Mixed Norm Bergman--Morrey-type Spaces on the Unit Disc

A. N. Karapetyants ; S. G. Samko

Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 47-58

Résumé

We introduce and study the mixed-norm Bergman–Morrey space $\mathscr A^{q;p,\lambda}(\mathbb D)$, mixed-norm Bergman–Morrey space of local type $\mathscr A_{\mathrm{loc}}^{q;p,\lambda}(\mathbb D)$, and mixed-norm Bergman–Morrey space of complementary type ${^{\complement}\!}\mathscr A^{q;p,\lambda}(\mathbb D)$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. The mixed norm Lebesgue–Morrey space $\mathscr L^{q;p,\lambda}(\mathbb D)$ is defined by the requirement that the sequence of Morrey $L^{p,\lambda}(I)$-norms of the Fourier coefficients of a function $f$ belongs to $l^q$ ($I=(0,1)$). Then, $\mathscr A^{q;p,\lambda}(\mathbb D)$ is defined as the subspace of analytic functions in $\mathscr L^{q;p,\lambda}(\mathbb D)$. Two other spaces $\mathscr A_{\mathrm{loc}}^{q;p,\lambda}(\mathbb D)$ and ${^{\complement}\!}\mathscr A^{q;p,\lambda}(\mathbb D)$ are defined similarly by using the local Morrey $L_{\mathrm{loc}}^{p,\lambda}(I)$-norm and the complementary Morrey ${^{\complement}\!}L^{p,\lambda}(I)$-norm respectively. The introduced spaces inherit features of both Bergman and Morrey spaces and, therefore, we call them Bergman–Morrey-type spaces. We prove the boundedness of the Bergman projection and reveal some facts on equivalent description of these spaces.

Détail

Keywords: Bergman–Morrey-type space, mixed norm.