Annales Academiĉ Scientiarum Fennicĉ

Mathematica

Volumen 35, 2010, 87-114

# A DECOMPOSITION TECHNIQUE FOR JOHN DOMAINS

## Lars Diening, Michael Ruzicka and Katrin Schumacher

Freiburg University, Section of Applied Mathematics

Eckerstrasse 1, D-79104 Freiburg, Germany;
diening 'at' mathematik.uni-freiburg.de

Freiburg University, Section of Applied Mathematics

Eckerstrasse 1, D-79104 Freiburg, Germany;
rose 'at' mathematik.uni-freiburg.de

Technische Universität Darmstadt, Department of Mathematics

Schlossgartenstrasse 7, D-64289 Darmstadt, Germany;
schumacher 'at' mathematik.tu-darmstadt.de

**Abstract.**
We develop a method to decompose functions with mean value zero that
are defined on a (possibly unbounded) John domain into
a countable sum of functions with mean value zero and support in
cubes or balls. This method enables us to generalize results known
for simple domains to the class of John domains and domains
satisfying a certain chain condition. As applications we present
the solvability of the divergence equation \div **u** = *f*, the
negative norm theorem, Korn's inequality, Poincaré's inequality
and a localized version of the Fefferman-Stein inequality. We
present the results for weighted Lebesgue spaces and Orlicz spaces.

**2000 Mathematics Subject Classification:**
Primary 26D10, 46E30, 35Q35.

**Key words:**
John domains, decomposition, divergence equation,
negative norm theorem, Korn's inequality, Poincaré's inequality,
Fefferman-Stein inequality, weighted Lebesgue spaces, Orlicz spaces,
Boman chain condition.

**Reference to this article:** L. Diening, M. Ruzicka and K. Schumacher:
A decomposition technique for John domains.
Ann. Acad. Sci. Fenn. Math. 35 (2010), 87-114.

Full document as PDF file

doi:10.5186/aasfm.2010.3506

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