We obtain certain sufficient conditions under which the couple of weighted Hardy spaces

on the two-dimensional torus 𝕋^{2} is *K*-closed in the couple (*L*_{r}(*w*_{1}( · , · )), *L*_{s}(*w*_{2}( · , · ))). For 0 < *r* < *s* < 1, the condition *w*_{1}, *w*_{2} ∈ *A*_{∞} suffices (A_{∞} is the Muckenhoupt condition over rectangles). For 0 < r < 1 < s < ∞, it suffices that *w*_{1} ∈ A_{∞} and *w*_{2} ∈ As. For 1 < *r* < *s* = ∞, we assume that the weights are of the form *w*_{i}(*z*_{1}, *z*_{2}) = *a*_{i}(z_{1})*ui*(z_{1}, z_{2})*b*_{i}(*z*_{2}), and then the following conditions suffice: *u*_{1} ∈ *A*_{p}, *u*_{2} ∈ A_{1}, \( {u}_2^p{u}_1\in {\mathrm{A}}_{\infty } \) , and log *a*_{i}, log *b*_{i} ∈ BMO. The last statement generalizes the previously known result for the case of *u*_{i} ≡ 1, *i* = 1, 2. Finally, for *r* = 1, *s* = ∞, the conditions *w*_{1}*, w*_{2} ∈ A_{1} and *w*_{1}*w*_{2} ∈ A_{∞} suffice.

## References

D. V. Rutsky, “Weighted Calderon–Zygmund decomposition with some applications to interpolation,”

*Zap. Nauchn. Semin. POMI*,**242**, 186–200 (2014).Quanhua Xu, “Some properties of the quotient space (

*L*^{1}(**T**^{d})*/H*^{1}(*D*^{d})),”*Illinois J. Math.*,**37**, 437–454 (1993).J. Garca-Cuerva and K. Kazarian, “Calderon–Zygmund operators and unconditional bases of weighted Hardy spaces,”

*Studia Mathematica*,**109**, 255–276 (1994).S. V. Kislyakov, “Interpolation involving bounded bianalytic functions,”

*Operator Theory: Advances and Applications*,**113**, 135–149 (2000).S. V. Kislyakov, “Interpolation of

*H*^{p}spaces: some recent developments,”*Israel Math. Conf. Proceedings*,**13**, 102–140 (1999).S. V. Kislyakov and Quan Hua Xu, “Real interpolation and singular integrals,”

*Algebra Analiz*,**8**, 75–109 (1996).D. Freitag, “Real interpolation of weighted

*L*_{p}-spaces,”*Math. Nachr.*,**86**, 15–18 (1978).E. M. Stein,

*Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals*, Princeton Univ. Press (1993).J. Bergh and J. Lofstrom,

*Interpolation Spaces*, Springer, Berlin-Heidelberg (1976).K. Hoffman,

*Banach Spaces of Analytic Functions*, Prentice Hall Inc. (1962).V. Borovitskiy, “K-closedness of weighted Hardy spaces on the two-dimensional torus,” https://arxiv.org/abs/1707.05239.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Translated from *Zapiski Nauchnykh Seminarov POMI*, Vol. 456, 2017, pp. 25–36.

## Rights and permissions

## About this article

### Cite this article

Borovitskiy, V.A.
*K*-Closedness for Weighted Hardy Spaces on the Torus 𝕋^{2}.
*J Math Sci* **234**, 282–289 (2018). https://doi.org/10.1007/s10958-018-4004-9

Received:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10958-018-4004-9