Square function characterizations of real and ergodic H^1 spaces

İngilizce (OpenAlex)

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.