Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)1742810.22363/2312-9735-2017-25-4-340-349Research ArticleIntegral Properties of Generalized Potentials of the Type Besseland Riesz TypeAlmohammadKh<p>Almohammad Kh. - student of Nonlinear Analysis and Optimization Department of Peoples’ Friendship University of Russia (RUDN University)</p>khaleel.almahamad1985@gmail.comAlkhalilN Kh<p>Alkhalil N. - student of Nonlinear Analysis and Optimization Department of Peoples’ Friendship University of Russia (RUDN University)</p>khaleel.almahamad1985@gmail.comDepartment of Nonlinear Analysis and Optimization Peoples’ Friendship University of Russia (RUDN university)1512201725434034910122017Copyright © 2017, Almohammad K., Alkhalil N.K.2017<p>In the paper we study integral properties of convolutions of functions with kernels generalizingthe classical Bessel-Macdonald kernels (), , 0 . The local behavior of Bessel-Macdonald kernels in the neighborhood of the origin are characterized by the singularity ofpower type ||-. 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.</p>Riesz potentialsLorentz spacesdecreasing rearrangementrearrangement-invariant spacesoptimal embeddingпотенциалы типа Риссапространства Лоренцаубывающиеперестановкиперестановочно-инвариантные пространстваоптимальные вложения[R. O’Neil, Convolution Operators and]