Geodesic
The variation operator of differences of averages over lacunary sequences maps $H^1_w(\mathbb{R})$ to $L^1_w(\mathbb{R})$
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 30-36 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $f$ be a locally integrable function defined on $\mathbb{R}$, and $(n_k)$ be a lacunary sequence. Define $$A_nf(x)=\frac{1}{n}\int_0^nf(x-t) dt,$$ and let $$\mathcal{V}_{\rho}f(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^{\rho}\right)^{1/\rho}.$$ Suppose that $w\in A_p$, $1\leq p<\infty$, and $\rho\geq 2$. Then, there exists a positive constant $C$ such that $$\|\mathcal{V}_{\rho}f\|_{L^1_w}\leq C\|f\|_{H^1_w}$$ for all $f\in H^1_w(\mathbb{R})$.
Keywords: variation operator, weighted Hardy space, $A_p$ weight. 
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     title = {The variation operator of differences of averages over lacunary sequences maps $H^1_w(\mathbb{R})$ to $L^1_w(\mathbb{R})$},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
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}
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S. Demir. The variation operator of differences of averages over lacunary sequences maps $H^1_w(\mathbb{R})$ to $L^1_w(\mathbb{R})$. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 30-36. http://geodesic.mathdoc.fr/item/IVM_2024_5_a2/

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