Duality and multipliers for mixed norm spaces
In the paper the authors consider a number of questions concerning duality and multipliers for the spaces \(A^{p,q,\alpha}\), where for \(p>0,\quad q>0,\quad \alpha<-1, A^{p,q,\alpha}\) denotes the space of functions f, holomorphic in the unit disc, for which \[ \int^{1}_{0}(1-r)^{\alpha}M_ q(r,f)^ pdr<\infty,\quad where \] \[ M_ q(r,f)^ q=(1/2\pi)\int^{2\pi}_{0}| f(re^{i\theta})|^ qd\theta, \] with the usual understanding if p or \(q=\infty.\) In the first section, the authors deal with the dual space of \(A^{p,q,\alpha}\), listing some known results and calculating the dual for the cases \(q=1\), \(1<p<\infty,\) and \(q=\infty, 1<p<\infty.\) In the second section, they use the duality results of the first section along with a method of \textit{J. M. Anderson} and \textit{A. L. Shields} [Trans. Am. Math. Soc. 224, 255-265 (1976; Zbl 0352.30032)] to calculate the coefficient multipliers of some of the spaces \(A^{p,q,\alpha}\). In the final section, the authors introduce some special mixed norm spaces \(D^{p,q}\) and give a number of results for these spaces, including results on duality, multipliers, and fractional derivatives.
scientific article; zbMATH DE number 3855462