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Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation. (English) Zbl 1124.31004

Let Ω n be a connected open subset, D ' (Ω,Λ 1 ) be the space of differential 1-forms (deRham currents) on Ω, L p (Ω,Λ 1 )D ' (Ω,Λ 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,hD ' (Ω,Λ 1 ) are given and operator A:Ω×Λ 1 Λ 1 satisfies

|A(x,ξ)|a|ξ| p-1 ,A(x,ξ),ξ|ξ| p (a>0,1<p<)·

Here d * :D ' (Ω,Λ l+1 )D ' (Ω,Λ 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 1 |B| B wdx1 |B| B 1 w 1/(r-1) dx r-1 <

for any ball BΩ. Applications to Sobolev-PoincarĂ© type embedding theorems and to L p -norm estimates of the homotopy operator are briefly mentioned.

31C05Generalizations of harmonic (subharmonic, superharmonic) functions
58A10Differential forms (global analysis)
58J05Elliptic equations on manifolds, general theory
31C12Potential theory on Riemannian manifolds