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The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights
Collectanea Mathematica ( IF 1.1 ) Pub Date : 2019-11-11 , DOI: 10.1007/s13348-019-00272-3
David Cruz-Uribe , Estefanía Dalmasso , Francisco J. Martín-Reyes , Pedro Ortega Salvador

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces \(L_w^{p(\cdot )}(0,\infty )\), assuming that the exponent function \({p(\cdot )}\) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals \(\{ (0,b) : b>0\}\) on \((0,\infty )\). Our results extend those in Duoandikoetxea et al. (Indiana Univ Math J 62(3):891–910, 2013) for the constant exponent \(L^p\) spaces with weights. We also give two applications: the first is a weighted version of Hilbert’s inequality on variable Lebesgue spaces, and the second generalizes the results in Soria and Weiss (Indiana Univ Math J 43(1):187–204, 1994) for integral operators to the variable exponent setting.

中文翻译:

Calderón运算符和Stieltjes对具有权重的可变Lebesgue空间进行变换

假设指数函数为\ {{p },我们表征了Stieltjes变换和Calderón运算符的权重,以加权变量Lebesgue空间\(L_w ^ {p(\ cdot)}(0,\ infty)\)为界(\ cdot)} \)在原点和无穷大处是对数连续的。我们借助于关于\((0,\ infty)\)上区间\(\ {(0,b):b> 0 \} \)定义的最大运算符来获得单个Muckenhoupt类型条件。我们的研究结果扩展了Duoandikoetxea等人的研究。(Indiana Univ Math J 62(3):891–910,2013)的常数指数\(L ^ p \)带重量的空间。我们还给出了两个应用程序:第一个是变量Lebesgue空间上希尔伯特不等式的加权版本,第二个是将Soria和Weiss的结果推广(印第安纳大学数学J 43(1):187–204,1994),以求积分算子对可变指数设置。
更新日期:2019-11-11
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