Estimates for the integral mean of harmonic functions on bounded domains in \(\mathbb{R}^ n\)
Let \(D\) be a bounded domain in \(\mathbb{R}^ n\) \((n\geq 2)\) given by a \(C^ \infty\) defining function \(\lambda\) satisfying \(|\nabla \lambda|\neq 0\) in a neighborhood of \(\partial D\). For \(r>0\) sufficiently small, let \(D_ r= \{x\in D\): \(\lambda(x)< -r\}\) and let \(\sigma_ r\) denote the \((n-1)\)-dimensional Hausdorff measure on \(\partial D_ r\). For a continuous function \(f\) on \(\partial D_ r\) and \(0<p <\infty\), the \(p\)-th integral mean of \(f\) on \(\partial D_ r\) is given by \(M_ p (f,r)= [\int_{\partial D_ r} | f|^ p d\sigma_ r ]^{1/p}\). In the paper the author proves that if \(f\) is harmonic in \(D\), then for \(\varepsilon>0\) sufficiently small, \(0<p\leq \infty\), \(0<q< \infty\), \(s>-1\), \(m\in \mathbb{N}\), and \(x_ 0\in D\), \[ \begin{aligned} \int_ 0^ \varepsilon r^{s+m} M_ p^ q( \text{grad}_ m f,r) dr &\leq C\int_ 0^ \varepsilon r^ s M_ p^ q (f,r) dr,\\ \text{and} \int_ 0^ \varepsilon r^ s M_ p^ q (f,r) dr &\leq C\Biggl\{ \sum_{j=0}^{m-1} | \text{ grad}_ j f(x_ 0)|^ q+ \int_ 0^ \varepsilon r^{s+mq} M_ p^ q (\text{grad}_ m f,r) dr \Biggr\}, \end{aligned} \] where \(\text{grad}_ j f\) is the gradient of \(f\) of order \(j\).
scientific article; zbMATH DE number 754162