Inequalities for the differences of averages on $H^1$ spaces

S. Demir

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2024), pp. 3-14

Résumé

Let $(x_n)$ be a sequence and $\{c_k\}\in \ell^\infty (\mathbb{Z})$ such that $\|c_k\|_{\ell^\infty}\leq 1$. Define $$\mathcal{G}(x_n)=\sup_j\left|\sum_{k=0}^j c_k(x_{n_{k+1}}-x_{n_k})\right|.$$ Let now $(X,\beta ,\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\beta ,\mu )$ a totally $\sigma$-finite measure space. Suppose that the sequence $(n_k)$ is lacunary. Then we prove the following results: (i) Define $\phi_n(x)=\dfrac{1}{n}\chi_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that $$\|\mathcal{G}(\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$$ for all $f\in H^1(\mathbb{R})$, (ii) Let $$A_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(\tau^kx)$$ be the usual ergodic averages in ergodic theory. Then $$\|\mathcal{G}(A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$$ for all $f\in H^1(X)$, (iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{G}(A_nf)$ is integrable.

Détail

Keywords: difference sequence, ergodic Hardy space, ergodic Average, lacunary sequence.

Bibliographie

[1] Calderón A.P., “Ergodic theory and translation-invariant operators”, Proc. Nat. Acad. Sci. USA, 59:2 (1968), 349–353 | DOI | MR | Zbl

[2] Demir S., “Variational inequalities for the differences of averages over lacunary sequences”, New York J. Math., 28 (2022), 1099–1111 | MR | Zbl

[3] Demir S., “A generalizaition of Calderón transfer principle”, J. Comp. Math. Sci., 9:5 (2018), 325–329

[4] Coifman R., Weiss G., “Maximal Function and $H^p$ Spaces Defined by Ergodic Transformations”, Proc. Natl. Acad. Sci. USA, 70:6 (1973), 1761–1763 | DOI | MR | Zbl

[5] Caballero R., de la Torre A., “An atomic theory of ergodic $H^p$ spaces”, Studia Math., 82:1 (1985), 39–69 | DOI | MR

[6] Ornstein D., “A remark on Birkhoff ergodic theorem”, Illinois J. Math., 15 (1971), 77–79 | MR | Zbl

[7] Demir S., $H^p$ Spaces and Inequalities in Ergodic Theory, Ph.D Thesis, University of Illinois at Urbana-Champaign, USA, May 1999 | MR | Zbl