This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\overline{\partial }$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^{-t\phi }$ where $\phi$ is a nondegenerate Morse function.

We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with $C^{\infty }$-bounded geometry and nonnegative Ricci curvature.

Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $-ℒ$, and let $[b,\infty )$ be the $L^2\left(X\right)$-spectrum of $ℒ$. For $\tau$ in $ℂ$ such that $\mathrm{Re}\tau \ge 0$, let ${𝒫}_{\tau }$ be the operator on $L^2\left(X\right)$ defined formally as $\exp (-\tau {(ℒ-b)}^{1/2})$. In this paper, we obtain $L^p-L^q$ operator norm estimates for ${𝒫}_{\tau }$ for all $\tau$, and show that these are optimal when $\tau$ is small and when $\left|\arg \tau \right|$ is bounded below $\pi /2$.

We extend some results by Gol${}^{\prime }$dshtein, Kuz${}^{\prime }$minov, and Shvedov about the $L_p$-cohomology of warped cylinders to $L_{p,q}$-cohomology for $p\ne q$. As an application, we establish some sufficient conditions for the nontriviality of the $L_{p,q}$-torsion of a surface of revolution.

Let $P_t$ be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with μ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of $P_t$ in L¹(μ) implies its hypercontractivity. Consequently, under this curvature condition L¹-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, L¹-convergence...

Soit $f:U\to V$ une application analytique propre entre des ouverts de ${\mathbf{C}}^n$, soit $X$ un sous-ensemble analytique de $U$ et soit $Z=f^{-1}\left(f\right(X\left)\right)$. On donne des conditions pour que $\overline{Z-X}\cap X$ soit de codimension 1 dans $X$.

Nel caso di una varietà di Banach complessa $X$, si costruisce una regolarizzata della metrica infinitesimale di Kobayashi. Se ne deduce una distanza integrata di Kobayashi e, se $X$ è iperbolica, si mostra che questa distanza è uguale alla distanza di Kobayashi.

Cet article est une présentation rapide, d’une part de résultats de l’auteur et Z. Lu [14], et d’autre part, de la résolution de la conjecture de l’écart fondamental par Andrews et Clutterbuck [1]. Nous commençons par rappeler ce qu’est la géométrie de Bakry-Émery, nous poursuivons en montrant les liens entre valeurs propres du laplacien de Dirichlet et de Neumann. Nous démontrons ensuite un rapport entre l’écart fondamental et la géométrie de Bakry-Émery, puis nous présentons les idées principales...