Self-improving properties of a generalized Muckenhoupt class
Given an interval \(J\subset\lbrack0,\infty)\) that contains the origin and a number \(p\in(1,\infty),\) a nonnegative measurable function \(w\) belongs to the Muckenhoupt class \(A^{p}(C)\) if there exists a positive constant \(C\) such that \[\left( \left( 1/\left\vert I\right\vert \right) \int_{I}w(t)dt\right)\left( \left( 1/\left\vert I\right\vert \right) \int_{I}w^{-1/(p-1)} (t)dt\right) ^{p-1}\leq C\] for every bounded subinterval \(I\subset J.\) \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 165, 207--226 (1972; Zbl 0236.26016)] has noticed the following self-improving property: if \(w\in A^{p}(C)\), then there exist two positive numbers \(\varepsilon\) and \(C_{1}\) (depending only on \(p\) and \(C)\) such that \(w\) \(\in\) \(A^{p-\varepsilon}(C_{1})\) and \(p-\varepsilon>1.\) The paper under review is aimed to study the structure of the weighted Muckenhoupt class \(A_{\lambda}^{p}(C)\) (whose definition is similar with that of \(A^{p}(C),\) with the difference that the Lebesgue measure \(dt\) is replaced by the weighted measure \(\lambda(t)dt\) and the normalization factor \(1/\left\vert I\right\vert \) is replaced by \(1/\Lambda(I)\) where \(\Lambda(I)=\int_{I} \lambda(t)dt\). Using an inequality of Hardy type, it is proved that every weight \(w\) in \(A_{\lambda}^{1}(C)\) belongs to a weighted Gehring class \(G_{\lambda}^{p}(K),\) that is, for some \(K\geq1\), one has \[\left( \left( 1/\Lambda([0,t])\right) \int_{0}^{t}w^{p}(t)\lambda (t)dt\right) ^{1/p}\leq K\left( \left( 1/\Lambda([0,t])\right) \int_{0}^{t}w(t)\lambda(t)dt\right),\] whenever \(t\in J\), \(t>0.\) The paper also includes the self-improving property of the weighted Muckenhoupt class \(A_{\lambda}^{p}(C)\) as well as of the weighted Gehring class \(G_{\lambda}^{p}(K),\) with a sharp value on the exponents.
scientific article; zbMATH DE number 7377532
scientific article