Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W-p(k)(R-s) ( 1 <= p <= infinity) 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)(R-s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L-p (1 <= p <= infinity) 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 is an element of L-p(R-s) (phi is an element of C(R-s) when p = infinity) satisfies a refinement equation with a finitely supported matrix mask, then all the components of phi must belong to a Lipschitz space Lip(nu, L-p(R-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. 7 ( 2001) 143 - 167) in the univariate setting to the multivariate setting.