Let \(\cal{X}\) represent either ℝn or a cube Q of ℝn with finite edge length. The authors prove that, for any p ∈ (1, ∞), the predual of the dyadic John-Nirenberg space \(JN_{p}^{{\rm{d}}}({\cal{X}})\) is a dyadic Hardy-type space and then establish some relationships among dyadic John-Nirenberg spaces, dyadic Hardy-type spaces, and their non-dyadic counterparts. Moreover, the authors prove that, for any α ∈ [0, n) and p, q ∈ (1, ∞) with \({1 \over q} = {1 \over p} - {\alpha \over n}\) , the fractional dyadic maximal operator \(M_{{\cal{X}}}^{{\rm{d}},\alpha}\) is bounded from \(JN_{p}^{\rm{d}}(\cal{X})\) to \(JN_{q}^{\rm{d}}(\cal{X})\) and maps \(VJN_{p}^{\rm{d}}(\cal{X})\) to \(VJN_{q}^{\rm{d}}(\cal{X})\) , where \(VJN_{p}^{\rm{d}}(\cal{X})\) and \(VJN_{q}^{\rm{d}}(\cal{X})\) are, respectively, the vanishing sub-spaces of \(JN_{p}^{\rm{d}}(\cal{X})\) and \(JN_{q}^{\rm{d}}(\cal{X})\) . As for the endpoint case p = 1 and α ∈ (0, n), the authors show \(M_{\cal{X}}^{{\rm{d,}}\alpha}:JN_1^{\rm{d}}(\cal{X}) \to JN_{{n \over {n - \alpha}}}^{\rm{d}}(\cal{X})\) , which improves the classical weak-type result \(M_{\cal{X}}^{{\rm{d}},\alpha}:{L^1}(\cal{X}) \to {L^{{n \over {n - \alpha}},\infty}}(\cal{X})\) because \(JN_{1}^{\rm{d}}(Q)=L^{1}(Q)\) and . Compared with recent corresponding results of J. Kinnunen and K. Myyryläinen, both the case \(\cal{X}=\mathbb{R}^{n}\) and the mapping result on vanishing subspaces are new. Also, an interesting phenomenon in the endpoint case p = 1 is that \(M_{\cal{X}}^{\rm{d},\alpha}\) maps \(JN_{1}^{\rm{d}}(\cal{X})\) to the convexification of the dyadic John-Nirenberg space when α = 0, but the convexification disappears when α > 0.
