Square function characterizations of real and ergodic $H^1$ spaces

S. Demir

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2023), pp. 15-26.

Résumé

Let $(n_k)$ be a lacunary sequence with no non-trivial common divisor and $f\in L^1(\mathbb{R})$. Define the square function $$Sf(x)=\left(\sum_{k=1}^{\infty}\left|\frac{1}{n_{k+1}}\int_{0}^{n_{k+1}}f(x-t) dt-\frac{1}{n_k}\int_{0}^{n_k}f(x-t) dt\right|^2\right)^{1/2}.$$ We show that there exist constants $A$ and $B$ such that $$\|f\|_{L^1(\mathbb{R})}\leq A\|Sf\|_{L^1(\mathbb{R})} \text{and} \|f\|_{H^1(\mathbb{R})}\leq B\|Sf\|_{L^1(\mathbb{R})}$$ for all $f\in L^1(\mathbb{R})$.\Let $(X,\mathscr{B} ,\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\mathscr{B} ,\mu )$ a totally $\sigma$-finite measure space. Let us consider the usual ergodic averages $$A_nf(x)=\frac{1}{n}\sum_{i=0}^{n-1}f(\tau^ix),$$ and define the ergodic square function $$\mathcal{S}f(x)=\left(\sum_{k=1}^{\infty}\left|A_{n_{k+1}}f(x)-A_{n_k}f(x)\right|^2\right)^{1/2}.$$ We also show that $$\|f\|_{L^1(X)}\leq A\|\mathcal{S}f\|_{L^1(X)} \text{and} \|f\|_{H^1(X)}\leq B\|\mathcal{S}f\|_{L^1(X)}$$ 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.

Détail

Keywords: ergodic square function, Hardy space, $H^1$ space, ergodic Hardy space, ergodic $H^1$ space, ergodic average, characterization.

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