Function spaces in the unit ball of \(\mathbb{C}^n\) (Q2738501)

This is a survey of various results about function spaces in the unit Euclidean ball of \(\mathbb{C}^n\) and the mappings between them. Many proofs are provided, sometimes simplifying the original ones, and a few small improvements are added. NEWLINENEWLINENEWLINESections 1 and 2 of the paper recall notations and definitions, all from \textit{W. Rudin} [`Function theory in the unit ball of \(\mathbb{C}^n\). Springer, New York (1980; Zbl 0495.32001)], about automorphisms of the ball and its Hardy and Bergman spaces. A holomorphic function \(f\) is in the Hardy space \(H^p\) if and only if the quantities NEWLINE\[NEWLINE M_p(r,f) := \int_S |f(r\zeta)|^p d\sigma(\zeta), 0\leq r <1, NEWLINE\]NEWLINE are uniformly bounded, where \(S\) stands for the unit sphere and \(\sigma\) for the normalized Lebesgue measure on \(S\) ; \(f\in A^p\), the Bergman space, if and only if \(M_p(.,f)\) is integrable. Expressions of the quantities \(M_p(r,f)\) in terms of the radial derivative and of the invariant gradient of \(f\) are given in Section 3, coming from \textit{F. Beatrous jun.} and \textit{J. Burbea} [Kodai Math. J. 8, 36-51 (1985; Zbl 0571.32005)], \textit{M. Stoll} [J. Lond. Math. Soc., II. Ser. 48, No. 1, 126-136 (1993; Zbl 0808.32006)], \textit{C. Ouyang, W. Yang} and \textit{R. Zhao} [Trans. Am. Math. Soc. 347, No. 11, 4301-4313 (1995; Zbl 0849.32005)], and \textit{M. Nowak} [Ann. Acad. Sci. Fenn., Math. 23, No. 2, 461-473 (1998; Zbl 0910.32006)]. Necessary conditions for the zero sets of Hardy or Bergman space functions are given. NEWLINENEWLINENEWLINESections 4 and 5 concern the Bloch space \(\mathcal B\) (resp. the little Bloch space \(\mathcal B_0\)), which can be defined as those holomorphic functions \(f\) such that \((1-|z|^2) |\nabla f(z)|\) is bounded, (resp. tends to zero as \(|z|\to 1\)) ; there are many equivalent ways of defining those spaces, for \(n>1\) as well as for the classical case \(n=1\) [\textit{R. M. Timoney}, Bull. Lond. Math. Soc. 12, 241-267 (1980; Zbl 0416.32010); \textit{K. T. Hahn} and \textit{E. H. Youssfi}, Complex Variables, Theory Appl. 17, No. 1-2, 89-104 (1991; Zbl 0770.47006)]. Other equivalent characterizations of the Bloch space are given, taken from the papers mentioned above of Ouyang, Yang and Zhao, and of the author herself. A theorem proved for the unit disk in \textit{S. Axler} [Duke Math. J. 53, 315-332 (1986; Zbl 0633.47014)] is proved in the case \(n>1\): given \(f\in A^2\), the norm of the Hankel operator \(H_{\bar f}\) is equivalent to the Bloch norm of \(f\). A result about the compactness of the product of two operators \(H_{\bar g}^*H_{\bar f}\) is mentioned from \textit{M. Nowak} [Proc. Am. Math. Soc. 126, 2005-2012 (1998; Zbl 0898.32005)]. NEWLINENEWLINENEWLINESection 6 is devoted to a characterization of the Bloch space and the wider class of Besov spaces (see the above mentioned paper of Hahn and Youssfi for a definition) in terms of divided differences, from \textit{M. Nowak} [Complex Variables, Theory Appl. 44, 1-12 (2001; Zbl 1026.32011)]. NEWLINENEWLINENEWLINESection 7 introduces a generalization of the BMOA space known as the \(Q_p\) spaces [see \textit{C. Ouyang, W. Yang} and \textit{R. Zhao}, Pac. J. Math. 182, 69-99 (1998; Zbl 0893.32005)], and characterizes them in terms of \(p\)-Carleson measures, as was done for \(n=1\) in \textit{R. Aulaskari, D. A. Stegenga} and \textit{X. Jie} [Rocky Mt. J. Math. 26, 485-506 (1996; Zbl 0861.30033)].NEWLINENEWLINEFor the entire collection see [Zbl 0957.00035].