Commutators of the Hilbert Transform Along a Parabola

Commutators of the Hilbert transform along a parabola

Tyler Bongers Joint with Zihua Guo (Monash University), Ji Li (Macquarie University), and Brett Wick (WUSTL)

Department of Mathematics and Statistics Washington University in St. Louis [email protected]

Ohio River Analysis Meeting, 2019

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 1 / 12 If the operator T has a kernel representation Tf (x) = K(x, y)f (y) dy, then this has the special form ´

[b, T ]f (x) = b(x) − b(y)K(x, y)f (y) dy. ˆ Note that b introduces cancellation not apparent from the kernel.

Question: Under what circumstances is the commutator bounded?

Commutators of operators

If T is an operator from an L2 space to itself and b is a function, then the commutator [b, T ] is formally deﬁned by

[b, T ]f := bT (f ) − T (bf )

with b interpreted as pointwise multiplication.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 2 / 12 Question: Under what circumstances is the commutator bounded?

Commutators of operators

If T is an operator from an L2 space to itself and b is a function, then the commutator [b, T ] is formally deﬁned by

[b, T ]f := bT (f ) − T (bf )

with b interpreted as pointwise multiplication.

If the operator T has a kernel representation Tf (x) = K(x, y)f (y) dy, then this has the special form ´

[b, T ]f (x) = b(x) − b(y)K(x, y)f (y) dy. ˆ Note that b introduces cancellation not apparent from the kernel.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 2 / 12 Commutators of operators

If T is an operator from an L2 space to itself and b is a function, then the commutator [b, T ] is formally deﬁned by

[b, T ]f := bT (f ) − T (bf )

with b interpreted as pointwise multiplication.

If the operator T has a kernel representation Tf (x) = K(x, y)f (y) dy, then this has the special form ´

[b, T ]f (x) = b(x) − b(y)K(x, y)f (y) dy. ˆ Note that b introduces cancellation not apparent from the kernel.

Question: Under what circumstances is the commutator bounded?

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 2 / 12 If H is the Hilbert transform, then

2 2 k[b, H]: L → L k ∼ kbkBMO (Coifman-Rochberg-Weiss).

Here, BMO is the space of bounded mean oscillation:

1 1 kbkBMO := sup b − b dx. Q cube |Q| ˆQ |Q| ˆQ L∞ ⊂ BMO, but the inclusion is proper.

Singular integral commutators

If T is bounded on L2 and b ∈ L∞, then [b, T ] is also L2 bounded. However, b introduces some extra cancellation.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 3 / 12 L∞ ⊂ BMO, but the inclusion is proper.

Singular integral commutators

If T is bounded on L2 and b ∈ L∞, then [b, T ] is also L2 bounded. However, b introduces some extra cancellation. If H is the Hilbert transform, then

2 2 k[b, H]: L → L k ∼ kbkBMO (Coifman-Rochberg-Weiss).

Here, BMO is the space of bounded mean oscillation:

1 1 kbkBMO := sup b − b dx. Q cube |Q| ˆQ |Q| ˆQ

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 3 / 12 Singular integral commutators

If T is bounded on L2 and b ∈ L∞, then [b, T ] is also L2 bounded. However, b introduces some extra cancellation. If H is the Hilbert transform, then

2 2 k[b, H]: L → L k ∼ kbkBMO (Coifman-Rochberg-Weiss).

Here, BMO is the space of bounded mean oscillation:

1 1 kbkBMO := sup b − b dx. Q cube |Q| ˆQ |Q| ˆQ L∞ ⊂ BMO, but the inclusion is proper.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 3 / 12 2 Hγ is not anti-self-adjoint, not an L isometry, and is “too singular” to be a Calder´on-Zygmundoperator - Hγf (x, y) depends only on the values of f on a strictly lower-dimensional object.

The curve introduces delicate geometry. For example, hHγχE , χF i can be zero depending on the arrangement of E and F .

p 2 p 2 By Littlewood-Paley theory, Hγ : L (R ) → L (R ) is bounded for 1 < p < ∞.

Hilbert transform along a Parabola

The Hilbert transform along a parabola γ(t) = (t, t2) is deﬁned for Schwartz functions by

2 dt Hγf (x, y) = p.v. f (x − t, y − t ) . ˆR t

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 4 / 12 The curve introduces delicate geometry. For example, hHγχE , χF i can be zero depending on the arrangement of E and F .

2 Hγ is not anti-self-adjoint, not an L isometry, and is “too singular” to be a Calder´on-Zygmundoperator - Hγf (x, y) depends only on the values of f on a strictly lower-dimensional object.

Hilbert transform along a Parabola

The Hilbert transform along a parabola γ(t) = (t, t2) is deﬁned for Schwartz functions by

2 dt Hγf (x, y) = p.v. f (x − t, y − t ) . ˆR t

p 2 p 2 By Littlewood-Paley theory, Hγ : L (R ) → L (R ) is bounded for 1 < p < ∞.

Hilbert transform along a Parabola

The Hilbert transform along a parabola γ(t) = (t, t2) is deﬁned for Schwartz functions by

2 dt Hγf (x, y) = p.v. f (x − t, y − t ) . ˆR t

p 2 p 2 By Littlewood-Paley theory, Hγ : L (R ) → L (R ) is bounded for 1 < p < ∞.

2 Hγ is not anti-self-adjoint, not an L isometry, and is “too singular” to be a Calder´on-Zygmundoperator - Hγf (x, y) depends only on the values of f on a strictly lower-dimensional object.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 4 / 12 Hilbert transform along a Parabola

The Hilbert transform along a parabola γ(t) = (t, t2) is deﬁned for Schwartz functions by

2 dt Hγf (x, y) = p.v. f (x − t, y − t ) . ˆR t

p 2 p 2 By Littlewood-Paley theory, Hγ : L (R ) → L (R ) is bounded for 1 < p < ∞.

2 Hγ is not anti-self-adjoint, not an L isometry, and is “too singular” to be a Calder´on-Zygmundoperator - Hγf (x, y) depends only on the values of f on a strictly lower-dimensional object.

The curve introduces delicate geometry. For example, hHγχE , χF i can be zero depending on the arrangement of E and F .

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 4 / 12 If b is bounded, then

p p p p k[b, Hγ]: L → L k . kbk∞kHγ : L → L k. But b is not necessarily bounded.

Boundedness of Commutators

The commutator [b, Hγ] has the form

2 b(x, y) − b(x − t, y − t ) 2 [b, Hγ]f (x, y) = f (x − t, y − t ) dt. ˆR t p 2 Question: When is [b, Hγ] bounded from L (R ) to itself?

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 5 / 12 Boundedness of Commutators

The commutator [b, Hγ] has the form

2 b(x, y) − b(x − t, y − t ) 2 [b, Hγ]f (x, y) = f (x − t, y − t ) dt. ˆR t p 2 Question: When is [b, Hγ] bounded from L (R ) to itself? If b is bounded, then

p p p p k[b, Hγ]: L → L k . kbk∞kHγ : L → L k. But b is not necessarily bounded.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 5 / 12 B.-Guo-Li-Wick ’19 p p If b is in parabolic BMO then k[b, Hγ]: L → L k . kbkBMOγ for all 1 < p < ∞. The implied constant depends only on p.

Note that unbounded functions with logarithmic growth can be in parabolic BMO.

Suﬃcient condition for boundedness

We say that b is in parabolic BMO if

kbkBMOγ := sup b − b dx < ∞ Q∈P Q Q

2 where P is the collection of all cubes in R with sides parallel to the axes and dimensions [`, `2].

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 6 / 12 Suﬃcient condition for boundedness

We say that b is in parabolic BMO if

kbkBMOγ := sup b − b dx < ∞ Q∈P Q Q

2 where P is the collection of all cubes in R with sides parallel to the axes and dimensions [`, `2]. B.-Guo-Li-Wick ’19 p p If b is in parabolic BMO then k[b, Hγ]: L → L k . kbkBMOγ for all 1 < p < ∞. The implied constant depends only on p.

Note that unbounded functions with logarithmic growth can be in parabolic BMO.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 6 / 12 Therefore, the commutator can be estimated through weighted bounds

1/2 −1/2 2 2 2 2 kw Hγ(w ·): L → L k = kHγ : L (w) → L (w)k.

with w = eb Rez. For operators like Hilbert and Riesz, L2 bounds hold for weights in the Muckenhoupt Ap class, and the logarithms of such weights are in BMO. Here, the situation is more complex.

Proof of boundedness

By the Cauchy integral trick,

zb/2 −zb/2 d zb/2 −zb/2 1 e Hγ(e f ) [b, Hγ]f = 2 e Hγ(e f ) = dz. dz πi ˆ z2 z=0 ∂D

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 7 / 12 For operators like Hilbert and Riesz, L2 bounds hold for weights in the Muckenhoupt Ap class, and the logarithms of such weights are in BMO. Here, the situation is more complex.

Proof of boundedness

By the Cauchy integral trick,

zb/2 −zb/2 d zb/2 −zb/2 1 e Hγ(e f ) [b, Hγ]f = 2 e Hγ(e f ) = dz. dz πi ˆ z2 z=0 ∂D Therefore, the commutator can be estimated through weighted bounds

1/2 −1/2 2 2 2 2 kw Hγ(w ·): L → L k = kHγ : L (w) → L (w)k.

with w = eb Rez.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 7 / 12 Proof of boundedness

By the Cauchy integral trick,

zb/2 −zb/2 d zb/2 −zb/2 1 e Hγ(e f ) [b, Hγ]f = 2 e Hγ(e f ) = dz. dz πi ˆ z2 z=0 ∂D Therefore, the commutator can be estimated through weighted bounds

1/2 −1/2 2 2 2 2 kw Hγ(w ·): L → L k = kHγ : L (w) → L (w)k.

with w = eb Rez. For operators like Hilbert and Riesz, L2 bounds hold for weights in the Muckenhoupt Ap class, and the logarithms of such weights are in BMO. Here, the situation is more complex.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 7 / 12 A result of Bernicot, Frey, and Petermichl gives weighted bounds for ΛS in terms of parabolic Ap and reverse-H¨oldercharacteristics of w.

For z suﬃciently small, these characteristics are controlled by the parabolic BMO norm of log w = log eb Re z .

In particular, Hγ is dominated by a sparse form

1/r 1/s X 1 1 Λ (f , g) = |Q| |f |r |g|s S,r,s |Q| ˆ |Q| ˆ Q∈S Q Q

where S is a sparse family of parabolic cubes.

Weighted estimates

Weighted estimates for Hγ pass through the sparse domination results of Cladek and Ou.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 8 / 12 For z suﬃciently small, these characteristics are controlled by the parabolic BMO norm of log w = log eb Re z .

A result of Bernicot, Frey, and Petermichl gives weighted bounds for ΛS in terms of parabolic Ap and reverse-H¨oldercharacteristics of w.

Weighted estimates

Weighted estimates for Hγ pass through the sparse domination results of Cladek and Ou. In particular, Hγ is dominated by a sparse form

1/r 1/s X 1 1 Λ (f , g) = |Q| |f |r |g|s S,r,s |Q| ˆ |Q| ˆ Q∈S Q Q

where S is a sparse family of parabolic cubes.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 8 / 12 For z suﬃciently small, these characteristics are controlled by the parabolic BMO norm of log w = log eb Re z .

Weighted estimates

Weighted estimates for Hγ pass through the sparse domination results of Cladek and Ou. In particular, Hγ is dominated by a sparse form

1/r 1/s X 1 1 Λ (f , g) = |Q| |f |r |g|s S,r,s |Q| ˆ |Q| ˆ Q∈S Q Q

where S is a sparse family of parabolic cubes.

A result of Bernicot, Frey, and Petermichl gives weighted bounds for ΛS in terms of parabolic Ap and reverse-H¨oldercharacteristics of w.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 8 / 12 Weighted estimates

Weighted estimates for Hγ pass through the sparse domination results of Cladek and Ou. In particular, Hγ is dominated by a sparse form

1/r 1/s X 1 1 Λ (f , g) = |Q| |f |r |g|s S,r,s |Q| ˆ |Q| ˆ Q∈S Q Q

where S is a sparse family of parabolic cubes.

A result of Bernicot, Frey, and Petermichl gives weighted bounds for ΛS in terms of parabolic Ap and reverse-H¨oldercharacteristics of w.

For z suﬃciently small, these characteristics are controlled by the parabolic BMO norm of log w = log eb Re z .

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 8 / 12 If E and F are superlevel and sublevel sets for b and the kernel does not change sign on E × F , this gives control on terms like

1/2 1/2 |E| |F | & |b(x) − b(y)| dx dy ˆE ˆF and bounds the BMO norm of b.

The median method fails for Hγ because the operator is too localized. Kernel estimates are not adapted to the underlying curve.

Necessary condition for boundedness

A typical method for ﬁnding a necessary condition for commutator bounds is the median method. If Tf (x) = K(x, y)f (y) dy, then ´ h[b, T ]χE , χF i = b(x) − b(y) K(x, y) dx dy. ˆE ˆF

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 9 / 12 The median method fails for Hγ because the operator is too localized. Kernel estimates are not adapted to the underlying curve.

Necessary condition for boundedness

A typical method for ﬁnding a necessary condition for commutator bounds is the median method. If Tf (x) = K(x, y)f (y) dy, then ´ h[b, T ]χE , χF i = b(x) − b(y) K(x, y) dx dy. ˆE ˆF

If E and F are superlevel and sublevel sets for b and the kernel does not change sign on E × F , this gives control on terms like

1/2 1/2 |E| |F | & |b(x) − b(y)| dx dy ˆE ˆF and bounds the BMO norm of b.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 9 / 12 Necessary condition for boundedness

A typical method for ﬁnding a necessary condition for commutator bounds is the median method. If Tf (x) = K(x, y)f (y) dy, then ´ h[b, T ]χE , χF i = b(x) − b(y) K(x, y) dx dy. ˆE ˆF

If E and F are superlevel and sublevel sets for b and the kernel does not change sign on E × F , this gives control on terms like

1/2 1/2 |E| |F | & |b(x) − b(y)| dx dy ˆE ˆF and bounds the BMO norm of b.

The median method fails for Hγ because the operator is too localized. Kernel estimates are not adapted to the underlying curve.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 9 / 12 If x ∈ Q, then Ix,Q := {t : x − γ(t) ∈ EQ }, and the testing norm of b is

dt kbk = sup b − b(x − γ(t)) dx. test t Q∈P Ix,Q

B.-Guo-Li-Wick ’19 p If the commutator [b, Hγ] is bounded on L for some 1 < p < ∞, then

p p kbktest . k[b, Hγ]: L → L k.

Proof: Direct estimation of [b, Hγ] on indicator functions.

Necessary condition for boundedness

Given a parabolic cube Q, let

EQ = {x − γ(t): x ∈ Q, 9`(Q) ≤ t ≤ 10`(Q)}

be the union of parabolic ﬂows of Q.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 10 / 12 B.-Guo-Li-Wick ’19 p If the commutator [b, Hγ] is bounded on L for some 1 < p < ∞, then

p p kbktest . k[b, Hγ]: L → L k.

Proof: Direct estimation of [b, Hγ] on indicator functions.

Necessary condition for boundedness

Given a parabolic cube Q, let

EQ = {x − γ(t): x ∈ Q, 9`(Q) ≤ t ≤ 10`(Q)}

be the union of parabolic ﬂows of Q. If x ∈ Q, then Ix,Q := {t : x − γ(t) ∈ EQ }, and the testing norm of b is

dt kbk = sup b − b(x − γ(t)) dx. test t Q∈P Ix,Q

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 10 / 12 Necessary condition for boundedness

Given a parabolic cube Q, let

EQ = {x − γ(t): x ∈ Q, 9`(Q) ≤ t ≤ 10`(Q)}

be the union of parabolic ﬂows of Q. If x ∈ Q, then Ix,Q := {t : x − γ(t) ∈ EQ }, and the testing norm of b is

dt kbk = sup b − b(x − γ(t)) dx. test t Q∈P Ix,Q

B.-Guo-Li-Wick ’19 p If the commutator [b, Hγ] is bounded on L for some 1 < p < ∞, then

p p kbktest . k[b, Hγ]: L → L k.

Proof: Direct estimation of [b, Hγ] on indicator functions.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 10 / 12 By analogy with the Coifman-Rochberg-Weiss results, one could reasonably expect that all three spaces are equivalent. There is some evidence in favor of this, but it is unclear.

Open Question

Therefore, we have the estimates

p p kbktest . k[b, Hγ]: L → L k . kbkBMOγ and their corresponding inclusions

BMOγ =⇒ bounded commutator =⇒ testing.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 11 / 12 Open Question

Therefore, we have the estimates

p p kbktest . k[b, Hγ]: L → L k . kbkBMOγ and their corresponding inclusions

BMOγ =⇒ bounded commutator =⇒ testing.

By analogy with the Coifman-Rochberg-Weiss results, one could reasonably expect that all three spaces are equivalent. There is some evidence in favor of this, but it is unclear.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 11 / 12 B.-Guo-Li-Wick ’19 If η is a monomial curve and b is in the BMO space adapted to p η-cubes, then [b, Hη] is bounded on L for all p ∈ (1, ∞). p If [b, Hη] is bounded on L , then b satisﬁes the testing condition relative to the curve η.

If η is not monomial but has non-vanishing torsion, then the above results apply for the local operator 1 dt Hηf = p.v. f (x − η(t)) . ˆ−1 t

Generalization

If η is a monomial curve, there is an associated family of η-cubes introduced by Cladek and Ou that generalize the dyadic (or parabolic) grids. There is an analogous Hilbert transform adapted to this curve, and the sparse domination results still hold.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 12 / 12 If η is not monomial but has non-vanishing torsion, then the above results apply for the local operator 1 dt Hηf = p.v. f (x − η(t)) . ˆ−1 t

Generalization

B.-Guo-Li-Wick ’19 If η is a monomial curve and b is in the BMO space adapted to p η-cubes, then [b, Hη] is bounded on L for all p ∈ (1, ∞). p If [b, Hη] is bounded on L , then b satisﬁes the testing condition relative to the curve η.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 12 / 12 Generalization

If η is not monomial but has non-vanishing torsion, then the above results apply for the local operator 1 dt Hηf = p.v. f (x − η(t)) . ˆ−1 t

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 12 / 12 References

R. Coifman, R. Rochberg, G. Weiss Factorization theorems for Hardy spaces in several variables. Ann. of Math., 103(3):611–635, 1976. F. Bernicot, D. Frey, S. Petermichl. Sharp weighted norm estimates beyond Calder´on-Zygmundtheory. Anal. PDE, 9:1079–1113, 2016. L. Cladek, Y. Ou. Sparse domination of Hilbert transforms along curves. Math. Res. Lett., 25:415–436, 2018.

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 12 / 12 Deﬁnitions

A weight is a nonnegative, locally integrable function.

A weight w is in the class of Muckenhoupt weights Ap if

p−1 1 1 1−p0 [w]Ap := sup w w < ∞. Q |Q| ˆQ |Q| ˆQ S The Ap classes are increasing in p, and A∞ = p>1 Ap.

A weight w is in the reverse H¨older class RH1+ if

−1 1/(1+) 1 1 1+ [w]RH1+ := sup w w < ∞. Q |Q| ˆQ |Q| ˆQ

Bongers, Guo, Li, Wick (WUSTL) Commutators of parabolic Hilbert ORAM 2019 12 / 12