Operator theoretic differences between Hardy and Dirichlet-type spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:418

Article

Angel Pelaez, Jose¹  Perez-Gonzalez, Fernando²  Rattya, Jouni³ 

[1] Univ Malaga, Dept Anal Matemat, E-29071 Malaga, Spain

[2] Univ La Laguna, Dept Anal Matemat, Tenerife 38271, Spain

[3] Univ Eastern Finland, Joensuu 80101, Finland

关键词: Operator theoretic differences;    Hardy spaces;    Spaces of Dirichlet type;    Integral operators;    Carleson measures;   

DOI  :  10.1016/j.jmaa.2014.03.058

来源: Elsevier

【 摘 要 】

For 0 < infinity, the Dirichlet-type space D-p-1(p) consists of the analytic functions f in the unit disc D such that integral(D) vertical bar f'(z)vertical bar(P) (1- vertical bar z vertical bar)(p-1) dA(z) < infinity. Motivated by operator theoretic differences between the Hardy space HP and /4_1, the integral operator Tg(f)(z) = integral(z)(0) f(zeta)g'(zeta) d zeta, z is an element of D, acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that Tg : D-p-1(p) -> H-P is bounded if and only if g E BMOA when 0 <= 2, and, on the other hand, that this equivalence is very far from being true if p > 2. Those symbols g such that Tg : D-p-1(p) -> H-q is bounded (or compact) when p < q are also characterized. Moreover, the best known sufficient L-infinity-type condition for a positive Borel measure mu, on D to be a p-Carleson measure for D-p-1,(p) p > 2, is significantly relaxed, and the established result is shown to be sharp in a very strong sense. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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