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Papers by Jan Kalis

Research paper thumbnail of On Whitney Sets and Their Generalization

Real Analysis Exchange, 2005

Research paper thumbnail of Whitney arcs and 1-critical arcs

Fundamenta Mathematicae, 2008

A simple arc γ ⊂ R n is called a Whitney arc if there exists a non-constant real function f on γ ... more A simple arc γ ⊂ R n is called a Whitney arc if there exists a non-constant real function f on γ such that limy→x, y∈γ |f (y) − f (x)|/|y − x| = 0 for every x ∈ γ; γ is 1-critical if there exists an f ∈ C 1 (R n) such that f (x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in R 2 each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.

Research paper thumbnail of Whitney Curves Revisited

Real Analysis Exchange, 2005

Research paper thumbnail of Symmetrization and sharp Sobolev inequalities in metric spaces

Revista Matemática Complutense, 2009

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtai... more We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.

Research paper thumbnail of On Whitney Sets and Their Generalization

Real Analysis Exchange, 2005

Research paper thumbnail of Whitney arcs and 1-critical arcs

Fundamenta Mathematicae, 2008

A simple arc γ ⊂ R n is called a Whitney arc if there exists a non-constant real function f on γ ... more A simple arc γ ⊂ R n is called a Whitney arc if there exists a non-constant real function f on γ such that limy→x, y∈γ |f (y) − f (x)|/|y − x| = 0 for every x ∈ γ; γ is 1-critical if there exists an f ∈ C 1 (R n) such that f (x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in R 2 each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.

Research paper thumbnail of Whitney Curves Revisited

Real Analysis Exchange, 2005

Research paper thumbnail of Symmetrization and sharp Sobolev inequalities in metric spaces

Revista Matemática Complutense, 2009

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtai... more We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.

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