Geodesic
Variation and $\lambda$-jump inequalities on $H^p$ spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2024), pp. 15-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\phi\in \mathscr{S}$ with $\displaystyle\int\phi (x) dx=1$, and define $$\phi_t(x)=\frac{1}{t^n}\phi \left(\frac{x}{t}\right),$$ and denote the function family $\{\phi_t\ast f(x)\}_{t>0}$ by $\Phi\ast f(x)$. Let $\mathcal{J}$ be a subset of $\mathbb{R}$ (or more generally an ordered index set), and suppose that there exists a constant $C_1$ such that $$\sum_{t\in\mathcal{J}} |\hat{\phi}_t(x)|^2<C_1$$ for all $x\in \mathbb{R}^n$. Then i) There exists a constant $C_2>0$ such that $$\|\mathscr{V}_2(\Phi\ast f)\|_{L^p}\leq C_2\|f\|_{H^p}, \frac{n}{n+1}<p\leq 1$$ for all $f\in H^p(\mathbb{R}^n)$, $\dfrac{n}{n+1}. ii) The $\lambda$-jump operator $N_{\lambda}(\Phi\ast f)$ satisfies $$\|\lambda [N_{\lambda}(\Phi\ast f)]^{1/2}\|_{L^p}\leq C_3\|f\|_{H^p}, \frac{n}{n+1}<p\leq 1,$$ uniformly in $\lambda >0$ for some constant $C_3>0$.
Keywords: Hardy space, variation operator, $\lambda$-jump operator. 
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     author = {S. Demir},
     title = {Variation and $\lambda$-jump inequalities on $H^p$ spaces},
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     pages = {15--19},
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}
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S. Demir. Variation and $\lambda$-jump inequalities on $H^p$ spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2024), pp. 15-19. http://geodesic.mathdoc.fr/item/IVM_2024_4_a1/

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