\(L^2\)-estimates for the \(d\)-equation and Witten's proof of the Morse inequalities
Let \(\varphi \) be a nondegenerate Morse function (i.e., the Hessian of \(\varphi\) is nondegenerate at each critical point) in \(\mathbb R^n\) and assume that \(|d\varphi |\) is bounded from below at infinity and that \(|D^2\varphi | \leq C |d\varphi |^2,\) outside of a compact subset, where \(D^2\) stands for any second order derivative. Suppose further that \(\varphi \) has no critical point of index 1 (i.e., the Hessian has one negative eigenvalue). Under these assumptions it is shown that for any \(d\)-closed 1-form \(f\) in \(\mathbb R^n\) and for any \(t>t_0\) one can solve the equation \(du=f\) with a function \(u\) satisfying the estimate \(\int u^2 \, e^{-2t\varphi } \leq C/t \, \int |f|^2\, e^{-2t\varphi }.\) In particular, if \(u\in L^2(e^{-2t\varphi } )\cap L^1(e^{-2t\varphi } ),\) then \(\int (u-u_t)^2 \, e^{-2t\varphi } \leq C/t \, \int |du|^2\, e^{-2t\varphi }\) holds where \(u_t=\int u \, e^{-2t\varphi } / \int e^{-2t\varphi }.\) The last statement is a weakened version of the so-called Brascamp-Lieb inequality for non-convex weights [\textit{H. J. Brascamp} and \textit{E. H. Lieb}, J. Funct. Anal. 22, 366--389 (1976; Zbl 0334.26009)]. For the proof of this result, the author uses the method of Hörmander's \(L^2\) estimates for the \(\overline \partial \)-equation. In the sequel, this result is used to explain Witten's proof of the Morse inequalities [\textit{E. Witten}, J. Differ. Geom. 17, 661--692 (1982; Zbl 0499.53056)].
scientific article; zbMATH DE number 5363339
scientific article