Bi-predual Spaces of Generalized Campanato Spaces with Variable Growth Condition

In this paper we extend the duality \(({\overline{C_{\rm comp}^{\infty}({\mathbb R}^{d})}}^{{\rm BMO}({\mathbb R}^{d})})^{\ast}=H^{1}({\mathbb {R}^{d}})\) to generalized Campanato spaces with variable growth condition \({\cal L}_{p,\phi}({\mathbb R}^{d})\) instead of BMO(ℝ^d). We also extend the characterization of \({\overline{C_{\rm comp}^{\infty}({\mathbb R}^{d})}}^{{\rm BMO}({\mathbb R}^{d})}\) by Uchiyama (1978) to \({\overline{C_{\rm comp}^{\infty}({\mathbb R}^{d})}}^{{\cal L}_{p,\phi}({\mathbb R}^{d})}\). Moreover, using this characterization, we prove the boundedness of singular and fractional integral operators on \({\overline{C_{\rm comp}^{\infty}({\mathbb R}^{d})}}^{{\cal L}_{p,\phi}({\mathbb R}^{d})}\). The function space \({\cal L}_{p,\phi}({\mathbb R}^{d})\) treated in this paper covers the case that it is coincide with Lip_α on one area, with BMO on another area and with the Morrey space on the other area, for example.