Weighted Morrey Type Triebel-Besov Spaces And Its Application

Function spaces have a long history. They play an important part in both classi-cal and modern mathematics. The operators obtaining from the study of PDE becomethe tool to study PDE, such as singular integral operator, pesudo-deferential operator,paradi?erential calculus. Our main purpose in this paper is to obtain the definitionof weighted Morrey type Triebel-Lizorkin spaces and Besov spaces and some proper-ties;secondly, get the boundedness of pesudo-deferential operator in weighted Morreytype Triebel-Lizorkin spaces and Besov spaces, including classical pesudo-deferential op-erator, Pseudo-di?erential operators with non-regular symbols and Bilinear pseudodi?er-ential operators;obtain the atomic decomposotion of function spaces, as an application,get the boundedness of pointwise multiplier operator and di?eomorphic property of thefunction space, finially , we study initial value problem about Euler equations for theinitial data belonging to Morrey type Triebel-Lizorkin space.Firstly, in first section, we presented our motivation, gist and signification for our pa-per and made a summarization of prior work on dispersal population dynamics. Secondly,some useful lemmas, definitions and fundamental theorems were given in the preliminaries.In the second section, we investigated the definition of weighted Morrey type Triebel-Lizorkin spaces and Besov spaces and some properties. main method is the use of maximalfunction, combining Littlewood-Paley decomposition. Because the Littlewood-Paley de-composition depend to the choice of function, so we need to prove definition of functionspace is right, get some results about multiplier operator, lift operator, discrete char-acterization of function spaces and Moser-type and Bilinear estimate, using the Bonydecomposition.In the third section, we study the boundedness of some pesudo-deferential operators.Firstly, using the discrete character of function space, we obtain the boundedness aboutclassical pesudo-deferential operator. However, The symbol class S1τ,δis a little bit toorestrictive for applications to non-linear equations , for example,the typically di?erentialoperators have the coe?cients that depend on the solution itself.Therefore, while theirsymbol is still a smooth function ofξ, in general it will have only limited regularity inx. We will concern ourselves with symbols for which x?regularity is measured in H¨olderspace or Zygmund space.WE will get their boundedness on weighted Morrey type Besovspace. Finally, we study the boundedness of several classes of bilinear pesudo-deferentialoperators. The bilinear pesudo-deferential operators are natural generalizations of thetranslation invariant bilinear operators, Another important example is provided by thebilinear partial di?erential operators with variable coe?cients. We propose the study of the boundedness properties on weighted Morrey type Besov space.It is well known, the atomic decomposition of function space is very important inthe theory and application of function space. in the fourth section, we obtain the atomicdecomposition about weighted Morrey type Triebel-Lizorkin space and Besov space. as anapplication, we get the boundedness of pointwise multiplier and di?eomorphic property.Furthermore,We are concerned with the Euler equations for the homogeneous incom-pressible uid oes.whereυ= (υ1,···,υn),υj =υj(x,t),j = 1,···,n, is the velocity of the ?uid ?ows,p =p(x,t) is the scalar pressure, andυ0 is the given initial velocity satisfying divυ0 = 0. westudy initial value problem for the initial data belonging to Morrey type Triebel-Lizorkinspace.On the Triebel-Lizorkin space associated to Hermite operator, we get the boundednessabout heat di?usion semigroup, poisson integral , multiplier operator and powers operatorassociated to the Hermite operator.