WEIGHTED ESTIMATES OF THE UNILATERAL POTENTIALS

We consider weighted unilateral ball potentials with radial quasi-monotone weights, as well as unilateral potentials related to the half-space R-+(n), with quasi-monotone weights depending on x(n) > 0. We give the sufficient and, in some cases, necessary conditions for the L-p -> L-q-boundedness of these potentials.

For some subclasses of quasi-monotone weights, we prove a pointwise estimate of the weighted potentials via the non-weighted potential and the weighted Hardy operator, which may be used for an arbitrary Banach function space with the lattice property.