Some estimates for commutators of Riesz transforms associated with Schrodinger operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:419

Article

Liu, Yu¹  Sheng, Jielai¹ 

[1] Univ Sci & Technol Beijing, Sch Math & Phys, Beijing 100083, Peoples R China

关键词: Commutator;    Hardy space;    Campanato space;    Reverse Holder inequality;    Riesz transform;    Schrodinger operator;   

DOI  :  10.1016/j.jmaa.2014.04.053

来源: Elsevier

【 摘 要 】

We consider the Schrodinger operator L = -Delta + V on R-n, where the nonnegative potential V belongs to the reverse Holder class B-q1 for some q(1) >= n/2. Let q(2) = 1 when q(1) >= n and 1/q(2) = 1 - 1/q(1) + 1/n when n/2 < q(1) < n. Set delta' = min{1, 2 - n/q(1)}. Let H-L(p)(R-n) be the Hardy space related to the Schrodinger operator L for n/n+delta' < p <= 1. The commutator [b, R] is generated by a function b is an element of Lambda(theta)(nu), where Lambda(theta)(nu) is a function space which is larger than the classical Companato space, and the Riesz transform R (=) over dot del(-Delta + V)(-1/2). We show that the commutator [b, R] is bounded from L-p(R-n) into L-q(R-n) for 1 < p < q(2)', where 1/q = 1/p - nu/n, and bounded from H-L(p)(R-n) into L-q (R-n) for n/n+nu < p <= 1, where 1/q = 1/p - nu/n. Moreover, we prove that the commutator [b, R] maps H-L(n/n+nu) (R-n) continuously into weak L-1 (R-n). At last, we give a characterization for the boundedness of the commutator [b, R] in an extreme case. (C) 2014 Elsevier Inc. All rights reserved.

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