Generalized Hilbert operators on weighted Bergman spaces

The main purpose of this paper is to study the generalized Hilbert operator Hg(f)(z)=∫01f(t)g′(tz)dt acting on the weighted Bergman space Aωp, where the weight function ωω belongs to the class RR of regular radial weights and satisfies the Muckenhoupt type condition equation(††)sup0≤r<1(∫r1(∫t1ω(s)ds)−p′pdt)pp′∫0r(1−t)−p(∫t1ω(s)ds)dt<∞. If q=pq=p, the condition on gg that characterizes the boundedness (or the compactness) of Hg:Aωp→Aωq depends on pp only, but the situation is completely different in the case q≠pq≠p in which the inducing weight ωω 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)=log11−z of HgHg) is bounded from L∫t1ω(s)dsp([0,1)) (the natural restriction of Aωp to functions defined on [0,1)[0,1)) to Aωp if and only if ωω satisfies the condition (††). On the way to these results decomposition norms for the weighted Bergman space Aωp are established.