K-Closedness for Weighted Hardy Spaces on the Torus ????2

Abstract

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

\( \left({H}_r\left({w}_1\left(\cdot, \cdot \right)\right),{H}_s\left({w}_2\left(\cdot, \cdot \right)\right)\right) \)

on the two-dimensional torus ????² is K-closed in the couple (L_r(w₁( · , · )), L_s(w₂( · , · ))). For 0 < r < s < 1, the condition w₁, w₂ ∈ A_∞ suffices (A_∞ is the Muckenhoupt condition over rectangles). For 0 < r < 1 < s < ∞, it suffices that w₁ ∈ A_∞ and w₂ ∈ As. For 1 < r < s = ∞, we assume that the weights are of the form w_i(z₁, z₂) = a_i(z₁)ui(z₁, z₂)b_i(z₂), and then the following conditions suffice: u₁ ∈ A_p, u₂ ∈ A₁, \( {u}_2^p{u}_1\in {\mathrm{A}}_{\infty } \) , 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 w₁, w₂ ∈ A₁ and w₁w₂ ∈ A_∞ suffice.