Solutions of the \(\bar\partial\)-equation with compact support on Stein and Kähler manifold (Q2040225)

Let \(M\) be a complex manifold and \(\Lambda_{p,q} ( \bar M)\) the set of \(\mathcal C^\infty \ (p,q)\)-forms on \(\bar M\). Let \(\mathcal H_{p,q}= \{ h \in \Lambda_{p,q} ( \bar M): \bar \partial h =\bar \partial^* h=0\}\). In the first part of this paper the following result is proved: let \(X\) be a Stein manifold and \(\omega\) be a \((p,q)\)-form in \(L^r(X)\), \(r>1\) with compact support in \(X\). Suppose in addition that \(\bar \partial \omega =0\) if \(1\le q <n\), and that for each \(V\subset X\) with \(\mathrm{Supp}\, \omega \subset V\) one has \(\omega \perp \mathcal H_{n-p,0}(V)\) if \(q=n\). Then there exists a \((p,q-1)\)-form \(u\in W^{1,r}(X)\) with compact support in \(X\) such that \(\bar \partial u= \omega\) and \(\| u \|_{W^{1,r}(\Omega)} \le C \| \omega \|_{L^r(\Omega)}\), where \(\Omega\) is a relatively compact domain in \(X\). Using completely different methods the author also shows that, if \(X\) is a complete oriented Riemannian manifold and \(\omega \in L^r_p(\Omega )\) with compact support being orthogonal to the harmonic \(p\)-forms, there exists a \(p\)-form \(u\in W^{2,r}_p(\Omega )\) with compact support in \(\Omega\) such that \(\triangle u =\omega\) and \(\| u \|_{W^{2,r}_p(\Omega)} \le C \| \omega \|_{L^r_p(\Omega)}\).