# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation. (English) Zbl 1124.31004

Let ${\Omega }\subset {ℝ}^{n}$ be a connected open subset, ${D}^{\text{'}}\left({\Omega },{{\Lambda }}^{1}\right)$ be the space of differential 1-forms (deRham currents) on ${\Omega }$, ${L}^{p}\left({\Omega },{{\Lambda }}^{1}\right)\subset {D}^{\text{'}}\left({\Omega },{{\Lambda }}^{1}\right)$ 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 ${d}^{*}v\right)$ for the solution of the equation

$A\left(x,g+du\right)=h+{d}^{*}v$

where $g,h\in {D}^{\text{'}}\left({\Omega },{{\Lambda }}^{1}\right)$ are given and operator $A:{\Omega }×{{\Lambda }}^{1}\to {{\Lambda }}^{1}$ satisfies

$|A\left(x,\xi \right)|\le {a|\xi |}^{p-1},〈A\left(x,\xi \right),\xi 〉\ge {|\xi |}^{p}\phantom{\rule{4pt}{0ex}}\left(a>0,1

Here ${d}^{*}:{D}^{\text{'}}\left({\Omega },{{\Lambda }}^{l+1}\right)\to {D}^{\text{'}}\left({\Omega },{{\Lambda }}^{1}\right)$ 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

$\underset{B}{sup}\left(\frac{1}{|B|}{\int }_{B}w\phantom{\rule{0.166667em}{0ex}}dx\right){\left(\frac{1}{|B|}{\int }_{B}{\left|\frac{1}{w}\right|}^{1/\left(r-1\right)}dx\right)}^{r-1}<\infty$

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.

##### MSC:
 31C05 Generalizations of harmonic (subharmonic, superharmonic) functions 58A10 Differential forms (global analysis) 58J05 Elliptic equations on manifolds, general theory 31C12 Potential theory on Riemannian manifolds
##### Keywords:
norm inequalities; harmonic equations; differential forms