JournalsrlmVol. 17 , No. 3DOI 10.4171/rlm/465

Local clustering of the non-zero set of functions in W1,1(E)W^{1,1}(E)

  • Emmanuele DiBenedetto

    Vanderbilt University, Nashville, United States
  • Ugo Gianazza

    Università di Pavia, Italy
  • Vincenzo Vespri

    Universita di Firenze, Italy
Local clustering of the non-zero set of functions in $W^{1,1}(E)$ cover

Abstract

We extend to the p=1p=1 case a measure theoretic result previously proved by DiBenedetto and Vespri for functions that belong to uW1,p(Kρ(x0))u\in W^{1,p}(K_\rho(x_0)) where Kρ(x0))K_\rho(x_0)) is a NN-dimensional cube of edge ρ\rho centered at x0x_0. It basically states that if the set where uu is bounded away from zero occupies a sizable portion of KρK_\rho, then the set where uu is positive clusters about at least one point of KρK_\rho.