Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 35, 2010, 87-114
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.
doi:10.5186/aasfm.2010.3506
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