A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations
A geometric and analytic approach to some problems associated with Emden equations
A geometric approach to on-diagonal heat kernel lower bounds on groups
We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product....
A Geometric Construction of the Discrete Series for Semisimple Lie Groups.
A geometric estimate for a periodic Schrödinger operator
We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator with potential given by the curvature of a closed curve.
A horizontal Lévy process on the bundle of orthonormal frames over a complete riemannian manifold
A K-theoretic relative index theorem and Callias-type Dirac operators.
A Lie Group Structure for Fourier Integral Operators.
A Lie Group Structure for Pseudodifferential Operators.
A Lie transformation group attached to a second order elliptic operator.
A local analytic splitting of the holonomy map on flat connections.
A local index theorem for non Kähler manifolds.
A logarithmic Sobolev form of the Li-Yau parabolic inequality.
We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities...
A lower bound for ... on manifolds with boundary.
A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds, II
This article is concerned with estimations from below for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Concentrating on the case of odd ℓ, it continues the work done in part I [21] which dealt with even ℓ.
A lower bound for the least eigenvalue of ... + V.
A lower bound on ... for geometrically finite hyperbolic n-manifolds.
A mean-value lemma and applications
We control the gap between the mean value of a function on a submanifold (or a point), and its mean value on any tube around the submanifold (in fact, we give the exact value of the second derivative of the gap). We apply this formula to obtain comparison theorems between eigenvalues of the Laplace-Beltrami operator, and then to compute the first three terms of the asymptotic time-expansion of a heat diffusion process on convex polyhedrons in euclidean spaces of arbitrary dimension. We also write...
A method of symmetrization ; Applications to heat and spectral estimates
A microlocal F. and M. Riesz theorem with applications.
Consider, by way of example, the following F. and M. Riesz theorem for Rn: Let μ be a finite measure on Rn whose Fourier transform μ* is supported in a closed convex cone which is proper, that is, which contains no entire line. Then μ is absolutely continuous (cf. Stein and Weiss [SW]). Here, as in the sequel, absolutely continuous means with respect to Lebesque measure. In this theorem one can replace the condition on the support of μ* by a similar condition on the wave front set WF(μ) of μ, while...