Weighted pointwise Hardy inequalities

• Autores: Pekka Koskela, Juha Lehrbäck
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 79, Nº 3, 2009, págs. 757-779
• Idioma: inglés
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• Resumen

  • We introduce the concept of a visual boundary of a domain �¶ �¼ Rn and show that the weighted Hardy inequality  �¶ |u|pd�¶ �À.p  C  �¶ |�Þu|pd�¶ �À, where d�¶(x) = dist(x, �Ý�¶), holds for all u �¸ C �� 0 (�¶) with exponents �À < �À0 when the visual boundary of �¶ is sufficiently large. Here �À0 = �À0(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 �À0 = p . 2 + �É, with �É = log 4/ log 3. These results are based on new pointwise Hardy inequalities.