Rearrangement invariant envelopes of generalized Bessel and Riesz potentials

The spaces of generalized Bessel and Riesz potentials in the n-dimensional Euclidean space is studied by constructing them on the basis of rearrangement invariant space (RIS). The equivalent characterizations of cones of decreasing rearrangements is established, sharp theorems on embeddings in RISes are deduced, and criteria for the boundedness of potentials is found. The two cones determine the global integral properties of potentials and their maximal functions respectively. Equivalences show that the cone determines the global integral properties of potentials and their maximal functions. The optimal RIS is also known as the rearrangement invariant envelope of the space of potentials.