Inequalities for mean values over quasiballs for functions defined on arbitrary open sets (Q2711477)

The authors bring a number of results concerning sharp two-sided estimates of mean values of functions over balls determined by a regular quasi-metric \(d\) (quasi-balls) in \(\mathbb R^n\) (here, the regularity of \(d\) means several quite natural restrictions including continuity of \(d\) considered as a function on \(\mathbb R^{2n}\) with respect to the Euclidean distance (the authors show on examples that this is not necessarily the case --- this function even does not have to be Lebesgue measurable)). A typical result states that, under certain natural mild restrictions on \(d\), given \(\varepsilon\), there is a pair of constants \(C_1,C_2\) depending only on \(d\) and \(\varepsilon\) such that NEWLINE\[NEWLINE C_1\int_\Omega|f|dx\leq\int_\Omega|f|_{d,\varepsilon} dx \leq C_2\int_\Omega|f|dx, NEWLINE\]NEWLINE where \(|f|_{d,\varepsilon}\) is the integral mean of \(|f|\) over the quasi-ball \(B_d(x, \varepsilon\varrho_d(x))\), \(\varrho_d(x)=\inf_{y\in\partial\Omega} d(x,y)\). \(\Omega\) is throughout supposed to be an open (in Euclidean sense) set in \(\mathbb R^n\). The results are illustrated with plenty of interesting examples and remarks.