Non-atomic Decomposition Of Hardy Spaces On Homogeneous Spaces And Certain Operator Boundedness

Let μ be a positive Radon measure which satisfiesfor any x∈ R^d and r > 0, where B(x, r) is the ball centered at x and having radius r, C₀ and n are positive constants with 0 < n ≤ d. The Euclidean space R^d with such a Radon measure is also called a non-homogeneous space.This dissertation is devoted to a new atomic decomposition of the Hardy space with measure μ and the L^p(μ) boundedness of the maximal operator associated with the commutators of singular integrals. It's also given the weighted inequality of the maximal operator associated with Orlicz function on those measure spaces. It consists of three parts.In the first part, we will establish a new atomic decomposition of the Hardy space with measure μ, which was introduced by in [18] and [21]. Firstly, the new atomic space, H_LlogL^1,ρ(μ), is defined and then showed that H_LlogL^1,ρ(μ) = H¹(μ). We also give two applications of their new characterization of H¹(μ). In the first application, we showed that if an operator T satisfies the size conditionfor some α with 0 < α ≤ n, and is bounded from H¹(μ) to L^n/(n-α)(μ), then T maps LlogL into L^n/(n-α)(μ) locally. In the second application, then showed that if a sublinear operator T is bounded on L^p(μ) for some 1 < p <∞ and is bounded from H¹(μ) to weak L¹(μ), then T is bounded from LlogL to weak L¹(μ). The results obtained in this part are interesting and are new even if μ is the Lebesgue measure.The second part deals with the L^p(μ) boundedness, when μ could be a non doubling measure, for the maximal operator associated with the commutators of singular integrals when the kernel satisfies a standard size condition and a certain minimum regularity condition. This improve a result due to Tolsa from two point of view. First one, in the sense that in this work the regularity condition on the kernel is weaker that in Tolsa's paper. The second one is due to the fact that in this case they study the k-th. order commutator and in Tolsa's paper only the case k = 1 is considered.The third part is concerned with the weighted inequality of the maximal operator associated with Orlicz function on non-homogeneous spaces. Applying the inequality, it is proved that the maximal operator satisfies the inequality which is similar to the condition of A\ weight.