Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Singular integrals of the compositions of LaplaceBeltrami and Green’s operators | Fang and Ding Journal of Inequalities and Applications 2011 2011 74 http content 2011 1 74 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Singular integrals of the compositions of LaplaceBeltrami and Green s operators Ru Fang 1 and Shusen Ding2 Correspondence fangr@. cn department of Mathematics Harbin Institute of Technology Harbin 150001 Full list of author information is available at the end of the article Springer Abstract We establish the Poincaré-type inequalities for the composition of the LaplaceBeltrami operator and the Green s operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density. Keywords Poincaré-type inequalities differential forms A-harmonic equations the Laplace-Beltrami operator Green s operator 1 Introduction The purpose of the article is to develop the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator A dd d d and Green s operator G over the d-John domain. Both operators play an important role in many fields including partial differential equations harmonic analysis quasiconformal mappings and physics 1-6 . We first give a general estimate of the composite operator A o G. Then we consider the composite operator with a singular factor. The consideration was motivated from physics. For instance when calculating an electric field we will deal with the integral p Í 1_ f Í1Í vt_ _4__fl A A crln o vo A Í í c A 11 A on ci A A 1 A í c rh o i nroncol AÍ ori . r 4ns0 Jd p x r-x 3 x where r x is a charge density and x is the integral variable. It is singular if r e D. Obviously the singular integrals are more interesting to us because of their wide applications in different fields of mathematics and physics. In this article we assume that M is a bounded convex domain and B is a ball .
