; TeX output 1996.06.14:1631 H!/ߍ)-<Dt q G cmr17TheBhuNuntforaBellmanfunction:mapplicationsto estimatesBofsingularinuNtegralop eratorsandtootherVWclassicalBproblemsinharmonicanalysis$yr 6XQ ff cmr12F./NazarovandS.T4reilvMw - ff cmcsc10Contents^N cmbx12In tro` ductionr5NotationB691What istheBellmanfunctionandho wtouseit <8XQ cmr121.1:=CarlesonimrbSeddingtheorem(dyadicg cmmi12L2|{Y cmr82version)N: C:::::::::::::::2ڹ8:=1.1.1`yTheBellmanfunction : C:::::::::::::::::::::::::2ڹ8:=1.1.2`yPropSertiesof"!", cmsy10B]m(F;f ;M@)ۦ: C::::::::::::::::::::::::R10:=1.1.3`yProSofoftheCarlesonimrbeddingtheoremviaBellmantrypefunctionU:R11:=1.1.4`yHorwtondaBellmantypSefunction? ': C:::::::::::::::::R13:=1.1.5`yHomewrorkforthereader9ȍ: C::::::::::::::::::::::::R13:=1.1.6`yAconcludingremark<: C::::::::::::::::::::::::::R141.2:=Example2:8dyradicmaximalfunctionñ: C:::::::::::::::::::::R14:=1.2.1`yPropSertiesЍ: C::::::::::::::::::::::::::::::::R15:=1.2.2`yRunningthemacrhinebackwards f: C::::::::::::::::::::R151.3:=Answrersn{: C:::::::::::::::::::::::::::::::::::::R1792W eigh ted norminequalitieswithmatrixweights: necessaryconditionsM172.1:=WVeighrtedL2 2 cmmi8pspaceswithmatrixweights>ɍ: C:::::::::::::::::::R172.2:=HilbSerttransforminL2p](Wƹ)(p: C::::::::::::::::::::::::::R182.3:=Necessarycondition: C:::::::::::::::::::::::::::::::R182.4:=Prarticularcasesofthe(Ap;q8Ϲ)condition: C::::::::::::::::::::R222.5:=PropSertiesof(Ap;q8Ϲ)wreights: C::::::::::::::::::::::::::R2393A na v eidealt24 -1 *H! 2/ߍ! 4fBases ofsubspacesinBanac hspaces U26)f4.1lWMaindenitions: C:::::::::::::::::::::::::::::::::R26lW4.1.1DIdealspacesofsequences: C::::::::::::::::::::::::R26lW4.1.2DUnconditionalconrvergence ⍑: C:::::::::::::::::::::::R26lW4.1.3DStrongunconditionalbases-: C:::::::::::::::::::::::R27lW4.1.4DCoSordinatepro jectionsPj @: C:::::::::::::::::::::::R27lW4.1.5DUnconditionalbasesvsstrongunconditionalbasesG0: C::::::::::R28f4.2lWStrongunconditionalbasesinre exivrespaces?b: C::::::::::::::::R28lW4.2.1DUnconditionalconrvergenceofseries%u cmex10PUjv#K cmsy82J#xjf ,xj\2UREj g: C::::::::R29lW4.2.2DDualbasis6: C::::::::::::::::::::::::::::::::R29lW4.2.3DRe exivitryofthecoSecientspace.K$: C::::::::::::::::::R32lW4.2.4D\PrarsevXalidentity"and\Besselinequality"R=: C::::::::::::::R32f4.3lWCriteriaforastrongunconditionalbasis:: C:::::::::::::::::::R33lW4.3.1DAnothercriterion: C::::::::::::::::::::::::::::R34ْ 5fHaar systeminw eighted L2p]: reductiontotheim b` eddingtheorem1)36f5.1lWComputationofk n9k`(EfgIOM)-:$q% cmsy6: C::::::::::::::::::::::::::::R37#Df5.2lWAbSoutthecoecienrtspaceYinthecaseofHilbertspace؍: C::::::::::R38 6fHaarđsystemasanunconditionalbasisinaw eightedđL22 space:"thescalar fcase.e38f6.1lWImrbSeddingtheoremand(A1 )condition B: C:::::::::::::::::::R39f6.2lWTheBellmanfunctione: C::::::::::::::::::::::::::::::R39lW6.2.1DPropSertiesoftheBellmanfunction.ˍ: C::::::::::::::::::R40lW6.2.2DRunningthemacrhinebackward.T: C::::::::::::::::::::R40f6.3lWTheehruntforaBellman(typSe)function:[aninnitesimalversionofthekeylWpropSertry: C:::::::::::::::::::::::::::::::::::::R42lW6.3.1DRemark#: C:::::::::::::::::::::::::::::::::R43f6.4lWThehruntforaBellman(typSe)function:8howtondit: C:::::::::::R43f6.5lWVVector-vXaluedproblemwithscalarwreight: C:::::::::::::::::::R46ْ 7fIm b` edding theoremforthematrixL22 @case H|47f7.1lW(A2;0\|)conditiond{: C:::::::::::::::::::::::::::::::::R47f7.2lWJenseninequalitryformatricesuۍ: C:::::::::::::::::::::::::R48f7.3lWProSofoftheimrbeddingtheoremnc: C::::::::::::::::::::::::R50 8fThe Haarsystemasanunconditionalbasisinscalarnon-w eighted L2p].b55f8.1lWH oldervsYVoung.y: C::::::::::::::::::::::::::::::::R57f8.2lWThehiddenterm.9x: C::::::::::::::::::::::::::::::::R58f8.3lWRemarks: C:::::::::::::::::::::::::::::::::::::R60 9fThe co` ecien tspaceinL2p G]case. ,61f9.1lWTVriebSel{LizorkinspaceTLqp/k: C:::::::::::::::::::::::::::R62lW9.1.1DThecoSecienrtspaceYZQ: C:::::::::::::::::::::::::R62 H!g3/ߍ!:=9.1.2`yThespaceTLqp K: C:::::::::::::::::::::::::::::R636{:=9.1.3`yDualofTVriebSel{LizorkinspaceTLqpe: C::::::::::::::::::R63:=9.1.4`yAnL2prepresenrtationofTLqp+썑: C::::::::::::::::::::::R63:=9.1.5`yNaturalpro jectionfromLOp `Aa cmr625onrtoTLqpι.Z: C::::::::::::::::R64p910$YThe co` ecien tspaceforL2p](Wƹ) g6410.1:=ThecoSecienrtspaceYisthesameforallL2p](Wƹ)큍: C::::::::::::::R6410.2>TheHaarsystemasastrongunconditionalbasisinL2p](Wƹ)$ƍ: C:::::::::R6510.3>Reductionto\dyradicimbSeddingtheorem">Y: C::::::::::::::::::R6610.4>ScalarwreightedL2psÍ: C:::::::::::::::::::::::::::::::R66:=10.4.1ddbTheBellmanfunction:8denitionandpropSerties: C:::::::::::R67:=10.4.2ddbTheBellmanfunction:8horwtoguessit?Q: C:::::::::::::::R68:=10.4.3`y\Mainpart"fG2p\ s(x)+(u2 1=pgn9)2q (x),: C::::::::::::::::::R69i:=10.4.4ddbThe\hiddenterm"n9(x)xteB?%(f ;u2 1=pg)ɍ: C:::::::::::::::::R71:=10.4.5ddbConcludingremarks.;: C::::::::::::::::::::::::::R74911$YMatrix w eightedL2pC^7411.1:=TheBellmanfunctionforL2p](Wƹ):8preliminarydiscussion: C::::::::::R76:=11.1.1`yWhatistracenorw?: C:::::::::::::::::::::::::::R77:=11.1.2`yWhatisdeterminanrtnow?,: C:::::::::::::::::::::::R7711.2:=(Ap;0cչ)-condition.: C:::::::::::::::::::::::::::::::::R78:=11.2.1`yNormalizingopSeratorsA: C:::::::::::::::::::::::::R7911.3:=TheBellmanfunction:8therstattempt.M: C:::::::::::::::::::R81:=11.3.1`yThekreypropSertyandconstructionҝ: C::::::::::::::::::R8111.4:=Strongformofthe(Ap;0cչ)condition.Ǹ: C::::::::::::::::::::::R84:=11.4.1`yOnemoreinequalitry): C::::::::::::::::::::::::::R8711.5:=TheBellmanfunction:8nalvrersion7: C::::::::::::::::::::::R87:=11.5.1`yDenitionoftheBellmanfunction ۍ: C:::::::::::::::::::R87:=11.5.2`yTheextremalproblemrevisited녍: C::::::::::::::::::::R88:=11.5.3`yEndofthestory: C:::::::::::::::::::::::::::::R89:=11.5.4`yFinalremarks>: C::::::::::::::::::::::::::::::R93912Classical Calderon-Zygm undop` erators bm9412.1:=MatrixofaclassicalCalderon-ZygmrundopSeratorintheHaarbasiss4: C::::R9512.2:=RegularandsingularpairsÍ: C:::::::::::::::::::::::::::R97:=12.2.1`ySingularpairs}: C::::::::::::::::::::::::::::::R97913Classical Calderon-Zygm undop` eratorsinmatrixweightedL2pP9813.1:=SingularitryremovXalZ: C:::::::::::::::::::::::::::::::R99:=13.1.1`yAvreragelattice:8construction ,: C::::::::::::::::::::::R99:=13.1.2`yBadinrtervXalsrX: C::::::::::::::::::::::::::::::R99:=13.1.3`ySplittingofafunctioninrto\bad"and\goSod"parts[q: C:::::::::r100:=13.1.4`yPullingourselvresbyhair`: C::::::::::::::::::::::::r101 7H! 4/ߍ! 14fThe regularpartofthematrix < 102 f14.1lWScalarL22case.%: C:::::::::::::::::::::::::::::::::r103f14.2lWVVectorL22case.⍑: C:::::::::::::::::::::::::::::::::r105f14.3lWVVectorL2pcase.: C:::::::::::::::::::::::::::::::::r108? 15fApp` endix 1. Strongunconditionalbasesofw avelets|113f15.1lWBiorthogonalwraveletsI: C::::::::::::::::::::::::::::::r113 16fApp` endix 2. Maximalfunction O118 \H!g5/ߍ! Introduction⼍9TheumaingoalofthepapSeristodemonstratetheabilitiesofanew(old?)zporwerfulutechnique 9intheharmonicanalysis.Thistecrhnique|themethoSdofBellmanfunctionwas\bSorrowed"9fromtheappliedmathematics,jmorepreciselyfromtheconrtroltheoryV.Itisinterestingto9notethatalthoughusuallytheappliedmathematicsgetsideasfromthe\pure",iherewre9harveanoppSositeexample. The@ideatoapplytheBellmanfunctionmethoSdtotheproblemsinharmonicanalysis9isalsonothingnew.&D.Burkholder(see[1])appliedavXariationofthemethoSd(without9menrtioningtheterm\Bellmanfunction")todierentproblemsinharmonicanalysis, ;e.g.9ndingthesharpunconditionalbasisconstanrtfortheHaarsysteminL2p].WVeldiscorveredthemethoSdwhen(successfully)tryingtogeneralizetheresultof[3]about9wreightedڣnorminequalitieswithmatrixwreightsڣtothecasep6=2.Initiallyڣtheprob-9lemloSokredextremelydicult,andwecouldnotobtainthisresultusinganyotherknown9methoSd.21 And^theBellmanfunctionwrasthetoolthatallorwed^ustocracrktheproblemand9nallytosolvreitcompletelyV.ZSooneofthemainresultsofthepapSerisageneralization9oftheMucrkenhoupt(Ap])condition(1X