### On boundedness of maximal functions in Sobolev spaces.

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We construct a bounded domain $\Omega \subset {\mathbb{R}}^{2}$ with the cone property and a harmonic function on Ω which belongs to ${W}_{0}^{1,p}\left(\Omega \right)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no ${L}^{p}$-Hodge decomposition in ${L}^{p}(\Omega ,{\mathbb{R}}^{2})$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in ${W}^{1,p}\left(\Omega \right)$ for all p > 4.

We find a condition for a Borel mapping $f:{\mathbb{R}}^{m}\to {\mathbb{R}}^{n}$ which implies that the Hausdorff dimension of ${f}^{-1}\left(y\right)$ is less than or equal to m-n for almost all $y\in {\mathbb{R}}^{n}$.

In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need...

The purpose of this paper is to provide a new characterization of the Sobolev space ${W}^{1,1}\left(\mathbb{R}\u207f\right)$. We also show a new proof of the characterization of the Sobolev space ${W}^{1,p}\left(\mathbb{R}\u207f\right)$, 1 ≤ p < ∞, in terms of Poincaré inequalities.

In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

We prove several results concerning density of ${C}_{0}^{\infty}$, behaviour at infinity and integral representations for elements of the space ${L}^{m,p}=\u2a0d|{\nabla}^{m}\u2a0d\in {L}^{p}$.

We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by ${C}^{m}$ functions both in norm and capacity.

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].

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 find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

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 $p=1$.

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