The A class of weights and some of its extensions Carlos P erez - PowerPoint PPT Presentation

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm If 1 ≤ q < ∞ , � Tf � L 1 ( w ) � [ w ] A q � Mf � L 1 ( w )

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm If 1 ≤ q < ∞ , � Tf � L 1 ( w ) � [ w ] A q � Mf � L 1 ( w ) • There is a much better result:

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm If 1 ≤ q < ∞ , � Tf � L 1 ( w ) � [ w ] A q � Mf � L 1 ( w ) • There is a much better result: Thm If p ∈ (0 , ∞ ) , � Tf � L p ( w ) � max { 1 , p } [ w ] A ∞ � Mf � L p ( w )

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm If 1 ≤ q < ∞ , � Tf � L 1 ( w ) � [ w ] A q � Mf � L 1 ( w ) • There is a much better result: Thm If p ∈ (0 , ∞ ) , � Tf � L p ( w ) � max { 1 , p } [ w ] A ∞ � Mf � L p ( w ) Recall, we are using here the following constant:

The C p class of weights cperez@bcamath.org Quantitative versions of the A ∞ thm If 1 ≤ q < ∞ , � Tf � L 1 ( w ) � [ w ] A q � Mf � L 1 ( w ) • There is a much better result: Thm If p ∈ (0 , ∞ ) , � Tf � L p ( w ) � max { 1 , p } [ w ] A ∞ � Mf � L p ( w ) Recall, we are using here the following constant: 1 � [ w ] A ∞ = sup Q M ( wχ Q ) dx w ( Q ) Q

The C p class of weights cperez@bcamath.org Key points

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then 1 � � 1 ≤ 2 1+ δ � � Q w 1+ δ Q w | Q | | Q | where

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then 1 � � 1 ≤ 2 1+ δ � � Q w 1+ δ Q w | Q | | Q | where 1 δ = c n [ w ] A ∞

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then 1 � � 1 ≤ 2 1+ δ � � Q w 1+ δ Q w | Q | | Q | where 1 δ = c n [ w ] A ∞ • 2) The local exponential decay

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then 1 � � 1 ≤ 2 1+ δ � � Q w 1+ δ Q w | Q | | Q | where 1 δ = c n [ w ] A ∞ • 2) The local exponential decay � � �� y ∈ Q : | Tf ( y ) | > 2 t, Mf ( y ) ≤ t ε � � � � ≤ cε | Q |

The C p class of weights cperez@bcamath.org Key points • 1) The quantitative RHI Thm T. Hyt¨ onen and C. P. Let w ∈ A ∞ , then 1 � � 1 ≤ 2 1+ δ � � Q w 1+ δ Q w | Q | | Q | where 1 δ = c n [ w ] A ∞ • 2) The local exponential decay � � �� y ∈ Q : | Tf ( y ) | > 2 t, Mf ( y ) ≤ t ε � � � � ≤ cε | Q | ≤ c e − c ε

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A 1 . a) Let 1 < p < ∞ . Then � T � L p ( w ) ≤ c pp ′ [ w ] A 1

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A 1 . a) Let 1 < p < ∞ . Then � T � L p ( w ) ≤ c pp ′ [ w ] A 1 b) � T � L 1 ( w ) → L 1 , ∞ ( w ) ≤ c [ w ] A 1 log( e + [ w ] A 1 )

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A 1 . a) Let 1 < p < ∞ . Then � T � L p ( w ) ≤ c pp ′ [ w ] A 1 b) � T � L 1 ( w ) → L 1 , ∞ ( w ) ≤ c [ w ] A 1 log( e + [ w ] A 1 ) • We thought that the correct result was linear, but it is false .

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A 1 . a) Let 1 < p < ∞ . Then � T � L p ( w ) ≤ c pp ′ [ w ] A 1 b) � T � L 1 ( w ) → L 1 , ∞ ( w ) ≤ c [ w ] A 1 log( e + [ w ] A 1 ) • We thought that the correct result was linear, but it is false . • Adam Ose ¸kowski found a different interesting argument

The C p class of weights cperez@bcamath.org More consequences: the A 1 theory • w ∈ A 1 if M ( w ) ≤ [ w ] A 1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A 1 . a) Let 1 < p < ∞ . Then � T � L p ( w ) ≤ c pp ′ [ w ] A 1 b) � T � L 1 ( w ) → L 1 , ∞ ( w ) ≤ c [ w ] A 1 log( e + [ w ] A 1 ) • We thought that the correct result was linear, but it is false . • Adam Ose ¸kowski found a different interesting argument • Lerner-Nazarov-Ombrosi: the result is sharp.

The C p class of weights cperez@bcamath.org More praises:

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j and � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p, ∞ ( w ) � L p, ∞ ( w ) � � j j

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j and � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p, ∞ ( w ) � L p, ∞ ( w ) � � j j • 2) Sawyer’s problem where one of the key results is

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j and � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p, ∞ ( w ) � L p, ∞ ( w ) � � j j • 2) Sawyer’s problem where one of the key results is Let u ∈ A 1 ( R n ) and v ∈ A ∞ ( R n ) . Then Thm

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j and � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p, ∞ ( w ) � L p, ∞ ( w ) � � j j • 2) Sawyer’s problem where one of the key results is Let u ∈ A 1 ( R n ) and v ∈ A ∞ ( R n ) . Then Thm � T ∗ ( fv ) � M ( fv ) � � � � � L 1 , ∞ ( uv ) ≤ c � � � � � L 1 , ∞ ( uv ) v v

The C p class of weights cperez@bcamath.org More praises: • 1) Vector-valued extensions Thm Let p, q ∈ (0 , ∞ ) and w ∈ A ∞ . Then � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p ( w ) � L p ( w ) � � j j and � 1 � 1 � � � � � � � � q q � � � � ( Tf j ) q ( Mf j ) q ≤ C � � � � � � � � � L p, ∞ ( w ) � L p, ∞ ( w ) � � j j • 2) Sawyer’s problem where one of the key results is Let u ∈ A 1 ( R n ) and v ∈ A ∞ ( R n ) . Then Thm � T ∗ ( fv ) � M ( fv ) � � � � � L 1 , ∞ ( uv ) ≤ c � � � � � L 1 , ∞ ( uv ) v v • (work with D. Cruz-Uribe, JM Martell).

The C p class of weights cperez@bcamath.org The C p condition

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w )

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation:

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p :

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q |

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q | • Compare with the A ∞ condition:

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q | � δ � | E | • Compare with the A ∞ condition: w ( E ) ≤ c w ( Q ) E ⊂ Q | Q |

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q | � δ � | E | • Compare with the A ∞ condition: w ( E ) ≤ c w ( Q ) E ⊂ Q | Q | • Hence:

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q | � δ � | E | • Compare with the A ∞ condition: w ( E ) ≤ c w ( Q ) E ⊂ Q | Q | • Hence: A ∞ ⊂ C p

The C p class of weights cperez@bcamath.org The C p condition Recall the A ∞ theorem � T ∗ f � L p ( w ) ≤ c � Mf � L p ( w ) • Key observation: If p > 1 , Muckenhoupt proved that then w ∈ C p : Definition w is in the C p class if there are constants c, δ > 0 such that � δ � � | E | R n ( Mχ Q ( x )) p w ( x ) dx w ( E ) ≤ C E ⊂ Q | Q | � δ � | E | • Compare with the A ∞ condition: w ( E ) ≤ c w ( Q ) E ⊂ Q | Q | • Hence: A ∞ ⊂ C p • Open problem, Is the C p condition sufficient?

The C p class of weights cperez@bcamath.org The C p theorems

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w )

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof • There is another interesting related result

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof • There is another interesting related result If p ∈ (1 , ∞ ) and w ∈ C p + ǫ Thm (K. Yabuta, 1990)

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof • There is another interesting related result If p ∈ (1 , ∞ ) and w ∈ C p + ǫ Thm (K. Yabuta, 1990) � Mf � L p ( w ) ≤ c � M # f � L p ( w )

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof • There is another interesting related result If p ∈ (1 , ∞ ) and w ∈ C p + ǫ Thm (K. Yabuta, 1990) � Mf � L p ( w ) ≤ c � M # f � L p ( w ) • Recall that

The C p class of weights cperez@bcamath.org The C p theorems Thm (E. Sawyer, 1984) If p ∈ (1 , ∞ ) and w ∈ C p + ǫ � Tf � L p ( w ) ≤ c � Mf � L p ( w ) • The proof is a sophisticated version of Coifman-Fefferman’s A ∞ ’s proof • There is another interesting related result If p ∈ (1 , ∞ ) and w ∈ C p + ǫ Thm (K. Yabuta, 1990) � Mf � L p ( w ) ≤ c � M # f � L p ( w ) M # f ( x ) = sup x ∈ Q 1 • Recall that � Q | f − f Q | | Q |

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li)

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ � Tf � L p ( w ) ≤ c T,p,ǫ � Mf � L p ( w )

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ � Tf � L p ( w ) ≤ c T,p,ǫ � Mf � L p ( w ) • The case of multilinear Calder´ on-Zygmund operators we obtained results

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ � Tf � L p ( w ) ≤ c T,p,ǫ � Mf � L p ( w ) • The case of multilinear Calder´ on-Zygmund operators we obtained results Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,mp } + ǫ

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ � Tf � L p ( w ) ≤ c T,p,ǫ � Mf � L p ( w ) • The case of multilinear Calder´ on-Zygmund operators we obtained results Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,mp } + ǫ � T ( � f ) � L p ( w ) ≤ c �M ( � f ) � L p ( w )

The C p class of weights cperez@bcamath.org Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,p } + ǫ � Tf � L p ( w ) ≤ c T,p,ǫ � Mf � L p ( w ) • The case of multilinear Calder´ on-Zygmund operators we obtained results Thm Let p ∈ (0 , ∞ ) and w ∈ C max { 1 ,mp } + ǫ � T ( � f ) � L p ( w ) ≤ c �M ( � f ) � L p ( w ) • Key point: the following pointwise inequality