  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.
