Geodesic
$K$-closedness for weighted Hardy spaces on the torus $\mathbb T^2$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 25-36 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 the condition $w_1,w_2\in A_\infty$ suffices ($A_\infty$ is the Muckenhoupt condition over rectangles). For $0 it suffices that $w_1\in A_\infty$, $w_2\in A_s$. For $1, 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. 
@article{ZNSL_2017_456_a2,
     author = {V. A. Borovitskiy},
     title = {$K$-closedness for weighted {Hardy} spaces on the torus~$\mathbb T^2$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {25--36},
     year = {2017},
     volume = {456},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a2/}
}
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V. A. Borovitskiy. $K$-closedness for weighted Hardy spaces on the torus $\mathbb T^2$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 25-36. http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a2/

[1] D. V. Rutskii, “Vesovoe razlozhenie Kalderona–Zigmunda i nekotorye ego prilozheniya k interpolyatsii”, Zap. nauchn. sem. POMI, 424, 2014, 186–200 | MR

[2] Xu Quanhua, “Some properties of the quotient space $(L^1(\mathbf T^d)/H^1(D^d))$”, Illinois J. Math., 37:3 (1993), 437–454 | MR | Zbl

[3] J. García-Cuerva, K. Kazarian, “Calderón–Zygmund operators and unconditional bases of weighted Hardy spaces”, Stud. Math., 109:3 (1994), 255–276 | DOI | MR | Zbl

[4] S. V. Kislyakov, “Interpolation involving bounded bianalytic functions”, Operator Theory: Advances and Applications, 113 (2000), 135–149 | MR | Zbl

[5] S. V. Kislyakov, “Interpolation of $H_p$ spaces: some recent developments”, Israel Math. Conf. Proceedings, 13 (1999), 102–140 | MR | Zbl

[6] S. V. Kislyakov, Shu Kuankhua, “Veschestvennaya interpolyatsiya i singulyarnye integraly”, Algebra i analiz, 8:4 (1996), 75–109 | MR | Zbl

[7] D. Freitag, “Real interpolation of weighted $L_p$-spaces”, Math. Nachrichten, 86:1 (1978), 15–18 | DOI | MR | Zbl

[8] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993 | MR | Zbl

[9] J. Bergh, J. Löfström, Interpolation Spaces, Springer, Berlin–Heidelberg, 1976 | Zbl

[10] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, “Nauka”, Moskva, 1984 | MR

[11] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall Inc., 1962 | MR | Zbl

[12] V. A. Borovitskii, K-zamknutost vesovykh prostranstv Khardi na $\mathbb T^2$, arXiv: 1707.05239