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.

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 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 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.

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 $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces

2020 ◽  
Vol 32 (2) ◽  
pp. 339-359 ◽  
Author(s):  
Yoshihiro Sawano ◽  
Masaki Shigematsu ◽  
Tetsu Shimomura
Keyword(s):  
Measure Space ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Doubling Condition ◽  
Riesz Potentials ◽  
General Measure ◽  
Open Set ◽  
Measure Spaces

AbstractThis paper proves the boundedness of the generalized Riesz potentials {I_{\rho,\mu,\tau}f} of functions in the Morrey space {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

2020 ◽  
Vol 126 (2) ◽  
pp. 259-275
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Vector Fields ◽  
Integral Functional ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Regular Measure ◽  
Local Boundedness ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Bounded Open Subset ◽  
Hörmander’S Condition

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

scholarly journals Poincaré inequality for abstract spaces

2005 ◽  
Vol 71 (2) ◽  
pp. 193-204 ◽  
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Ricci Curvature ◽  
Poincaré Inequality ◽  
Strong Version ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Poincare Inequality ◽  
Contraction Property ◽  
Measure Spaces ◽  
Measure Contraction Property ◽  
Abstract Spaces

The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

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