Piotr Hajłasz

Sobolev Spaces on an Arbitrary Metric Space

We define Sobolev space W1,p for 1<p≤∞ on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.

Hölder quasicontinuity of Sobolev functions on metric spaces

We prove that every Sobolev function defined on a metric space coincides with a Holder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Holder 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].

Pointwise Hardy inequalities

If Ω ⊂ Rn is an open set with the sufficiently regular boundary, then the Hardy inequality ∫ Ω |u|p%−p ≤ C ∫ Ω |∇u|p holds for u ∈ C∞ 0 (Ω) and 1 < p < ∞, where %(x) = dist(x, ∂Ω). The main result of the paper is a pointwise inequality |u| ≤ %M2%|∇u|, where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy– Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.

Approximation of Sobolev mappings

where 1 sp < co. This definition is far from being intrinsic. For an intrinsic definition of WrTp(Mm, N”) see [ 11. In this space, beside the standard topology induced by the norm (1. ]ll,p, we also have weak topology and weak convergence. Let fk, f E W’~p(Mm), where 1 < p < co. We say that fk converges to f in weak topology iff fk + f in Lp and the set (lIVfk[lp)k is bounded. Weak convergence is denoted by fk f. It is not difficult to prove that our definition is equivalent to the standard definition of weak convergence in Banach space. We aim to prove the following theorem.

Measure density and extendability of Sobolev functions

We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.

Sobolev Mappings between Manifolds and Metric Spaces

In connection with the theory of p-harmonic mappings, Eells and Lemaire raised a question about density of smooth mappings in the space of Sobolev mappings between manifolds. Recently Hang and Lin provided a complete solution to this problem. The theory of Sobolev mappings between manifolds has been extended to the case of Sobolev mappings with values into metric spaces. Finally analysis on metric spaces, the theory of Carnot– Caratheodory spaces, and the theory of quasiconformal mappings between metric spaces led to the theory of Sobolev mappings between metric spaces. The purpose of this paper is to provide a self-contained introduction to the theory of Sobolev spaces between manifolds and metric spaces. The paper also discusses new results of the author.

Whitney's example by way of Assouad's embedding

In this note we show how to use the Assouad embedding theorem (about almost bi-Lipschitz embeddings) to construct examples of C m functions which are not constant on a critical set homeomorphic to the n-dimensional cube. This extends the famous example of Whitney. Our examples are shown to be sharp.

A note on weak approximation of minors

Abstract Let A p , q (Ω), where Ω ⊆ R n is a bounded open domain, be a set of all mappings u ∈ W 1, p (Ω, R n ) such that adj Du ∈ L q . Among other results we prove that if n − 1 ≤ p n , 1 q n /( n − 1), then the subclass of A p , q mappings, which consists of mappings with bounded ( n − 1)-dimensional image, is dense in the sequential weak topology of A p , q . We also extend this result to other A p , q type spaces.

On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

We study the question: when are Lipschitz mappings dense in the Sobolev space W (M, H)? Here M denotes a compact Riemannian manifold with or without boundary, while H denotes the nth Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W (M, H) for all 1 ≤ p < ∞ if dim M ≤ n, but that Lipschitz maps are not dense in W (M, H) if dim M ≥ n + 1 and n ≤ p < n + 1. The proofs rely on the construction of smooth horizontal embeddings of the sphere S into H. We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the nth Lipschitz homotopy group of H. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.

On approximate differentiability of the maximal function

We prove that if f ∈ L 1 (ℝ n ) is approximately differentiable a.e., then the Hardy-Littlewood maximal function Mf is also approximately differentiable a.e. Moreover, if we only assume that f ∈ L 1 (ℝ n ), then any open set of ℝ n contains a subset of positive measure such that Mf is approximately differentiable on that set. On the other hand we present an example of f ∈ L 1 (ℝ) such that Mf is not approximately differentiable a.e.