洛伦茨-穆肯霍普特类的性质与应用
Properties and applications of Lorentz--Muckenhoupt classes
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中文总结 AI 辅助
本文研究洛伦茨-穆肯霍普特类的性质，分析极大算子在乘子加权洛伦茨空间的有界性，应用其结果刻画分数次积分交换子、建立临界情形的哈代不等式并求解分数阶薛定谔方程，部分回答开放问题。
中文摘要 AI 辅助
本文通过引入洛伦茨-穆肯霍普特（Lorentz–Muckenhoupt）类，系统研究了极大算子在乘子加权洛伦茨空间上的有界性。作为应用，给出了分数次积分交换子的刻画，部分回答了D. Cruz-Uribe提出的开放问题；建立了洛伦茨空间中的哈代不等式，其中临界情形为p=d，此时经典哈代不等式不成立；最后将洛伦茨估计应用于带有奇异位势的分数阶薛定谔方程。
英文摘要
In this paper, through the introduction of Lorentz--Muckenhoupt classes, we systematically investigate the boundedness of maximal operators on multiplier weighted Lorentz spaces. As applications, we give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe. Second, Hardy inequalities in Lorentz spaces are established, with the critical case $p=d$, in which the classical Hardy inequality fails. Finally, we apply the Lorentz estimates to fractional Schrödinger equations with singular potentials.
