scholarly journals Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces

2012 ◽  
Vol 64 (4) ◽  
pp. 892-923 ◽  
Author(s):  
Tuomas Hytönen ◽  
Suile Liu ◽  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Polynomial Growth ◽  
Metric Spaces ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Zygmund Operator ◽  
Borel Measures ◽  
Measure Spaces ◽  
Atomic Condition ◽  
Complex Valued ◽  
Polynomial Growth Condition

Abstract Let (𝒳, d, μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛x﹜) = 0 for all x ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to that of T on Lp(μ) for some p ∈ (1,∞), and that of T from L1(μ) to L1,∞(μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L2(μ), then its maximal operator is bounded on Lp(μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

scholarly journals Boundedness of maximal Calderón–Zygmund operators on non-homogeneous metric measure spaces

2014 ◽  
Vol 144 (3) ◽  
pp. 567-589 ◽  
Author(s):  
Suile Liu ◽  
Yan Meng ◽  
Dachun Yang
Keyword(s):  
Singular Integral ◽  
Measure Space ◽  
Type Inequality ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Standard Size ◽  
Hörmander Condition ◽  
Zygmund Operator ◽  
Measure Spaces ◽  
Size Condition

Let (X, d, μ) be a metric measure space and let it satisfy the so-called upper doubling condition and the geometrically doubling condition. We show that, for the maximal Calderón–Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its Lp(μ)-boundedness with p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into L1,∞(μ). Moreover, applying this, together with a new Cotlar-type inequality, the authors show that if the Calderón–Zygmund operator T is bounded on L2(μ), then the corresponding maximal Calderón–Zygmund operator is bounded on Lp(μ) for all p ∈ (1, ∞), and bounded from L1(μ) into L1,∞ (μ). These results essentially improve the existing results.

scholarly journals Dichotomy of global capacity density in metric measure spaces

2018 ◽  
Vol 11 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Exact Formula ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Geodesic Metric ◽  
Measure Spaces ◽  
Variational Capacity ◽  
John Domains ◽  
Superlevel Sets ◽  
Sobolev Capacity

AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

scholarly journals Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

2013 ◽  
Vol 1 ◽  
pp. 232-254 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Jeremy T. Tyson ◽  
Kevin Wildrick
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Metric Space ◽  
Metric Spaces ◽  
Higher Dimension ◽  
Metric Measure Spaces ◽  
Regular Mapping ◽  
Sobolev Mappings ◽  
Measure Spaces ◽  
Left And Right

Abstract We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Regularity of minimizers of the area functional in metric spaces

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Heikki Hakkarainen ◽  
Juha Kinnunen ◽  
Panu Lahti
Keyword(s):  
Linear Growth ◽  
Metric Spaces ◽  
Metric Measure Spaces ◽  
Local Boundedness ◽  
Regularity Of Minimizers ◽  
Jump Discontinuities ◽  
Area Functional ◽  
Measure Spaces

AbstractIn this article we study minimizers of functionals of linear growth in metric measure spaces. We introduce the generalized problem in this setting, and prove existence and local boundedness of the minimizers. We give counterexamples to show that in general, minimizers are not continuous and can have jump discontinuities inside the domain.

scholarly journals Boundedness of Marcinkiewicz Integrals on RBMO Spaces over Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Ji Cheng ◽  
Guanghui Lu
Keyword(s):  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Doubling Condition ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Dominating Function

LetX,d,μbe a metric measures space satisfying the upper doubling conditions and the geometrically doubling conditions in the sense of Hytönen. Under the assumption that the dominating function satisfies the weak reverse doubling condition, the authors prove that Marcinkiewicz integral with kernel satisfying certain stronger Hörmander-type condition is bounded on RBMOμspace.

scholarly journals Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Measure Space ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Lebesgue Spaces ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Weak Lebesgue Spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p<n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

scholarly journals Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Sylvester Eriksson-Bique ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam
Keyword(s):  
Bounded Variation ◽  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
Measure Spaces

Abstract We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.

Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces

2007 ◽  
Vol 59 (6) ◽  
pp. 1135-1153 ◽  
Author(s):  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Continuous Functions ◽  
Characteristic Functions ◽  
Metric Measure Spaces ◽  
Hölder Continuous ◽  
Geometric Conditions ◽  
Sobolev Functions ◽  
Measure Spaces ◽  
Hölder Continuous Functions

AbstractWe study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

scholarly journals Existence of parabolic minimizers to the total variation flow on metric measure spaces

2022 ◽  
Author(s):  
Vito Buffa ◽  
Michael Collins ◽  
Cintia Pacchiano Camacho
Keyword(s):  
Total Variation ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Parabolic Boundary ◽  
Open Set ◽  
Variational Solutions ◽  
Total Variation Flow ◽  
Parabolic Function ◽  
Measure Spaces ◽  
Abstract Notion

AbstractWe give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

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