}_+\). In another paper of the same author [Chin. Ann. Math., Ser. A 16, No. 2, 150-163 (1995)] there was introduced the space \(EH^p (\mathbb{R}^n \times\mathbb{R}_+) = \{h(x,y)\); \(x \in \mathbb{R}^n, y > 0\), \(\exists f_\beta \in H^p\), \(\beta = (\beta_0, \beta_1, \ldots, \beta_n) \in (\mathbb{Z}_+)^{n + 1}\), \(|\beta |\leq k\), such that \(h(x,y) = \sum_{|\beta |\leq k} D^\beta f_\beta\}\). In the article under review the author states the Theorem: \(EH^p = \{h(x,y)\); \(h(x,y)\) is a harmonic function in \(\mathbb{R}^n \times \mathbb{R}_+ \), and for each \(y > 0\), \(h(x,y) \in L^p (\mathbb{R}^n)\), and there is some \(\alpha = \alpha (h) \geq 0\), \(C = C(h, \alpha)\), such that \[ |h(x,y) |_{L^p (\mathbb{R}^n)} \leq C \text{ if } y \geq 1,\text{ or } \leq C/y^\alpha \text{ if } 0 < y \leq 1\}. \]