\(L^ p\) estimates for local solutions of \({\bar \partial}_ b\) on strongly pseudo-convex CR-manifolds
Let M be a strongly pseudo-convex hypersurface in \({\mathbb{C}}^ n\), \(n\geq 3\), and \(z_ 0\in M\). We prove that the tangential Cauchy-Riemann operators are locally solvable with \(L^ p\) estimates away from the top degree (the Lewy case). We show that there exists neighborhood base \(\omega\) of \(z_ 0\) such that for any \({\bar \partial}_ b\)-closed (0,q) form \(\alpha\) on \(\omega\), \(1\leq q\leq n-2\), with \(L^ p\)- coefficients, \(1<p<\infty\), there exists a (0,q-1) form u on \(\omega\) with \(L^ p\)-coefficients satisfying \({\bar \partial}_ bu=\alpha\) in the distribution sense and \[ \| u\|_{L^ p_{(0,q- 1)}(\omega)}\leq c_ p\| \alpha \|_{L^ p_{(0,q)}(\omega)} \] where \(c_ p\) is independent of \(\alpha\). In particular, the range of \({\bar \partial}_ b\) is closed in \(L^ p_{(0,q)}(\omega)\) when \(1\leq q\leq n-2\). This result is in sharp contrast to the top degree case (when \(q=n-1)\) where the range of \({\bar \partial}_ b\) is not closed in \(L^ p_{(0,n-1)}(\omega)\) for any neighborhood \(\omega\) of \(z_ 0\). The \(L^ p\)-estimates are sharp and cannot be improved.
scientific article; zbMATH DE number 4125789
scientific article