Poincar inequalities that fail to constitute an open-ended - PowerPoint PPT Presentation

ﯲ ﯰ ﯱ ﯰ Poincaré inequalities that fail to constitute an open-ended condition Lukáš Malý Workshop on Geometric Measure Theory July 14, 2017

ﯰ ﯱ Poincaré inequalities Setting Let ( X , d, µ ) be a complete metric space endowed with a doubling measure . Definition (Heinonen–Koskela, 1996) A B or e l f un c tion g ∶ X → [ 0 , ∞ ] is a n upper gradient o f u ∶ X → R i f ∣ u ( γ ( 0 )) − u ( γ ( l γ ))∣ ≤ ∫ γ g ds f or eve r y r ec ti fiab l e c ur ve γ ∶ [ 0, l γ ] → X . Definition (Heinonen–Koskela, 1996) T h e sp ace X ad mits a ( 1 , p ) -Poincaré inequality w ith p ∈ [ 1 , ∞ ) i f 1 / p ∫ B ∣ u − u B ∣ d µ ≤ c PI d i a m ( B )( ∫ λ B g p d µ ) lo c ( X ) , its upp e r g r ad i e nt g , a n d eve r y ba ll B ⊂ X , w h e r e c PI , λ ≥ 1. f or eve r y u ∈ L 1 Lukáš Malý Poincaré inequalities that fail to be open-ended 1/17

ﯰ ﯱ Self-impro v ement of Poincaré inequalities Theorem (Keith–Zhong, 2 00 8) If X ad mits a p- P oin ca r é in e qu a lit y w ith p ∈ ( 1, ∞ ) , th e n it ad mits a ( p − ε ) - P oin ca r é in e qu a lit y f or som e ε > 0 . Re m a rk : B oth c ompl e t e n e ss o f X a n d d ou b lin g o f µ a r e us ed! E x ample (Koskela, 1999) F or eve r y 1 < p ≤ n , th e r e is a c los ed s e t E ⊂ R n o f ze ro L n -m ea sur e su c h th a t X = R n \ E supports a p - PI, b ut not q - PI f or a n y q < p . Re m a rk : X in this exa mpl e is lo ca ll y c omp ac t a n d µ ≔ L n ∣ X is d ou b lin g . Question D o e s a n A - PI w ith a n ave r ag in g op e r a tor A on th e RHS o f PI ( A is c los e to th e L p - ave r age ) impro ve to a ( p − ε ) - PI? Lukáš Malý Poincaré inequalities that fail to be open-ended 2/17

ﯰ ﯱ M otiv a tion T h e or e m ( D ur a n d - Ca rt age n a, Ja r a millo , S h a nmu ga lin ga m , 2013) Assume that µ is A hl f ors s-r eg ul a r (i. e . , µ ( B ( x , r )) ≈ r s ). Le t p > s. T h e n , TFAE: ▶ X supports a p- PI ; ▶ E v e r y u ∈ N 1, p ( X ) is ( 1 − s p ) - H öl de r c ontinuous a n d ∣ u ( x ) − u ( y )∣ ≲ d ( x , y ) 1 − s / p ∥ g u ∥ L p ( B ( x , λ d ( x , y ))) ▶ T h e r e is C ≥ 1 su c h th a t f or a ll x , y ∈ X Mod p ({ γ ∈ Γ x , y ∶ len ( γ ) ≤ C d ( x , y )}) ≈ d ( x , y ) s − p Lukáš Malý Poincaré inequalities that fail to be open-ended 3/17

ﯰ ﯱ M otiv a tion Idea f or th e c riti ca l e xpon e nt Assume that µ is Ahlfors s -regular, s ≥ 1 (i.e., µ ( B ( x , r )) ≈ r s ). Then, TFAE: 1. X supports a Lorentz-type L s ,1 - PI 2. Eve r y u ∈ N 1 L s , 1 ( X ) is c ontinuous a n d ∣ u ( x ) − u ( y )∣ ≲ ∥ g u ∥ L s , 1 ( B ( x , λ d ( x , y ))) 3. T h e r e is C ≥ 1 su c h th a t f or a ll x , y ∈ X M o d L s , 1 ({ γ ∈ Γ x , y ∶ l e n ( γ ) ≤ C d ( x , y )}) ≈ 1 W orks f or s = 1 F or s > 1 , th e f ollo w in g ca n be pro v e n : 1. ⇐ 2. ⇔ 3. p - PI w ith p < s ⇒ 2. Lukáš Malý Poincaré inequalities that fail to be open-ended 4/17

ﯰ ﯱ L or e ntz spaces L p , q Norm in the Lebesgue L p -spaces 1 / p 1 / p ∥ u ∥ L p ( X ) = ( ∫ X ∣ u ( x )∣ p d µ ( x )) = ( ∑ k ∈ Z ∫ { x ∶ 2 k < ∣ u ( x )∣ ≤ 2 k + 1 } ∣ u ( x )∣ p d µ ( x )) = � � 1 / p � � � ∞ � = ( ∑ ∥ u χ { 2 k < ∣ u ∣ ≤ 2 k + 1 } ∥ L p ( X ) ) p �{∥ u χ { 2 k < ∣ u ∣ ≤ 2 k + 1 } ∥ L p ( X ) } � � � � � ℓ p ( Z ) k = −∞ k ∈ Z Definition (functional comparable w ith the Lorent z norm) Let p ∈ [ 1, ∞ ) and q ∈ [ 1, ∞ ] . Then, we define ∣∣∣ u ∣∣∣ L p , q ( X ) = � � � � � ∞ � �{∥ u χ { 2 k < ∣ u ∣ ≤ 2 k + 1 } ∥ L p ( X ) } � � � � � ℓ q ( Z ) k = −∞ Re m ark: L p ,1 ( X ) ↪ L p ( X ) = L p , p ( X ) ↪ L p , ∞ ( X ) for 1 ≤ p < ∞ . Remark: L p log α L ( X ) ↪ L p , q ( X ) , whene v er α > ( p / q ) − 1 ≥ 0 . Lukáš Malý Poincaré inequalities that fail to be open-ended 5/17

ﯰ ﯱ K nown r esults on Orlic z –Poincaré inequalities Definition (Orlic z –Poincaré inequalit y ) Let Ψ ∶ [ 0, ∞ ) → [ 0, ∞ ) be strictly increasing, continuous, and con v e x w ith Ψ ( 0 + ) = 0 and Ψ ( ∞ − ) = ∞ . The space X admits a Ψ - P oin ca r é in e qu a lit y if ∫ B ∣ u − u B ∣ d µ ≤ c P I d i a m ( B ) Ψ − 1 ( ∫ λ B Ψ ( g ) d µ ) lo c ( X ) , its upp e r g r ad i e nt g , a n d eve r y ba ll B ⊂ X , w h e r e c PI , λ ≥ 1. f or eve r y u ∈ L 1 Re m a rk : If Ψ ( t ) = t p , th e n Ψ - PI is just a p - PI . S om e s e l f -impro ve m e nt o f Ψ - PI to ( p − ε ) - PI ca n be ex p ec t ed i f Ψ ( t ) ∼ t p η ( t ) , w h e r e η ( t ) g ro w s (or decay s) slo we r th a n a n y po we r t δ (or t − δ ) Lukáš Malý Poincaré inequalities that fail to be open-ended 6/17

ﯰ ﯱ K nown r esults on Orlic z –Poincaré inequalities Theorem (J. B jörn , 2010) T h e E u c li dea n sp ace ( R , µ ) ad mits a Ψ - PI if and only if th e Ha r d y –L ittl e w oo d m a x im a l op e r a tor M ∶ L Ψ ( R , µ ) → L Ψ ( R , µ ) is b oun ded . T h e or e m ( B loom –Ke rm a n , 199 4 ) Le t Ψ ( t ) = t p log α ( e + t ) w ith p > 1 a n d α ∈ R . If M ∶ L Ψ ( R , µ ) → L Ψ ( R , µ ) is b oun ded, th e n M ∶ L p − ε ( R , µ ) → L p − ε ( R , µ ) is b oun ded f or som e ε > 0 . C oroll a ry S uppos e th a t ( R n , µ ) , w h e r e µ = µ 1 × µ 2 × ⋯ × µ n , ad mits a Ψ - PI, w h e r e Ψ ( t ) = t p log α ( e + t ) f or som e p > 1 a n d α ∈ R . T h e n , ( R n , µ ) ad mits a ( p − ε ) - PI f or som e ε > 0 . Lukáš Malý Poincaré inequalities that fail to be open-ended 7/17

ﯰ ﯱ L or e ntz-typ e P oin ca r é in e qu a liti es Definition (Lorent z -t y pe Poincaré inequalit y ) Let p ∈ [ 1, ∞ ) and q ∈ [ 1, ∞ ] . The space X admits an L p , q - P oin ca r é in e qu a lit y if ∥ gχ λ B ∥ L p , q ( X ) ∥ gχ λ B ∥ L p , q ( X ) ∫ B ∣ u − u B ∣ d µ ≤ c P I d i a m ( B ) = c PI d i a m ( B ) ∥ χ λ B ∥ L p , q ( X ) µ ( λ B ) 1 / p lo c ( X ) , its upp e r g r ad i e nt g , a n d eve r y ba ll B ⊂ X , w h e r e c PI , λ ≥ 1. f or eve r y u ∈ L 1 Re m a rk : If p = q , th e n L p , q - PI is just a p - PI . Obser v ation D u e to th e e m bedd in g be t wee n L or e nt z sp ace s , we h ave L p , Q - PI L p , q - PI p - PI ⇒ ⇒ w h e n eve r 1 ≤ q < p < Q ≤ ∞ . Lukáš Malý Poincaré inequalities that fail to be open-ended 8/17

ﯰ ﯱ L or e ntz-typ e P oin ca r é in e qu a liti es Comparison to Muckenhoupt w eights on R ▶ ( R , µ ) admits an L p , q - PI, p ∈ ( 1 , ∞ ) , q ∈ [ 1 , ∞ ] , if and onl y if th e HL m ax im a l op e r a tor M ∶ L p , q ( R , µ ) → L p , q ( R , µ ) is b oun ded . ▶ M ∶ L p , q ( R , µ ) → L p , q ( R , µ ) is b oun ded if and onl y if M ∶ L p ( R , µ ) → L p ( R , µ ) is b oun ded, p ∈ ( 1 , ∞ ) , q ∈ ( 1 , ∞ ] . ( C hun g–H unt –K urt z, 19 8 2) ▶ If ( R n , µ ) , w h e r e µ = µ 1 × ⋯ × µ n , ad mits a n L p , q - PI w ith p ∈ ( 1 , ∞ ) a n d q ∈ ( 1 , ∞ ] , th e n it ad mits a ( p − ε ) - PI f or som e ε > 0 . Question D o e s a n L p , 1 - PI un de r g o s e l f -impro ve m e n t? Lukáš Malý Poincaré inequalities that fail to be open-ended 9/17

ﯰ ﯱ Ge n e r a l P oin ca r é in e qu a liti es + ( X ) × B ( X ) → [ 0, ∞ ] be an a v erag in g op e r a tor , i.e., Let A ∶ L 0 ▶ A ( c , B ) ≈ c for e v er y constant c ≥ 0 and e v er y ball B ⊂ X ; ▶ if 0 ≤ g 1 ≤ g 2 a.e. in B , then A ( g 1 , B ) ≤ A ( g 2 , B ) . Let M A be the associated (non-centered) ma x imal operator M A f ( x ) = sup A (∣ f ∣ , B ) , x ∈ X . B ∋ x Definition (Generali z ed Poincaré inequalit y ) The space X admits an A - P oin ca r é in e qu a lit y if ∫ B ∣ u − u B ∣ d µ ≤ c P I d i a m ( B ) A ( g , λ B ) lo c ( X ) , its upp e r g r ad i e nt g , a n d eve r y ba ll B ⊂ X , w h e r e c PI , λ ≥ 1. f or eve r y u ∈ L 1 Lukáš Malý Poincaré inequalities that fail to be open-ended 10/17

ﯰ ﯱ Ge n e r a l P oin ca r é in e qu a liti es Lemma (M.) Assume that X supports a n A - PI . S upp ose that 1 / p A A ( g , B ) ≲ ( ∫ B g p A dµ ) for all g , B . The n , X ad mits a ( p A − ε ) - PI f or som e ε > 0 . Corollar y Le t p > 1 . S uppos e th a t X supports ▶ a n L p , q - PI w ith q ∈ [ p , ∞ ] , OR ▶ a Ψ - PI w ith Ψ ( t ) ∼ t p log α ( e + t ) , w h e r e α ∈ R . T h e n , X ad mits a ( p − ε ) - PI f or som e ε > 0 . Lukáš Malý Poincaré inequalities that fail to be open-ended 11/17

ﯰ ﯱ Ge n e r a l P oin ca r é in e qu a liti es Theorem (M.) Assume that X supports a n A - PI a n d th a t M A ∶ RI 1 ( X ) → w- RI 2 ( X ) is b oun ded, w h e r e RI 1 a n d RI 2 a r e r ea rr a n ge m e nt in v a ri a nt sp ace s , su c h th a t : ▶ RI 1 a n d RI 2 a r e c los e to eac h oth e r , v i z . , th e ir f un da m e nt a l f un c tions s a tis f y 2 ( t )) ≤ t ( 1 + lo g − t ) β , t ≤ φ 1 ( φ − 1 β ≥ 0 ; ∥ f χ ⋃ k E k ∥ ∥ f χ E k ∥ q q RI 1 ≤ C ∑ ▶ RI 1 s a tis fie s a n upp e r q- e stim a t e, i. e . , RI 1 , q ≥ 1; k φ 1 ( t p A ) ▶ RI 1 li e s c los e to L p A , sp ec i fica ll y , t ( log − t ) 1 − ( β + 1 / q ) → 0 a s t → 0 . T h e n , X ad mits a ( p A − ε ) - PI f or som e ε > 0 . Lukáš Malý Poincaré inequalities that fail to be open-ended 12/17