The monotonic property of \(L^s(\mu)\)-averaging domains and weighted weak reverse Hölder inequality
Let \(\Omega\) be a bounded domain of \(\mathbb{R}^n\) and \(\mu\) a (weak) doubling measure in \(\Omega\). The domain \(\Omega\) is called an \(L^s(\mu)\)-averaging domain, \(s\geq 1\), if for all \(u\in L^s_{\text{loc}}(\Omega; \mu)\) \[ \Biggl(-\hskip-1,1em\int_\Omega|u- u_{B_0}|^s d\mu\Biggr)^{1/s}\leq C\sup_{2B\subset \Omega} \Biggl(-\hskip-1,1em\int|u- u_B|^s d\mu\Biggr)^{1/s} \] for some \(C<\infty\) and for some \(B_0\subset \Omega\). The averaging domains are intimately connected to the Poincaré inequality [\textit{S. G. Staples}, Ann. Acad. Sci. Fenn., Ser. A I Math. 14, No. 1, 103-127 (1989; Zbl 0706.26010)]; John domains are examples of \(L^s(\mu)\)-averaging domains. First, the authors prove a local weak reverse Hölder inequality for \(A\)-harmonic tensors and then a monotonic property of \(L^s(\mu)\)-averaging domains which is the local weighted reverse Hölder inequality in \(L^s(\mu)\)-averaging domains.
scientific article; zbMATH DE number 1346388