Some results on weighted Bergman spaces (Q2704592)

For \(0<p< \infty\) and \(0<\alpha <\infty\) let \(A^{p,\alpha}\) be the Fréchet (resp. Banach) spaces of all analytic functions on the unit disk satisfying NEWLINE\[NEWLINE\|f\|_{A^{p, \alpha}}= \left(\int^1_0 (1-r)^{p\alpha-1}M^p_p (r,f)dr \right)^{1/p}< \infty,NEWLINE\]NEWLINE where \(M_p(r,f)=({1\over 2\pi}\int^{2\pi}_0 |f(re^{it}) |^pdt)^{1/p}\) are the usual \(p\)-th-integral means of \(f\). In section 2 of the paper, the authors give estimates for the Taylor coefficients of functions in \(A^{p,\alpha}\). In section 3 a necessary and sufficient condition for membership in \(A^{p,\alpha}\) is derived. Finally, in section 4, they study the multipliers for \(A^{p,\alpha}\). Here they identify the elements in the function spaces with their Taylor coefficients and call a sequence \((\lambda_n)\) of complex numbers a multiplier from a sequence space \(X\) to a sequence space \(Y\) if \((\lambda_n a_n)\in Y\) whenever \((a_n)\in X\). The space of all such multipliers is denoted by \((X,Y)\). For example it is shown that if \(X=A^{p, \alpha}\), \(0<p\leq 1\), \(\alpha>0\) and \(Y=\ell^\infty\), then \((A^{p, \alpha},\ell^\infty) =\{(\lambda_n) :\lambda_n= O(n^{1- \alpha-1/p})\}\). Special cases are pointed out separately. For related results, it is referred to papers of \textit{D. M. Campbell} and \textit{R. Leach} [Complex Variables, Theory Appl. 3, 85-111 (1984; Zbl 0562.30025)] and to \textit{X. Yue} [ibid. 40, 163-172 (1999)].