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Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. (English) Zbl 1298.42021

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 \[ \|w\chi_Q\|_{L^{p(\cdot)}}\|w^{-1}\chi_Q\|_{L^{p'(\cdot)}}\leq K|Q| \] 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 \[ Mf(x)=\sup_{Q\owns x}\frac1{|Q|}\int_Q|f(y)|\,dy \] satisfies the strong type estimate \[ \|(Mf)w\|_{L^{p(\cdot)}}\leq C \|fw\|_{L^{p(\cdot)}} \] for \(\text{ess\,inf}\,p>1\) and the weak type estimate \[ \|t\chi_{\{Mf>t\}}w\|_{L^{p(\cdot)}}\leq C \|fw\|_{L^{p(\cdot)}} \] 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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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