On the smoothness in the weighted Triebel-Lizorkin and Besov spaces via the continuous wavelet transform with rotations

The main goal of this paper is to show that if u is an element of W-m,W- p (R-n) is a weak solution of Qu = f where f is an element of X-p,k(r,q) (R-n), then u is an element of X-p,k(m+r,q) (R-n) with 1 < p, q < infinity, 0 < r < 1, k is a temperate weight function in the Hormander sense, Q = Sigma (|beta|<= m) c(beta)(partial derivative beta) is a linear partial differential operator of order m >= 0 with non-zero constant coefficients c(beta), and where X-p,X-k (r,q) (R-n) is either the weighted Triebel-Lizorkin or the weighted Besov space. The way to prove this result is based on the boundedness of the continuous wavelet transform with rotations.