Related papers: Sharp microlocal Kakeya--Nikodym estimates for eig…

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the…

We study the boundedness of some sublinear operators on weighted Morrey spaces under certain size conditions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator,…

We develop a generalized Floquet-Bloch theory for discrete torsion-free nilpotent groups by exploiting their Malcev completions. Our main result is a branching formula that relates finite-dimensional representations of a discrete nilpotent…

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds…

We consider eigenfunction estimates in $L^p$ for Schr\"odinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge…

The Hardy--Littlewood inequalities on $\ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $\ell _{p}$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood…

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel…

We investigate the concentration of eigenfunctions for the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^n$ by establishing local $L^p$ bounds over the compact sets with arbitrary dilations and translations. These new results extend the…

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these…

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless…

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\lambda_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla…

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov…

This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…

We prove the self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion…

We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…

On compact Riemannian manifolds with non-negative Ricci curvature and smooth (possibly empty), convex (or mean convex) boundary, if the sharp Li-Yau type gradient estimate of an Neumann (or Dirichlet) eigenfunction holds at some…

We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the non-sharp admissible region of exponents, covering wave, Klein-Gordon, and fractional Schr\"odinger equations. Our approach combines the…

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers,…