AI 中文总结
本文引入与线性典型邓克尔变换相关的伪微分算子，证明其在施瓦茨空间、缓增分布上的连续性及L²有界性，建立核的性质并将其应用于非齐次与非线性抛物型偏微分方程研究。
AI 中文摘要
本文引入一种与线性典型邓克尔变换相关的伪微分算子。对于一类特定符号，我们证明该算子定义了从施瓦茨空间到其自身的连续线性映射。我们进一步建立该算子的振幅积分与核表示，并研究相关核的光滑性与衰减性质。随后，我们在线性典型邓克尔索伯列夫空间上建立该伪微分算子的L¹范数不等式。此外，我们将该伪微分算子扩展到缓增分布，证明其连续性。接着，我们研究符号类S^m_0下该算子的L²有界性。最后，作为应用，我们利用该伪微分算子研究非齐次与非线性抛物型偏微分方程。
英文摘要
In this paper, we introduce a pseudo-differential operator associated with the linear canonical Dunkl transform. For a particular class of symbols, we prove that this operator defines a continuous linear mapping from the Schwartz space into itself. We further establish an amplitude integral and kernel representation of the operator and investigate the smoothness and decay properties of the associated kernel. Subsequently, we establish an $L^1$-norm inequality for the pseudo-differential operator on the linear canonical Dunkl Sobolev spaces. Moreover, we extend the pseudo-differential operator to tempered distributions, proving its continuity. We then investigate the $L^2$-boundedness of the operator for the symbol class $S^m_0$. Finally, as an application, we employ the pseudo-differential operator to study non-homogeneous and nonlinear parabolic partial differential equations.
