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

The boundedness of the small Hankel operator $h_f^{\omega }(g)=\overline{P_{\omega }}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_{\omega }$ associated to a radial weight $\omega$ acting from the weighted Bergman space $A_{\omega }^p$ to its conjugate analytic counterpart $\overline{A_{\omega }^p}$ is characterized on the range $1<p<\mathrm{\infty }$ 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.