Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

Abstract

The boundedness of the small Hankel operator hfν(g)=Pν(fg‾), induced by an analytic symbol f and the Bergman projection Pν associated to ν, acting from the weighted Bergman space Aωp to Aνq is characterized on the full range 0<p,q<∞ of parameters when ω and ν belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of hfν is established in terms of a suitable norm of f(n) depending upon the inducing weights and parameters. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization Aηq=Aωp1⊙Aνp2, where 1<q,p1,p2<∞ such that q−1=p1−1+p2−1 and η˜1q≍ω˜1p1ν˜1p2. Here τ˜(r)=∫r1τ(t)dt/(1−r) for all 0≤r<1.