Some properties of elliptic Riesz means at critical index on H^p(T^H)

Abstract

Suppose f∈H^p(Tⁿ), 0<p<1. The elliptic Riesz means of f at critical index are denoted by σ ^δ _r , δ=n/p−(n+1)/2. In this paper we eastablish the following inequality

$$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$

It implies that

$$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$

Moreover we obtain the same conclusion when p=1 and n=1.