We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

We obtain sufficient conditions for nonexistence of nontrivial solutions for some classes of nonlinear partial differential inequalities containing the fractional powers of the Laplace operator.