Sobolev spaces on metric-measure spaces

Hajłasz, Piotr

doi:10.1090/conm/338/06074

Cited by 180 publications

(199 citation statements)

References 29 publications

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“…The purpose of this paper is to show that the methods used in [8] can be applied in the setting of Q-Ahlfors regular metric measure spaces and that the obtained generalized Hausdorff dimension estimates are essentially sharp for each Q > 1 and each modulus of continuity satisfying (3). Our main result is the following.…”

Section: Introductionmentioning

confidence: 94%

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Generalized dimension estimates for images of porous sets in metric spaces

Kauranen¹

2016

Ann. Acad. Sci. Fenn. Math.

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Section: Introductionmentioning

confidence: 94%

“…A Hajłasz gradient does not directly correspond to the classical gradient but instead essentially to the maximal function of the gradient (see [3]). Another common choice for Sobolev spaces in metric spaces is the so-called Newtonian Sobolev space N 1,p .…”

Section: Preliminariesmentioning

confidence: 99%

Generalized dimension estimates for images of porous sets in metric spaces

Kauranen¹

2016

Ann. Acad. Sci. Fenn. Math.

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“…Finally (3) follows from a general fact that a recti able curve in any metric space admits an arc-length parametrization with respect to which the curve is -Lipschitz, see e.g. [4, Proposition 2.5.9], [8,Theorem 3.2].…”

Section: Classi Cation Of Non-unique Geodesicsmentioning

confidence: 99%

Geodesics in the Heisenberg Group

Hajłasz¹,

Zimmerman²

2015

Analysis and Geometry in Metric Spaces

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“…Since a horizontal path γ : I → H n is assumed to be Euclidean absolutely continuous, and its length calculated with respect to sub-Riemannian metric is no greater than its Euclidean length, such paths are also rectifiable in H n . Every rectifiable curve in a metric space admits an arc-length parameterization [13,Theorem 3.2]. With this parameterization the curve is 1-Lipschitz.…”

Section: The Heisenberg Groupmentioning

confidence: 99%

Weak contact equations for mappings into Heisenberg groups

Balogh¹,

Hajłasz²,

Wildrick³

2014

Indiana Univ. Math. J.

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Abstract. Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1 ) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

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Sobolev spaces on metric-measure spaces

Cited by 180 publications

References 29 publications

Generalized dimension estimates for images of porous sets in metric spaces

Generalized dimension estimates for images of porous sets in metric spaces

Geodesics in the Heisenberg Group

Weak contact equations for mappings into Heisenberg groups

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