Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type

Abstract

Abstract In this paper, we introduce the mixed radial-angular central Campanato space ℭ ⁢ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\mathfrak{C}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(\mathbb% {R}^{n})}} and the mixed radial-angular central Morrey space 𝔐 ⁢ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\mathfrak{M}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(\mathbb% {R}^{n})}} , respectively. Characterizations of the mixed radial-angular central Campanato space are established via the boundedness of the commutators of Hardy type on the mixed radial-angular central Morrey space. Moreover, some characterizations of the mixed radial-angular λ-central bounded mean oscillation space CMO ⁡ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\operatorname{CMO}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(% \mathbb{R}^{n})}} are also obtained.