Search results for "ta111"
showing 10 items of 251 documents
The Choquet and Kellogg properties for the fine topology when $p=1$ in metric spaces
2017
In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincar´e inequality, we prove the fine Kellogg property, the quasi-Lindel¨of principle, and the Choquet property for the fine topology in the case p = 1. Dans un contexte d’espace m´etrique complet muni d’une mesure doublante et supportant une in´egalit´e de Poincar´e, nous d´emontrons la propri´et´e fine de Kellogg, le quasi-principe de Lindel¨of, et la propri´et´e de Choquet pour la topologie fine dans le cas p = 1. peerReviewed
Remarks about the Besicovitch Covering Property in Carnot groups of step 3 and higher
2016
International audience
Hardy-Orlicz Spaces of conformal densities
2014
We define and prove characterizations of Hardy-Orlicz spaces of conformal densities.
Cheeger-harmonic functions in metric measure spaces revisited
2014
Abstract Let ( X , d , μ ) be a complete metric measure space, with μ a locally doubling measure, that supports a local weak L 2 -Poincare inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on ( X , d , μ ) . Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.
A density problem for Sobolev spaces on Gromov hyperbolic domains
2017
We prove that for a bounded domain $\Omega\subset \mathbb R^n$ which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when $\Omega$ is a finitely connected planar domain, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover if $\Omega$ is also Jordan or quasiconvex, then $C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(\Omega)$ for $1\le p<\infty$.
Dynamics of the scenery flow and geometry of measures
2015
We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a n…
Quasispheres and metric doubling measures
2018
Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking.
$C^{1,��}$ regularity for the normalized $p$-Poisson problem
2017
We consider the normalized $p$-Poisson problem $$-��^N_p u=f \qquad \text{in}\quad ��.$$ The normalized $p$-Laplacian $��_p^{N}u:=|D u|^{2-p}��_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,��}_{loc}$ regularity with nearly optimal $��$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory.
On the representation of integers by indefinite binary Hermitian forms
2011
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f, as s tends to infinity.
Assouad dimension, Nagata dimension, and uniformly close metric tangents
2013
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular subRiemannian manifolds and show that locally their Nagata dimension equals the to…