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 \(u\in W^{m,p}(\mathbb R^n)\) is a weak solution of \(Qu = f\) where \(f \in X^{r,q}_{p,k}(\mathbb R^n)\), then \(u \in X^{m+r,q}_{p,k}(\mathbb R^n)\) with \(1< p,q < \infty \), \(0< r < 1\), k is a temperate weight function in the Hörmander sense, \(Q = \sum _{|\beta | \le m} c_{\beta }\partial ^{\beta }\) is a linear partial differential operator of order \(m \ge 0\) with non-zero constant coefficients \(c_{\beta }\), and where \(X^{r,q}_{p,k}(\mathbb R^n)\) 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.