A Derivative Hilbert operator acting from Bergman spaces to Hardy spaces

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ \mathcal{H}_{\mu} = (\mu_{n, k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $, where $ \mu_{n} = \int_{[0, 1)}t^nd\mu(t) $, formally induces the operator as follows: $ \mathcal{DH}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , \; z\in \mathbb{D}, $ where $ f(z) = \sum_{n = 0}^\infty a_nz^n $ is an analytic function in $ \mathbb{D} $. In this article, we characterize those positive Borel measures on $ [0, 1) $ such that $ \mathcal{DH}_\mu $ is bounded (resp., compact) from Bergman spaces $ \mathcal{A}^p $ into Hardy spaces $ H^q $, where $ 0 < p, q < \infty $.