Let be a connected open subset, be the space of differential 1-forms (deRham currents) on , be the Banach subspace (with the standard norm). The article involves a large number of local and global -norm inequalities (the inequalities between norm of and for the solution of the equation
where are given and operator satisfies
Here is the adjoint operator to the exterior derivative . The inequalities are extended to involve the -weighted versions with weights satisfying
for any ball . Applications to Sobolev-Poincaré type embedding theorems and to -norm estimates of the homotopy operator are briefly mentioned.