Two characterizations of central BMO space via the commutators of Hardy operators

Abstract

Abstract This article addresses two characterizations of BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -closure of C c ∞ ⁢ ( ℝ n ) {C_{c}^{\infty}(\mathbb{R}^{n})} 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.