Related papers: A note on measure-geometric Laplacians

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics.…

Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(M),W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein…

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a…

In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=\mu,$$ in a bounded domain $\Omega\subset\mathbb{R}^{N}$, where $A(x,\nabla…

We prove the quasi-invariance of gaussian measures (supported by functions of increasing Sobolev regularity) under the flow of one dimensional Hamiltonian PDE's such as the regularized long wave (BBM) equation.

In 1961 G.Polya published a paper about the eigenvalues of vibrating membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first eigenvalue…

In this paper, we consider a certain convolutional Laplacian for metric measure spaces and investigate its potential for the statistical analysis of complex objects. The spectrum of that Laplacian serves as a signature of the space under…

We consider the generalised Krein-Feller operator $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ with the natural restrictions that $\mu$ is atomless, the supp$(\nu)\subseteq$supp$(\mu)$…

Bounded symmetric domains carry several natural invariant metrics, for example the Carath\'eodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the…

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -\Delta_g e_\lambda = \lambda^2…

Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $ M \setminus \text{supp}(\pi_* \mu) \neq \emptyset$…

Let $\mu$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(\mu)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with $\mathcal{C}^\alpha$ metric). These coordinates are…

We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal…

We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn…

We obtain characterizations of positive Borel measures $\mu$ on $\B^n$ so that some weighted Hardy-Sobolev are imbedded in $L^p(d\mu)$, where $w$ is an $A_p$ weight in the unit sphere of $\C^n$.

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.),…

We prove that a probability measure on a compact non-singular lamination by hyperbolic Riemann surfaces is harmonic if and only if it is the projection of a measure on the unit tangent bundle such that it is invariant under both the…

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$,…