Generalized structural conditions for a Calderón-Zygmund theory on double phase problems

In this paper, we considered a general class of double-phase problems of the form

$$-\mathrm{d}\mathrm{i}\mathrm{v}(A(x,Du))=-\mathrm{d}\mathrm{i}\mathrm{v}(B(x,F)).$$

Here, the ellipticity, growth, and continuity assumptions on the main operator $A$ were described using an auxiliary vector field $G$. The proposed structural condition allowed us to treat, in a single setting, different structural forms that have appeared in earlier studies. Under this generalized structure, we derived Calderón-Zygmund estimates for the gradients of solutions.