The canonical solution operator to \(\overline{\partial}\) restricted to Bergman spaces (Q2731927)

Let \(\Omega\) be a bounded domain in \(\mathbb{C}^n\) and let \(A^2(\Omega)\) denote the Bergman space of all holomorphic functions \(f:\Omega\rightarrow \mathbb{C}\) such that NEWLINE\[NEWLINE \int_\Omega |f(z)|^2 dm(z)<\infty NEWLINE\]NEWLINE where \(dm\) denotes the Lebesgue measure in \(\mathbb{C}^n\). The author constructs an integral kernel to solve the inhomogeneous Cauchy-Riemann equation \(\overline\partial u=g\), where \(g=\sum_{j=1}^n g_jd\overline z_j\) is a \((0,1)\)-form with coefficients \(g_j\in A^2(\Omega)\), \(j=1,\dots,n\). In this case, the coefficients \(g_j\), \(j=1,\dots,n\) can be expressed by an integral operator using the Bergman kernel. The author uses this result to prove that in the case of the unit disc in \(\mathbb{C}\), the canonical solution operator to \(\overline\partial\) restricted to \((0,1)\)-forms with holomorphic coefficients is a Hilbert-Schmidt operator.