Characterizations of Morrey type spaces

• Published in: Canadian mathematical bulletin Vol. 65; no. 2; pp. 328 - 344
• Main Authors: Sun, Fangmei, Wulan, Hasi
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical Society 01.06.2022; Cambridge University Press
• Subjects: Fractional calculus; Mathematical analysis; Partial differential equations; Morrey type space; 30D99; 47B38; fractional order derivative; embedding map; 30H25; K-Carleson measure; 30D45
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439521000308

For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $ , we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$ , denoted by $\mathcal {D}^s_K$ . Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$ , we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.