On the necessity of bump conditions for the two-weighted maximal inequality (Q2832811)

Let \(1<p<\infty\). Let us consider the two-weighted maximal inequalities, i.e. to characterize those couples \((w,\sigma)\) of nonnegative locally integrable functions, called weights, which satisfy the inequality (1) \(\int_{\mathbb R^n}(M(f\sigma))^pwdx\leq C\int_{\mathbb R^n}|f|^p\sigma dx\), where \(M(f)\) is the Hardy-Littlewood maximal function of \(f\). E. Sawyer gave a necessary and sufficient condition (Sawyer' condition) for (1). C. Pérez and Rela gave a sufficient condition in terms of Banach function spaces \(X\). They define the maximal operator \(M_X\). Then, if (2) \(\sup_{Q:cubes}\|w^{1/p}\|_{L^p,Q}\|\sigma^{1/p'}\|_{X,Q}<\infty\) (bump condition), and (3) \(\int_Q (M_{X'}(\sigma^{1/p}\chi_Q))^pdx\leq C\int_Q \sigma dx\) (\(X'\) is the associate space of \(X\)), it follows (1).NEWLINENEWLINEIf \(\sigma\) is a Muckenhoupt's \(A_\infty\) weight, (1) implies (2) and (3) for \(X=L^{p'}\). The author gave a couple \((w,\sigma)\) satisfying (1) but there is no Banach function space \(X\) for which (2) and (3) hold simultaneously. The author's example is as follows: \(w(x)=\frac{|x|_{\max}^{n(p-1)}}{(1+\log^+|x|_{\max}^{n})^p}\) and \(\sigma(x)=\frac{1}{|x|_{\max}^{n}(1+\log^+(1/|x|_{\max}^{n}))^{p'}}\), where \(|x|_{\max}=\max_{1\leq j\leq n}|x_j|\). The author remarked that \(\frac{1}{|x|_{\max}^{\alpha}(1+\log^+(1/|x|_{\max}^{n}))^{\beta}}\in A_{\infty}\) if \(0<\alpha<n\) and \(\beta\in\mathbb R\).