Serre duality and Hörmander's solution of the \(\overline{\partial}\)-equation (Q310071)

Let \(L^{2,c}_{(p,q)}(\Omega,e^\phi)\) denote the compactly supported weighted square integrable \((p,q)\)-forms with weight \(e^\phi\) and \(H_p(\Omega, e^{-\phi})\subset L^2_{(p,0)}(\Omega,e^{-\phi})\) denote the \((p,0)\)-forms with holomorphic coefficients on \(\Omega\). In the theorem below \(\omega\in L^2_{(p,n)}(\Omega,e^\phi)\) is said to be orthogonal to \(H_{n-p}(\Omega, e^{-\phi})\) if \(\int_\Omega\omega\wedge f=0\) for all \(f\in H_{n-p}(\Omega, e^{-\phi})\). The author proves the following theorem generalizing \textit{H. Hedenmalm}'s result on \(\mathbb C\) [Math. Z. 281, No. 1, 349--355 (2015; Zbl 1351.30041)].NEWLINENEWLINETheorem. Let \(\Omega\) be a pseudoconvex domain in \(\mathbb C^n\), \(\phi\) be a real-valued \(C^2\)-smooth strictly plurisubharmonic function on \(\Omega\), and \(c_\phi(z)\) denote the smallest eigenvalue of the complex Hessian of \(\phi\) and \(z\). Let \(\omega\in L^{2,c}_{(p,q)}(\Omega,e^{\phi})\) satisfy the following property: \(\omega\) is \(\overline{\partial}\)-closed if \(1\leq q\leq n-1\) and \(\omega\) is orthogonal to \(H_{n-p}(\Omega,e^{-\phi})\) if \(q=n\). Then there exists \(u\in L^2_{(p,q-1)}(\Omega,c_{\phi}e^\phi)\) such that \(\overline{\partial} u=\omega\) and \(\| u\|_{L^2(\Omega,c_\phi e^\phi)}\leq C \|\omega\|_{L^2(\Omega,e^\phi)}\) where \(C>0\) is independent of \(u\) and \(\omega\).