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An inverse Sobolev lemma.

Pekka Koskela — 1994

Revista Matemática Iberoamericana

We establish an inverse Sobolev lemma for quasiconformal mappings and extend a weaker version of the Sobolev lemma for quasiconformal mappings of the unit ball of R to the full range 0 < p < n. As an application we obtain sharp integrability theorems for the derivative of a quasiconformal mapping of the unit ball of R in terms of the growth of the mapping.

Quasiconformal mappings and Sobolev spaces

Pekka KoskelaPaul MacManus — 1998

Studia Mathematica

We examine how Poincaré change under quasiconformal maps between appropriate metric spaces having the same Hausdorff dimension. We also show that for many metric spaces the Sobolev functions can be identified with functions satisfying Poincaré, and this allows us to extend to the metric space setting the fact that quasiconformal maps from Q onto Q preserve the Sobolev space L 1 , Q ( Q ) .

Mappings of finite distortion: formation of cusps.

Pekka KoskelaJuhani Takkinen — 2007

Publicacions Matemàtiques

In this paper we consider the extensions of quasiconformal mappings f: B → Ω to the whole plane, when the domain Ω is a domain with a cusp of degree s > 0 and thus not an quasidisc. While these mappings do not have quasiconformal extensions, they may have extensions that are homeomorphic mappings of finite distortion with an exponentially integrable distortion, but in such a case ∫ exp(λK(x)) dx = ∞ for all λ > 1/s. Conversely, for a given s > 0 such a mapping is constructed...

On the fusion problem for degenerate elliptic equations II

Stephen M. BuckleyPekka Koskela — 1999

Commentationes Mathematicae Universitatis Carolinae

Let F be a relatively closed subset of a Euclidean domain Ω . We investigate when solutions u to certain elliptic equations on Ω F are restrictions of solutions on all of Ω . Specifically, we show that if F is not too large, and u has a suitable decay rate near F , then u can be so extended.

Metric Sobolev spaces

Koskela, Pekka — 2003

Nonlinear Analysis, Function Spaces and Applications

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.

Subelliptic Poincaré inequalities: the case p < 1.

Stephen M. BuckleyPekka KoskelaGuozhen Lu — 1995

Publicacions Matemàtiques

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.

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