Weighted Boundedness in Morrey Spaces for.doc
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1、精品论文推荐Weighted Boundedness in Morrey Spaces forSublinear Operators1Hou Weijie, Liu MingjuBeijing University of Aeronautics and Astronautics (100191)AbstractThe classical Morrey spaces were introduced by Morrey to study the local behaviour of solutions tosecond order elliptic partial differential equ
2、ations. Since then these spaces play a very import role in studying the regularity of solutions to second order elliptic partial differential equations.As Morrey spaces may be considered as an extension of Lebesgue spaces, it is natural and important to study the weighted boundedness for operaters i
3、n Morrey spaces. Much work in this direction has been done.The studying of sublinear operators is very active these years, in this paper, the authors introduce a type of topological structure in the Cartesian product and a set function, and in advance discuss weighted boundedness of sublinear operat
4、ors in Morrey spaces. The result improve and extend the known results. Keywords: Sublinear operator; Morrey spaces; weight functionCLC Number: O177.31. Introduction and the main resultsThe classical Morrey spaces were introduced by Morrey to study the local behaviour of solutions to second order ell
5、iptic partial differential equations. As Morrey spaces may be considered as an extension of Lebesgue spaces, it is natural and important to study the weightedboundedness for operaters in Morrey spaces. Much work in this direction has been done- 4 -(145), our results extend that in this papers. In th
6、e following,X E denotes the characteristicfunction of the set E , C is a constant, not necessarily the same in each line.Let be a positively growth function on (0, )and satisfy a doubling condition, thatis (2r ) D (r) , where D is a constant independent of r , is a weight function onRn .AssumethatVi
7、sasetandRn Vhassometopologicalstructure.LetRn V = ( x, t ) : x Rn , t V , is the Borel measure on Rn V . is a funtionmapping the balls inRn into the Borel sets innRn Vand satisfies:(1) IfB1 , B2are balls inRwith B1 I B2 = , then (B1 ) I (B2 ) = ;(2) IfB1 B2 , then ( B1 ) (B2 ) ;(3) For anyx Rn ,U (B
8、( x, r ) = Rn V .r 0*Let be a Young function and satisfy the conditon 2 or p , that is for anyt 0 , (2t ) C (t ) , or there existsk 1 , such that (2t) 2k p (t)for anyt 0and0 p 0 (r)We define the generalized Morrey spaces onRn Vas follows1 本课题得到国家自然科学基金(10726008)的资助。L(RV , ) = f :| f | , . , nL ( )Le
9、t g be a locally integrable function on Rn , ifRn V = Rn ,d ( y, t ) = ( x)dx , then weobtain the generalized Morrey spaces onRn .L , (Rn , ) = g :| g |= sup1 (| g ( y) |) ( y)d 0 (r) B ( x,r )If (r) = r , 0 , (t) = t pspace(see3)., thenLp , = Lp , which is the classical MorreyIn the following we wi
10、ll give the main results of this paper.TheoremSuppose that T is a sublinear operator and(, ) C1 ( ) , that issup ( (B) C ( x)a.e.xB (B)If T is bounded fromL (Rn , ) toL (Rn V , ) , i.e. (| Tf ( y, t ) |)d ( y, t) C (| f ( y) |) ( y)d ,Rn VRnthen T is also bounded fromL , (Rn , )to L , (Rn V , ) , th
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