AN HARDY ESTIMATE FOR COMMUTATORS OF PSEUDO-DIFFERENTIAL OPERATORS

• Published in: Taiwanese journal of mathematics Vol. 19; no. 4; pp. 1097 - 1109
• Main Authors: Hung, Ha Duy, Ky, Luong Dang
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.08.2015
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.19.2015.5003

LetTbe a pseudo-differential operator whose symbol belongs to the Hörmander class S p , δ m with 0 ≤δ< 1, 0 <ρ≤ 1,δ≤ρand −(n+ 1) <m≤ −(n+ 1)(1 −ρ). In present paper, we prove that ifbis a locally integrable function satisfying sup balls   B ⊂ ℝ n log ( e + 1 / | B | ) ( 1 + | B | ) θ 1 | B | ∫ B | f ( x ) − 1 | B | ∫ B f ( y ) d y | d x < ∞ for someθ∈ [0, ∞), then the commutator [b,T] is bounded on the local Hardy spaceh 1(ℝ n ) introduced by Goldberg [9]. As a consequence, whenρ= 1 andm= 0, we obtain an improvement of a recent result by Yang, Wang and Chen [21]. 2010Mathematics Subject Classification: 47G30, 42B35. Key words and phrases: Pseudo-differential operators, Hardy spaces, BMO spaces, LMO spaces, Commutators.