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

The boundedness of the small Hankel operator h(f)(nu) (g) = P-nu(fg), induced by an analytic symbol f and the Bergman projection P(nu )associated to nu, acting from the weighted Bergman space A(omega)(p) to A(nu)(q) is characterized on the full range 0 < p, q < infinity of parameters when omega and nu belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of h(f)(nu) 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 A(eta)(q) = A(omega)(1)(p) circle dot A(nu)(2)(p), where 1 < q, p(1), p(2) < infinity such that q(-1) = p(1)(-1) + p(2)(-1) and eta(1/q) x omega(1)/p(1) nu(1)/p(2). Here tau(r) = integral(1 )(r)tau(t) dt/(1 - r) for all 0 <= r < 1.(c) 2023 Elsevier Inc. All rights reserved.