Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that \(X^{(\alpha)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.
Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $\Delta f$ (Assumptions H1-H3), with growth exponent $\alpha>0$. We study the drift Laplacian $L_f=\Delta-\langle\nabla f,\nabla\cdot\rangle$ on $L^2(M,...
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on e...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
For a (not necessarily smooth) bounded domain Ω$\Omega$ of RN$\mathbb {R}^N$ , N⩾2$N \geqslant 2$ and a Carathéodory vector‐valued function a:Ω×RN→RN$a:\Omega \times \mathbb {R}^N \rightarrow \mathbb {R}^N$ , we study the compactness of the inverse of the Leray–Lions operator A(u)=−div(a(x,∇u))$A(u)=-\text{div}(a(x, \n...
D. Arcoya, M. C. Rezende, E. A. Silva· Journal of the London Mathem...· 0 citations
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