We consider the action of the operator ℒg(z) = (1 − z)−1∫z 1f(ζ)dζ on a class of “mixed norm” spaces of analytic functions on the unit disk, X = H α,ν p,q, defined by the requirement g ∈ X⇔r ↦ (1 − r)αMp(r, g(ν)) ∈ Lq([0,1], dr/(1 − r)), where 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 0, and ν is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expression ℒg need not be defined for g analytic in the unit disk, even for g ∈ X. A sufficient, but not necessary, condition is that . We identify the indices p, q, α, and ν for which 1°ℒ is well defined on X, 2°ℒ acts from X to X, 3° the implication holds. Assertion 2° extends some known results, due to Siskakis and others, and contains some new ones. As an application of 3° we have a generalization of Bernstein's theorem on absolute convergence of power series that belong to a Hölder class.
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机译:我们考虑算子ℒg(z)=(1 − z) −1 sup>∫z 1 sup> f(ζ)dζ的作用单位磁盘上的解析函数的总和X = Hα,ν p,q sup>,由需求g∈X⇔r↦(1 − r em>)定义α em> sup> M em> p em>( r em>, g em> ( ν em>) sup>)∈ L em> q em> sup>([0,1], dr < / em> /(1 − r em>)),其中1≤ p em>≤∞,0 em> q em>≤∞,α< / em 0,并且ν em>是一个非负整数。此类包含Besov空间,加权Bergman空间,Dirichlet类型空间,Hardy-Sobolev空间等。无需为单位磁盘中的 g em>分析定义表达式ℒg em>,即使是 g em>∈ X em>。一个充分但不是必需的条件是展开▼