Integral Properties of Generalized Potentials of the Type Besseland Riesz Type
In the paper we study integral properties of convolutions of functions with kernels generalizingthe classical Bessel-Macdonald kernels 𝐺𝛼(𝑥), ∈ 𝑅𝑛, 0 < < 𝑛. The local behavior of Bessel-Macdonald kernels in the neighborhood of the origin are characterized by the singularity ofpower type |𝑥|𝛼-𝑛. The kernels of generalized Bessel-Riesz potentials may have non-powersingularities in the neighborhood of the origin. Their behavior at the inﬁnity is restricted onlyby the integrability condition, so that the kernels with compact support are included too. In thepaper the general criteria for the embedding of potentials into rearrangement invariant spacesare concretized in the case when the basic space coincides with the weighted Lorentz space.We obtain the explicit descriptions for the optimal rearrangement invariant space for such anembedding.