Small Hankel operator induced by measurable symbol acting on weighted Bergman spaces

The boundedness of the small Hankel operator $h^\omega_{f}(g)=\overline{P_\omega}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_\omega$ associated to a radial weight $\omega$ acting from the weighted Bergman space $A^p_\omega$ to its conjugate analytic counterpart $\overline{A^p_\omega}$ is characterized on the range $1<p<\infty$ when $\omega$ belongs to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. On the way to the proof a sharp integral estimate for certain modified Bergman kernels is obtained.