Some Research Of POINCAR(?)-Type Inequalities On Riemannian Manifold

A manifold is an abstract mathematical space in which every point has a neigh-borhood which resembles Euclidean space, but in which the global structure may bemore complicated. Manifolds are very important objects in mathematics and physicsbecause they allow more complicated structures to be expressed and understood interms of the relatively well-understood properties of simpler spaces. So it is importantto study various concepts and theoretics on manifold.In this paper, we consider the Poincarétype inequality on manifold. By aPoincarétype inequality, in the widest sense, we mean a norm inequality in whichthe variation of a function from its average value on a domain is in some way con-trolled by its gradient or higher gradients on that domain. This class of inequalityincludes Poincaréinequality, Caccioppli inequality, Hardy-Littewood inequality andreverse Ho¨lder inequality. Such inequalities play an important role in some fields,such as partial differential equations, potential theory. Because of the complicationof manifold, we choose Riemannian manifold to search. In particular, we choose flatRiemann. We extend the Poincaréinequality on Euclid space to some Riemannianmanifold and then to Flat Riemannian manifold. At last, on Flat Riemannian, wediscuss the two-weighted Poincaréinequality on F(δ,N) domain.