The Boundedness Of Operators On Herz-type Spaces On Locally Compact Vilenkin Groups

It is well-known that harmonic analysis is always one of the most important areas in studying of analysis. Accompanying with the development and perfection of its theories, the effectiveness of harmonic analysis is seems more outstanding in application in partial differential equation, representations of groups, the theory of numbers and also in engineering such as signal processing, nerwork safety.The main tools in the applications are function spaces and operator theory.The dissertation is devoted to investigate the boundedness of operators on Herz type spaces, the decomposition and the basic properties of Herz-type Besov space on locally compact Vilenkin groups. Our research is divided into two parts:On the one hand, we give the atomic decomposition of Herz-type Besov space. It's a trend to use an unified frame to describe smooth properties of functions. J.Peetre firstly applied the Littlewood-Paley theory to deal with Besov space in 1976. Subsequently, choosing atoms and molecules being of various character to construct function spaces became a new developmental orientation after the Littlewood-Paley theory. The general function space on Rn usually have their own atomic decomposition. Some people extended the works to locally compact Vilenkin groups G. Fox example, C. W. Onneweer and Su Weiyi treated the atomic decomposition of the Besov space on G. Motivated by their work, the authors introduce the Herz- type Besov spaces on G and give the atomic characterizations above all in charpter I. In the stuation, we substitute the i7-norm with that of ka,pq in orginal definition of Besov space, redefine the norm of the Herz-type Besov space Ka,pq BBs There are relations among the indices such as a, p, q, B and s which are pointed out precisely in charpter I. If a = 0 and q = p in our situation, then ka'pBB(G) = Bs,pB(G), which concludes that the atomic chacterization of Besov space given by Su and Onneweer is a special case of our dissertation . Meanwhile, the embedding properties and the lifting properties in Herz-type Besov spaceare also discussed.On the other hand, the theory of operator is another emphasis in this paper. The authors discuss the boundedness of some sublinear operators and commutators of singular integral operators on Herz type spaces in following charpters. In Charpter 3, we show that some sunlinear operators are bounded operators from Herz-type Hardy space to Herz space or Herz-type Hardy space when they satisify a certain size condition. Beside Herz-type Besov space, Herz-type Triebel-Lizorkin space is also introduced. When we consider the boundedness of commutator ofsingular integral operators, the boundedness on Hardy space and the boundedness from IP space to Triebel-Lizorkin space FB,p(G) are firstly given in Charpter 4 and Charpter 5. Furthermore, we can conclude the commutator is exactly mapped from Herz space into Herz-type Triebel-Lizorkin space.In the final charpter of this dissertation, the boundedness of bilinear operator on Herz-type space on G is given. If the bilinear operator B(f, g) satisfiesthe vanishing moment condition then B(f,g) is bounded from to HKa,pq(G). The boundedness from HKa1,p1q1(G) x HKa2,p2 q2(G) to HKa,p q(G) is also obtained in this charpter.