Weighted Hardy inequalities

Abstract

For bounded Lipschitz domains D in R n it is known that if 1 < p < ∞ , then for all β ∈ [ 0 , β 0 ) , where β 0 = p − 1 > 0 , there is a constant c < ∞ with ∫ D | u ( x ) | p dist ( x , ∂ D ) β − p d x ⩽ c ∫ D | ∇ u ( x ) | p dist ( x , ∂ D ) β d x for all u ∈ C 0 ∞ ( D ) . We show that if D is merely assumed to be a bounded domain in R n that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β 0 now taken to be p ( n − λ ) ( n + p ) n ( p + 2 n ) . 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 W 1 , p ( Ω ) , Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality ∫ D | u ( x ) | p dist ( x , ∂ D ) − p d x + | D | − p / n ∫ D | u ( x ) | p d x ⩽ c ∫ D | ∇ u ( x ) | p d x , 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.