赋以混合范数的各向异性besov类

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1、492Vol.49,No.22006
3ACTAMATHEMATICASINICA,ChineseSeriesMar.,2006:0583-1431(2006)02-0381-10:AÆ Besov"#$%&()*)+)),$%100875E-mail:fanggs@bnu.edu.cnrd-./0235679:<=>@BesovABpθ(R):BDEFG,rdrdHIJDEFG,KL3MNOP
RS:BTUFGBpθ(R)→Bqθ(R)

2、,VW1≤p≤q≤∞,r=(1−d(1/p−1/q)1/r)r.j=1jjjXYDEFGTUFG<=>BesovAMR(2000)Z[]46E35,46B03,41A17,30D15^_]O174.41EmbeddingTheoremonAnisotropicBesovClassesinDifferentMetricswithMixedNormLiXinQIANGenSunFANGDepartmentofMathematics,BeijingNormalUniversity,Beijing100875,P.R.ChinaE-mai

3、l:fanggs@bnu.edu.cnAbstractInthispaper,weestablisharepresentationtheoremonanisotropicBesovclasswithmixednormBr(Rd).Usingthistheorem,weproveanembeddingtheorempθrdrdontheBesovclasswithmixednormindifferentmetrics:B(R)→B(R),wherepθqθd1≤p≤q≤∞,r=(1−j=1(1/pj−1/qj)1/rj)r.Keyword

4、srepresentationtheorem;embeddingtheorem;anisotropicBesovclassMR(2000)SubjectClassification46E35,46B03,41A17,30D15ChineseLibraryClassification0174.411`abcdefa=(a1,...,ad),b=(b1,...,bd),a≤b(a

5、p2pd1p1pd−1pdf:=dxdx···

6、f(x,...,x)

7、p1dx···<∞Lp(Rd)dd−11d1RRR:2004-07-08; :2005-02-18:(10371009);(20050027003)382ghi49dfLp(R)(pid).!,·p:=·Lp(Rd).k∈N,t∈R,kΔkf(x):=(−1)+kCf(x+t)tk=0kkfx

8、!tk"",t=(0,...,0,tj,0,...,0),Δtf(x)Δtjf(x)k[1,2]#Δtj,jf(x).#$%
'Besov$.ddjk1k=(k1,...,kd)∈N,r=(r1,...,rd)∈R,ki>ri>0,i=1,...,d,r(Rd),d1≤p≤∞,1≤θ≤∞.%f∈Bpθ&(f∈Lp(R)')
!⎧1⎪⎪Δkif(·)θθ⎪⎪tipdti⎨<∞,1≤θ<∞,R

9、ti

10、ri

11、ti

12、fbri(Rd):=xipθ⎪⎪Δkif(·)⎪⎪tip⎩sup<∞,θ=∞,

13、

14、t

15、ti

16、rii

17、=0"kikiki+jkiΔtif(x)=(−1)f(x1,...,xi−1,xi+jti,xi+1,...,xd)jj=0)f(x)x*xi!tiki"".+p=(p,...,p)*,%Br(Rd):=Br(Rd).pθpθ,+#[2],-$ki(ki>ri)*-$,
,%.*ki(ki>ri)&'-)/(r(Rd),')-.BpθdfBr(Rd)=fp+fbri(Rd)pθxipθi=1
01Banach-.,
'Besov$.

18、,r1=···=rd=r,rd'$,2*Bpθ(R).'$-$+3-$,(,%.')'$-/.5.[1].rdrd