Poincar e, Sobolev and Rubio de Francia Ezequiel Rela Departamento - PowerPoint PPT Presentation

Poincar´ e, Sobolev and Rubio de Francia Ezequiel Rela Departamento de Matem´ atica Facultad de Ciencias Exactas y Naturales - Universidad de Buenos Aires CONICET Joint work with Carlos P´ erez Moreno (BCAM) XIV Encuentro Nacional de Analistas Alberto P. Calder´ on 22 de noviembre de 2018

Outline What? Why? How? Ezequiel Rela Poincar´ e-Sobolev-RdF

Outline What? � 1 � 1 � 1 � � 1 � q p | f − f Q | q w |∇ f | p w ≤ C w ℓ ( Q ) w ( Q ) w ( Q ) Q Q Why? A ( x ) ξ.ξ ≈ | ξ | 2 w ( x ) div( A ( x ) ∇ u ) = 0 , How? Unweighted L 1 inequalities involving “Self-improving functionals” � − | f − f Q | dx ≤ a ( Q ) Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Main problem � 1 � 1 � 1 � � 1 � q p | f − f Q | q w |∇ f | p w ≤ C w ℓ ( Q ) w ( Q ) w ( Q ) Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Main problem � 1 � 1 � 1 � � 1 � q p | f − f Q | q w ≤ C w ℓ ( Q ) |∇ f | p w w ( Q ) w ( Q ) Q Q For a given p ≥ 1. Ezequiel Rela Poincar´ e-Sobolev-RdF

Main problem � 1 � 1 � 1 � � 1 � q p | f − f Q | q w ≤ C w ℓ ( Q ) |∇ f | p w w ( Q ) w ( Q ) Q Q For a given p ≥ 1. There is a natural choice for a class A p of weights. Ezequiel Rela Poincar´ e-Sobolev-RdF

Main problem � 1 � 1 � 1 � � 1 � q p | f − f Q | q w ≤ C w ℓ ( Q ) |∇ f | p w w ( Q ) w ( Q ) Q Q For a given p ≥ 1. There is a natural choice for a class A p of weights. We try to reach the best possible q = p ∗ w . Ezequiel Rela Poincar´ e-Sobolev-RdF

Main problem � 1 � 1 � 1 � � 1 � ≤ C w ℓ ( Q ) q p | f − f Q | q w |∇ f | p w w ( Q ) w ( Q ) Q Q For a given p ≥ 1. There is a natural choice for a class A p of weights. We try to reach the best possible q = p ∗ w . Keeping track of the constant C w ! Ezequiel Rela Poincar´ e-Sobolev-RdF

e in ( R n , dx ) Unweighted Poincar´ (1 , 1) Poincar´ e inequality 1 � | f − f Q | dx � ℓ ( Q ) 1 � |∇ f | dx | Q | | Q | Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

e in ( R n , dx ) Unweighted Poincar´ (1 , 1) Poincar´ e inequality 1 � | f − f Q | dx � ℓ ( Q ) 1 � |∇ f | dx | Q | | Q | Q Q ( p , p ) Poincar´ e inequality, 2 ≤ n , 1 ≤ p < n . � 1 � 1 � 1 � 1 � � p p | f − f Q | p dx |∇ f | p � ℓ ( Q ) | Q | | Q | Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

e in ( R n , dx ) Unweighted Poincar´ (1 , 1) Poincar´ e inequality 1 � | f − f Q | dx � ℓ ( Q ) 1 � |∇ f | dx | Q | | Q | Q Q ( p , p ) Poincar´ e inequality, 2 ≤ n , 1 ≤ p < n . � 1 � 1 � 1 � 1 � � p p | f − f Q | p dx |∇ f | p � ℓ ( Q ) | Q | | Q | Q Q Higher order Poincar´ e inequality with polynomials, m ∈ N | f ( y ) − π Q ( y ) | dy � ℓ ( Q ) m 1 � � |∇ m f | dy | Q | | Q | Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Poincar´ e - Sobolev Poincar´ e-Sobolev inequality � 1 � 1 � 1 � 1 � � p ∗ p | f − f Q | p ∗ dx |∇ f | p � ℓ ( Q ) | Q | | Q | Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Poincar´ e - Sobolev Poincar´ e-Sobolev inequality � 1 � 1 � 1 � 1 � � p ∗ p | f − f Q | p ∗ dx |∇ f | p � ℓ ( Q ) | Q | | Q | Q Q np p ∗ = n − p Ezequiel Rela Poincar´ e-Sobolev-RdF

Weights w 1 − p ′ � p − 1 � � � � � [ w ] A p := sup − w − Q Q Q � w ( E ) � 1 | E | 1 p p | Q | ≤ [ w ] A p w ( Q ) � � � � w − 1 � L ∞ ( Q ) [ w ] A 1 := sup − w Q Q Equivalently: Mw ( x ) ≤ Cw ( x ) a.e. x ∈ R n Ezequiel Rela Poincar´ e-Sobolev-RdF

Hardy-Littlewood maximal function � Mf ( x ) = sup − | f ( y ) | dy , Q ∋ x Q M : L p ( w dx ) → L p ( w dx ) ⇐ ⇒ w ∈ A p 1 < p < ∞ M : L 1 ( w dx ) → L 1 , ∞ ( w dx ) ⇐ ⇒ w ∈ A 1 1 p − 1 � M � L p ( w ) � p ′ [ w ] A p , 1 < p < ∞ 1 p � M � L p , ∞ ( w ) ≈ [ w ] A p , 1 ≤ p < ∞ Ezequiel Rela Poincar´ e-Sobolev-RdF

Fractional integrals and Poincar´ e The following are equivalent � � 1) − | f ( x ) − f Q | dx � ℓ ( Q ) − |∇ f ( x ) | dx Q Q � ( |∇ f | χ Q )( y ) 2) | f ( x ) − f Q | � I 1 ( |∇ f | χ Q )( x ) = | x − y | n − 1 dy R n As a consequence of 2), | f ( x ) − f Q | � I 1 ( |∇ f | χ Q )( x ) � ℓ ( Q ) M ( |∇ f | )( x ) 1 p � f − f Q � L p , ∞ Q , w � ℓ ( Q ) � M ( |∇ f | ) � L p , ∞ Q , w � ℓ ( Q )[ w ] A p �∇ f � L p Q ( w ) Ezequiel Rela Poincar´ e-Sobolev-RdF

Fractional integrals and Poincar´ e Truncation or weak implies strong lemma: Lemma Let g ≥ 0 , Lipschitz. Suppose a weak (1 , p ) -type estimate for the measures µ, ν and p > 1 : � t µ ( { x ∈ R n : g ( x ) > t } ) 1 / p � sup R n |∇ g ( x ) | d ν t > 0 Then the strong estimate also holds, namely � � g � L p µ � R n |∇ g ( x ) | d ν Ezequiel Rela Poincar´ e-Sobolev-RdF

Fractional integrals and Poincar´ e Theorem Let w ∈ A p , then � 1 � 1 � 1 � 1 � 1 � p p | f − f Q | p w |∇ f | p w p ≤ [ w ] A p ℓ ( Q ) w ( Q ) w ( Q ) Q Q How to deal with higher order Poincar´ e inequality with polynomials? No truncation... | f ( y ) − π Q ( y ) | dy � ℓ ( Q ) m 1 � � |∇ m f | dy | Q | | Q | Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Self improving functionals Starting point � − | f − f Q | d µ ≤ a ( Q ) , a : Q → (0 , ∞ ) Q Ezequiel Rela Poincar´ e-Sobolev-RdF

Self improving functionals Starting point � − | f − f Q | d µ ≤ a ( Q ) , a : Q → (0 , ∞ ) Q Hypothesis on the functional a � a ( P ) p w ( P ) ≤ � a � p a ( Q ) p w ( Q ) P ∈ Λ a ∈ D p ( w ) (1) Ezequiel Rela Poincar´ e-Sobolev-RdF

Self improving functionals Theorem (Franchi-Perez-Wheeden - 1998) Let w ∈ A ∞ and a ∈ D p ( w ) for some p > 0 . Let f such that 1 � | f − f Q | ≤ a ( Q ) . | Q | Q Then � f − f Q � L p , ∞ ≤ C � a � a ( Q ) . w Q , w ( Q ) Only for the weak norm C depends exponentially on [ w ] ∞ . Ezequiel Rela Poincar´ e-Sobolev-RdF

New D p -type condition Small families A family of pairwise disjoint subcubes { Q i } ⊂ D ( Q ) is in S ( L ) , L > 1 if | Q i | ≤ | Q | � L i Smallness preserving functionals a ∈ SD s p ( w ) for 0 ≤ p < ∞ and s > 1 if � p � 1 s � a ( Q i ) p w ( Q i ) ≤ � a � p a ( Q ) p w ( Q ) L i whenever { Q i } ∈ S ( L ) Ezequiel Rela Poincar´ e-Sobolev-RdF

Main Theorem Theorem (A) Let w be any weight, p ≥ 1 , s > 1 and a ∈ SD s p ( w ) . If 1 � | f − f Q | ≤ a ( Q ) , | Q | Q then � 1 � 1 � p | f − f Q | p w ≤ C n s � a � s a ( Q ) w ( Q ) Q Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof � | f − f Q | a ∈ SD s Hypothesis: − ≤ 1, p ( w ) a ( Q ) Q Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof � | f − f Q | a ∈ SD s Hypothesis: − ≤ 1, p ( w ) a ( Q ) Q Calderon - Zygmund decomposition � � | f − f Q | � � � x ∈ Q : M d Ω L := a ( Q ) χ Q ( x ) > L = Q j Q j � | f − f Q | dy ≤ L 2 n L < − a ( Q ) Q j Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof � | f − f Q | a ∈ SD s Hypothesis: − ≤ 1, p ( w ) a ( Q ) Q Calderon - Zygmund decomposition � � | f − f Q | � � � x ∈ Q : M d Ω L := a ( Q ) χ Q ( x ) > L = Q j Q j � | f − f Q | dy ≤ L 2 n L < − a ( Q ) Q j Key step: Go from ( · ) Q to ( · ) Q j Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof � | f − f Q | a ∈ SD s Hypothesis: − ≤ 1, p ( w ) a ( Q ) Q Calderon - Zygmund decomposition � � | f − f Q | � � � x ∈ Q : M d Ω L := a ( Q ) χ Q ( x ) > L = Q j Q j � | f − f Q | dy ≤ L 2 n L < − a ( Q ) Q j Key step: Go from ( · ) Q to ( · ) Q j Triangular inequality is not a good idea �� � � � | f − f Q | p wdx ≤ 2 p − 1 | f − f Q j | p wdx + | f Q j − f Q | p wdx Q j Q j Q j Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof Calder´ on - Zygmund decomposition into good and bad parts | g ( x ) | ≤ 2 n L  f − f Q  f ( x ) − f Q i a ( Q ) = g Q + b Q , � b Q ( x ) = χ Q i ( x ) a ( Q )  i 1   � 1 p | f − f Q | p � 1 � 1 � p | b Q j | p wdx ≤ 2 n L + � wdx   a ( Q ) p w ( Q ) w ( Q ) Q Ω L j Ezequiel Rela Poincar´ e-Sobolev-RdF

About the proof � � � p wdx b Q j | p wdx � � � � | ≤ � b Q j Ω L Q i j i p a ( Q i ) p w ( Q i ) � � 1 � f − f Q i � � � = wdx � � a ( Q ) p w ( Q i ) a ( Q i ) � � Q i i X p � a ( Q i ) p w ( Q i ) , ≤ a ( Q ) p i where X is the quantity defined by p � 1 / p � � � 1 � f − f Q � � X = sup wdx . � � w ( Q ) a ( Q ) � � Q Q Ezequiel Rela Poincar´ e-Sobolev-RdF