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Journal of the London Mathematical Society Advance Access published online on April 28, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp013
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© 2009 London Mathematical Society

Weighted pointwise Hardy inequalities

Pekka Koskela and Juha Lehrbäck

Department of Mathematics and Statistics
University of Jyväskylä
PO Box 35 (MaD)
FIN-40014
Finland
pkoskela@maths.jyu.fi

We introduce the concept of a visual boundary of a domain {Omega} sub Rn and show that the weighted Hardy inequality Formula , where d{Omega}(x) = dist(x, {partial}{Omega}), holds for all Formula with exponents β < β0 when the visual boundary of {Omega} is sufficiently large. Here Formula is explicit, essentially sharp, and may even be greater than p – 1, which is the known bound for smooth domains. For instance, in the case of the usual von Koch snowflake domain the sharp bound is shown to be β0 = p – 2 + {lambda}, with {lambda} = log 4/log 3. These results are based on new pointwise Hardy inequalities.

2000 Mathematics Subject Classification 46E35, 26D15.

Both authors were supported in part by the Academy of Finland.

Received February 13, 2007; revised April 22, 2008;
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