Besov capacity and Hausdorff measures in metric measure spaces

Costea, Şerban

doi:10.5565/publmat_53109_07

Cited by 24 publications

(28 citation statements)

References 17 publications

(11 reference statements)

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“…From this equivalence we also conclude that in R n the above 'Lipschitz' capacity agrees with the 'smooth' capacity defined in (1).…”

Section: Juha Lehrbäcksupporting

confidence: 57%

“…For more information on capacities we refer to [1] and the book [7, Ch. 2] by Heinonen, Kilpeläinen, and Martio.…”

Section: Juha Lehrbäckmentioning

confidence: 99%

“…If the space X supports a (1, p)-Poincaré inequality (and µ is doubling, as usual), the above definition of the capacity coincides with the more abstract and more general definition of the relative (Newtonian) Sobolev capacity; see Costea [1] for the definitions and the details, and in particular his Remark 3.4 for the above-mentioned equivalence of the definitions. From this equivalence we also conclude that in R n the above 'Lipschitz' capacity agrees with the 'smooth' capacity defined in (1).…”

Section: Juha Lehrbäckmentioning

confidence: 99%

“…Here the (variational) p-capacity of a compact set E with respect to an open set Ω ⊃ E is defined as (1) cap p (E, Ω) = inf…”

Section: Dist(x E) < T} and Call E T The (Open) T-neighbourhood Of Ementioning

confidence: 99%

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Neighbourhood capacities

Lehrbäck¹

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study the behaviour of the p-capacity of a compact set E with respect to the t-neighbourhoods of E as t varies. We establish sharp upper and lower bounds for these capacities in terms of Minkowski and Hausdorff type contents of E, respectively, and our results hold in both Euclidean and more general metric spaces. In our lower bounds the porosity of the set E plays an important role, and it is shown by examples that an assumption like this is in general necessary. In addition, we present a self-contained approach to the theory of sets of zero capacity in metric spaces.

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“…From this equivalence we also conclude that in R n the above 'Lipschitz' capacity agrees with the 'smooth' capacity defined in (1).…”

Section: Juha Lehrbäcksupporting

confidence: 57%

“…For more information on capacities we refer to [1] and the book [7, Ch. 2] by Heinonen, Kilpeläinen, and Martio.…”

Section: Juha Lehrbäckmentioning

confidence: 99%

Section: Juha Lehrbäckmentioning

confidence: 99%

“…Here the (variational) p-capacity of a compact set E with respect to an open set Ω ⊃ E is defined as (1) cap p (E, Ω) = inf…”

Section: Dist(x E) < T} and Call E T The (Open) T-neighbourhood Of Ementioning

confidence: 99%

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Neighbourhood capacities

Lehrbäck¹

2012

Ann. Acad. Sci. Fenn. Math.

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“…e.g. the works of Hajłasz [13], Costea [7] or the monograph of A. Björn and J. Björn [2]; our investigations are concentrated to the above-mentioned nonlinear capacity.…”

Section: Introductionmentioning

confidence: 99%

p-Transfinite diameter and p-Chebyshev constant in locally compact spaces

Horváth¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. We extend the notion of transfinite diameter and Chebyshev constant to p-potential theory in locally compact spaces and study their relations. As in the classical case, it turns out that provided that the kernel satisfies a certain condition, for any compact sets the energy, the Chebyshev constant and the transfinite diameter are coincide. The investigations follow the linear method developed by e.g. Choquet, Fuglede, Ohtsuka, Farkas and Nagy. Taking into consideration the significance of finite sets of the minimal and almost minimal energy, we examine Fekete and greedy energy sets as well.

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The equivalence between pointwise Hardy inequalities and uniform fatness

2010

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We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.

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Besov capacity and Hausdorff measures in metric measure spaces

Cited by 24 publications

References 17 publications

Neighbourhood capacities

Neighbourhood capacities

p-Transfinite diameter and p-Chebyshev constant in locally compact spaces

The equivalence between pointwise Hardy inequalities and uniform fatness

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