Generalized Hilbert operators on weighted Bergman spaces

The main purpose of this paper is to study the generalized Hilbert operator H-g(f)(z) = integral(1)(0) f(t)g' (tz) dt acting on the weighted Bergman space A(omega)(p), where the weight function omega belongs to the class R of regular radial weights and satisfies the Muckenhoupt type condition sup(0 <= r<1) (integral(1)(t)(integral(1)(t) omega(s)ds)(-p/p)dt)(p/p') integral(r)(0) (1 - t)(-p) (integral(1)(t) omega(s)ds) dt < infinity. (dagger) If q = p, the condition on g that characterizes the boundedness (or the compactness) of H-g : A(omega)(p) -> A(omega)(q) depends on p only, but the situation is completely different in the case q not equal p in which the inducing weight omega plays a crucial role. The results obtained also reveal a natural connection to the Muckenhoupt type condition (dagger). Indeed, it is shown that the classical Hilbert operator (the case g(z) = log 1/1-z of H-g) is bounded from L-integral t1 omega(s)ds(p), ([0, 1)) (the natural restriction of A(omega)(p) functions defined on [0, 1)) to A(omega)(p) if and only if a) satisfies the condition (l.). On the way to these results decomposition norms for the weighted Bergman space A(omega)(p) are established. (C) 2013 Elsevier Inc. All rights reserved.