Optimal Embeddings of Riesz Type Potentials

We study Riesz potentials in n-dimensional Euclidean space. They are constructed on rearrangement-invariant spaces as convolutions with kernels with general form, their description of the class of kernels is based by means of some non-negative, decreasing function Φ. Generalized Riesz potentials include classical Riesz potentials spaces. Here we consider as a “base” space RIS Lorentz type space Λ p, 1 < p < ∞. During consideration of the question of finding conditions for embeddings of Riesz type potentials in RIS we used criteria stated by M.L. Goldman, where the operator of Hardy type and inequalities for operators of this type are playing the key role. For the case of Riesz potentials, 1 < p < ∞, the condition of optimal embedding in RIS is established. The case of Riesz type potentials based on space L p, 1 < p < ∞, considered by the authors M.L. Goldman and O.M. Guselnikova, corresponds with the result of this work.

Riesz potentials, Lorentz spaces, decreasing rearrangement, rearrangement-invariant spaces, optimal embeddings