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

Abstract

In this paper we extend the duality $$({\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^d)}}^{\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^d)}{)}^{\ast }=H^1({\mathbb{R}}^d)$$ to generalized Campanato spaces with variable growth condition $${\mathcal{L}}_{p,\varphi }({\mathbb{R}}^d)$$ instead of BMO(ℝ^d). We also extend the characterization of $${\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^d)}}^{\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^d)}$$ by Uchiyama (1978) to $${\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^d)}}^{{\mathcal{L}}_{p,\varphi }({\mathbb{R}}^d)}$$. Moreover, using this characterization, we prove the boundedness of singular and fractional integral operators on $${\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^d)}}^{{\mathcal{L}}_{p,\varphi }({\mathbb{R}}^d)}$$. The function space $${\mathcal{L}}_{p,\varphi }({\mathbb{R}}^d)$$ 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.