If Ω ⊂ R n is an open set with the sufficiently regular boundary, then the Hardy inequality Ω |u|... more If Ω ⊂ R n 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| ≤ M 2 |∇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.
Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric... more Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the measure is doubling.
We find a condition for a Borel mapping f : R m → R n which implies that the Hausdorff dimension ... more We find a condition for a Borel mapping f : R m → R n which implies that the Hausdorff dimension of f −1 (y) is less than or equal to m − n for almost all y ∈ R n .
We construct a bounded domain Ω ⊂ R 2 with the cone property and a harmonic function on Ω which b... more We construct a bounded domain Ω ⊂ R 2 with the cone property and a harmonic function on Ω which belongs to W 1,p 0 (Ω) for all 1 ≤ p < 4/3. As a corollary we deduce that there is no L p-Hodge decomposition in L p (Ω, R 2) for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in W 1,p (Ω) for all p > 4.
In the paper we investigate continuity of Orlicz-Sobolev mappings W 1,P (M, N) of finite distorti... more In the paper we investigate continuity of Orlicz-Sobolev mappings W 1,P (M, N) of finite distortion between smooth Riemannian n-manifolds, n ≥ 2, under the assumption that the Young function P satisfies the so-called divergence condition ∞ 1 P(t)/t n+1 dt = ∞. We prove that if the manifolds are oriented, N is compact, and the universal cover of N is not a rational homology sphere, then such mappings are continuous. That includes mappings with D f ∈ L n and, more generally, mappings with D f ∈ L n log −1 L. On the other hand, if the space W 1,P is larger than W 1,n (for example if D f ∈ L n log −1 L), and the universal cover of N is homeomorphic to S n , n = 4, or is diffeomorphic to S n , n = 4, then we construct an example of a mapping in W 1,P (M, N) that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: Both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold N .
We prove that if Φ : X → Y a mapping of weak bounded length distortion from a quasiconvex and com... more We prove that if Φ : X → Y a mapping of weak bounded length distortion from a quasiconvex and complete metric space X to any metric space Y , then for any Lipschitz mapping f : R k ⊃ E → X we have that H k (f (E)) = 0 in X if and only if H k (Φ(f (E))) = 0 in Y. This generalizes an earlier result of Haj lasz and Malekzadeh where the target space Y was a Euclidean space Y = R N .
We construct a large class of pathological n-dimensional topological spheres in R n+1 by showing ... more We construct a large class of pathological n-dimensional topological spheres in R n+1 by showing that for any Cantor set C ⊂ R n+1 there is a topological embedding f : S n → R n+1 of the Sobolev class W 1,n whose image contains the Cantor set C.
We prove that for n ≥ 2, the Lipschitz homotopy group π Lip n+1 (H n) = 0 of the Heisenberg group... more We prove that for n ≥ 2, the Lipschitz homotopy group π Lip n+1 (H n) = 0 of the Heisenberg group H n is nontrivial.
We construct an almost everywhere approximately differentiable, orientation and measure preservin... more We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit n-dimensional cube onto itself, whose Jacobian is equal to −1 a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.
Annales de l'Institut Henri Poincare (C) Non Linear Analysis
L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/lo... more L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
The Morse-Sard theorem requires that a mapping v : R n → R m is of class C k , k > max(n − m, 0).... more The Morse-Sard theorem requires that a mapping v : R n → R m is of class C k , k > max(n − m, 0). In 1957 Dubovitskiȋ generalized this result by proving that almost all level sets for a C k mapping have H s-negligible intersection with its critical set, where s = max(n − m − k + 1, 0). Here the critical set, or m-critical set is defined as Z v,m = {x ∈ R n : rank ∇v(x) < m}. Another generalization was obtained independently by Dubovitskiȋ and Federer in 1966, namely for C k mappings v : R n → R d and integers m ≤ d they proved that the set of m-critical values v(Z v,m) is H q•-negligible for q • = m − 1 + n−m+1 k. They also established the sharpness of these results within the C k category. Here we prove that Dubovitskiȋ's theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class W k p,1 (R n , R d), p = n k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. This result is new also for smooth mappings, but is presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015). Note, that in this paper some result concerning the Coarea formula was not formulated accurately. Now we put an Addendum consisting of three parts: first, we describe the accurate formulation of this result, then we give some historical remarks, and finally its relation to other results of the paper.
We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. ... more We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.
We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacob... more We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis' method which has not been used for such systems so far.
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