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A counterexample to the L p -Hodge decomposition

Piotr Hajłasz — 1996

Banach Center Publications

We construct a bounded domain Ω 2 with the cone property and a harmonic function on Ω which belongs to W 0 1 , p ( Ω ) for all 1 ≤ p < 4/3. As a corollary we deduce that there is no L p -Hodge decomposition in L p ( Ω , 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.

Change of variables formula under minimal assumptions

Piotr Hajłasz — 1993

Colloquium Mathematicae

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...

A new characterization of the Sobolev space

Piotr Hajłasz — 2003

Studia Mathematica

The purpose of this paper is to provide a new characterization of the Sobolev space W 1 , 1 ( ) . We also show a new proof of the characterization of the Sobolev space W 1 , p ( ) , 1 ≤ p < ∞, in terms of Poincaré inequalities.

The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

Piotr HajłaszJacob Mirra — 2013

Analysis and Geometry in Metric Spaces

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.

Geodesics in the Heisenberg Group

Piotr HajłaszScott Zimmerman — 2015

Analysis and Geometry in Metric Spaces

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.

On Conditions for Unrectifiability of a Metric Space

Piotr HajłaszSoheil Malekzadeh — 2015

Analysis and Geometry in Metric Spaces

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.

Hölder quasicontinuity of Sobolev functions on metric spaces.

Piotr HajlaszJuha Kinnunen — 1998

Revista Matemática Iberoamericana

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].

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