arXiv subjects

Tomáš Roskovec

Publications and source records attributed to Tomáš Roskovec.

5 recordsLinked to original sources

arXiv2025

Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form

We obtain the inequalities of the form $$\int_Ω|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_Ω \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +Θ,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(Ω)$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\cal T}_{H}(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot)$, which is the primitive of the weight $h(\cdot)$, and $Θ$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae.

math.AP↗

arXiv2021

Regularity of the inverse mapping in Banach function spaces

We study the regularity properties of the inverse of a bilipschitz mapping $f$ belonging $W^m X_{\text{loc}}$, where $X$ is an arbitrary Banach function space. Namely, we prove that the inverse mapping $f^{-1}$ is also in $W^m X_{\text{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in $W^m X_{\text{loc}}$ is closed with respect to composition and multiplication.

math.FA↗

arXiv2020

Hausdorff measure of critical set for Luzin $N$ condition

It is well-known that there is a Sobolev homeomorphism $f\in W^{1,p}([-1,1]^n,[-1,1]^n)$ for any $p<n$ which maps a set $C$ of zero Lebesgue $n$-dimensional measure onto the set of positive measure. We study the size of this critical set $C$ and characterize its lower and upper bounds from the perspective of Hausdorff measures defined by a general gauge function.

math.FA↗

arXiv2018

Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks

A carefully written Nirenberg's proof of the well known Gagliardo-Nirenberg interpolation inequality for intermediate derivatives in $\mathbb{R}^n$ seems, surprisingly, to be missing in literature. In our paper we shall first introduce this fundamental result and provide information about it's historical background. Afterwards we present a complete, student-friendly proof. In our proof we use the architecture of Nirenberg's proof, the proof is, however, much more detailed, containing also some differences. The reader can find a short comparison of differences and similarities in the final chapter.

math.FA↗

arXiv2018

Interpolation between H\" older and Lebesgue spaces with applications

Classical interpolation inequality of the type $\|u\|_{X}\leq C\|u\|_{Y}^θ\|u\|_{Z}^{1-θ}$ is well known in the case when $X$, $Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing norms $\|\cdot\|_{Y}$ or $\|\cdot\|_{X}$ by suitable H\" older semi-norm. We shall even prove sharper version involving weak Lorentz norm. We apply this result to prove the Gagliardo--Nirenberg inequality for a wider scale of parameters.

math.FA↗