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)1788910.22363/2312-9735-2018-26-1-3-12Research ArticleDiﬀerential Properties of Generalized Potentialsof the Type Bessel and Riesz TypeAlkhalilN Kh<p>Alkhalil N. - student of Nonlinear Analysis and Optimization Department of Peoples’ Friendship University of Russia (RUDN University)</p>khaleel.almahamad1985@gmail.comAlmohammadKh<p>Almohammad Kh. - student of Nonlinear Analysis and Optimization Department of Peoples’ Friendship University of Russia (RUDN University)</p>khaleel.almahamad1985@gmail.comPeoples’ Friendship University of Russia (RUDN University)1512201826131228022018Copyright © 2018, Alkhalil N.K., Almohammad K.2018<p>In this paper we study diﬀerential properties of convolutions of functions with kernels thatgeneralize the classical Bessel-Macdonald kernels ... The theory ofclassical Bessel potentials is an important section of the general theory of spaces of diﬀerentiablefunctions of fractional smoothness and its applications in the theory of partial diﬀerentialequations. The properties of the classical Bessel-Macdonald kernels are studied in detail in thebooks of Bennett and Sharpley, S. M. Nikolskii, I. M. Stein, V. G. Mazya. The local behavior ofthe Bessel-Macdonald kernels in the neighborhood of the origin is characterized by the presenceof a power-type singularity ||-. At inﬁnity, they tend to zero at an exponential rate. Therecent work of M. L. Goldman, A. V. Malysheva, and D. Haroske was devoted to the investigationof the diﬀerential properties of generalized Bessel-Riesz potentials.In this paper we study the diﬀerential properties of potentials that generalize the classicalBessel-Riesz potentials. Potential kernels can have nonpower singularities in the neighborhoodof the origin. Their behavior at inﬁnity is related only to the integrability condition, so thatkernels with a compact support are included. In this connection, the spaces of generalized Besselpotentials generated by them belong to the so-called spaces of generalized smoothness. The casewith the satisﬁed criterion for embedding potentials in the space of continuous bounded functionsis considered. In this case, the diﬀerential properties of the potentials are expressed in termsof the behavior of their module of continuity in the uniform metric. Criteria for embedding ofpotentials in Calderon spaces are established and explicit descriptions of the module of continuityof potentials and optimal spaces for such embeddings are obtained in the case when the basespace for potentials is the Lorentz weight space. These results specify the general constructionsestablished in previous works.</p>Bessel potentialsLorentz spacesCalderon spacesrearrangement-invariant spacesoptimal embeddingsпотенциалы Бесселяпространства Лоренцапространства Каль-деронамодули непрерывностиперестановочно инвариантные пространстваоптимальныевложения[C. Bennett, R. Sharpley, Interpolation of Operators, Vol. 129, Academic Press, New York, 1988.][S. M. Nikolsky, Approximation of Functions of Several Variables and Embedding Theorems, Nauka, Moscow, 1977, in Russian.][E. M. Stein, Singular Integrals and Diﬀerentiability Properties of Functions, Mir, Moscow, 1973, in Russian.][V. G. Mazya, Sobolev Spaces, LSU, Leningrad, 1985, in Russian.][M. L. Goldman, The Cone of Permutations for Generalized Bessel Potentials, Vol. 260, 2008, pp. 151–163, in Russian.][M. L. Goldman, On Optimal Investment Potentials of the Generalized Bessel and Riesz, Vol. 269, 2010, pp. 91–111, in Russian.][Kh. Almohammad, N. Alkhalil, Integral properties of generalized bessel and riesz potentials, Bulletin of RUDN University. Series: Mathematics. Information Sciences. Physics 25 (4) (2017) 331–340, in Russian. doi:10.22363/2312-9735-2017-25-4-340-349.][A. Gogatishvili, M. Johansson, C. A. Okpoti, L. E. Persson, Characterization of Embeddings in Lorentz Spaces Using a Method of Discretization and Anti- Discretization, Bulletin of the Australian Mathematical Society 76 (2007) 69–92.][M. L. Goldman, A. V. Malysheva, Two-Sided Estimate for the Modulus of Continuity of a Convolution, Diﬀerential Equations 49 (5) (2013) 557–568.][M. L. Goldman, A. V. Malysheva, An Estimate of the Uniform Modulus of the Generalized Bessel Potential Continuity, Proceedings of Steklov Mathematical Institute 283 (2013) 1–12, in Russian.][M. L. Goldman, D. Haroske, Optimal Calderon Spaces for Generalized Bessel Potentials, Doklady Mathematics 492 (1) (2015) 404–407, in Russian.]