Weighted norm inequalities for conjugate \(A\)-harmonic tensors
\noindent Conjugate harmonic tensors generalize conjugate harmonic functions and \(p\)-harmonic functions which arise in non-linear potential theory. In this paper the authors extend two recent results of Nolder (to appear in Illinois J. Math) to analogous weighted versions. Let \(u\) and \(v\) denote conjugate \(A\)-harmonic tensors in a domain \(\Omega \subset R^{n}\) and \(s\) and \(t\) be any constants, \(0<s,t <\infty\). The first inequality is a local result which holds for all cubes \(\sigma Q \subset \Omega , \sigma >1\) and relates the weighted \(L^{s}\)-norm of \(u-u_{Q}\) over \(Q\) ( or \(|v- v_{Q}|_{s, Q}\) ) to the \(L^{t}\)-norm of \(v -c_{1}\) over \(\sigma Q\) (or respectively \(|u -c _{2} |_{t,\sigma Q}\)). Here \(c_{1}\) is any form in the Sobolev space \(W_{q,loc}^{1} (\Omega , \Lambda) \) with the Hodge star operator satisfying \(d\star c_{1} = 0\). The second inequality holds globally over John domains \(\Omega\). Here \(s\) and \(t\) are related by a specific algebraic formula in terms of the conjugate exponents \(p\) and \(q\) tied to \(u\) and \(v\). In that setting the authors prove a weighted form of \(|u- u_{Q}|_{s, \Omega} \leq C |v -c_{1} |_{t, \Omega} ^{q/p}\), with \(c_{1}\) as above . The weights the authors use in each of their theorems bear some resemblance to Muckenhoupt's \(A^{p}\)-weights.
scientific article; zbMATH DE number 944388
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