scholarly journals An alternative capacity in metric measure spaces

2021 ◽  
Vol 18 (2) ◽  
pp. 196-208
Author(s):  
Olli Martio
Keyword(s):  
Measure Space ◽  
Lipschitz Functions ◽  
Metric Measure Space ◽  
Good Measure ◽  
Metric Measure Spaces ◽  
Doubling Property ◽  
Condenser Capacity ◽  
Regularity Properties ◽  
Measure Spaces ◽  
Curve Family

A new condenser capacity $\CMp(E,G)$ is introduced as an alternative to the classical Dirichlet capacity in a metric measure space $X$. For $p>1$, it coincides with the $M_p$-modulus of the curve family $\Gamma(E,G)$ joining $\partial G$ to an arbitrary set $E \subset G$ and, for $p = 1$, it lies between $AM_1(\Gamma(E,G))$ and $M_1(\Gamma(E,G))$. Moreover, the $\CMp(E,G)$-capacity has good measure theoretic regularity properties with respect to the set $E$. The $\CMp(E,G)$-capacity uses Lipschitz functions and their upper gradients. The doubling property of the measure $\mu$ and Poincar\'e inequalities in $X$ are not needed.

scholarly journals The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Marcello Lucia ◽  
Michael J. Puls
Keyword(s):  
Real Number ◽  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Type Problem ◽  
Locally Compact ◽  
Dirichlet Type ◽  
Algebra Of Functions ◽  
Measure Spaces ◽  
Dirichlet Type Problem

Abstract Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

scholarly journals Boundedness of Lusin-area andgλ*functions on localized Morrey-Campanato spaces over doubling metric measure spaces

2011 ◽  
Vol 9 (3) ◽  
pp. 245-282 ◽  
Author(s):  
Haibo Lin ◽  
Eiichi Nakai ◽  
Dachun Yang
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Admissible Function ◽  
Regularity Property ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Area Function ◽  
Measure Spaces ◽  
Lusin Area Function ◽  
Dimensional Lebesgue Measure

Letχbe a doubling metric measure space andρan admissible function onχ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spacesερα,p(χ)and Morrey-Campanato-BLO spacesε̃ρα,p(χ)whenα∈(-∞,0)andp∈[1,∞). Ifχhas the volume regularity Property(P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, fromερa,p(χ)toε̃ρa,p(χ)without invoking any regularity of considered kernels. The same is true for thegλ*function and, unlike the Lusin-area function, in this case,χis even not necessary to have Property(P). These results are also new even forℝdwith thed-dimensional Lebesgue measure and have a wide applications.

scholarly journals The Measure Preserving Isometry Groups of Metric Measure Spaces

2020 ◽  
Author(s):  
Yifan Guo ◽  
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Measure Space ◽  
Isometry Group ◽  
Compact Riemannian Manifold ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Isometry Groups ◽  
Measure Spaces ◽  
Effective Estimate

Bochner's theorem says that if M is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso(M) is finite. In this article, we show that if (X,d,m) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group Iso(X,d,m) is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Emery Ricci curvature except for small portions.

scholarly journals Boundedness for Commutators of Bilinearθ-Type Calderón-Zygmund Operators on Nonhomogeneous Metric Measure Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Rulong Xie ◽  
Lisheng Shu ◽  
Aiwen Sun
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Measure Spaces

Let(X,d,μ)be a nonhomogeneous metric measure space. In this paper, the boundedness for commutators generated by bilinearθ-type Calderón-Zygmund operators andRBMO(μ)functions on(X,d,μ)is obtained.

scholarly journals A remark on Poincaré inequalities on metric measure spaces

2004 ◽  
Vol 95 (2) ◽  
pp. 299 ◽  
Author(s):  
Stephen Keith ◽  
Kai Rajala
Keyword(s):  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Regular Measure ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Lipschitz Constants

We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

scholarly journals Generic Quantum Metric Rigidity

2020 ◽  
Author(s):  
Alexandru Chirvasitu
Keyword(s):  
Automorphism Group ◽  
Measure Space ◽  
Baire Category ◽  
Finite Graph ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Isomorphism Classes ◽  
Quantum Symmetries ◽  
Measure Spaces

Abstract We introduce the coherent algebra of a compact metric measure space by analogy with the corresponding concept for a finite graph. As an application we show that upon topologizing the collection of isomorphism classes of compact metric measure spaces appropriately, the subset consisting of those with trivial compact quantum automorphism group is of 2nd Baire category. The latter result can be paraphrased as saying that “most” compact metric measure spaces have no (quantum) symmetries; in particular, they also have trivial ordinary (i.e., classical) automorphism group.

scholarly journals Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Space ◽  
Lebesgue Space ◽  
Measure Space ◽  
Metric Measure Space ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Measure Spaces ◽  
Atomic Hardy Space

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

Functions of bounded variation and curves in metric measure spaces

2016 ◽  
Vol 9 (4) ◽  
pp. 305-322 ◽  
Author(s):  
Olli Martio
Keyword(s):  
Bounded Variation ◽  
Measure Space ◽  
Metric Measure Space ◽  
Functions Of Bounded Variation ◽  
Metric Measure Spaces ◽  
Function Of Bounded Variation ◽  
Measure Spaces

AbstractAn approximation modulus, the $\mathrm{AM}$-modulus, for a family of paths is introduced in a metric measure space ${(X,d,\nu)}$. It is shown that a function of bounded variation in X is a BV function on almost every path with respect to the $\mathrm{AM}$-modulus. Properties of the $\mathrm{AM}$-modulus, its relations to the usual ${M_{1}}$-modulus and to the BV-capacities of condensers are studied. Only minimal assumptions on X and the measure ν are employed.

The John–Nirenberg Inequality for the Regularized BLO Space on Non-homogeneous Metric Measure Spaces

2019 ◽  
Vol 63 (3) ◽  
pp. 643-654
Author(s):  
Haibo Lin ◽  
Zhen Liu ◽  
Chenyan Wang
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Nirenberg Inequality ◽  
Measure Spaces

AbstractLet $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

scholarly journals On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

2019 ◽  
Vol 31 (3) ◽  
pp. 785-801
Author(s):  
Dariusz Kosz
Keyword(s):  
Special Class ◽  
Measure Space ◽  
Necessary Conditions ◽  
Metric Measure Space ◽  
Analogous Problem ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Strong Type ◽  
Measure Spaces

Abstract In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} , the sets of all {p\in[1,\infty]} for which the weak and strong type {(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators {M^{{\mathrm{c}}}_{k}} and {M_{k}} , {k\geq 1} . For any fixed k we describe the necessary conditions that {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k.

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