Two weight L 2 inequality for the Hilbert transform Eric T. Sawyer - PowerPoint PPT Presentation

Toward a geometric characterization The one weight inequality for the maximal function In 1972 B. Muckenhoupt showed that the ‘poor cousin’ maximal function De…nition (maximal function) Z 1 Mf ( x ) � sup Q j f ( y ) j dy , j Q j intervals Q : x 2 Q Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 7 / 49

Toward a geometric characterization The one weight inequality for the maximal function In 1972 B. Muckenhoupt showed that the ‘poor cousin’ maximal function De…nition (maximal function) Z 1 Mf ( x ) � sup Q j f ( y ) j dy , j Q j intervals Q : x 2 Q satis…es the L 2 weighted norm inequality with weight w , Z Z Mf ( x ) 2 w ( x ) dx � C j f ( x ) j 2 w ( x ) dx , Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 7 / 49

Toward a geometric characterization The one weight inequality for the maximal function In 1972 B. Muckenhoupt showed that the ‘poor cousin’ maximal function De…nition (maximal function) Z 1 Mf ( x ) � sup Q j f ( y ) j dy , j Q j intervals Q : x 2 Q satis…es the L 2 weighted norm inequality with weight w , Z Z Mf ( x ) 2 w ( x ) dx � C j f ( x ) j 2 w ( x ) dx , if and only if w satis…es the ‘ A 2 condition’ Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 7 / 49

Toward a geometric characterization The one weight inequality for the maximal function In 1972 B. Muckenhoupt showed that the ‘poor cousin’ maximal function De…nition (maximal function) Z 1 Mf ( x ) � sup Q j f ( y ) j dy , j Q j intervals Q : x 2 Q satis…es the L 2 weighted norm inequality with weight w , Z Z Mf ( x ) 2 w ( x ) dx � C j f ( x ) j 2 w ( x ) dx , if and only if w satis…es the ‘ A 2 condition’ De…nition ( A 2 condition) � 1 � � 1 � Z Z 1 Q w ( y ) dy � C . w ( y ) dy j Q j j Q j Q Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 7 / 49

The one weight inequality for the Hilbert transform In 1973 R. Hunt, B. Muckenhoupt and R. L. Wheeden showed that De…nition (Hilbert transform) Z ∞ f ( x � y ) Hf ( x ) � p . v . dy , y � ∞ Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 8 / 49

The one weight inequality for the Hilbert transform In 1973 R. Hunt, B. Muckenhoupt and R. L. Wheeden showed that De…nition (Hilbert transform) Z ∞ f ( x � y ) Hf ( x ) � p . v . dy , y � ∞ satis…es the L 2 weighted norm inequality with weight w , Z Z j Hf ( x ) j 2 w ( x ) dx � C j f ( x ) j 2 w ( x ) dx , Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 8 / 49

The one weight inequality for the Hilbert transform In 1973 R. Hunt, B. Muckenhoupt and R. L. Wheeden showed that De…nition (Hilbert transform) Z ∞ f ( x � y ) Hf ( x ) � p . v . dy , y � ∞ satis…es the L 2 weighted norm inequality with weight w , Z Z j Hf ( x ) j 2 w ( x ) dx � C j f ( x ) j 2 w ( x ) dx , if and only if w satis…es the A 2 condition. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 8 / 49

The two weight Hilbert transform inequality a function theoretic characterization analogous to Helson-Szegö In 1979 Cotlar and Sadosky showed that Z Z T j Hf j 2 d ω 1 � A T j f j 2 d ω 2 , f 2 C ∞ ( T ) , if and only if Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 9 / 49

The two weight Hilbert transform inequality a function theoretic characterization analogous to Helson-Szegö In 1979 Cotlar and Sadosky showed that Z Z T j Hf j 2 d ω 1 � A T j f j 2 d ω 2 , f 2 C ∞ ( T ) , if and only if d ω 1 � d θ , d ω 1 � Ad ω 2 , and there exists a holomorphic function h 2 H 1 ( D ) , i.e. Z � re i θ �� � � � k h k H 1 ( D ) � sup � h � d θ < ∞ , T 0 < r < 1 such that j Ad ω 2 + d ω 1 � hd θ j � j Ad ω 2 � d ω 1 j . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 9 / 49

Toward a geometric characterization The two weight inequality for the maximal function In 1981 Sawyer showed that the maximal function Mf satis…es the L 2 two weight norm inequality with weight pair ( ω , σ ) , Z Z M ( f σ ) ( x ) 2 d ω ( x ) � C j f ( x ) j 2 d σ ( x ) , (in the one weight setting σ � ω � 1 ) Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 10 / 49

Toward a geometric characterization The two weight inequality for the maximal function In 1981 Sawyer showed that the maximal function Mf satis…es the L 2 two weight norm inequality with weight pair ( ω , σ ) , Z Z M ( f σ ) ( x ) 2 d ω ( x ) � C j f ( x ) j 2 d σ ( x ) , (in the one weight setting σ � ω � 1 ) if and only if the pair of weights ( ω , σ ) satis…es the testing condition: Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 10 / 49

Toward a geometric characterization The two weight inequality for the maximal function In 1981 Sawyer showed that the maximal function Mf satis…es the L 2 two weight norm inequality with weight pair ( ω , σ ) , Z Z M ( f σ ) ( x ) 2 d ω ( x ) � C j f ( x ) j 2 d σ ( x ) , (in the one weight setting σ � ω � 1 ) if and only if the pair of weights ( ω , σ ) satis…es the testing condition: De…nition (maximal testing condition) Z � � ( x ) 2 d ω ( x ) � C j Q j σ . Q M χ Q σ Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 10 / 49

Toward a geometric characterization The two weight inequality for fractional and Poisson integrals In 1986 Sawyer showed that Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 11 / 49

Toward a geometric characterization The two weight inequality for fractional and Poisson integrals In 1986 Sawyer showed that De…nition (fractional integral) Z R n j x � y j α � n f ( y ) dy I α f ( x ) � Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 11 / 49

Toward a geometric characterization The two weight inequality for fractional and Poisson integrals In 1986 Sawyer showed that De…nition (fractional integral) Z R n j x � y j α � n f ( y ) dy I α f ( x ) � satis…es the two weight norm inequality Z Z j I α ( f σ ) j 2 d ω � C j f j 2 d σ Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 11 / 49

Toward a geometric characterization The two weight inequality for fractional and Poisson integrals In 1986 Sawyer showed that De…nition (fractional integral) Z R n j x � y j α � n f ( y ) dy I α f ( x ) � satis…es the two weight norm inequality Z Z j I α ( f σ ) j 2 d ω � C j f j 2 d σ if and only if the following two testing conditions hold: Z Z � � 2 d ω � C j Q j σ and � � 2 d σ � C j Q j ω . Q I α χ Q σ Q I α χ Q ω Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 11 / 49

Toward a geometric characterization The two weight inequality for fractional and Poisson integrals In 1986 Sawyer showed that De…nition (fractional integral) Z R n j x � y j α � n f ( y ) dy I α f ( x ) � satis…es the two weight norm inequality Z Z j I α ( f σ ) j 2 d ω � C j f j 2 d σ if and only if the following two testing conditions hold: Z Z � � 2 d ω � C j Q j σ and � � 2 d σ � C j Q j ω . Q I α χ Q σ Q I α χ Q ω and a similar result for the Poisson integral Z t P f ( x , t ) = t 2 + x 2 f ( t ) dt . R Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 11 / 49

Toward a geometric characterization The T1 theorem for Calderón-Zygmund kernels In 1984 David and Journé showed that if K ( x , y ) is a standard kernel on R n , C j x � y j � n , j K ( x , y ) j � � j x 0 � x j � δ � � � � K ( x , y ) � � K � + ... C j x � y j � n x 0 , y � , j x � y j Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 12 / 49

Toward a geometric characterization The T1 theorem for Calderón-Zygmund kernels In 1984 David and Journé showed that if K ( x , y ) is a standard kernel on R n , C j x � y j � n , j K ( x , y ) j � � j x 0 � x j � δ � � � � K ( x , y ) � � K � + ... C j x � y j � n x 0 , y � , j x � y j and if Tf ( x ) � R R n K ( x , y ) f ( y ) dy for x / 2 supp f , then T is bounded on L 2 ( R n ) if and only if T 2 WBP and Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 12 / 49

Toward a geometric characterization The T1 theorem for Calderón-Zygmund kernels In 1984 David and Journé showed that if K ( x , y ) is a standard kernel on R n , C j x � y j � n , j K ( x , y ) j � � j x 0 � x j � δ � � � � K ( x , y ) � � K � + ... C j x � y j � n x 0 , y � , j x � y j and if Tf ( x ) � R R n K ( x , y ) f ( y ) dy for x / 2 supp f , then T is bounded on L 2 ( R n ) if and only if T 2 WBP and De…nition ( T 1 or testing conditions) � � Z � � � 2 � C j Q j � T χ Q T 1 2 BMO , , Q � � Z � � � 2 � C j Q j � T � χ Q T � 1 2 BMO , . Q Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 12 / 49

Toward a geometric characterization In 2004 Nazarov, Treil and Volberg showed that if a weight pair ( ω , σ ) satis…es the pivotal condition Z ∞ j I j j I r j ω P ( I r , χ I 0 σ ) 2 � P 2 ∑ � j I 0 j σ ; P ( I , ν ) = j I j 2 + x 2 d ν ( x ) , r = 1 for all decompositions of an interval I 0 into subintervals I r , 1.0 y -4 -2 0 2 4 x Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 13 / 49

Toward a geometric characterization In 2004 Nazarov, Treil and Volberg showed that if a weight pair ( ω , σ ) satis…es the pivotal condition Z ∞ j I j j I r j ω P ( I r , χ I 0 σ ) 2 � P 2 ∑ � j I 0 j σ ; P ( I , ν ) = j I j 2 + x 2 d ν ( x ) , r = 1 for all decompositions of an interval I 0 into subintervals I r , 1.0 y -4 -2 0 2 4 x then the Hilbert transform H satis…es the two weight L 2 inequality Z Z j H ( f σ ) j 2 d ω � C j f j 2 d σ , uniformly for all smooth truncations of the Hilbert transform, Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 13 / 49

Toward a geometric characterization The NTV conditions if and only if the weight pair ( ω , σ ) satis…es Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 14 / 49

Toward a geometric characterization The NTV conditions if and only if the weight pair ( ω , σ ) satis…es De…nition ( A 2 condition on steroids) P ( I , ω ) � P ( I , σ ) � A 2 sup 2 < ∞ , I Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 14 / 49

Toward a geometric characterization The NTV conditions if and only if the weight pair ( ω , σ ) satis…es De…nition ( A 2 condition on steroids) P ( I , ω ) � P ( I , σ ) � A 2 sup 2 < ∞ , I as well as the two interval testing conditions Z I j H ( χ I σ ) j 2 d ω T 2 j I j σ , � Z ( T � ) 2 j I j ω . I j H ( χ I ω ) j 2 d σ � Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 14 / 49

Maximal inequalities and doubling Nazarov, Treil and Volberg showed that the pivotal conditions are implied by the boundedness of the maximal operator and its ‘dual’: M : L 2 ( σ ) ! L 2 ( ω ) and M : L 2 ( ω ) ! L 2 ( σ ) . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 15 / 49

Maximal inequalities and doubling Nazarov, Treil and Volberg showed that the pivotal conditions are implied by the boundedness of the maximal operator and its ‘dual’: M : L 2 ( σ ) ! L 2 ( ω ) and M : L 2 ( ω ) ! L 2 ( σ ) . They also showed that the pivotal conditions are implied by the testing conditions and the A 2 condition if the measures σ and ω are both doubling: Z Z Z Z 2 Q d σ . 2 Q d ω . Q d σ and Q d ω for all intervals Q . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 15 / 49

The role of cancellation In 2002 Nazarov showed that the strengthened A 2 condition alone is not enough for the two weight inequality to hold. The reason for the failure lies in the fact that this condition is a consequence solely of the size and smoothness of the kernel. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 16 / 49

The role of cancellation In 2002 Nazarov showed that the strengthened A 2 condition alone is not enough for the two weight inequality to hold. The reason for the failure lies in the fact that this condition is a consequence solely of the size and smoothness of the kernel. Indeed, strengthened A 2 follows from the ‘kernel’ inequality tested j I j over f ( y ) = 1 ( a � r , a ) ( y ) j I j + j y � x I j : Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 16 / 49

The role of cancellation In 2002 Nazarov showed that the strengthened A 2 condition alone is not enough for the two weight inequality to hold. The reason for the failure lies in the fact that this condition is a consequence solely of the size and smoothness of the kernel. Indeed, strengthened A 2 follows from the ‘kernel’ inequality tested j I j over f ( y ) = 1 ( a � r , a ) ( y ) j I j + j y � x I j : De…nition (kernel inequality) Z Z R n support f j H ( f σ ) j 2 d ω . N 2 R j f j 2 d σ Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 16 / 49

The role of cancellation In 2002 Nazarov showed that the strengthened A 2 condition alone is not enough for the two weight inequality to hold. The reason for the failure lies in the fact that this condition is a consequence solely of the size and smoothness of the kernel. Indeed, strengthened A 2 follows from the ‘kernel’ inequality tested j I j over f ( y ) = 1 ( a � r , a ) ( y ) j I j + j y � x I j : De…nition (kernel inequality) Z Z R n support f j H ( f σ ) j 2 d ω . N 2 R j f j 2 d σ It is the pair of testing conditions that encode the cancellation required for the L 2 norm inequality. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 16 / 49

Energy and hybrid conditions Two years ago, Lacey Sawyer and Uriarte-Tuero showed that the pivotal conditions are not necessary, that the following energy condition is, � 2 ! 1 / 2 � j x � x 0 j E ω ( dx 0 ) E ω ( dx ) E ( I , ω ) � , I I j I j ∞ ω ( I r )[ E ( I r , ω ) P ( I r , χ I 0 σ )] 2 � E 2 σ ( I 0 ) , ∑ r = 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 17 / 49

Energy and hybrid conditions Two years ago, Lacey Sawyer and Uriarte-Tuero showed that the pivotal conditions are not necessary, that the following energy condition is, � 2 ! 1 / 2 � j x � x 0 j E ω ( dx 0 ) E ω ( dx ) E ( I , ω ) � , I I j I j ∞ ω ( I r )[ E ( I r , ω ) P ( I r , χ I 0 σ )] 2 � E 2 σ ( I 0 ) , ∑ r = 1 and that the following hybrid condition is ‘su¢cient’ for 0 � γ < 1 (but still not necessary): ∞ ω ( I r )[ E ( I r , ω ) γ P ( I r , χ I 0 σ )] 2 � E 2 ∑ γ σ ( I 0 ) , r = 1 for all intervals I 0 , and decompositions f I r : r � 1 g of I 0 into disjoint intervals I r ( I 0 . Note that for γ = 0 this is the pivotal condition, while for γ = 1 it is the energy condition. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 17 / 49

Bounded ‡uctuation characterization Last year Lacey Sawyer Shen and Uriarte-Tuero showed the Hilbert transform two weight inequality is equivalent to the A 2 condition and the bounded ‡uctuation conditions taken over all dyadic grids D : � � Z Z I H ( 1 I f σ ) 2 d ω I j f j 2 d σ � C j I j σ + , (1) � � Z Z I H ( 1 I g ω ) 2 d σ I j g j 2 d ω � C j I j ω + , for all I 2 D and all functions f , g of unit D -‡uctuation on I . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 18 / 49

Bounded ‡uctuation characterization Last year Lacey Sawyer Shen and Uriarte-Tuero showed the Hilbert transform two weight inequality is equivalent to the A 2 condition and the bounded ‡uctuation conditions taken over all dyadic grids D : � � Z Z I H ( 1 I f σ ) 2 d ω I j f j 2 d σ � C j I j σ + , (1) � � Z Z I H ( 1 I g ω ) 2 d σ I j g j 2 d ω � C j I j ω + , for all I 2 D and all functions f , g of unit D -‡uctuation on I . A function f 2 L 2 ( σ ) is of unit D - ‡uctuation on I, written R 1 f 2 BF σ ( I ) , if it is supported in I and K j f j d σ � 1 for all j K j σ dyadic subintervals K of I on which f is not constant. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 18 / 49

Bounded ‡uctuation characterization Last year Lacey Sawyer Shen and Uriarte-Tuero showed the Hilbert transform two weight inequality is equivalent to the A 2 condition and the bounded ‡uctuation conditions taken over all dyadic grids D : � � Z Z I H ( 1 I f σ ) 2 d ω I j f j 2 d σ � C j I j σ + , (1) � � Z Z I H ( 1 I g ω ) 2 d σ I j g j 2 d ω � C j I j ω + , for all I 2 D and all functions f , g of unit D -‡uctuation on I . A function f 2 L 2 ( σ ) is of unit D - ‡uctuation on I, written R 1 f 2 BF σ ( I ) , if it is supported in I and K j f j d σ � 1 for all j K j σ dyadic subintervals K of I on which f is not constant. Such functions are special cases of dyadic BMO D ( σ ) functions of norm 1, and include functions bounded by 1 in modulus. They arise as the good functions in a Calderón-Zygmund decomposition. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 18 / 49

The Nazarov Treil Volberg conjecture A question raised in Volberg’s 2003 CBMS book, which we refer to as the NTV conjecture , is whether or not Z Z R j H ( f σ ) j 2 ω � N R j f j 2 σ , f 2 L 2 ( σ ) , (2) is equivalent to the A 2 condition and the two interval testing conditions. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 19 / 49

The Nazarov Treil Volberg conjecture A question raised in Volberg’s 2003 CBMS book, which we refer to as the NTV conjecture , is whether or not Z Z R j H ( f σ ) j 2 ω � N R j f j 2 σ , f 2 L 2 ( σ ) , (2) is equivalent to the A 2 condition and the two interval testing conditions. A weaker conjecture, that we refer to as the indicator/interval NTV conjecture, is that (2) is equivalent to the A 2 condition and the two indicator/interval testing conditions, Z Z I j H ( 1 E σ ) j 2 ω � A j I j σ , I j H ( 1 E ω ) j 2 σ � A � j I j ω , (3) for all intervals I and closed subsets E of I . Note that E does not appear on the right side of these inequalities, and that if H were a positive operator we could take E = I . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 19 / 49

Our characterization of the two weight Hilbert transform inequality The indicator/interval NTV conjecture Theorem The best constant N in the two weight inequality (2) for the Hilbert transform satis…es p A 2 + A + A � , N � i.e. H σ is bounded from L 2 ( σ ) to L 2 ( ω ) if and only if the strong A 2 and indicator/interval testing conditions hold. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 20 / 49

Our characterization of the two weight Hilbert transform inequality The indicator/interval NTV conjecture Theorem The best constant N in the two weight inequality (2) for the Hilbert transform satis…es p A 2 + A + A � , N � i.e. H σ is bounded from L 2 ( σ ) to L 2 ( ω ) if and only if the strong A 2 and indicator/interval testing conditions hold. Corollary The Hilbert transform H σ is bounded from L 2 ( σ ) to L 2 ( ω ) if and only if both it and its dual H ω are weak type ( 2 , 2 ) , i.e. Z Z λ 2 jfj H σ f j > λ gj ω . j f j 2 d σ and λ 2 jfj H ω g j > λ gj σ . j g j 2 d ω . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 20 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Restricted bounded ‡uctuation 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Restricted bounded ‡uctuation 1 Minimal bounded ‡uctuation 2 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Restricted bounded ‡uctuation 1 Minimal bounded ‡uctuation 2 The far term 5 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Restricted bounded ‡uctuation 1 Minimal bounded ‡uctuation 2 The far term 5 The functional energy inequality 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Outline of Part II: the proof of the theorem The Haar decomposition 1 The random grids of NTV 1 Interval size splitting of the bilinear form 2 Triple corona decomposition of the functions 2 Parallel corona splitting of the bilinear form 3 The near term 4 Restricted bounded ‡uctuation 1 Minimal bounded ‡uctuation 2 The far term 5 The functional energy inequality 1 The two weight norm inequality for the Poisson operator 2 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 21 / 49

Haar functions adapted to a measure The Haar function h σ I adapted to a positive measure σ and a dyadic interval I 2 D is a positive (negative) constant on the left (right) child, has vanishing mean R h σ I d σ = 0, and is normalized 1 1 k h σ I k L 2 ( σ ) = 1. For example if j [ 2 , 3 ] j σ = 15 and j [ 3 , 4 ] j σ = 10 , then 4 y 2 -1 1 2 3 4 5 -2 The Haar function h σ [ 2 , 4 ] Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 22 / 49

Haar functions adapted to a measure The Haar function h σ I adapted to a positive measure σ and a dyadic interval I 2 D is a positive (negative) constant on the left (right) child, has vanishing mean R h σ I d σ = 0, and is normalized 1 1 k h σ I k L 2 ( σ ) = 1. For example if j [ 2 , 3 ] j σ = 15 and j [ 3 , 4 ] j σ = 10 , then 4 y 2 -1 1 2 3 4 5 -2 The Haar function h σ [ 2 , 4 ] The supremum norm of h σ I is quite large if σ is very unbalanced (not doubling). Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 22 / 49

The good dyadic grids of NTV For any β = f β l g 2 f 0 , 1 g Z , de…ne the dyadic grid D β to be the collection of intervals ( !) [ 0 , 1 ) + k + ∑ 2 n 2 i � n β i D β = i < n n 2 Z , k 2 Z and place the usual uniform probability measure P on the space f 0 , 1 g Z . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 23 / 49

The good dyadic grids of NTV For any β = f β l g 2 f 0 , 1 g Z , de…ne the dyadic grid D β to be the collection of intervals ( !) [ 0 , 1 ) + k + ∑ 2 n 2 i � n β i D β = i < n n 2 Z , k 2 Z and place the usual uniform probability measure P on the space f 0 , 1 g Z . For weights ω and σ , consider random choices of dyadic grids D ω and D σ . Fix ε > 0 and for a positive integer r , an interval J 2 D ω is said to be r-bad if there is an interval I 2 D σ with j I j � 2 r j J j , and 2 j J j ε j I j 1 � ε . dist ( e ( I ) , J ) � 1 where e ( I ) is the set of the three discontinuities of h σ I . Otherwise, J is said to be r-good . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 23 / 49

The good dyadic grids of NTV For any β = f β l g 2 f 0 , 1 g Z , de…ne the dyadic grid D β to be the collection of intervals ( !) [ 0 , 1 ) + k + ∑ 2 n 2 i � n β i D β = i < n n 2 Z , k 2 Z and place the usual uniform probability measure P on the space f 0 , 1 g Z . For weights ω and σ , consider random choices of dyadic grids D ω and D σ . Fix ε > 0 and for a positive integer r , an interval J 2 D ω is said to be r-bad if there is an interval I 2 D σ with j I j � 2 r j J j , and 2 j J j ε j I j 1 � ε . dist ( e ( I ) , J ) � 1 where e ( I ) is the set of the three discontinuities of h σ I . Otherwise, J is said to be r-good . We have P ( J is r -bad ) � C 2 � ε r . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 23 / 49

Reduction to good projections Let D σ be randomly selected with parameter β , and D ω with parameter β 0 . De…ne a projection P σ ∑ ∆ σ good f � I f , I is r -good 2D σ and likewise for P ω good g . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 24 / 49

Reduction to good projections Let D σ be randomly selected with parameter β , and D ω with parameter β 0 . De…ne a projection P σ ∑ ∆ σ good f � I f , I is r -good 2D σ and likewise for P ω good g . De…ne P σ bad f � f � P σ good f . Then bad f k L 2 ( σ ) � C 2 � ε r E β 0 k P σ 2 k f k L 2 ( σ ) . and likewise for P ω bad g . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 24 / 49

Reduction to good projections Let D σ be randomly selected with parameter β , and D ω with parameter β 0 . De…ne a projection P σ ∑ ∆ σ good f � I f , I is r -good 2D σ and likewise for P ω good g . De…ne P σ bad f � f � P σ good f . Then bad f k L 2 ( σ ) � C 2 � ε r E β 0 k P σ 2 k f k L 2 ( σ ) . and likewise for P ω bad g . There is an absolute choice of r so that if T : L 2 ( σ ) ! L 2 ( ω ) is a bounded linear operator, then � � T P σ good f , P ω k T k L 2 ( σ ) ! L 2 ( ω ) � 2 sup sup E β E β 0 j good g ω j . k f k L 2 ( σ ) = 1 k g k L 2 ( ω ) = 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 24 / 49

The Haar expansion Let D σ and D ω be an r -good pair of grids, and let f h σ I g I 2D σ and f h ω J g J 2D ω be the corresponding Haar bases, so that I f = ∑ I = ∑ b ∑ 4 σ h f , h σ I i h σ f ( I ) h σ f = I , I 2D σ I 2D σ I 2D σ J g = ∑ J = ∑ 4 ω h g , h ω J i h ω g ( J ) h ω ∑ = b g J , J 2D ω J 2D ω J 2D ω where the appropriate grid is understood in the notation b f ( I ) and b g ( J ) . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 25 / 49

The Haar expansion Let D σ and D ω be an r -good pair of grids, and let f h σ I g I 2D σ and f h ω J g J 2D ω be the corresponding Haar bases, so that I f = ∑ I = ∑ b ∑ 4 σ h f , h σ I i h σ f ( I ) h σ f = I , I 2D σ I 2D σ I 2D σ J g = ∑ J = ∑ 4 ω h g , h ω J i h ω g ( J ) h ω ∑ = b g J , J 2D ω J 2D ω J 2D ω where the appropriate grid is understood in the notation b f ( I ) and b g ( J ) . Inequality (2) is equivalent to boundedness of the bilinear form h H ( σ 4 σ I f ) , 4 ω ∑ H ( f , g ) � h H ( f σ ) , g i ω = J g i ω I 2D σ and J 2D ω on L 2 ( σ ) � L 2 ( ω ) , i.e. jH ( f , g ) j � N k f k L 2 ( σ ) k g k L 2 ( ω ) . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 25 / 49

Splitting of the form by interval size Virtually all attacks on the two weight inequality (2) to date have proceeded by …rst splitting the bilinear form H into three natural forms determined by the relative size of the intervals I and J in the inner product: H = H lower + H diagonal + H upper ; (4) ∑ h H ( σ 4 σ I f ) , 4 ω H lower ( f , g ) � J g i ω , I 2D σ and J 2D ω j J j < 2 � r j I j h H ( σ 4 σ I f ) , 4 ω ∑ H diagonal ( f , g ) � J g i ω , I 2D σ and J 2D ω 2 � r j I j�j J j� 2 r j I j h H ( σ 4 σ I f ) , 4 ω ∑ H upper ( f , g ) � J g i ω , I 2D σ and J 2D ω j J j > 2 r j I j and then continuing to establish boundedness of each of these three forms. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 26 / 49

Boundedness of the split forms Now the boundedness of the diagonal form H diagonal is an automatic consequence of that of H since it is shown by NTV that �p A 2 + T + T � � . jH diagonal ( f , g ) j k f k L 2 ( σ ) k g k L 2 ( ω ) . N k f k L 2 ( σ ) k g k L 2 ( ω ) . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 27 / 49

Boundedness of the split forms Now the boundedness of the diagonal form H diagonal is an automatic consequence of that of H since it is shown by NTV that �p A 2 + T + T � � . jH diagonal ( f , g ) j k f k L 2 ( σ ) k g k L 2 ( ω ) . N k f k L 2 ( σ ) k g k L 2 ( ω ) . However, it is not known if the boundedness of H lower and H upper follow from that of H , which places in jeopardy the entire method of attack based on the splitting (4) of the form H . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 27 / 49

Circumventing the obstacles The triple coronas The triple corona decomposition consists of a series of three reductions performed with two Calderón-Zygmund corona decompositions, followed by an energy corona decomposition, in order to identify the extremal functions that fail to yield to the standard analyses. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 28 / 49

Circumventing the obstacles The triple coronas The triple corona decomposition consists of a series of three reductions performed with two Calderón-Zygmund corona decompositions, followed by an energy corona decomposition, in order to identify the extremal functions that fail to yield to the standard analyses. These extremals are certain bounded functions, and functions of minimal bounded ‡uctuation , occurring in a corona with energy control. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 28 / 49

Circumventing the obstacles The triple coronas The triple corona decomposition consists of a series of three reductions performed with two Calderón-Zygmund corona decompositions, followed by an energy corona decomposition, in order to identify the extremal functions that fail to yield to the standard analyses. These extremals are certain bounded functions, and functions of minimal bounded ‡uctuation , occurring in a corona with energy control. In the end, the standard NTV methodology is, to some extent, decisive when used on these extremal functions with very special structure. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 28 / 49

Circumventing the obstacles Parallel coronas We use parallel corona splittings of the bilinear form, followed by an analysis of the extremal functions that fail both the energy and Calderón-Zygmund stopping time methodology. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 29 / 49

Circumventing the obstacles Parallel coronas We use parallel corona splittings of the bilinear form, followed by an analysis of the extremal functions that fail both the energy and Calderón-Zygmund stopping time methodology. The parallel corona splitting involves de…ning upper and lower and diagonal forms relative to the tree of triple corona stopping time intervals, rather than the full tree of dyadic intervals. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 29 / 49

Circumventing the obstacles Parallel coronas We use parallel corona splittings of the bilinear form, followed by an analysis of the extremal functions that fail both the energy and Calderón-Zygmund stopping time methodology. The parallel corona splitting involves de…ning upper and lower and diagonal forms relative to the tree of triple corona stopping time intervals, rather than the full tree of dyadic intervals. The enemy of Calderón-Zygmund stopping times is degeneracy of the doubling property, while the enemy of energy stopping times is degeneracy of the energy functional (since nondegenerate doubling implies nondegenerate energy, it is really the failure of doubling in both weights that is the common enemy). Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 29 / 49

CZ stopping trees In order to improve on the splitting in (4), we introduce stopping trees F and G for the functions f 2 L 2 ( σ ) and g 2 L 2 ( ω ) . Let F be a collection of Calderón-Zygmund stopping intervals for f , and let [ D σ = C F be the associated corona decomposition of the dyadic F 2F grid D σ . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 30 / 49

CZ stopping trees In order to improve on the splitting in (4), we introduce stopping trees F and G for the functions f 2 L 2 ( σ ) and g 2 L 2 ( ω ) . Let F be a collection of Calderón-Zygmund stopping intervals for f , and let [ D σ = C F be the associated corona decomposition of the dyadic F 2F grid D σ . For I 2 D σ let π D σ I be the D σ -parent of I in the grid D σ , and let π F I be the smallest member of F that contains I . For F , F 0 2 F , we say that F 0 is an F -child of F if π F ( π D σ F 0 ) = F , and we denote by C ( F ) the set of F -children of F . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 30 / 49

CZ stopping trees In order to improve on the splitting in (4), we introduce stopping trees F and G for the functions f 2 L 2 ( σ ) and g 2 L 2 ( ω ) . Let F be a collection of Calderón-Zygmund stopping intervals for f , and let [ D σ = C F be the associated corona decomposition of the dyadic F 2F grid D σ . For I 2 D σ let π D σ I be the D σ -parent of I in the grid D σ , and let π F I be the smallest member of F that contains I . For F , F 0 2 F , we say that F 0 is an F -child of F if π F ( π D σ F 0 ) = F , and we denote by C ( F ) the set of F -children of F . For F 2 F , de…ne the projection P σ C F onto the linear span of the Haar functions f h σ I g I 2C F by C F f = ∑ I f = ∑ f = ∑ P σ 4 σ h f , h σ I i σ h σ P σ I ; C F f , I 2C F I 2C F F 2F Z � � � � L 2 ( σ ) = ∑ � 2 k f k 2 � P σ P σ C F f σ = 0 , C F f L 2 ( σ ) . F 2F Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 30 / 49

The triple corona decomposition We perform a corona decomposition three times on each grid D σ and D ω , improving the upper blocks of functions as follows: Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 31 / 49

The triple corona decomposition We perform a corona decomposition three times on each grid D σ and D ω , improving the upper blocks of functions as follows: P σ C F f is of bounded ‡uctuation after the …rst CZ decomposition, 1 Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 31 / 49

The triple corona decomposition We perform a corona decomposition three times on each grid D σ and D ω , improving the upper blocks of functions as follows: P σ C F f is of bounded ‡uctuation after the …rst CZ decomposition, 1 � � P σ P σ C F f is of minimal bounded ‡uctuation or simply bounded 2 C K appropriately after a complicated second CZ decomposition, Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 31 / 49

The triple corona decomposition We perform a corona decomposition three times on each grid D σ and D ω , improving the upper blocks of functions as follows: P σ C F f is of bounded ‡uctuation after the …rst CZ decomposition, 1 � � P σ P σ C F f is of minimal bounded ‡uctuation or simply bounded 2 C K appropriately after a complicated second CZ decomposition, � � �� P σ P σ P σ C F f is as in step 2 but with additional energy control 3 C S C K after the third energy decomposition, analogous to the pivotal stopping time corona of NTV , but using the necessary energy condition instead. Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 31 / 49

The triple corona decomposition We perform a corona decomposition three times on each grid D σ and D ω , improving the upper blocks of functions as follows: P σ C F f is of bounded ‡uctuation after the …rst CZ decomposition, 1 � � P σ P σ C F f is of minimal bounded ‡uctuation or simply bounded 2 C K appropriately after a complicated second CZ decomposition, � � �� P σ P σ P σ C F f is as in step 2 but with additional energy control 3 C S C K after the third energy decomposition, analogous to the pivotal stopping time corona of NTV , but using the necessary energy condition instead. This is called the triple corona decomposition for f , and there is an analogous decomposition for g . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 31 / 49

The parallel corona splitting Consider the following parallel corona splitting of the inner product h H ( f σ ) , g i ω that involves the projections P σ C F acting on f and the projections P ω C G acting on g . We have � � � � �� σ P σ P ω ∑ h H ( f σ ) , g i ω = H C F f , C G g (5) ω ( F , G ) 2F�G ( ) ∑ ∑ ∑ = + + ( F , G ) 2 Near ( F�G ) ( F , G ) 2 Disjoint ( F�G ) ( F , G ) 2 Far ( F�G ) � � � � �� σ P σ P ω � H C F f , C G g ω � H near ( f , g ) + H disjoint ( f , g ) + H far ( f , g ) . Two weight L 2 inequality E. Sawyer (McMaster University) August 24 2012 32 / 49