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In the setting of quaternionic Heisenberg group Hn−1$\mathcal H^{n-1}$, we characterize the boundedness and compactness of commutator [b,C]$[b,\mathcal {C}]$ for the Cauchy–Szegö operator C$\mathcal {C}$ on the weighted Morrey space Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ with p∈(1,∞)$p\in (1, \infty)$, κ∈(0,1)$\kappa \in (0, 1)$, and w∈Ap(Hn−1)$w\in A_p(\mathcal H^{n-1})$. More precisely, we prove that [b,C]$[b,\mathcal {C}]$ is bounded on Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ if and only if b∈BMO(Hn−1)$b\in {\rm BMO}(\mathcal H^{n-1})$. And [b,C]$[b,\mathcal {C}]$ is compact on Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ if and only if b∈VMO(Hn−1)$b\in {\rm VMO}(\mathcal H^{n-1})$. [ABSTRACT FROM AUTHOR]

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]

This article addresses two characterizations of BMO(ℝn)-type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO(ℝn) space by the boundedness of the commutators; (ii) characterization of the central BMO(ℝn)-closure of Cc∞(ℝn) space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function. [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]

Some boundedness results are established for commutators generated by classical fractional Hardy operators and one-sided CMO functions on the homogeneous Herz spaces, which extends some known results. [ABSTRACT FROM AUTHOR]

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones. [ABSTRACT FROM AUTHOR]