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On a metric measure space (X,ϱ,μ), consider the weight functions
$w_{α}(x) = ϱ(x,z₀)^{-α₀}$ if ϱ(x,z₀) < 1,
$w_{α}(x) = ϱ(x,z₀)^{-α₁}$ if ϱ(x,z₀) ≥ 1,
$w_{β}(x) = ϱ(x,z₀)^{-β₀}$ if ϱ(x,z₀) < 1,
$w_{β}(x) = ϱ(x,z₀)^{-β₁}$ if ϱ(x,z₀) ≥ 1,
where z₀ is a given point of X, and let $κ_{a}: X×X → ℝ₊$ be an operator kernel satisfying
$κ_{a}(x,y) ≤ cϱ(x,y)^{a-d}$ for all x,y ∈ X such that ϱ(x,y) < 1,
$κ_{a}(x,y) ≤ cϱ(x,y)^{a-D}$ for all x,y ∈ X such that ϱ(x,y)≥ 1,
where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev operator with kernel $κ(x,y) = w_{β}(x)κ_{a}(x,y)w_{α}(y)$ to be bounded from $L^{p}(X)$ to $L^{q}(X)$ for 1 < p ≤ q < ∞.
$w_{α}(x) = ϱ(x,z₀)^{-α₀}$ if ϱ(x,z₀) < 1,
$w_{α}(x) = ϱ(x,z₀)^{-α₁}$ if ϱ(x,z₀) ≥ 1,
$w_{β}(x) = ϱ(x,z₀)^{-β₀}$ if ϱ(x,z₀) < 1,
$w_{β}(x) = ϱ(x,z₀)^{-β₁}$ if ϱ(x,z₀) ≥ 1,
where z₀ is a given point of X, and let $κ_{a}: X×X → ℝ₊$ be an operator kernel satisfying
$κ_{a}(x,y) ≤ cϱ(x,y)^{a-d}$ for all x,y ∈ X such that ϱ(x,y) < 1,
$κ_{a}(x,y) ≤ cϱ(x,y)^{a-D}$ for all x,y ∈ X such that ϱ(x,y)≥ 1,
where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev operator with kernel $κ(x,y) = w_{β}(x)κ_{a}(x,y)w_{α}(y)$ to be bounded from $L^{p}(X)$ to $L^{q}(X)$ for 1 < p ≤ q < ∞.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
77-104
Opis fizyczny
Daty
wydano
2006
Twórcy
autor
- Département de Mathématiques, Université de Cergy-Pontoise, 95 302 Cergy-Pontoise Cedex, France
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-cm105-1-9