Bounded compact and dual compact approximation properties of Hardy spaces: new results and open problems

TY - JOUR

T1 - Bounded compact and dual compact approximation properties of Hardy spaces: new results and open problems

AU - Shargorodsky, Eugene

AU - Karlovych, Oleksiy

N1 - Funding Information: This work is funded by national funds through the FCT - Fundação para a Ciẽncia e a Tecnologia, Portugal , I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications). Publisher Copyright: © 2023 The Author(s)

PY - 2024/1/5

Y1 - 2024/1/5

N2 - The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H[X(w)] built upon translation-invariant Banach function spaces X with weights w such that w∈X and w −1∈X ′, where X ′ is the associate space of X. We prove that if X is separable, then H[X(w)] has the BCAP with the approximation constant M(H[X(w)])≤2. Moreover, if X is reflexive, then H[X(w)] has the BCAP and the DCAP with the approximation constants M(H[X(w)])≤2 and M ∗(H[X(w)])≤2, respectively. In the case of classical weighted Hardy space H p(w)=H[L p(w)] with 1p(w))≤2 |1−2/p| and M ∗(H p(w))≤2 |1−2/p|.

AB - The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H[X(w)] built upon translation-invariant Banach function spaces X with weights w such that w∈X and w −1∈X ′, where X ′ is the associate space of X. We prove that if X is separable, then H[X(w)] has the BCAP with the approximation constant M(H[X(w)])≤2. Moreover, if X is reflexive, then H[X(w)] has the BCAP and the DCAP with the approximation constants M(H[X(w)])≤2 and M ∗(H[X(w)])≤2, respectively. In the case of classical weighted Hardy space H p(w)=H[L p(w)] with 1p(w))≤2 |1−2/p| and M ∗(H p(w))≤2 |1−2/p|.

KW - Bounded compact and dual compact approximation properties

KW - translation-invariant Banach function space

KW - weighted Hardy space

UR - http://www.scopus.com/inward/record.url?scp=85175318139&partnerID=8YFLogxK

U2 - 10.1016/j.indag.2023.10.004

DO - 10.1016/j.indag.2023.10.004

M3 - Article

SN - 0019-3577

VL - 35

SP - 143

EP - 158

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 1

ER -