5.
For bounded Lipschitz domains
D in it is known that if 1<
p<∞, then for all
β[0,
β0), where
β0=
p−1>0, there is a constant
c<∞ with
for all . We show that if
D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent
λ and has plump complement, then the same inequality holds with
β0 now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for
W1,p(
Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality
c=
c(
n,
p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.
相似文献