Outline Commutators, paraproducts and BMO in non-homogeneous martingale harmonic analysis Sergei Treil Department of Mathematics Brown University August 22, 2012 Abel Symposium Oslo, Norway 1
Outline Main objects 1 Martingale difference decomposition Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO Bounds on paraproducts and commutators 2 Paraproducts Extended paraproducts and commutators Basis properties of martingale difference spaces From dyadic to classical 3 Random dyadic lattices Averaging of dyadic shifts H 1 vs. dyadic H 1 2
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Haar system: For an interval I let h I be the Haar function h I := | I | − 1 / 2 ( 1 I + − 1 I − ) , where I ± are the right and left halves of I respectively. 3
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Haar system: For an interval I let h I be the Haar function h I := | I | − 1 / 2 ( 1 I + − 1 I − ) , where I ± are the right and left halves of I respectively. Dyadic lattice D := { 2 k ( j + [0 , 1)) : j, k ∈ Z } 3
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Haar system: For an interval I let h I be the Haar function h I := | I | − 1 / 2 ( 1 I + − 1 I − ) , where I ± are the right and left halves of I respectively. Dyadic lattice D := { 2 k ( j + [0 , 1)) : j, k ∈ Z } { h I } I ∈D is and orthonormal basis in L 2 ( R ) 3
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Haar system: For an interval I let h I be the Haar function h I := | I | − 1 / 2 ( 1 I + − 1 I − ) , where I ± are the right and left halves of I respectively. Dyadic lattice D := { 2 k ( j + [0 , 1)) : j, k ∈ Z } { h I } I ∈D is and orthonormal basis in L 2 ( R ) It is also an unconditional basis in L p ( R ) , 1 < p < ∞ : � f = � f, h I � h I I ∈D and the series converges unconditionally. 3
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Martingale difference decomposition: � � ˆ | I | − 1 E I F := fdx 1 I I � ∆ I := − E I + E J J ∈ child( I ) � So f = ∆ I f ; I ∈D Note that on R we have ∆ I f = � f, h I � h I ; Advantage of MDD notation: the same notation in R n , where there are 2 n − 1 Haar functions for each cube. Can use arbitrary measure. 4
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Martingale transforms and martingale multipliers R N , Radon measure µ : E Q = E µ Q , ∆ Q = ∆ µ Q are with respect to µ . Q := ∆ Q L p — martingale difference space; D Q = D p Martingale multiplier T α , α = { α Q } Q ∈D , � T α f := α Q ∆ Q f. Q ∈D Martingale transform T is a diagonal operator in the basis { D Q : Q ∈ D} : � Tf = T Q (∆ Q f ) , T Q : D Q → D Q . Q ∈D For doubling µ , T is bounded in L p , 1 < p < ∞ iff T Q are uniformly bounded: not true in general case for p � = 2 . 5
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Paraproducts Let b locally integrable. Let M b be the multiplication operator, M b f = bf . Decompose M b in the basis { D Q : Q ∈ D} (recall D Q = ∆ Q L 2 ), � Tf = ∆ Q M b ∆ R f Q,R ∈D � � � . . . = π b f + Λ b f + π ∗ = . . . + . . . + b f Q � R Q = R R � Q π b f = � Q ∈D (∆ Q b )( E Q f ) — paraproduct Λ b is a martingale transform, commutes with all martingale multipliers. � � b f = � Q ∈D E Q ( b ∆ Q f ) = � π ∗ Q ∈D E Q (∆ Q b )(∆ Q f ) 6
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Paraproducts: another point of view Decompose � bf = (∆ Q b )(∆ R f ) Q,R ∈D � � � b f + π ( ∗ ) . . . = π b f + Λ 0 = . . . + . . . + b f Q � R R � Q Q = R b f = � Λ 0 R ∈D ( E R b )(∆ R f ) — martingale multiplier, commutes with all martingale transforms. b f = � π ( ∗ ) Q ∈D (∆ Q b )(∆ Q f ) � � b f = � (recall that π ∗ Q ∈D E Q (∆ Q b )(∆ Q f ) ). b = π ( ∗ ) For classical Haar system (Lebesgue measure) on R , π ∗ b (because h 2 I ≡ Const on I ). 7
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Paraproducts and commutators M b = π b + π ∗ b + Λ b , where Λ b is a martingale transform, and so commutes with all martingale multipliers. Therefore, if π b is bounded, then [ M b , T ] := M b T − TM b is bounded for any martingale multiplier T = T α M b = π b + π ( ∗ ) + Λ 0 b where Λ 0 b is a martingale multiplier, and so b commutes with all martingale transforms. Therefore, if π b and π ( ∗ ) are bounded, then [ M b , T ] := M b T − TM b b is bounded for any martingale transform T = diag { T Q : Q ∈ D} . In fact, if π ( ∗ ) is bounded, then π b is bounded b � � b = � π ( ∗ ) − π ∗ b = Λ b − Λ 0 Q ∈D ∆ Q (∆ Q b )(∆ Q f ) — one can split b this term between π b and π ∗ b . 8
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Paraproducts and Calder´ on–Zygmund operators For CZO paraproducts catch some hidden oscillation ( T (1) theorem). If Q ∩ R = ∅ then � T ∆ Q f, ∆ R g � is easy to estimate using only smoothness of the kernel. If R ⊂ Q one needs to estimate � T 1 Q ′ , ∆ R g � ; here Q ′ is the child of Q , Q ′ ⊃ R . if T 1 = 0 this is equivalent the estimating � T 1 R N \ Q ′ , ∆ R g � , which can be done standard way If T 1 � = 0 one needs to replace T by T − π b , b = T 1 Condition b ∈ BMO implies that π b is bounded Case Q ⊂ R is treated symmetrically. Condition T ∗ 1 ∈ BMO is used. 9
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Let child( Q ) = child 1 ( Q ) denote the children of Q , and let child n ( Q ) be the grandchildren of order n . Definition (Dyadic (Haar) shift) with parameters m and n is an operator T = � Q ∈D A I , where A I : ⊕ R ∈ child m ( Q ) D R → ⊕ R ∈ child n ( Q ) D R where A I can be represented as an integral operator with kernel a Q ( x, y ) , � a Q � ∞ ≤ | Q | − 1 . Complexity of T is r = max { m, n } . Dyadic shift is not a martingale transform. But T can be decomposed T = T 1 + T 2 + . . . + T r , T k = � � Q ∈D : ℓ ( Q )=2 k + rj A Q ; j ∈ Z Each T k can be treated as a martingale transform if one goes r steps at a time. 10
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Definition (Dyadic (Haar) shift) with parameters m and n is an operator T = � Q ∈D A I , where A I : ⊕ R ∈ child m ( Q ) D R → ⊕ R ∈ child n ( Q ) D R where A I can be represented as an integral operator with kernel a Q ( x, y ) , � a Q � ∞ ≤ | Q | − 1 . Complexity of T is r = max { m, n } . Bound � a Q � ∞ ≤ | Q | − 1 means simply that after “renormalization” bilinear form of the operator A Q is bounded on L 1 × L 1 . 11
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Square function and Hardy space H 1 d �� Q ∈D | ∆ Q f ( x ) | 2 � 1 / 2 Square function: Sf ( x ) = . 12
Main objects Martingale difference decomposition Bounds on paraproducts and commutators Martingale transforms, paraproducts and dyadic shifts Square function, H 1 and BMO From dyadic to classical Square function and Hardy space H 1 d �� Q ∈D | ∆ Q f ( x ) | 2 � 1 / 2 Square function: Sf ( x ) = . Linearized (vector) square function ( � Sf ( x ) ∈ ℓ 2 ): � Sf ( x ) = { ∆ Q ( x ) : Q ∋ x, ℓ ( Q ) = 2 − k } k ∈ Z . | Sf ( x ) | = � � Sf ( x ) � ℓ 2 . 12
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