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Approximation of critical regularity functions on stratified homogeneous groups
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-09-03 , DOI: 10.1142/s0219199720500595 Eduard Curcă 1
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-09-03 , DOI: 10.1142/s0219199720500595 Eduard Curcă 1
Affiliation
Let G be a stratified homogeneous group with homogeneous dimension Q and whose Lie algebra is generated by the left-invariant vector fields X 1 , … , X d 1 . Let 1 < p , q < ∞ , α : = Q / p and δ > 0 . We prove that for any function f ∈ Ḟ q α , p ( G ) there exists a function F ∈ L ∞ ( G ) ∩ Ḟ q α , p ( G ) such that
∑ i = 1 k ∥ X i ( f − F ) ∥ Ḟ q α − 1 , p ( G ) ≤ δ ∥ f ∥ Ḟ q α , p ( G ) ,
∥ F ∥ L ∞
( G ) + ∥ F ∥ Ḟ q α , p ( G ) ≤ C δ ∥ f ∥ Ḟ q α , p ( G )
where k is the largest integer smaller than min ( p , d 1 ) and C δ is a positive constant depending only on δ . Here, F ̇ q α , p ( G ) is a homogeneous Triebel–Lizorkin type space adapted to G .
This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9 (2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal.
276 (5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.
中文翻译:
分层同质组的临界正则函数逼近
让G 是具有同质维度的分层同质组问 并且其李代数由左不变向量场生成X 1 , … , X d 1 . 让1 < p , q < ∞ ,α : = 问 / p 和δ > 0 . 我们证明对于任何函数F ∈ F q α , p ( G ) 存在一个函数F ∈ 大号 ∞ ( G ) ∩ F q α , p ( G ) 这样
∑ 一世 = 1 ķ ∥ X 一世 ( F - F ) ∥ F q α - 1 , p ( G ) ≤ δ ∥ F ∥ F q α , p ( G ) ,
∥ F ∥ 大号 ∞
( G ) + ∥ F ∥ F q α , p ( G ) ≤ C δ ∥ F ∥ F q α , p ( G )
在哪里ķ 是小于的最大整数分钟 ( p , d 1 ) 和C δ 是一个正常数,仅取决于δ . 这里,F ̇ q α , p ( G ) 是一个齐次 Triebel-Lizorkin 型空间,适用于G . 这概括了 Bourgain, Brezis [椭圆方程和霍奇类型系统的新估计,J.欧元。数学。社会党。 9 (2) (2007) 277–315] 和 Bousquet, Russ, Wang, Yung [在分数 Sobolev 空间和 Hodge 系统中的近似,J. 功能。肛门。
276 (5) (2019) 1430–1478] 在欧几里得案例中,并在后一个参考文献中回答了一个悬而未决的问题。
更新日期:2020-09-03
中文翻译:
分层同质组的临界正则函数逼近
让