## Abstract

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space $W_p^k(\RR^s) (1\le p \le \infty)$ to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space $W_p^k(\RR^s)$ will be investigated. When the dilation matrix is isotropic, a characterization will be given for the $L_p (1\le p \le \infty)$ critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector $\phi$ in $L_p(\RR^s)$ satisfies a refinement equation with a finitely supported mask, then all the components of $\phi$ must belong to a Lipschitz space $\hbox{Lip}(\nu, L_p(\RR^s))$ for some $\nu>0$. This paper generalizes the results in [R.~Q.~Jia, K.~S.~Lau, and D.~X.~Zhou, J. Fourier Anal. Appl., {\bf 7} (2001), pp. 143--167] in the univariate setting to the multivariate setting.

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