Global integral gradient bounds for quasilinear equations below or near the natural exponent

Abstract

We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for $\mathcal{A}$-superharmonic functions in the whole space ℝⁿ. This answers a question raised in our earlier work (On Calderón–Zygmund theory for p- and $\mathcal{A}$-superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.