The authors study the regularity of weak solutions of the equation \[ \operatorname{div}(A(\nabla u)):=\operatorname{div}(|\nabla u|^{p-2} \nabla u)=\operatorname{div} F\text{ in }\Omega, \] where \(1 < p < \infty\), \(\Omega\) is an open set in \(\mathbb{R}^d\), \(u:\Omega \to \mathbb{R}^n\), and \(d, n \in \mathbb{N}\). In one of the main results it is shown that for \(p \geq 2\), \(d=2\), \(n=1\), and \(F \in L^{p'}(\Omega)\), a weak solution \(u \in W^{1,p}(\Omega)\) of the considered equation satisfies \[ |A(\nabla u)|_{\mathbf{B}^s_{\varrho,q}(B)}\lesssim |F|_{\mathbf{B}^s_{\varrho,q}(2B)} + |2B|^{-\frac{1}{p'}} \left\|A(\nabla u) - |2B|^{-1} \int_{2B}A(\nabla u)\,dx\right\|_{L^{p'}(2B)}, \] provided \(s>0\) and \(\varrho, q \in (0,\infty]\) are such that \(2\left(\frac{1}{\varrho}-\frac{1}{p'}\right)_+ < s < 1\), and a ball \(B\) is such that \(2B \subset \Omega\). Here \(\mathbf{B}^s_{\varrho,q}\) is the Besov space. If, in addition, \(\varrho<\infty\) and \(2\left(\frac{1}{q}-\frac{1}{p'}\right)_+ < s < 1\), then the same estimate on \(A(\nabla)\) is valid with \(\mathbf{B}^s_{\varrho,q}\) replaced by the Triebel-Lizorkin space \(\mathbf{F}^s_{\varrho,q}\).