Not Get Sub-spaces Of Homogeneous Type Boundedness Of Some Research

In 1950's, Calderon and Zygmund established the first generation of Calderon-Zygmund operators. From then on, it has developed into a very wide research field. Just com-parig the prototypical book "Singular integrals and differentiability properties of functions" which Stein wrote in 1970 with his book " Harmonic analysis:Real-Variable methods, Orthogonality; and Oscillatory integrals"published in 1993, we find the only unchanged matter is that the underlying measure u, satisfies the doubling property i.e. there exists a constants C>0 such that u(B(x,2r)} < Cu(B(x,r)) for every x X, r>0, where X is a metric space with a Borel measure u.X. Tolsa established Littlewood-Paley theory and T1 theory on non-homogeneous space in [19]; F.Nazanov, S.Treil and A.Volberg obtained Tb theory on non-homogeneous in [8] and [9] which built up the theory frame of non-homogeneous. But there are many difficult problems still unsolved. The main one is to estabish Hardy space theory and weighted norm inequalities on non-homogeneous spaces. The solution to these problmes will not only enrich singular integral operator theory and function space theory, but also push on the research of partial differential equations. This paper includes the preliminary research of singular integral operators and related operators. In [18], X. Tolsa introduced RBMO(u) space and a suitable Sharp maximal function enjoying properties similar to ones of the classical Sharp function: i.e., RBMO(u) M 6 L(u). About commutators [b,t],b RBMO(u), T is a C-Z operator with measure u, Tolsa obtained the boundedness on Lp(u). In the first chapter, we obtained weighted norm inequality on the commutators, by proving a variant of Sharp function estimates; J.Orobitg and C.Perez introduced Ap weights for nondoubing measures and proved weighted norm inequality of C-Z singular integrals. In the second chapter, we obtained weighted inequalities of fractional integral and its maximal function with Ap(u) weights for nondoubing measure; E. Sawyer obtained a weak type double weights inequality for fractional integrals in [13]. In the third chapter, we generalized the Sawyer's result for non-doubling measures.