Restricted weak-type Rubio de Francia extrapolation for \(p>p_{0}\) with applications to exponential integrability estimates
Let \(p\in[1,\infty)\) and \(\widehat{A}_p\) be the class of weights \(\omega\) of the form \(\omega:=(Mg)^{1-p}u\) for some \(g\in L^1_{\mathrm{loc}}(\mathbb{R}^n)\) and \(u\in A_1\), where \(A_1\) represents the class of \(A_1\) weights and \(M\) is the Hardy-Littlewood maximal operator. For any \(p\in[1,\infty)\) and \(q\in[1,\infty]\), denote the Lorentz spaces associated with weights \(v\in\widehat{A}_p\) by \(L^{p,q}(v)\). The authors show that, if \(p_0\in(1,\infty)\) and \(T\) is a sublinear operator such that, for every \(v\in\widehat{A}_p\), \(T\) is bounded from \(L^{p_0,1}(v)\) into \(L^{p_0,\infty}(v)\) with a constant less than or equal to \(\varphi(\| v\|_{\widehat{A}_{p_0}})\) for some increasing function \(\varphi\) on \((0,\infty)\), then for every \(p\in(p_0,\infty)\) and every \(v\in\widehat{A}_p\), \(T\) is bounded from \(L^{p,1}(v)\) into \(L^{p,\infty}(v)\) with a precise control of the constant depending on \(\|v\|_{\widehat{A}_p}\) and \(\varphi\). As applications, exponential integrability estimates are proved for several classes of operators appearing in harmonic analysis, such as square functions, Haar shift operators, maximal Calderón-Zygmund operators, etc.
scientific article; zbMATH DE number 6538735
scientific article