2012/03/28 by Luong Dang Ky, Ky, Luong Dang
Mathematics · #35J10 (Primary) #42B20 (Secondary) #42B35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1203.6335
openalex publication_date 2012/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L= -\Δ+ V be a Schr "odinger operator on mathbb Rd, d\≥ 3,\nwhere V is a nonnegative potential, V\≠ 0, and belongs to the reverse\nH "older class RHd/2. In this paper, we study the commutators [b,T] for\nT in a class mathcal KL of sublinear operators containing the fundamental\noperators in harmonic analysis related to L. More precisely, when T\∈\n mathcal KL, we prove that there exists a bounded subbilinear operator\n mathfrak R= mathfrak RT: H1L( mathbb Rd)\× BMO( mathbb Rd)\→\nL1( mathbb Rd) such that |T( mathfrak S(f,b))|- mathfrak R(f,b)\≤\n|[b,T](f)|\≤ mathfrak R(f,b) + |T( mathfrak S(f,b))|, where mathfrak S\nis a bounded bilinear operator from H1L( mathbb Rd)\× BMO( mathbb Rd)\ninto L1( mathbb Rd) which does not depend on T. The subbilinear\ndecomposition ( refabstract 1) explains why commutators with the fundamental\noperators are of weak type (H1L,L1), and when a commutator [b,T] is of\nstrong type (H1L,L1). Also, we discuss the H1L-estimates for\ncommutators of the Riesz transforms associated with the Schr "odinger operator\nL.\n