Hausdorff算子在函数空间上的有界性.docx

Hausdorff算子在函数空间上的有界性
Hausdorffoperatorisalinearoperatordefinedonthespaceofcontinuousfunctions,whichmapsagivenfunctiontoitssetofoscillationvalues.TheboundednessoftheHausdorffoperatoronfunctionspaceisanimportanttopicinthetheoryoffunctionalanalysis.Inthispaper,wewilldiscusstheboundednessoftheHausdorffoperatorandprovideaproofforitsboundednessonfunctionspaces.
Tobeginwith,weintroducetheconceptoftheHausdorffoperator.LetXbeacompactmetricspaceandC(X)bethespaceofcontinuousfunctionsonX.ForanyfunctionfinC(X),theHausdorffoperatorisdefinedasfollows:
H(f)={osc(f;K)|KisacompactsubsetofX},
whereosc(f;K)denotestheoscillationoffonthecompactsetK,definedby
osc(f;K)=sup{f(x)-f(y)|x,y∈K}.
Inotherwords,theHausdorffoperatormapseachfunctiontoasetofoscillationvaluesonallpossiblecompactsubsetsofX.
Next,wewillprovetheboundednessoftheHausdorffoperatoronthespaceC(X).Todothis,weneedtoshowthatforanyfinC(X),thereexistsaconstantMsuchthat
|osc(f;K)|≤M,forallcompactsubsetsKofX.
Toprovethis,wewillusethefactthatfisacontinuousfunctiononthecompactmetricspaceX.ThisimpliesthatfisuniformlycontinuousonX,i.e.,foranyε>0,thereexistsaδ>0suchthat
|f(x)-f(y)|<ε,forallx,y∈Xwith|x-y|<δ.
Now,letKbeacompactsubsetofXandletε>0begiven.SinceKiscompact,itcanbecoveredbyafinitenumberofopenballsofradiusδ/2,i.e.,
K⊆⋃_{i=1}^{n}B(x_i,δ/2),
whereB(x_i,δ/2)denotestheopenballcenteredatx_iwithradiusδ/2.SincefisuniformlycontinuousonX,wehave
|f(x)-f(y)|<ε,forallx,y∈Xwith|x-y|<δ.
Thisimpliesthat
|f(x)-f(y)|<ε,forallx,y∈B(x_i,δ/2),
foralli=1,2,...,n.Therefore,wehave
osc(f;K)=sup{f(x)-f(y)|x,y∈K}≤sup{|f(x)-f(y)||x,y∈B(x_i,δ/2),i=1,2,...,n}≤ε,
forallε>0.Hence,wecanconcludethat
|osc(f;K)|≤ε,forallcompactsubsetsKofX.
ThisprovestheboundednessoftheHausdorffoperatoronthespaceC(X),aswehaveshownthatthereexistsaconstantM=εsuchthat|osc(f;K)|≤M,forallcompactsubsetsKofXandforallfinC(X).
Inconclusion,wehavediscussedtheboundednessoftheHausdorffoperatoronthespaceofcontinuousfunctions.Wehaveprovidedaproofforitsboundednessonfunctionspacesbyutilizingtheun