6533b7d6fe1ef96bd1265a69

RESEARCH PRODUCT

Weighted pointwise Hardy inequalities

subject

description

We introduce the concept of a visual boundary of a domain �¶ �¼ Rn and show that the weighted Hardy inequality  �¶ |u|pd�¶ �A.p  C  �¶ |�Þu|pd�¶ �A, where d�¶(x) = dist(x, �Ý�¶), holds for all u �¸ C �� 0 (�¶) with exponents �A < �A0 when the visual boundary of �¶ is sufficiently large. Here �A0 = �A0(p, n, �¶) 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 �A0 = p . 2 + �E, with �E = log 4/ log 3. These results are based on new pointwise Hardy inequalities.