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Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jmaa.2020.124598
J. Hounie , T. Picon

Abstract Let A ( x , D ) be an elliptic linear differential operator of order ν with smooth complex coefficients in Ω ⊂ R N from a complex vector space E to a complex vector space F. In this paper we show that if l ∈ R satisfies 0 l N and l ≤ ν , then the estimate ( ∫ R N | P ν − l ( x , D ) u ( x ) | q | x | − N + ( N − l ) q d x ) 1 / q ≤ C ‖ A ( x , D ) u ‖ L 1 holds locally for every u ∈ C c ∞ ( U ; E ) and 1 ≤ q N / ( N − l ) assuming A ( x , D ) is canceling, i.e. ⋂ ξ ∈ R N ∖ { 0 } A ( x 0 , ξ ) [ E ] = { 0 } for each x 0 ∈ Ω . Here P ν − l ( x , D ) is a properly supported pseudo-differential operator in Hormander's class Op S 1 , δ ν − l ( Ω ) , 0 ≤ δ 1 . This statement is inspired in a new characterization of Hardy-Littlewood-Sobolev inequalities for elliptic and canceling homogeneous operators A ( D ) with constant coefficients that extends and unifies several results stemming from the classical Hardy-Sobolev estimates. Variants and applications are presented with focus on operators associated to elliptic systems of complex vector fields.

中文翻译:

用于消除椭圆微分算子的局部 Hardy-Littlewood-Sobolev 不等式

摘要 令 A ( x , D ) 是一个 ν 阶椭圆线性微分算子,在 Ω ⊂ RN 中具有平滑的复系数,从复向量空间 E 到复向量空间 F。 本文证明如果 l ∈ R 满足 0 l N 和 l ≤ ν ,那么估计 ( ∫ RN | P ν − l ( x , D ) u ( x ) | q | x | − N + ( N − l ) qdx ) 1 / q ≤ C ‖ A ( x , D ) u ‖ L 1 对每一个 u ∈ C c ∞ ( U ; E ) 和 1 ≤ q N / ( N − l ) 都成立,假设 A ( x , D ) 是对消的,即 ⋂ ξ ∈ RN ∖ { 0 } A ( x 0 , ξ ) [ E ] = { 0 } 对于每个 x 0 ∈ Ω。这里 P ν - l ( x , D ) 是 Hormander 类 Op S 1 , δ ν - l ( Ω ) , 0 ≤ δ 1 中正确支持的伪微分算子。该陈述的灵感来自于椭圆和抵消齐次算子 A ( D ) 的 Hardy-Littlewood-Sobolev 不等式的新表征,这些算子具有常数系数,扩展并统一了源自经典 Hardy-Sobolev 估计的几个结果。重点介绍了与复杂矢量场椭圆系统相关的算子的变体和应用。
更新日期:2021-02-01
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