标题： 环面上的伪微分算子的估计重新审视。 III 显示英文标题

标题： Estimates for pseudo-differential operators on the torus revisited. III

摘要： 本文完成了作者在之前两篇手稿中开始的目标，这些手稿致力于重新审视Hörmander类符号的环面伪微分算子的连续性性质。 在这里，我们证明了基于Fefferman-Stein尖锐极大函数和Hardy-Littlewood极大函数的逐点估计。 结合这些估计与Muckenhoupt权类$A_p$的性质，我们得到了环面上加权Lebesgue空间之间的伪微分算子的有界性定理$L^p(w)$。 这些结果是在Ruzhansky和Turunen通过使用离散傅里叶分析在$\mathbb{T}^n\times \mathbb{Z}^n$上定义的全局符号分析框架中给出的，并扩展了Park和Tomita在欧几里得情况下的结果。 此外，我们还包括了环面上Sobolev空间$W^s_p$和Besov空间$B^s_{p,q}$的连续性结果。 我们的技术来自Park和Tomita\cite{park-tomita}，我们在此考虑其环面扩展，以确保当前文献中环面伪微分算子有界性的完整性。 显示英文摘要

摘要： This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the H\"ormander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature.