Non-doubling Measures Of Generalized Calder¨®n-zygmund Operator

As we all know,the doubling measure is used widely in harmonic analysis, the proofs of many results depend on the doubling condition of the measure. But in some situations, the doubling condition of the measure is needless to many results in harmonic analysis. In recent years, there are many papers focusing on the representation of the function spaces and the boundedness of the Calderon-Zgymund operators on these spaces.X.Tolsa studied the boundcdess of the Calderon-Zgymund operators and the commutators of the Calderon-Zgymund operator with the RBMO function for the non-doubling measure. But the boundedness of the 0(t) type Calderon-Zgymund operator and its commutator are not studied by now.The operator has a profound partial differential coefficient background. The reason is that it is relatively complex and there is a little difficulty.In this thesis, we first prove that the θ(t) type Calderon-Zgymund operator is bounded from H1,∞ atb(μ) to L1(μ) and from L∞(μ) to RBMO(μ) for non-doubling measure.For doubling measure, in order to remedy the boundedness from H1 to L1 of the Calderon-Zgymund operator, a new space H1b is introduced.So the Calderon-Zgymund operator is bounded from H1b to L1 .In this thesis , according to the idea,we get that the commutator of the RBMO(μ) with the θ(t) type Calderon-Zgymund operator is bounded from L∞(μ) to RBMO(μ), and the boundedness of (H1b(μ), L1 (μ))).For doubling measure,the boundedness of the sub-linear operator plays an important role in many problems.In this thesis,at, last,according to the boundedness of the sub-linear operator on Herz space to doubling measure,we study the boundedness of the sub-linear operator on the Herz space and the (θ,0) type fractional operator's boundedness of (H1(μ), Lq(μ)) for non-doubling measure.