A Hardy type inequality for W0m,1(Ω)W^{m,1}_0(\Omega) functions

  • Juan Dávila

    Universidad de Chile, Santiago, Chile
  • Hernán Castro

    Rutgers University, Piscataway, USA
  • Hui Wang

    Rutgers University, Piscataway, USA


We consider functions uW0m,1(Ω)u\in W^{m,1}_0(\Omega), where ΩRN\Omega\subset \mathbb R^N is a smooth bounded domain, and m2m\geq 2 is an integer. For all j0j\geq 0, 1km11\leq k\leq m-1, such that 1j+km1\leq j+k\leq m, we prove that ju(x)d(x)mjkW0k,1(Ω)\frac{\partial^ju(x)}{d(x)^{m-j-k}}\in W^{k,1}_0(\Omega) with

k(ju(x)d(x)mjkL1(Ω)CuWm,1(Ω),\| \partial^k ( {\frac{\partial^ju(x)}{d(x)^{m-j-k}}}|_ {L^1(\Omega)}\leq C\|u\|_{W^{m,1}(\Omega)},

where dd is a smooth positive function which coincides with dist(x,Ω)(x,\partial \Omega) near Ω\partial \Omega, and l\partial^l denotes any partial differential operator of order ll.

Cite this article

Juan Dávila, Hernán Castro, Hui Wang, A Hardy type inequality for W0m,1(Ω)W^{m,1}_0(\Omega) functions. J. Eur. Math. Soc. 15 (2013), no. 1, pp. 145–155

DOI 10.4171/JEMS/357