Theory of Ba Spaces and Applications

Abstract

The concept of Ba spaces was first coined by Ding Xiaxi and Luo Peizhu in 1980. The Ba space is a class of new function spaces extracted from a series of important research works by Ding and Luo themselves. It is a very natural generalization of the L P spaces and includes some important Orlicz spaces, Sobolev spaces and Orlicz-Sobolev spaces, etc. As the investigation is based on L P , the methods are more natural, the results are more concrete. It possesses not only many properties which are similar to those in Orlicz spaces but also the refined results which are different from those of Orlicz spaces. In the 14 years period since Ding-Luo paper appeared, several authors have found the Ba space to be a useful and interesting concept. This class of spaces has been effectively applied to partial differential equations, to harmonic analysis and function theory. By using the theory of Ba spaces Ding et al. proved the existence of the generalized solution for the strong non-linear parabolic equations, and they also found out that the solution for the strong non-linear variational problem just belongs to such a new function space. The boundedness of some operators on Ba spaces has yielded deep and complete results. In the function theory, some interesting relations between Ba space and BMOA (or Bloch space) were pointed out, some results on the estimate of approximation order in Ba space were obtained. In this survey we’ll introduce the theory of Ba spaces with its applications which has been achieved by the Chinese scholars.