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We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest. [ABSTRACT FROM AUTHOR]

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions. [ABSTRACT FROM AUTHOR]

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case. [ABSTRACT FROM AUTHOR]

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp ( p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L(ℍ) is still p/( p-1). This goes some way to imply that the L norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝ, or 'ellipsoids' on the Heisenberg group ℍ. By constructing a special function, we find the best constant in the weak type (1, 1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H to L. Moreover, we describe the difference between M weights and A weights and obtain the characterizations of such weights using the weighted Hardy inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we study the weighted norm inequalities for commutators formed by a class of one-sided oscillatory integral operators and functions in one-sided BMO spaces. [ABSTRACT FROM AUTHOR]

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ n)) on Morrey spaces. [ABSTRACT FROM AUTHOR]