Calderón–Zygmund type estimates for a class of obstacle problems with p(x)p(x) growth

For local minimizers u∈Wloc1,p(⋅)(Ω) of quasiconvex integral functionals of the typeF[u]:=∫Ωf(x,Du(x))dx with p(x)p(x) growth in the class K:={u∈Wloc1,p(⋅)(Ω):u⩾ψ}, where ψ∈Wloc1,p(⋅)(Ω) is a given obstacle function, we show estimates of Calderón–Zygmund type, i.e.|Dψ|p(⋅)∈Llocq⇒|Du|p(⋅)∈Llocq, for any q>1q>1, provided that the modulus of continuity ω of the exponent function p satisfies the conditionω(ρ)log1ρ→0as ρ→0.