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

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 Caldern-Zygmund type estimates below the natural exponent are also obtained for -superharmonic functions in the whole space a"e (n) . This answers a question raised in our earlier work (On Caldern-Zygmund theory for p- and -superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.