$K$-closedness for weighted Hardy spaces on the torus $\mathbb T^2$

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.