Volterra type integration operators between weighted Bergman spaces and Hardy spaces

Let $\mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $\mathbb{D}$. We characterize bounded and compact Volterra type integration operators $$J_{g}(f)(z)=\int_{0}^{z}f(\lambda)g'(\lambda)d\lambda$$ between weighted Bergman spaces $L_{a}^{p}(\omega )$ induced by $\mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.