Preduals of variable Morrey–Campanato spaces and boundedness of operators

Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (1,\infty )\) be a variable exponent, such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space \(L^{p(\cdot )}({\mathbb {R}}^n),\) and \(\phi :\ {\mathbb {R}}^n\times (0,\infty )\rightarrow (0,\infty )\) be a function satisfying some conditions. In this article, we give some properties of variable Campanato spaces \({\mathcal {L}}_{p(\cdot ),\phi ,d}({\mathbb {R}}^n),\) with a non-negative integer d,  and variable Morrey spaces \(L_{p(\cdot ),\phi }({\mathbb {R}}^n),\) and then establish their predual spaces. As an application of duality obtained in this article, we consider the boundedness of singular integral operators on variable Morrey spaces and their predual spaces.