Weighted integral inequalities for solutions of the \(A\)-harmonic equation
The parametric version of the Cacciopoli inequality \[ \Biggl( \int_B |du|^s w^\beta dx \Biggr)^{1/s}\leq C\dim (B)^{-1} \Biggl( \int_{\rho B} |u-c|^s w^{\beta\lambda} dx\Biggr)^{1/s} \] and of the weak reverse Hölder inequality \[ \Biggl( \int_B |u|^s w^\beta dx\Biggr)^{1/2}\leq C|B|^{(t-s)/st} \Biggl(\int_{\rho B}|u|^t w^{\beta\lambda t/s} dx\Biggr)^{1/2} \] with weight \(w(x)\in A_r^\lambda(\Omega)\) for differential forms \(u\in D'(\Omega, \Lambda^l)\) satisfying the \(A\)-harmonic equation \(d^* A(x,du)= 0\) in a domain \(\Omega \subset \mathbb{R}^n\) are proved. Here \(B\) are balls with \(\rho B\subset \Omega\) (\(\rho> 1\) fixed), \(c\) are arbitrary closed differential forms, \(A:\Omega\times \Lambda^l (\mathbb{R}^n)\to \Lambda^l \mathbb{R}^n\) satisfies \(A(x,\xi)\leq a|\xi|^{p-1}\) and \((A(x,\xi),\xi)\geq |\xi|^p\) (\(1<p <\infty\) fixed), \(D'(\Omega,\Lambda^l)\) denotes the deRham currents and the class \(A_r^\lambda(\Omega)\) of weights is defined by the inequality \[ \sup_B \Biggl( \frac{1}{|B|} \int_B w dx\Biggr) \Biggl( \frac{1}{|B|}\int_B w^{-1/(r-1)} dx \Biggr)^{\lambda(r-1)}< \infty. \]
scientific article; zbMATH DE number 1915754