BMO functions generated by \(A_X(\mathbb{R}^n)\) weights on ball Banach function spaces (Q2046591)

Summary: Let \(X\) be a ball Banach function space on \(\mathbb{R}^n\). We introduce the class of weights \(A_X(\mathbb{R}^n)\). Assuming that the Hardy-Littlewood maximal function \(M\) is bounded on \(X\) and \(X^\prime \), we obtain that \(\mathrm{BMO}(\mathbb{R}^n)=\{\alpha \ln \omega : \alpha \geq 0\), \(\omega \in A_X(\mathbb{R}^n)\} \). As a consequence, we have \(\mathrm{BMO}(\mathbb{R}^n)=\{\alpha \ln \omega: \alpha \geq 0\), \(\omega \in A_{L^{p (\cdot)}(\mathbb{R}^n)} (\mathbb{R}^n)\}\), where \(L^{p( \cdot)}(\mathbb{R}^n)\) is the variable exponent Lebesgue space. As an application, if a linear operator \(T\) is bounded on the weighted ball Banach function space \(X(\omega)\) for any \(\omega\in A_X(\mathbb{R}^n)\), then the commutator \([b,T]\) is bounded on \(X\) with \(b\in\mathrm{BMO}(\mathbb{R}^n)\).