Holomorphic Triebel-Lizorkin spaces
Let \(B\) be the unit ball of \(\mathbb C^n\), \(S\) its boundary, \(R\) the radial derivative, \(I\) the identity operator, and \([s]^+\) the integer part of \(s\) plus 1. For \(0<p<\infty\), \(0<q\leq\infty\) and \(s\geq0\), the holomorphic Triebel-Lizorkin space \(H F^{pq}_s\) on the unit ball is the space of all holomorphic functions \(f\) on \(B\) such that \((1-r^2) ^{[s]^+ -s} (I+R)^{[s]^+} f(r\zeta)\) belongs to the mixed norm space \(L^p(L^q(d\mu),d\sigma)\) where \(d\mu\) is the measure \(d\mu(r)=2n r^{2n-1} (1-r^2) dr\) on \([0,1]\) and \(d\sigma(\zeta)\) is the normalized surface measure on \(S\). Generalizing the results of \textit{V. S. Guliev} and \textit{P. I. Lizorkin} for \(n=1\) [ Sov. Math., Dokl. 44, No. 1, 215-219 (1992); translation from Dokl. Akad. Nauk SSSR 319, No. 4, 806-810 (1991; Zbl 0784.46020)] and parallelling the developments for other function spaces (Sobolev, Besov, Hardy, BMOA, etc.), the authors obtain characterizations of the spaces \(H F^{pq}_s\) in terms of Littlewood-Paley type functions, admissible area functions (with admissible regions \(\Gamma_\alpha(\zeta)= \{z\in B: |1-\bar\zeta z|<\alpha(1-|z|) \}\)), complex tangential derivatives, and boundary values (both an isotropic and an anisotropic version). Further results are given concerning duality (the dual of \(H F^{pq}_s\) is \(H F^{p'q'}_s\) if \(1<p<\infty, 1\leq q<\infty, s\geq0\)), embeddings between \(H F^{pq}_s\) and the holomorphic Besov spaces, complex interpolation, traces on submanifolds transversal to \(S\), and traces on simple complex tangential curves in \(S\).
scientific article; zbMATH DE number 1106594
scientific article