17 Monotonicity Formula And Basic Consequences

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Παρθενορή Κουρμούλης
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1 Lectues o Vaifols Leo Sio Zhag Zui 7 Mootoicity Foula A Basic Cosequeces I this sectio we assue that U is oe i R, V v( M,θ) has the geealize ea cuvatue H i U ( see 6.5), a we wite µ fo µ V ( H θ as i 5.). Ou ai is to obtai ifoatio about V by aig aoiate choice of X i the foula (see 6.5) M i R c 7., ( ; iv Xµ XHµ X C U ) Fist we choose X γ()( ), whee U is fie,, a γ is a C ( R ) fuctio with γ' () t 0 t; γ() t fot / ; γ() t 0 fot> whee > 0 is such that B( ) U. ( Hee a subsequetly B ( ) eotes the oe ball i R with cete a aius. ) Fo ay f C ( U) a ay M such that TM eists M l (see.4.6) we have (by.) f( ) edf( e ) whee Df l eotes the atial eivative, l, f l i U a whee ( e ) is the ati of the othogoal l l of f tae oectio of R oto TM. Thus witig M M ei ( as i 7

2 Lectues o Vaifols Leo Sio Zhag Zui Sectio 6), with the above choice of X we euce iv X X e e M Moe ecise, a l l M l () '() γ γ i l, M f( ) P ( ) TM ga f P TM Df l el R l l l Df l PTMe l Df l ee edf l e l l, l, iv X M ei X M M l ei edxe l l, l edx l l, X l l l e γ' () i( ) γ() δ l l, l l l γ' () e i γ() e l, l l l γ' () e e, l γ' P DD, γ () () TM γ' D γ () () e γ () l Sice ( e ) eesets the othogoal oectio oto TM 8

3 Lectues o Vaifols Leo Sio Zhag Zui we have e a i l, l l l e D, whee D eotes the othogoal oectio of D (which is a vecto of legth TM. ) oto The foula 7. thus yiels (*) () '() γ µ γ µ Hi( ) γ () µ γ '() D µ ovie B( ) U. Now tae φ such that φ() t fo t, φ () t 0 fo t a φ't () 0 fo all t. The we ca use (*) with γ() φ. Sice γ' () φ' φ this gives whee I I' J' L I φ µ, L φ ( ) Hµ J φ D µ Thus, ultily by a eaagig we have 7. I J' L Thus lettig φ icease to the chaacteistic fuctio of the iteval (, ), we obtai, i the istibutio sese, 9

4 Lectues o Vaifols Leo Sio Zhag Zui 7.3 µ ( B() ) D µ ( ) Hµ B( ) i B( ) ( B ) This is the fuaetal ootoicity ietity. Sice µ () a D µ ae iceasig i it also hols i the B( ) classical sese fo ae.. > 0 such that B( ) U. Moe ecise, µ ( B() ) µ ( B() ) µ ( B() ) ' a li li B( ) B( ) B ( ) B( ) B ( ) D µ D µ D µ whee we ca estiate the last quatity as follows, B ( ) B ( ) B ( ) B ( ) B ( ) B ( ) D µ D µ D µ 0

5 Lectues o Vaifols Leo Sio Zhag Zui Notice that if H 0 the 7.3 tells us that the atio µ ( B () ) is o-eceasig i. Geeally, by itegatig with esect to i 7.3 we get the ietity 7.4 µ ( B ()) µ B() D B ( ) Β ( ) ( ) H µ i µ fo all 0< with B( ) U, whee a {, }, so that if H 0 we have the aticulaly iteestig ietity 7.5 µ ( B ()) µ B() D B ( ) B ( ) µ Moe ecise, itegatig 7.3 with esect to, we get µ ( B() ) µ B() D µ ( ) Hµτ B( ) B( ) τ i Bτ( ) D µ χb ( ) ( ) H τ µτ B( ) B( ) τ i B( ) χb ( ) τ i B( ) B ( ) B ( ) D µ ( ) H τ µ B( ) B( ) i B( ) a {, } τ B( ) B ( ) D µ H τ µ τ D µ ( ) H i µ B( ) We ow wat to eaie the iotat questio of what 7.3

6 Lectues o Vaifols Leo Sio Zhag Zui tells us i case we assue boueess a L coitios o H. 7.6 Theoe If U,0< α, Λ 0, a if α (*) Hµ Λ µ ( B() ) foall ( 0R, ) α B( ) R α αµ ( B () R Λ ) whee BR() U, the e is a o-eceasig fuctio of ( 0R, ), a i fact () Wheeve 0 () α α ( ()) () R B α α B D Λ µ µ R Λ e e µ B( ) B( ) < < R. Also, α α ( ()) () R B α α B D Λ µ µ R Λ e e µ B( ) Poof Fo 7.3, ultily by the itegatig facto get e α α ΛR we D α α D ΛR e B ( ) B ( ) µ µ ( ()) R α α µ B α α Λ ΛR e e ( ) Hµ i B( ) α α ( B ()) R µ α α Λ ΛR e e Hµ B( ) α α α ( B ()) R µ α α Λ ΛR e e αλ µ ( B () ) R ( ()) ( () R α α µ B α α µ B R Λ Λ α α ) e e ΛR α i

7 Lectues o Vaifols Leo Sio Zhag Zui α αµ ( B () R ) e Λ Thus itegate with esect to o the iteval (, ), () follows, B ( ) µ ( B() ) α α ΛR µ B() e α α ΛR µ e D Fo (), we ust use the sae etho, D α α D ΛR e B ( ) B ( ) µ µ ( ()) R α α µ B α α Λ ΛR e e ( ) Hµ i B( ) α α ( B ()) R µ α α Λ ΛR e e Hµ B( ) α α ( B ()) R µ α α α Λ ΛR e e αλ ( ()) µ B R ( ()) ( () R α α µ B α α µ B R ) Λ Λ α α e e ( ΛR α ) i α αµ ( B () R ) e Λ 7.7 Theoe If U a BR( ) H µ Γ, whee BR( ) U a >, the µ ( B () ) B () µ Γ 3

8 Lectues o Vaifols Leo Sio Zhag Zui wheeve 0 < < R. Poof Fo 7.3, we get µ ( B() ) ( ) Hµ i B( ) Hµ i B( ) B ( ) µ ( B() ) Γ H µ µ ( B() ) That is, µ B() Γ Itegatig with esect to, we have µ ( B ()) ( ()) µ B Γ 7.8 Coollay If H L ( µ ) i U fo soe >, the the loc µ B esity Θ ( µ, ) li eists at evey oit U, 0 ω a Θ ( µi,) is a ue-sei-cotiuous fuctio i U : Θ ( µ, ) lisu Θ ( µ, y), U y Poof The iequality 7.7 tells us that ( ()) µ B Γ 4

9 Lectues o Vaifols Leo Sio Zhag Zui µ ( B () ) is a o-eceasig fuctio of, hece li eists 0 µ ( B () ) µ ( B() ) µ B( ) (a is the sae as li ). [ li li µ B () ( ()) li µ B li, 0 0 N ] ( ) Now fo the oof of coollay, Θ ( µ, ) li su Θ ( µ, y) y Θ ( µ, ) li su Θ ( µ, y) ε 0 y < ε Θ ( µ, ) δ ( 0, ), > lisu y < εθ ( µ, y) δ ε 0 Θ ( µ, ) δ ( 0, ), ε> 0st,.. if y < ε, theθ ( µ, y) < δ Θ ( µ, ) δ ( 0, ), ε> 0st,.. if y < ε, theθ ( µ, y) < δ get Thus we ove the last assetio, to this e, let 0, we µ B ( y) Γ Θ ( µ, y) ω ω µ ε c (,, ) ω ε Θ ( µ, ) < δ ( B ε ) ( ) if we tae ε sall eough. 5

10 Lectues o Vaifols Leo Sio Zhag Zui 7.9 Reas () If θ µ ae.. U, H L ( U), > as i 7.8, the loc Θ ( µ, ) at each oit of stµ U, a hece we ca wite (, ) V U v M θ whee M* stµ U, θ* Θ ( µ, ), U. * * Thus V U is eesete i tes of a elatively close coutably ectifiable set with ue-sei-cotiuous ultilicity fuctio. Obseve that H ( A M) H A M A M ( θ θ ) A Mθ( µ H ) fo ay H easuable subset of µ ae is equivalet R... to H ae.. o M, the Rea is uestaable. Fo esity µ B Θ ( µ, ) li li 0 ω 0 θh B ( ) N li 0 ω B M ω θh whee N is C subaifol of R with N, such N eists by.7. Fo vaifol ( ( )) M U stµ µ 0 θ H H M U st µ U stµ 6

11 Lectues o Vaifols Leo Sio Zhag Zui () If U, Θ ( µ, ), a H µ Γ ω BR( ), whee B () U a >, the both iequalities 7.6(), () R hol with Λ ΓR a α, ovie ΓR. Poof Notice fist that Hµ H µ µ ( B() ) Γ ω µ B( ) B ( ) B ( ) ( B ) We estiate µ () usig 7.7 as follows (lettig 0 ) Γ ω µ ( B() ) ω ω µ ( B ()) µ ( B() ) Γ ω ω The µ B() ω [ ], hece, Hµ Γ ω ω µ B i i i B ( ) ( ) [ ] () Γ µ ( B() ) Γ R µ B() i R Thus the hyothesis of 7.6 hol with α. Λ ΓR a (3) Notice that eithe 7.6() o 7.7 give bous of the fo µ ( () ) β, 0< < R fo suitable costat β. Such a B 7

12 Lectues o Vaifols Leo Sio Zhag Zui iequality ilies B ( ) α β µ α α fo ay ( 0R, ) a 0 < α<. Poof We ust aly the follow lea. α µ α ( B () α α ) B( ) B( ) µ µ α ( α ) t µ ; t > t µ ( B () α ) α ( α ) t µ B() t µ B () α α β β α ( ) t t α t α ( α) β β α α α t β α 7.0 Lea If X is a abstact sace, µ is a easue o X, { } α> 0. f L( µ ), f 0 a if A X; f( ) > t, the Moe geeally α t ( At) t 0 A α α µ f µ α µ f t µ t t α α α t At t 0 A 0 t0 Poof It is a by-ouct of Fubii s theoe. Moe ecise, we have t0 8

13 Lectues o Vaifols Leo Sio Zhag Zui α α ( t) ( ) α α A t ( At ) 0 t0 t0 0 f( ) α α α ( α t t ) µ ( f t0) µ α t µ A t α t µ t t t A 0 0 t α t χ µ t α t χ tµ t A A t A t t0 0 t0 A 7. Lea Suose θ µ ae.. i U, H L ( µ ) i U loc fo soe >. If the aoiate taget sace TV ( see Sectio 5) eists at a give oit U, the TV is a classical taget lae fo stµ i the sese that ist( ytv, ) li suy B( ) st 0 0 µ Poof Fo sufficietly sall R (with B R U ), 7.7,7.8 (with 0 ) evietly ily () µ ( B ()),, ω 0< < R B stµ Moe ecise, sice H L ( µ ), we choose R, Γ> 0 such that B R U a µ,0 B ( ) st B R loc H µ Γ. The fo ay < < R, H µ H µ Γ B B R a by vitue of 7.7, 7.8, 7.9(), we have ( ) B ω µ Γ,, thus µ ( B ) ω if 9

14 Lectues o Vaifols Leo Sio Zhag Zui Γ R, this is ossible sice we ca fist fi soe Γ fo soe R 0, the let R0> R 0 with Γ uchage. Usig this we ae goig to ove that if α 0, a ( 0R, ), the () ( B { yist ; ( ytv, ) }) µ < ε < ωα {;, } B stµ yist ytv < ε α µ ε α {;, } Iee if B ( ) st yist( ytv) ( ) {;, ε} the B () B ( ) yist( ytv) α < a hece the hyothesis of () ilies µ Bα() ωα ha, () ilies µ Bα() ωα, so we have a. O the othe cotaictio. Thus () is ove, a () evietly leas ieiately to the equie esult. Moe ecise, by (5) i the Poof of Theoe.8, we ve fo, ay ε 0,, ( B X (, ) ) µ π ε li 0 ω 0 [This is coe coitio] whee π ( P). Thus we ca tae sall eough such that 0

15 Lectues o Vaifols Leo Sio Zhag Zui ( B { yist( ytv ) < }) µ ;, ε ε < ω [This is cylie coitio, you ay tae a ictue to see it clealy.] Hece ist ytv µ ε ε ε, y B ( ) st < ( ) 4

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