Search results for "Bounded function"
showing 10 items of 508 documents
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
2013
We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.
Tensor tomography on Cartan–Hadamard manifolds
2017
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. This work extends the results of Lehtonen (2016) to dimensions $n \geq 3$ and to the case of tensor fields of any order.
Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
2011
We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincar\'e inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincar\'e inequality is implied by the weak displacement convexity of the functional.
The ends of manifolds with bounded geometry, linear growth and finite filling area
2002
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Mappings of finite distortion: the degree of regularity
2005
This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)⩾1 be a measurable function defined on a domain Ω⊂Rn,n⩾2, and such that exp(βK(x))∈Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|n⩽K(x)J(x,f) for a.e. x∈Ω and such that the Jacobian determinant J(x,f) is locally in L1log−c1(n)βL. Then automatically J(x,f) is locally in L1logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite disto…
Geometry and analysis of Dirichlet forms
2012
Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Ne…
Codimensions of algebras and growth functions
2008
Abstract Let A be an algebra over a field F of characteristic zero and let c n ( A ) , n = 1 , 2 , … , be its sequence of codimensions. We prove that if c n ( A ) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α > 1 , an F-algebra A α such that lim n → ∞ c n ( A α ) n exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.
Noncommutative Davis type decompositions and applications
2018
We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in $\H_1^c \cap \H_q^c$ when $q\geq 2$. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space $E$ that is either an interpolation of the couple $(L_p, L_2)$ for some $1<p<2$ or is an interpolation of the couple $(L_2, L_q)$ for some $2<q<\infty$. We also obtain the corresponding $\Phi$-moment Burkholder/Rosenthal inequality for Orlicz functions that…
M-bornologies on L-valued Sets
2017
We develop an approach to the concept of bornology in the framework of many-valued mathematical structures. It is based on the introduced concept of an M-bornology on an L-valued set (X, E), or an LM-bornology for short; here L is an iccl-monoid, M is a completely distributive lattice and \(E: X\times X \rightarrow L\) is an L-valued equality on the set X. We develop the basics of the theory of LM-bornological spaces and initiate the study of the category of LM-bornological spaces and appropriately defined bounded “mappings” of such spaces.
Ein Kriterium f�r die Approximierbarkeit von Funktionen aus sobolewschen R�umen durch glatte Funktionen
1981
The present paper provides a necessary and sufficient criterion for an element of a Sobolev space W k p (Ω) to be approximated in the Sobolev norm by Ck(En)-smooth functions. Here Ω is a bounded open set of n-dimensional Euclidean space En with convex closure $$\bar \Omega$$ and boundary ∂Ω having n-dimensional Lebesgue measure zero. No further boundary regularity (such as e.g. the segment property) is required.Our main tools are the Hardy-Littlewood maximal functions and a slightly strengthened version of a well-known extension theorem of Whitney.This work was inspired by and is very close in spirit to the pertinent parts of Calderon-Zygmund [6].