Local Calderón–Zygmund estimates for parabolic minimizers

We establish local Calderón–Zygmund estimates for parabolic minimizers uu to certain variational inequalities. The formal corresponding parabolic partial differential equation would be ∂tu−div∂ζf(z,u,Du)=−∂uf(z,u,Du)+div(|F|p−2F)+h, with p≥2n/(n+2)p≥2n/(n+2). However, since we do not impose any condition on the existence of the derivative ∂uf∂uf, the PDE above may not have any meaning at all. For proving the Calderón–Zygmund estimates, we thus remain solely on the level of parabolic minimizers. Precisely, for q>pq>p we prove that Du∈Llocq if uu is locally Hölder continuous and F∈Llocq and h∈LlocQ for a QQ depending on qq.