Characrterization of H ^p in terms of Riesz transforms

Abstract

Theorem 17.1. Let f ∈ L ²(R ⁿ). Let

$$ m\left\{ {\begin{array}{*{20}c} { \in {\text{N }}if{\text{ }}n \in \left\{ {2,3,4, \cdots } \right\}} \\ { = 1{\text{ }}if{\text{ }}n = 1,} \\ \end{array} } \right. $$

(17.1)

and

$$ p \in \left( {\frac{{n - 1}} {{n + m - 1}},1} \right]. $$

(17.2)

Then

$$ \left\| {\vec R^m f} \right\|_{H^p } \leqslant C\left( {p,m,n} \right)\left\| {\vec R^m f} \right\|_{L^{p.} } $$

(17.3)

Corollary 17.1. Let f, m and p be as in Theorem 17.1. Then

$$ \left\| f \right\|_{H^p } \leqslant C\left( {p,m,n} \right)\left\{ {\left\| f \right\|_{L^p } + \sum\limits_{k = 1}^m {\sum\limits_{j_1 = 1}^n { \cdots \sum\limits_{j_k = 1}^n {\left\| {R_{j_1 } R_{j_2 } \cdots R_{j_k } f} \right\|_{L^p } } } } } \right\} $$

.