WeightedBoundednessinMorreySpacesfor.doc

精品論文推薦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 equations. 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 in 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 operators 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 elliptic 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 the 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 .assumethatvisasetandrn 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( 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 本課題得到國(guó)家自然科學(xué)基金(10726008)的資助。l(rv , ) = f :| f | , . , nl ( )let 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 will 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 , ) , that is| tf | , c | f | , .l ( )l ( )where c is a constant, not necessarily the same in each line.2. proof of the theoremlet us first give two lemmas before we prove our theorem.nlemma 1 let(, ) c1 ( ) , 1 d 2, for a locally integrable function g onrn v, letg* ( x) = sup1| g ( y, t) | d ( y, t ), then for anyb = b( x, r )andxb (b)young function ，we have ( b ) (| f ( y) |) x *( y)d c (r) | f |.rn ( b )l , ( )lemma 2 letg 0be a locally integrable function onrn vand be a positiveborel measure onrn v . if for any(, ) c1 ( ) , the sublinear operator t is boundedfrom l ( rn , ) tol (rn v , ) ,then (| tf ( y, t ) |) g ( y, t)d ( y, t) c (| f ( y) |) g* ( y)d .rn vrnproof of lemma 1 by(, ) c1 ( )x *( y) = sup1| x( x, t ) | d ( x, t) ( b )yq (q) (q ) ( b )= sup1 ( (q) i ( b)yq (q) sup1 ( (q)yq (q) c ( y)then we have (| f ( y) |) x *( y)d rn= b ( b )* (| f ( y) |) x ( b )( y)d +* 2k +1 b / 2k bk =0 (| f ( y) |) x ( b )( y)d c b (| f ( y) |) ( y)d + c 2k +1 b / 2k bk = 0 (| f ( y) |) ( y)d c (| f ( y) |) ( y)d + c 2 kn (| f ( y) |) ( y)d b c2 kn2k bk =1 (| f ( y) |) ( y)d 2k bk =0 c 2 kn (2kr) | f | , l ( )k =0 c 2 knkd (r ) | f | , l ( )k =0 c (r ) | f | , l ( )the last inequality holds because of the fact that the series0 2 n d 1 . (2 n d)kk = 0is covergent sinceproof of lemma 2 let haved ( x, t) = g ( x, t )d ( x, t ), ( x) = g * ( x) ,by the definition ofg* , weg* ( y) = sup1| g ( y, t ) | d ( y, t )yb (b) ( b )= sup1 ( (b)yb (b) c ( y)(c 1)fromit is obvious that(, ) c1 ( ) ,then by the assumption of lemma 2,t is boundedl ( rn , ) tol (rn v , ) ,we get (| tf ( y, t) |) g ( y, t)d ( y, t ) = (| tf ( y, t) |)d ( y, t)rn v rn v . c rn (| f ( y) |) ( y)d crn (| f ( y) |) g* ( y)d now let us turn to the proof of our theorem.fix a ballb = b( x, r) rn , and takingg ( y, t ) = x ( b )( y, t)in lemma 2 and by usinglemma 1, we have ( b ) (| tf ( y, t ) |)d ( y, t )= rn v (t | f ( y, t ) |) x ( b ) ( y, t)d ( y, t ) c rn (| f ( y) |) x *( y)d c (r ) | f | , ( b )l ( )that is| tf | , c | f | , .l ( )l ( )the proof of the theorem is complete.+remark.(1) ifrn v = rn+1 , (t ) = t p , (b) = b% = ( y, t ) rn+1 : y b, 0 t r ,we get the theorem in 4.+(2) ifrn v = rn+1 , (b) = b% , our result is the same as that in 5.(3) as maximal operators are included in sublinear operators, so our results partly extend the one in 6references1 mizuhara l t. boundedness of some classical operators on generalized morrey spaces in harmonic analysis, icm-90 conference proceedings(ed.s.igari), tokyo:springer-verlag, 1991:183-1892 j.o.stromberg. bounded mean oscillation with orlicz