Weighted Hardy inequalities

AbstractFor 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)|pdist(x,∂D)β−pdx⩽c∫D|∇u(x)|pdist(x,∂D)βdx for all u∈C0∞(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)|pdist(x,∂D)−pdx+|D|−p/n∫D|u(x)|pdx⩽c∫D|∇u(x)|pdx, 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