Generalized Hilbert operators on weighted Bergman spaces

By J. A. Pel'aez, Jouni Rattya. 2012

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Abstract The main purpose of this paper is to study the generalized Hilbert operator H g ( f ) ( z ) = ∫ 0 1 f ( t ) g ′ ( t z ) d t acting on the weighted Bergman space A ω p , where the weight function ω belongs to the class R of regular radial weights and satisfies the Muckenhoupt type condition ( † ) sup 0 ≤ r 1 ( ∫ r 1 ( ∫ t 1 ω ( s ) d s ) − p ′ p d t ) p p ′ ∫ 0 r ( 1 − t ) − p ( ∫ t 1 ω ( s ) d s ) d t ∞ . If q = p , the condition on g that characterizes the boundedness (or the compactness) of H g : A ω p → A ω q depends on p only, but the situation is completely different in the case q ≠ p in which the inducing weight ω plays a crucial role. The results obtained also reveal a natural connection to the Muckenhoupt type condition ( † ) . Indeed, it is shown that the classical Hilbert operator (the case g ( z ) = log 1 1 − z of H g ) is bounded from L ∫ t 1 ω ( s ) d s p ( [ 0 , 1 ) ) (the natural restriction of A ω p to functions defined on [ 0 , 1 ) ) to A ω p if and only if ω satisfies the condition ( † ) . On the way to these results decomposition norms for the weighted Bergman space A ω p are established.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/315b3683791ce9f5d95eb74c648d7f0ca81f503a

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