Characterization of multi parameter BMO spaces through commutators Stefanie Petermichl Universit´ e Paul Sabatier IWOTA Chemnitz August 2017 S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 1 / 30
history Hankel vs. Toeplitz on T P ± projection operator onto non-negative and negative frequencies. A Hankel operator with symbol b is H b : L 2 + → H 2 − , f �→ P − bP + f b ∈ BMO characterises boundedness. A Toeplitz operator with symbol b is T b : L 2 + → H 2 + , f �→ P + bP + f b ∈ L ∞ characterises boundedness. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 2 / 30
history Hankel Operators P − bP + � H b � = sup � g � H 2 − =1 sup � f � H 2 =1 | ( H b f , g ) | = sup � g � H 2 − =1 sup � f � H 2 =1 | ( P − ( bf ) , g ) | = sup � g � H 2 − =1 sup � f � H 2 + =1 | ( P − bf , g ) | = sup � g � H 2 =1 sup � f � H 2 =1 | ( P − b , ¯ f g ) | Anti-analytic part of b defines bounded linear functional on H 1 ⊂ L 1 . Extend by Hahn Banach to all of L 1 , i.e. a bounded function with the same anti-analytic part as b . Using H 1 − BMO duality, we get the characterisation of BMO. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 3 / 30
history Toeplitz operators P + bP + Clearly || T b || ≤ || b || ∞ But L ∞ also characterises boundedness: It is easy to see that λ n P + λ n → I ¯ in L 2 in SOT. Nothing happens to such f with FS cut off at − n . So λ n P + bP + λ n → b ¯ in L 2 in SOT. Now as multiplication operators: � b � ≤ sup n � ¯ λ n P + bP + λ n � ≤ � P + bP + � S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 4 / 30
Commutators with the Hilbert transform Commutators [ H , b ] [ b , H ] f = b · Hf − H ( bf ) where H is the Hilbert transform and b is multiplication by the function b . If we write H = P + − P − and I = P + + P − then [ b , H ] = [( P + + P − ) b , ( P + − P − )] = 2 P − bP + − 2 P + bP − , two Hankel operators with orthogonal ranges. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 5 / 30
Commutators with the Hilbert transform Extensions of [ H , b ] Riesz transform commutators and similar. Coifman, Rochberg, Weiss, Uchiyama, Lacey ... The passage to several parameters, initiation. Cotlar, Ferguson, Sadosky ... Thiele, Muscalu, Tao, Journe, Holmes, Lacey, Pipher, Strouse, Wick, P. ... S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 6 / 30
Commutators with the Hilbert transform Commutators [ H 1 H 2 , b ] with b ( x 1 , x 2 ) Tensor product one-parameter case. Theorem (Ferguson, Sadosky) [ H 1 H 2 , b ] bounded in L 2 iff b ∈ bmo ‘little BMO’ || b || bmo = max { sup || b ( x 1 , · ) || BMO , sup || b ( · , x 2 ) || BMO } x 1 x 2 i.e. b uniformly in BMO in each variable separately or of bounded mean oscillation on rectangles. more Hilbert transforms [ H 1 H 2 H 3 , b ] etc implicit. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 7 / 30
Commutators with the Hilbert transform Commutators [ H 1 , [ H 2 , b ]] with b ( x 1 , x 2 ) Theorem (Ferguson, Lacey) [ H 1 , [ H 2 , b ]] bounded in L 2 iff b ∈ BMO ‘product BMO’ 1 || b || 2 � | ( b , h R ) | 2 BMO = sup | O | O R ⊂ O more iterations [ H 1 , [ H 2 , [ H 3 , b ]]] ‘not’ implicit, but Terwilliger, Lacey. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 8 / 30
Commutators with the Hilbert transform Lower estimates for Hilbert commutators Ferguson-Sadosky: elegant ’soft’ argument, based on Toeplitz forms. Ferguson-Lacey: extremely technical ’hard’ real analysis argument based on Hankel forms. Using scale analysis, Schwarz tail estimates, geometric facts on distribution of rectangles in the plane... S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 9 / 30
Commutators with the Hilbert transform Commutators [ H 2 , [ H 3 H 1 , b ]] with b ( x 1 , x 2 , x 3 ) Theorem (Ou, Strouse, P.) [ H 2 , [ H 3 H 1 , b ]] bounded in L 2 iff b ∈ BMO (13)2 ‘little product BMO’ || b || BMO (13)2 = max { sup || b ( x 1 , · , · ) || BMO , sup || b ( · , · , x 3 ) || BMO } x 1 x 3 b uniformly in product BMO when fixing variables x 3 and x 1 . more Hilbert transforms and more iterations implicit. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 10 / 30
Commutators with the Hilbert transform Commutators [ H 2 , [ H 3 H 1 , b ]] with b ( x 1 , x 2 , x 3 ) Infact TFAE: 1 b ∈ BMO (13)2 2 [ H 2 , [ H 1 , b ]] and [ H 2 , [ H 3 , b ]] bounded in L 2 ( T 3 ) 3 [ H 2 , [ H 3 H 1 , b ]] bounded in L 2 ( T 3 ). 1 eq 2: Wiener’s theorem and Ferguson, Lacey: [ H 2 , [ H 1 , b ]] f ( x 1 , x 2 ) g ( x 3 ) = g ( x 3 )[ H 2 , [ H 1 , b ]] f ( x 1 , x 2 ) 2 eq 3: Toeplitz argument: typical terms that arise: P + 1 P + 2 bP − 1 P − 2 and P + 1 P + 2 P + 3 bP − 1 P − 2 P + 3 S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 11 / 30
Commutators with Riesz transforms Riesz Riesz commutators and the absence of Hankel and Toeplitz. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 12 / 30
Commutators with Riesz transforms One parameter: [ R i , b ] It is a classical result by Coifman, Rochberg and Weiss, that the Riesz transform commutators classify BMO . For each symbol b ∈ BMO we may choose the worst Riesz transform. In this sense � [ b , R i ] � 2 → 2 � � b � BMO But � b � BMO � sup � [ b , R i ] � 2 → 2 i Testing class for CZOs. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 13 / 30
Commutators with Riesz transforms One parameter tensor product: [ R 1 , i 1 R 2 , i 2 , b ] Through use of the little BMO norm, one sees that � [ b , R 1 , i 1 R 2 , i 2 ] � 2 → 2 � � b � bmo Through a direct calculation using the little BMO norm one also sees � b � bmo � sup � [ b , R 1 , i 1 R 2 , i 2 ] � 2 → 2 i 1 , i 2 S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 14 / 30
Commutators with Riesz transforms multi-parameter: [ R 2 , j 2 , [ R 1 , j 1 , b ]] Theorem (Lacey, Pipher, Wick, P.) sup � [ R 2 , j 2 , [ R 1 , j 1 , b ]] � ∼ � b � BMO . j 1 , j 2 By BMO, we mean Chang–Fefferman product BMO. Implicit generalizations with similar proof. Testing for CZOs. This means we test a symbol on Riesz transforms. The estimate then self improves to all operators of the same type as Riesz transforms. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 15 / 30
Commutators with Riesz transforms multi-parameter tensor product: [ R 2 , j 2 , [ R 1 , j 1 R 3 , j 3 , b ]] Theorem (Ou, Strouse, P.) � b � BMO (13)2 ∼ � [ R 2 , j 2 , [ R 1 , j 1 R 3 , j 3 , b ]] � where we mean little product BMO. Implicit generalizations but proof is substantially more difficult when dimensions are greater than 2 or when slots contain tensor products of more than 2. Testing for (paraproduct-free) Journ´ e operators - we will see later what these are. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 16 / 30
Commutators with Riesz transforms Cones Riesz transforms do not have the same relation to projections as the Hilbert transform does. Replace by well chosen, smooth half plane projections that are CZO. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 17 / 30
Commutators with Riesz transforms Cones Depending on the make of the symbol function, a multi parameter skeleton of cones of large aperture is chosen via a probabilistic procedure. The others are filled in using tiny cones via a Toeplitz argument. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 18 / 30
Commutators with Riesz transforms Polynomials We have now showed that there is a lower estimate if one can choose from cone operators. How does this help for Riesz transforms? Observe: [ T 1 T 2 , b ] = T 1 [ T 2 , b ] + [ T 1 , b ] T 2 If the commutator with T 1 T 2 is large, then one of the commutators with T i has to be large. Riesz transforms have Fourier symbols ξ i on S n (monomials) well adapted for polynomial approximation. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 19 / 30
Commutators with Riesz transforms Passage to tensor products of Riesz transforms What goes wrong: If [ R 2 1 , i 1 R 2 , i 2 , b ] large, cannot say [ R 1 , i 1 R 2 , i 2 , b ] remains large. It is too wasteful to just have any lower estimates of tensor products of cone operators. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 20 / 30
Commutators with Riesz transforms Passage to tensor products of Riesz transforms These strip operators work well on products of S 1 and a deep generalization using a probabilistic construction and zonal harmonics will work on higher dimensional spheres. S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 21 / 30
Commutators with Riesz transforms Passage to tensor products of Riesz transforms Looking like this: S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 22 / 30
Commutators with Riesz transforms Passage to tensor products of Riesz transforms and this: S. Petermichl (Universit´ e Paul Sabatier) Commutators and BMO Chemnitz 23 / 30
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