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Zapiski Nauchnykh Seminarov POMI, 2017, Volume 456, Pages 25–36
(Mi znsl6419)
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This article is cited in 6 scientific papers (total in 6 papers)
$K$-closedness for weighted Hardy spaces on the torus $\mathbb T^2$
V. A. Borovitskiyab a St. Petersburg State University, St. Petersburg, Russia
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
Abstract:
Certain sufficient conditions are established for the couple of weighted Hardy spaces $(H_r(w_1(\cdot,\cdot)),H_s(w_2(\cdot,\cdot)))$ on the two-dimensional torus $\mathbb T^2$ to be $K$-closed in the couple $(L_r(w_1(\cdot,\cdot)),L_s(w_2(\cdot,\cdot)))$. For $0<r<s<1$ the condition $w_1,w_2\in A_\infty$ suffices ($A_\infty$ is the Muckenhoupt condition over rectangles). For $0<r<1<s<\infty$ it suffices that $w_1\in A_\infty$, $w_2\in A_s$. For $1<r<s=\infty$, we assume that the weights are of the form $w_i(z_1,z_2)=a_i(z_1)u_i(z_1,z_2)b_i(z_2)$, and then the following conditions suffice: $u_1\in A_p$, $u_2\in A_1$, $u_2^pu_1\in A_\infty$, $\log a_i,\log b_i\in BMO$. The last statement generalizes the previously known result for the case of $u_i\equiv1$, $i=1,2$. Finally, for $r=1$, $s=\infty$, the conditions $w_1,w_2\in A_1$, $w_1w_2\in A_\infty$ suffice.
Key words and phrases:
Hardy classes, $K$-closedness, the space BMO, Muckenhoupt conditions.
Received: 05.06.2017
Citation:
V. A. Borovitskiy, “$K$-closedness for weighted Hardy spaces on the torus $\mathbb T^2$”, Investigations on linear operators and function theory. Part 45, Zap. Nauchn. Sem. POMI, 456, POMI, St. Petersburg, 2017, 25–36; J. Math. Sci. (N. Y.), 234:3 (2018), 282–289
Linking options:
https://www.mathnet.ru/eng/znsl6419 https://www.mathnet.ru/eng/znsl/v456/p25
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