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Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces

100%

Pietruska-Pałuba K.

nr 3

257-299

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 $1/C ℰ(f,f) ≤ lim inf_{σ↗1} (1−σ)||f||_{W^{σ,2}} ≤ lim sup_{σ↗1}(1−σ) ||f||_{W^{σ,2}} ≤ Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.

Walk dimension and function spaces on self-similar fractals

100%

Pietruska-Pałuba K.

Banach Center Publications

2009

tom 84

nr 1

63-73

We outline the construction of Brownian motion on certain self-similar fractals and introduce the notion of walk dimension. We then show how the probabilistic approach relates to the theory of function spaces on fractals.

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

63%

Kałamajska A. , Pietruska-Pałuba K.

tom 10

nr 6

2033-2050

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ 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.

Gagliardo-Nirenberg inequalities in weighted Orlicz spaces

63%

Kałamajska A. , Pietruska-Pałuba K.

nr 1

49-71

We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.