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Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation. (English) Zbl 1124.31004
Let \(\Omega\subset\mathbb{R}^n\) be a connected open subset, \(D' (\Omega,\Lambda^1)\) be the space of differential 1-forms (deRham currents) on \(\Omega\), \(L^p(\Omega, \Lambda^1)\subset D'(\Omega, \Lambda^1)\) be the Banach subspace (with the standard \(L^p\) norm). The article involves a large number of local and global \(L^r\)-norm inequalities (the inequalities between norm of \(du\) and \(d^*v)\) for the solution of the equation
\[ A(x,g+du)=h+d^*v \] where \(g,h\in D' (\Omega,\Lambda^1)\) are given and operator \(A:\Omega\times \Lambda^1\to\Lambda^1\) satisfies
\[ |A(x,\xi)|\leq a|\xi|^{p-1},\langle A(x,\xi),\xi\rangle\geq|\xi|^p\;(a>0,1<p<\infty). \] Here \(d^*:D'(\Omega, \Lambda^{l+1})\to D'(\Omega,\Lambda^1)\) is the adjoint operator to the exterior derivative \(d\). The inequalities are extended to involve the \(A_r\)-weighted versions with weights \(w>0\) satisfying
\[ \sup_B\left( \frac{1}{|B|}\int_Bw\,dx\right) \left(\frac{1}{|B|} \int_B\left|\frac{1}{w}\right|^{1/(r-1)}dx\right)^{r-1}<\infty \]
for any ball \(B\subset \Omega\). Applications to Sobolev-Poincaré type embedding theorems and to \(L^p\)-norm estimates of the homotopy operator are briefly mentioned.

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
58A10 Differential forms in global analysis
58J05 Elliptic equations on manifolds, general theory
31C12 Potential theory on Riemannian manifolds and other spaces
Full Text: DOI
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