Boundedness Of Some Commutators On Non-Homogeneous Spaces

It is well known that the doubling condition on the underlying measure is a key assumption in the harmonic analysis on Euclidean space or more general spaces of homogeneous type. We recall that the measure # is said to satisfy the doubling condition if there exist a constant C＞0 such thatμ(B(x, 2r))≤Cμ(B(x,r)) for all x∈supp(μ) and r＞0, where we denote by B(x, r)the open ball centered at x and having the radius r. However, some recent research has revealed that the most results of classical Calderon-Zygmund operator theory are still true with the assumption that the underlying measure only satisfies the following growth condition, namely, there exists a constant C₀＞0 such that for all x∈R^d and r＞0, where n is a fixed number and 0＜n≤d. Here, the measureμis not necessary to satisfy the doubling condition. So we call the Euclidean space R^d, which is endowed with the usual Euclidean distance and a non-negative Radon measureμonly satisfying the above growth condition, a nonhomogeneous space. The analysis on non-homogeneous spaces was proved to play a striking role in solving the long open Painleve's problem and in the solution of Vitushkin's conjecture.In this dissertation, we obtain the boundedness of maximal multilinear commutators generated by Calderon-Zygmund operator with standard kernel and RBMO(μ) functions or Osc_{exp L^Ï„}(μ) functions; and also, such a type commutator generated by multilinear Calderon-Zygmund operator and RBMO(μ) functions on product Lebesgue spaces.In Chapter 2, we discuss the L^p(μ) (1＜p＜∞) boundedness of the maximal multilinear commutators which is generated by Calderon-Zygmund operator with standard kernel and RBMO(μ) functions. As an Corollary, we get the same estimate for maximal higher order commutators.Moreover, we establish the weak LlogL type endpoint estimate for f maximal multilinear commutators generated by Calderon-Zygmund operators with standard kernel and Osc_{exp L^Ï„}(μ) functions. The corresponding results are also proved for maximal higher order commutators. In the end, we proceed to investigate the boundedness of commutator generated by multilinear Calderon-Zygmund integral operator and RBMO(μ) functions on product Lebesgue spaces. This type is a vector extension of usual commutators.The results of us can be seen as the natural extension of classical Calderon-Zygmund commutator theory to the non-homogeneous space. However, since the measure in non-homogeneous space only satisfies the growth condition, we need to overcome some essential difficulties. And some methods in this dissertation are different from the classical case and the estimates we need are more subtle.