arXiv research

We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm pro…

New method deforms function algebras on manifolds using spectral decomposition.

New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact …

New tensorization theorem for Sobolev spaces on product spaces.

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are…

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…

Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic …

Global convergence proved for Gursky-Malchiodi $Q$-curvature flow in dimensions $n \geq 5$.

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…

The gradient shrinking $ρ$-Einstein soliton is a triple $(M^n,g,f)$ such that $R_{ij}+f_{ij}=(ρR+λ) g_{ij},$ where $(M^n,g)$ is a Riemannian manifold, $λ>0, ρ\in\mathbb{R}\setminus\{0\}$ and $f$ is the potential function on $M^n$. In this paper, using algebraic curvature estimates and the Yamabe-Sobolev inequality, w…

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.

Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge…

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow …

Proves Sobolev inequality on manifolds with specific curvature properties.

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

Study shows how close functions are to optimal in Riemannian manifolds.

New Sobolev inequalities found for curved spaces.

Study noncommutative Sobolev inequalities using quantum state metrics.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

Improved CR Sobolev inequalities on CR sphere established.

Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.

The paper improves CR Sobolev inequalities and classifies minimizers.

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

Completeness of surface metrics established for Sobolev spaces.

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We fin…

Study on completeness of Sobolev metrics on manifold-valued curves.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Uniform Sobolev inequality for Kähler metrics with entropy bound.

Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their …

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.

This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.

The paper shows how Sobolev maps affect currents in metric spaces.

New proof of Sobolev inequality with constraints on sphere.

Sharp fractional Sobolev inequalities on closed manifolds identified.