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arXiv · 2006.13216

Oscillation inequalities on real and ergodic $H^1$ spaces

Sakin Demir
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Abstract

Let $(x_n)$ be a sequence and $\rho\geq 1$. For a fixed sequences $n_1 0$ such that $\|\mathcal{O}_\rho (\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$ for all $f\in H^1(\mathbb{R})$. (ii) Let $A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(\tau^kx)$ be the usual ergodic averages in ergodic theory. Then $\|\mathcal{O}_\rho (A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$ for all $f\in H^1(X)$. (iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{O}_\rho (A_nf)$ is integrable.

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Sakin Demir. 2020-06-23. Oscillation inequalities on real and ergodic $H^1$ spaces. https://arxiv.org/abs/2006.13216

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