Some remarks on L1 embeddings in the subelliptic setting. (January 2021)

• Record Type: Journal Article
• Authors: Krantz, Steven G.; Peloso, Marco M.; Spector, Daniel
• Is Part Of: Nonlinear analysis. Volume 202(2021)
• Publication Date: 2021-01
• Subjects: Sobolev embeddings -- Lorentz spaces -- L1 regime -- Stratified group -- Subelliptic estimates; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.112149
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L 1 regime in the setting of a stratified group G : Let Q ≥ 2 be the homogeneous dimension of G and I α denote the Riesz potential of order α on G . Then, for every α ∈ ( 0, Q ), there exists a constant C = C ( α, Q ) > 0 such that (0.1) ‖ I α f ‖ L Q ∕ ( Q − α ), 1 ( G ) ≤ C ‖ X I 1 f ‖ L 1 ( G ) for all f ∈ C c ∞ ( G ) such that X I 1 f ∈ L 1 ( G ), where X denotes the horizontal gradient.