Weighted estimates for commutators on nonhomogeneous spaces
- PDF / 267,030 Bytes
- 14 Pages / 467.717 x 680.315 pts Page_size
- 19 Downloads / 175 Views
Let μ be a Borel measure on Rd which may be nondoubling. The only condition that μ must satisfy is μ(Q) ≤ c0 l(Q)n for any cube Q ⊂ Rd with sides parallel to the coordinate axes and for some fixed n with 0 < n ≤ d. This paper is to establish the weighted norm ´ inequality for commutators of Calderon-Zygmund operators with RBMO(μ) functions by an estimate for a variant of the sharp maximal function in the context of the nonhomogeneous spaces. Copyright © 2006 W. Chen and B. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let μ be some nonnegative Borel measure on Rd satisfying μ(Q) ≤ c0 l(Q)n
(1.1)
for any cube Q ⊂ Rd with sides parallel to the coordinate axes, where l(Q) stands for the side length of Q and n is a fixed real number such that 0 < n ≤ d. Throughout this paper, all cubes we will consider will be those with sides parallel to the coordinate axes. For r > 0, rQ will denote the cube with the same center as Q and with l(rQ) = rl(Q). Moreover, Q(x,r) will be the cube centered at x with side length r. The classical theory of harmonic analysis for maximal functions and singular integrals on (Rd ,μ) has been developed under the assumption that the underlying measure μ satisfies the doubling property, that is, there exists a constant c > 0 such that μ(B(x,2r)) ≤ cμ(B(x,r)) for every x ∈ Rd and r > 0. But recently, many classical results have been proved still valid without the doubling condition; see [1–18] and their references. Orobitg and P´erez [11] have studied an analogue of the classical theory of A p (μ) weights in Rd without assuming that the underlying measure μ is doubling. Then, they Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 89396, Pages 1–14 DOI 10.1155/JIA/2006/89396
2
Commutator on nonhomogeneous space
obtained weighted norm inequalities for the (centered) Hardy-Littlewood maximal func´ tion and corresponding weighted estimates for nonclassical Calderon-Zygmund oper´ ators. They also considered commutators of those Calderon-Zygmund operators with BMO(μ) functions. The purpose of this paper is to establish weighted estimates for com´ mutators of those nonclassical Calderon-Zygmund operators with RBMO(μ) in this new setting. Let us introduce some notations and definitions. Given two cubes Q ⊂ R in Rd , we set μ 2k Q n ,
NQ,R
KQ,R = 1 +
k =1
l 2k Q
(1.2)
where NQ,R is the first integer k such that l(2k Q) ≥ l(R). KQ,R was introduced by Tolsa in [15]. Given βd (depending on d) big enough (e.g., βd > 2n ), we say that some cube Q ⊂ Rd is doubling if μ(2Q) ≤ βd μ(Q). Given a cube Q ⊂ Rd , let N be the smallest integer ≥ 0 such that 2N Q is doubling. We denote this cube by Q. Let η > 1 be some fixed constant. We say that a function b(x) is in RBMO(μ) if there exists some constant c1 such that for any cube Q, 1 μ(ηQ)
mQ b − mR b ≤ c1 K
Data Loading...
