Square Function Characterizations of Real and Ergodic H¹ Spaces

Abstract

Let $({{n}_{k}})$ be a lacunary sequence with no nontrivial common divisor and $f \in {{L}^{1}}(\mathbb{R})$. Define the square function

$$Sf(x) = {{\left( {\sum\limits_{k = 1}^\infty {{{\left| {\frac{1}{{{{n}_{{k + 1}}}}}\int\limits_0^{{{n}_{{k + 1}}}} {f(x - t)dt} - \frac{1}{{{{n}_{k}}}}\int\limits_0^{{{n}_{k}}} {f(x - t)dt} } \right|}}^{2}}} \right)}^{{1/2}}}.$$

We show that there exist constants$A$ and $B$ such that

$${{\left\| f \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant A{{\left\| {Sf} \right\|}_{{{{L}^{1}}(\mathbb{R})}}}\quad {\text{and}}\quad {{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}} \leqslant B{{\left\| {Sf} \right\|}_{{{{L}^{1}}(\mathbb{R})}}},$$

for all $f \in {{L}^{1}}(\mathbb{R})$. Let $(X,\mathcal{B},\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\mathcal{B},\mu )$ a totally $\sigma$-finite measure space. Let us consider the usual ergodic averages

$${{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{i = 0}^{n - 1} \,f({{\tau }^{i}}x),$$

and define the ergodic square function

$$\mathcal{S}f(x) = {{\left( {\sum\limits_{k = 1}^\infty {{{\left| {{{A}_{{{{n}_{{k + 1}}}}}}f(x) - {{A}_{{{{n}_{k}}}}}f(x)} \right|}}^{2}}} \right)}^{{1/2}}}.$$

We also show that

$${{\left\| f \right\|}_{{{{L}^{1}}(X)}}} \leqslant A{{\left\| {\mathcal{S}f} \right\|}_{{{{L}^{1}}(X)}}}\;\;{\text{and}}\;\;{{\left\| f \right\|}_{{{{H}^{1}}(X)}}} \leqslant B{{\left\| {\mathcal{S}f} \right\|}_{{{{L}^{1}}(X)}}},$$

for all $f \in {{L}^{1}}(X)$, where ${{H}^{1}}(X)$ denotes the ergodic Hardy space. Combining these results with the author’s earlier results we also conclude that the square function $Sf$ characterizes the real Hardy space ${{H}^{1}}(\mathbb{R})$, and the ergodic square function $\mathcal{S}f$ characterizes the ergodic Hardy space ${{H}^{1}}(X)$ when the sequence $({{n}_{k}})$ is lacunary.