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1
Content available remote

Two-weight weak type maximal inequalities in Orlicz classes

100%
Pick L.
Studia Mathematica
|
1991
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tom 100
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nr 3
207-218
EN
Necessary and sufficient conditions are shown in order that the inequalities of the form $ϱ({M_μ f > λ})Φ(λ) ≤ C ʃ_X Ψ(C|f(x)|) σ(x)dμ$, or $ϱ({M_μ f > λ}) ≤ C ʃ_X Φ(Cλ^{-1}|f(x)|) σ(x)dμ$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_μ$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.
2
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Optimal Sobolev imbedding spaces

64%
Kerman R., Pick L.
Studia Mathematica
|
2009
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tom 192
|
nr 3
195-217
EN
This paper continues our study of Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. In it we characterize when the norms considered are optimal. Explicit expressions are given for the optimal partners corresponding to a given domain or range norm.
3
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Compactness of Sobolev imbeddings involving rearrangement-invariant norms

64%
Kerman R., Pick L.
Studia Mathematica
|
2008
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tom 186
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nr 2
127-160
EN
We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m,ϱ}(Ω)$ be compactly imbedded into the rearrangement-invariant space $L_{σ}(Ω)$, where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{ϱ}(0,|Ω|)$ into $L_{σ}(0,|Ω|)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.
4
Content available remote

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

64%
Kerman R., Pick L.
Studia Mathematica
|
2011
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tom 206
|
nr 2
97-119
EN
We study imbeddings of the Sobolev space $W^{m,ϱ}(Ω)$: = {u: Ω → ℝ with $ϱ(∂^{α}u/∂x^{α})$ < ∞ when |α| ≤ m}, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_{ϱ}$ and $σ_{ϱ}$, that are optimal with respect to the inclusions $W^{m,ϱ}(Ω) ⊂ W^{m,τ_{ϱ}}(Ω) ⊂ L_{σ_{ϱ}}(Ω)$. General formulas for $τ_{ϱ}$ and $σ_{ϱ}$ are obtained using the 𝓚-method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.
5
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Characterization of associate spaces of weighted Lorentz spaces with applications

51%
Gogatishvili A., Pick L., Soudský F.
Studia Mathematica
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2014
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tom 224
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nr 1
1-23
EN
We characterize associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into $L^{∞}$.
6
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Compactness of Hardy-type integral operators in weighted Banach function spaces

51%
E. Edmunds D., Gurka P., Pick L.
Studia Mathematica
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1994
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tom 109
|
nr 1
73-90
EN
We consider a generalized Hardy operator $Tf(x) = ϕ(x) ʃ_{0}^{x} ψfv$. For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition $ℬ = sup_{R>0} ∥ϕχ_{(R,∞)}∥_{Y}∥ψχ_{(0,R)}∥_{X'} < ∞$ be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize pairs of Lorentz spaces which fall into M (T).
7
Content available remote

A sharp iteration principle for higher-order Sobolev embeddings

51%
Cianchi A., Pick L., Slavíková L.
Banach Center Publications
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2014
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tom 101
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nr 1
37-58
EN
We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving arbitrary rearrangement-invariant norms on open sets in ℝⁿ, possibly endowed with a measure density and satisfying an isoperimetric inequality of fairly general type, to considerably simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a direct application of the reduction principle we determine the optimal target space in the relevant Sobolev embeddings both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, the results apply to any-order Sobolev embedding on regular (John) domains, on Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and on families of product probability spaces, of which the Gauss space and the exponential measure space are classical instances.
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