Displaying similar documents to “Is the maximal function of a Lipschitz function continuous?”

Subelliptic Poincaré inequalities: the case p < 1.

Stephen M. Buckley, Pekka Koskela, Guozhen Lu (1995)

Publicacions Matemàtiques

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We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p < 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.

Metric Sobolev spaces

Koskela, Pekka

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We describe an approach to establish a theory of metric Sobolev spaces based on Lipschitz functions and their pointwise Lipschitz constants and the Poincaré inequality.

Bilipschitz embeddings of metric spaces into euclidean spaces.

Stephen Semmes (1999)

Publicacions Matemàtiques

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When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat...

Definitions of Sobolev classes on metric spaces

Bruno Franchi, Piotr Hajłasz, Pekka Koskela (1999)

Annales de l'institut Fourier

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There have been recent attempts to develop the theory of Sobolev spaces W 1 , p on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case p = 1 .

Hölder quasicontinuity of Sobolev functions on metric spaces.

Piotr Hajlasz, Juha Kinnunen (1998)

Revista Matemática Iberoamericana

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We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].

Some remarks about metric spaces, spherical mappings, functions and their derivatives.

Stephen Semmes (1996)

Publicacions Matemàtiques

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If p ∈ R, then we have the radial projection map from R {p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In...