$p$ -Adic多维小波及其对$p$ -adic伪微分算子的应用
$p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators
摘要：
本文结合多维$p$ -adic Lizorkin空间中的$p$-adic伪微分算子理论，研究了与多维$p$ -adic小波理论有关的一些问题.引入了一类新的$n$ -维$p$ -进紧支集小波.在一维的情况下，这类包括Kozyrev $p$ -进小波。这些小波（和它们的傅里叶变换）在${\cL}^2(\bQ_p^n)$中形成一个正交完全基。给出了多维$p$ -进小波是伪微分算子本征函数的一个判据。我们证明了这些小波是Taieton分数次算子的特征函数。由于许多$p$ -adic模型使用伪微分算子（分数阶算子），这些结果可以在应用中得到广泛的应用。此外，$p$ -进小波被用来构造线性和{ \it半线性}伪微分方程的解。
英文摘要：
In this paper we study some problems related with the theory of multidimensional $p$-adic wavelets in connection with the theory of multidimensional $p$-adic pseudo-differential operators (in the $p$-adic Lizorkin space). We introduce a new class of $n$-dimensional $p$-adic compactly supported wavelets. In one-dimensional case this class includes the Kozyrev $p$-adic wavelets. These wavelets (and their Fourier transforms) form an orthonormal complete basis in ${\cL}^2(\bQ_p^n)$. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the Taibleson fractional operator. Since many $p$-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in applications. Moreover, $p$-adic wavelets are used to construct solutions of linear and {\it semi-linear} pseudo-differential equations.
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