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Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces

Katarzyna Pietruska-Pałuba — 2008

Bulletin of the Polish Academy of Sciences. Mathematics

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms | | f | | W σ , 2 of a function f ∈ L²(E,μ) have the property 1 / C ( f , f ) l i m i n f σ 1 ( 1 σ ) | | f | | W σ , 2 l i m s u p σ 1 ( 1 σ ) | | f | | W σ , 2 C ( f , f ) , where ℰ is the Dirichlet form relative to the fractional diffusion.

Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka KałamajskaKatarzyna Pietruska-Pałuba — 2006

Colloquium Mathematicae

We obtain interpolation inequalities for derivatives: M q , α ( | f ( x ) | ) d x C [ M p , β ( Φ ( x , | f | , | ( 2 ) f | ) ) d x + M r , γ ( Φ ( x , | f | , | ( 2 ) f | ) ) d x ] , and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ) , where | | · | | ( s , κ ) is the Orlicz norm relative to the function M s , κ ( t ) = t s ( l n ( 2 + t ) ) κ . The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher...

On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka KałamajskaKatarzyna Pietruska-Pałuba — 2009

Studia Mathematica

Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities M ( ω ( x ) | u ( x ) | ) e x p ( - φ ( x ) ) d x C M ( | u ' ( x ) | ) e x p ( - φ ( x ) ) d x , where u belongs to some set of locally absolutely continuous functions containing C ( ) . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Agnieszka KałamajskaKatarzyna Pietruska-Pałuba — 2012

Open Mathematics

We obtain Hardy type inequalities 0 M ω r u r ρ r d r C 1 0 M u r ρ r d r + C 2 0 M u ' r ρ r d r , and their Orlicz-norm counterparts ω u L M ( + , ρ ) C ˜ 1 u L M ( + , ρ ) + C ˜ 2 u ' L M ( + , ρ ) , with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Poincaré inequality and Hajłasz-Sobolev spaces on nested fractals

Katarzyna Pietruska-PałubaAndrzej Stós — 2013

Studia Mathematica

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

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