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Coarea integration in metric spaces

Malý, Jan — 2003

Nonlinear Analysis, Function Spaces and Applications

Let X be a metric space with a doubling measure, Y be a boundedly compact metric space and u : X → Y be a Lebesgue precise mapping whose upper gradient g belongs to the Lorentz space L m , 1 , m ≥ 1 . Let E ⊂ X be a set of measure zero. Then ℋ ^ m ( E ∩ u - 1 ( y ) ) = 0 for ℋ m -a.e. y ∈ Y , where ℋ m is the m -dimensional Hausdorff measure and ℋ ^ m is the m -codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...

Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points

Jan Malý — 1996

Commentationes Mathematicae Universitatis Carolinae

Let u be a weak solution of a quasilinear elliptic equation of the growth p with a measure right hand term μ . We estimate u ( z ) at an interior point z of the domain Ω , or an irregular boundary point z ∈ ∂ Ω , in terms of a norm of u , a nonlinear potential of μ and the Wiener integral of 𝐑 n ∖ Ω . This quantifies the result on necessity of the Wiener criterion.

The area formula for W 1 , n -mappings

Jan Malý — 1994

Commentationes Mathematicae Universitatis Carolinae

Let f be a mapping in the Sobolev space W 1 , n ( Ω , 𝐑 n ) . Then the change of variables, or area formula holds for f provided removing from counting into the multiplicity function the set where f is not approximately Hölder continuous. This exceptional set has Hausdorff dimension zero.

L p -approximation of Jacobians

Jan Malý — 1991

Commentationes Mathematicae Universitatis Carolinae

The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from Cart p ( Ω , 𝐑 m ) is approximated by 𝒞 1 functions strongly in 𝒜 q ( Ω , 𝐑 m ) whenever q < p . An example is shown of a function which is in cart p ( Ω , 𝐑 2 ) but not in cart p ( Ω , 𝐑 2 ) .

Absolutely continuous functions of several variables and diffeomorphisms

Stanislav Hencl;  Jan Malý — 2003

Open Mathematics

In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.

On a generalization of Henstock-Kurzweil integrals

Jan Malý;  Kristýna Kuncová — 2019

Mathematica Bohemica

We study a scale of integrals on the real line motivated by the M C α integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

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