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