Weighted norm inequalities for the maximal operator on variable Lebesgue spaces (Q442500)

Let \(p(\cdot):\mathbb R^ n\to[1,\infty)\) be the exponent function and let \(L^{p(\cdot)}\) be the variable Lebesgue space, that is, the set of all measurable functions \(f\) on \(\mathbb R^ n\) such that \(\int_{\mathbb R^ n}|\alpha f(x)|^{p(x)}\,dx<\infty\) for some \(\alpha>0\). A weight \(w\) is said to satisfy the \(A_{p(\cdot)}\) condition if NEWLINE\[NEWLINE \|w\chi_Q\|_{L^{p(\cdot)}}\|w^{-1}\chi_Q\|_{L^{p'(\cdot)}}\leq K|Q| NEWLINE\]NEWLINE for some constant \(K\) and every cube \(Q\). The authors prove that under the usual logarithmic-type restrictions on the function \(p\), the maximal operator \(M\) defined by NEWLINE\[NEWLINE Mf(x)=\sup_{Q\owns x}\frac1{|Q|}\int_Q|f(y)|\,dyNEWLINE\]NEWLINE satisfies the strong type estimate NEWLINE\[NEWLINE \|(Mf)w\|_{L^{p(\cdot)}}\leq C \|fw\|_{L^{p(\cdot)}} NEWLINE\]NEWLINE for \(\text{ess\,inf}\,p>1\) and the weak type estimate NEWLINE\[NEWLINE \|t\chi_{\{Mf>t\}}w\|_{L^{p(\cdot)}}\leq C \|fw\|_{L^{p(\cdot)}} NEWLINE\]NEWLINE for \(\text{ess\,inf}\,p\geq1\), provided \(w\) satisfies the \(A_{p(\cdot)}\) condition and, conversely, if either of the above inequalities is true, then \(w\) satisfies the \(A_{p(\cdot)}\) condition.