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Two weighted norm inequalities for Riesz potentials and uniform $$L^ p$$- weighted Sobolev inequalities. (English) Zbl 0736.42015
The author proves a two-weighted norm inequality for the fractional maximal operator and the corresponding inequality for Riesz potentials of fractional order. These results also yield some inequalities coupling a weighted $$L^ p$$-norm of a function and a weighted $$L^ p$$-norm of its derivative.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 31B35 Connections of harmonic functions with differential equations in higher dimensions
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