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1
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Weighted $L_{Φ}$ integral inequalities for operators of Hardy type

100%
Bloom S., Kerman R.
Studia Mathematica
|
1994
|
tom 110
|
nr 1
35-52
EN
Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^{-1} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_{1}^{-1}(ʃΦ_{1}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.
2
Content available remote

Optimal Sobolev imbedding spaces

100%
Kerman R., Pick L.
Studia Mathematica
|
2009
|
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

100%
Kerman R., Pick L.
Studia Mathematica
|
2008
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tom 186
|
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
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Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

100%
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.
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