K-Closedness for Weighted Hardy Spaces on the Torus 𝕋2

Autor: Viacheslav Borovitskiy
Rok vydání: 2018
Předmět:
Zdroj: Journal of Mathematical Sciences. 234:282-289
ISSN: 1573-8795
1072-3374
DOI: 10.1007/s10958-018-4004-9
Popis: 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 𝕋2 is K-closed in the couple (Lr(w1( · , · )), Ls(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 wi(z1, z2) = ai(z1)ui(z1, z2)bi(z2), and then the following conditions suffice: u1 ∈ Ap, u2 ∈ A1, $$ {u}_2^p{u}_1\in {\mathrm{A}}_{\infty } $$ , and log ai, log bi ∈ BMO. The last statement generalizes the previously known result for the case of ui ≡ 1, i = 1, 2. Finally, for r = 1, s = ∞, the conditions w1, w2 ∈ A1 and w1w2 ∈ A∞ suffice.
Databáze: OpenAIRE
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