论文标题
$ l^p $ - $ l^q $伪划分的操作员在光滑的歧管上及其应用于非线性方程的应用
$L^p$-$L^q$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations
论文作者
论文摘要
在本文中，我们研究了整体伪差异操作员对平滑流形的界限。通过使用全球符号的概念，我们扩展了HörmanderType的经典条件，以确保全球运营商的$ l^p $ - $ l^q $结合。首先，鉴于Hörmander-Mihlin条件，我们调查了伪差异操作员的$ l^p $结合。我们还证明$ l^\ infty $ - $ bmo $ $ $ $ $ $ differentic operators。后来，我们将调查集中在解决$ l^p $ - $ l^q $定居下来的傅立叶乘数和伪差异操作员的范围$ 1 <p \ leq 2 \ leq 2 \ leq q <\ iftty。 Hausdorff-Young-paley不平等，用于平滑歧管。最后，我们介绍了界定定理对不同类型的非线性偏微分方程的适当性能的应用。
In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of Hörmander type to guarantee the $L^p$-$L^q$-boundedness of global operators. First we investigate $L^p$-boundedness of pseudo-differential operators in view of the Hörmander-Mihlin condition. We also prove $L^\infty$-$BMO$ estimates for pseudo-differential operators. Later, we concentrate our investigation to settle $L^p$-$L^q$ boundedness of the Fourier multipliers and pseudo-differential operators for the range $1<p \leq 2 \leq q<\infty.$ On the way to achieve our goal of $L^p$-$L^q$ boundedness we prove two classical inequalities, namely, Paley inequality and Hausdorff-Young-Paley inequality for smooth manifolds. Finally, we present the applications of our boundedness theorems to the well-posedness properties of different types of the nonlinear partial differential equations.
