Weighted Hardy inequalities

For bounded Lipschitz domains D in Rn it is known that if 1< p <∞, then for all β ∈ [0,β0), where β0 = p −1 > 0, there is a constant c <∞with D u(x) p dist(x, ∂D)β−p dx c D ∇u(x) p dist(x, ∂D)β dx for all u ∈ C∞0 (D). We show that if D is merely assumed to be a bounded domain in Rn 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 p(n−λ)(n+p) n(p+2n) . 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 D u(x) p dist(x, ∂D)−p dx + |D|−p/n D u(x) p dx c D ∇u(x) p dx,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.  2005 Elsevier Inc. All rights reserved