Inégalités à poids pour l'opérateur de Hardy-Littlewood-Sobolev dans les espaces métriques mesurés à deux demi-dimensions

David Mascré

Colloquium Mathematicae, Tome 106 (2006), p. 77-104 / Harvested from The Polish Digital Mathematics Library

Résumé

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 < ∞.

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Publié le : 2006-01-01

Zbl 1119.42007

EUDML-ID : urn:eudml:doc:286131