Homotopy formulas for \(\overline{\partial}_ b\) in \(CR\) manifolds with mixed Levi signatures
We obtain local homotopy formulas for the tangential Cauchy-Riemann complex \(\overline \partial_b\) in a neighborhood \(\omega\) of a hypersurface in \(\mathbb{C}^n\) when the Levi form has both positive and negative eigenvalues. If the Levi form has \(q+2\) pairs of positive and negative eigenvalues at some point, we construct explicit integral operators \(E\) and \(F\) in a neighborhood base such that the homotopy formula \(\alpha(z)= \overline \partial_b E\alpha+F \overline \partial_b \alpha\) holds for any \((p,q)\) form \(\alpha\). We first construct the formula on the boundary of a model domain without shrinkage and then on general hypersurfaces. Regularity for the local solution of \(\overline \partial_b\) is obtained in the model case. Examples are given to show that the assumption on the Levi form is also necessary for the existence of homotopy formulas.
scientific article; zbMATH DE number 1028144