![Part 4 — Twisted paraproduct || |||| |||| |||| |||| ||||
Dyadic version
Td(F, G) :=
k∈Z
(E
(1)
k F)(∆
(2)
k G)
Continuous version
Tc(F, G) :=
k∈Z
(P(1)
ϕk
F)(P
(2)
ψk
G)
Bilinear multipliers from our theorems reduce to these
using cone decomposition of the symbol:
m =
j
m[j]
from the Fourier series
m[j]
(ξ1, η2) =
k∈Z
ϕ
[j]
k (ξ1) ψ
[j]
k (η2)](https://image.slidesharecdn.com/kovaccrm2013-190108093130/85/Estimates-for-a-class-of-non-standard-bilinear-multipliers-84-320.jpg)
Estimates for a class of non-standard bilinear multipliers
- 1. Estimates for a class of non-standard bilinear multipliers Vjekoslav Kovaˇc (University of Zagreb) Joint work with Fr´ed´eric Bernicot (Universit´e de Nantes) and Christoph Thiele (Universit¨at Bonn) Joint CRM-ISAAC Conference on Fourier Analysis and Approximation Theory Bellaterra, November 6, 2013
- 2. Talk outline | Part 1 — Introduction Motivation for multilinear estimates Concrete examples and some results
- 3. Talk outline | Part 1 — Introduction Motivation for multilinear estimates Concrete examples and some results Part 2 — The “entangled” structure The scope of our techniques
- 4. Talk outline | Part 1 — Introduction Motivation for multilinear estimates Concrete examples and some results Part 2 — The “entangled” structure The scope of our techniques Part 3 — Dyadic model operators Formulating a T(1)-type theorem Setting up the Bellman function scheme
- 5. Talk outline | Part 1 — Introduction Motivation for multilinear estimates Concrete examples and some results Part 2 — The “entangled” structure The scope of our techniques Part 3 — Dyadic model operators Formulating a T(1)-type theorem Setting up the Bellman function scheme Part 4 — Transition to continuous-type operators Back to bilinear multipliers “Entangled” operators with continuous kernels
- 6. Part 1 — Multilinear estimates || T = a multilinear integral operator T is “singular” in some sense
- 7. Part 1 — Multilinear estimates || T = a multilinear integral operator T is “singular” in some sense We are interested in Lp estimates: T(F1, F2, . . . , Fk) Lp(Rn) ≤ Cp,p1,...,pk k j=1 Fj L pj (R nj ) in a subrange of 0 < p, p1, . . . , pk < ∞ Possibly also some Sobolev norm estimates, etc.
- 8. Part 1 — Multilinear estimates, motivation ||| Motivation for multilinear estimates:
- 9. Part 1 — Multilinear estimates, motivation ||| Motivation for multilinear estimates: paraproducts J.-M. Bony (1981) — paradifferential operators G. David and J.-L. Journ´e (1984) — T(1) theorem
- 10. Part 1 — Multilinear estimates, motivation ||| Motivation for multilinear estimates: paraproducts J.-M. Bony (1981) — paradifferential operators G. David and J.-L. Journ´e (1984) — T(1) theorem multilinear expansions of nonlinear/“curved” operators A. Calder´on (1960s) — Cauchy integral on Lipschitz curves M. Christ and A. Kiselev (2001) — Hausdorff-Young inequalities for the Dirac scattering transform J. Bourgain and L. Guth (2010) — restriction estimates, oscillatory integrals
- 11. Part 1 — Multilinear estimates, motivation ||| Motivation for multilinear estimates: paraproducts J.-M. Bony (1981) — paradifferential operators G. David and J.-L. Journ´e (1984) — T(1) theorem multilinear expansions of nonlinear/“curved” operators A. Calder´on (1960s) — Cauchy integral on Lipschitz curves M. Christ and A. Kiselev (2001) — Hausdorff-Young inequalities for the Dirac scattering transform J. Bourgain and L. Guth (2010) — restriction estimates, oscillatory integrals recurrence in ergodic theory J. Bourgain (1988) — return times theorem C. Demeter, M. Lacey, T. Tao, and C. Thiele (2008) — extending the exponent range
- 12. Part 1 — A basic example |||| Bilinear case only (for simplicity)
- 13. Part 1 — A basic example |||| Bilinear case only (for simplicity) From the viewpoint of bilinear singular integrals: T(F, G)(x) = p.v. (Rn)2 K(s, t)F(x − s)G(x − t) ds dt K = translation-invariant Calder´on-Zygmund kernel Generalized by L. Grafakos and R. H. Torres (2002): multilinear C-Z operators Take m = K
- 14. Part 1 — A basic example |||| Bilinear case only (for simplicity) From the viewpoint of bilinear multipliers: Coifman-Meyer multipliers, R. Coifman and Y. Meyer (1978) T(F, G)(x) = (Rn)2 m(ξ, η)e2πix·(ξ+η) F(ξ)G(η)dξdη m∈C∞ R2{(0, 0)} ∂α1 ξ ∂α2 η m(ξ, η) ≤ Cα1,α2,n(|ξ| + |η|)−α1−α2 Note: m(ξ, η) is singular only at the origin ξ = η = 0
- 15. Part 1 — More singular examples, 1D |||| 1D example: bilinear Hilbert transform Suggested by A. Calderon (Cauchy integral on Lipschitz curves) Bounded by M. Lacey and C. Thiele (1997)
- 16. Part 1 — More singular examples, 1D |||| 1D example: bilinear Hilbert transform Suggested by A. Calderon (Cauchy integral on Lipschitz curves) Bounded by M. Lacey and C. Thiele (1997) As a singular integral: T(f , g)(x) = p.v. R f (x − t)g(x + t) dt t
- 17. Part 1 — More singular examples, 1D |||| 1D example: bilinear Hilbert transform Suggested by A. Calderon (Cauchy integral on Lipschitz curves) Bounded by M. Lacey and C. Thiele (1997) As a singular integral: T(f , g)(x) = p.v. R f (x − t)g(x + t) dt t As a multiplier: T(f , g)(x) = R2 πi sgn(η − ξ)e2πix(ξ+η) f (ξ)g(η)dξdη Note: m(ξ, η) = πi sgn(η − ξ) is singular along the line ξ = η
- 18. Part 1 — More singular examples, 2D | |||| 2D example: a variant of the 2D bilinear Hilbert transform Introduced by Demeter and Thiele and bounded for “most” cases of A, B ∈ M2(R) (2008)
- 19. Part 1 — More singular examples, 2D | |||| 2D example: a variant of the 2D bilinear Hilbert transform Introduced by Demeter and Thiele and bounded for “most” cases of A, B ∈ M2(R) (2008) As a singular integral: T(F, G)(x, y) = p.v. R2 K(s, t)F (x, y)−A(s, t) G (x, y)−B(s, t) ds dt
- 20. Part 1 — More singular examples, 2D | |||| 2D example: a variant of the 2D bilinear Hilbert transform Introduced by Demeter and Thiele and bounded for “most” cases of A, B ∈ M2(R) (2008) As a singular integral: T(F, G)(x, y) = p.v. R2 K(s, t)F (x, y)−A(s, t) G (x, y)−B(s, t) ds dt Essentially the only case that was left out: A = 1 0 0 0 and B = 0 0 0 1
- 21. Part 1 — More singular examples, 2D | |||| 2D example: a variant of the 2D bilinear Hilbert transform As a multiplier: T(F, G)(x, y) = R4 µ(ξ1, ξ2, η1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2 µ(ξ1, ξ2, η1, η2) = m Aτ (ξ1, ξ2) + Bτ (η1, η2) , m = K m ∈ C∞ R2{(0, 0)} ∂α1 τ1 ∂α2 τ2 m(τ1, τ2) ≤ Cα1,α2 (|τ1| + |τ2|)−α1−α2 Note: µ(ξ1, ξ2, η1, η2) is singular along the 2-plane Aτ (ξ1, ξ2) + Bτ (η1, η2) = (0, 0)
- 22. Part 1 — Remaining case of 2D BHT || |||| T(F, G)(x, y) = R4 m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2
- 23. Part 1 — Remaining case of 2D BHT || |||| T(F, G)(x, y) = R4 m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2 Theorem. Lp estimate — F. Bernicot (2010), V. K. (2010) T(F, G) Lr ≤ Cp,q,r F Lp G Lq for 1 < p, q < ∞, 0 < r < 2, 1 p + 1 q = 1 r .
- 24. Part 1 — Remaining case of 2D BHT || |||| T(F, G)(x, y) = R4 m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2 Theorem. Lp estimate — F. Bernicot (2010), V. K. (2010) T(F, G) Lr ≤ Cp,q,r F Lp G Lq for 1 < p, q < ∞, 0 < r < 2, 1 p + 1 q = 1 r . Theorem. Sobolev estimate — F. Bernicot and V. K. (2013) If supp m ⊆ (ξ1, η2) : |ξ1| ≤ c |η2| , then T(F, G) Lr y (Ws,r x ) ≤ Cp,q,r,s F Lp G Ws,q for s ≥ 0, 1 < p, q < ∞, 1 < r < 2, 1 p + 1 q = 1 r .
- 25. Part 1 — A warning example ||| |||| Bi-parameter bilinear Hilbert transform
- 26. Part 1 — A warning example ||| |||| Bi-parameter bilinear Hilbert transform T(F, G)(x, y) = p.v. R2 F(x − s, y − t) G(x + s, y + t) ds s dt t
- 27. Part 1 — A warning example ||| |||| Bi-parameter bilinear Hilbert transform T(F, G)(x, y) = p.v. R2 F(x − s, y − t) G(x + s, y + t) ds s dt t T(F, G)(x, y) = R4 π2 sgn(ξ1 + ξ2)sgn(η1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2
- 28. Part 1 — A warning example ||| |||| Bi-parameter bilinear Hilbert transform T(F, G)(x, y) = p.v. R2 F(x − s, y − t) G(x + s, y + t) ds s dt t T(F, G)(x, y) = R4 π2 sgn(ξ1 + ξ2)sgn(η1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2 Satisfies no Lp estimates! C. Muscalu, J. Pipher, T. Tao, and C. Thiele (2004) Note: the symbol is singular along the union of two 3-planes, ξ1 + ξ2 = 0 and η1 + η2 = 0
- 29. Part 1 — Open problem #1 |||| |||| Triangular Hilbert transform
- 30. Part 1 — Open problem #1 |||| |||| Triangular Hilbert transform T(F, G)(x, y) = p.v. R F(x−t, y)G(x, y −t) dt t
- 31. Part 1 — Open problem #1 |||| |||| Triangular Hilbert transform T(F, G)(x, y) = p.v. R F(x−t, y)G(x, y −t) dt t T(F, G)(x, y) = R4 −πisgn(ξ1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2
- 32. Part 1 — Open problem #1 |||| |||| Triangular Hilbert transform T(F, G)(x, y) = p.v. R F(x−t, y)G(x, y −t) dt t T(F, G)(x, y) = R4 −πisgn(ξ1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2)) F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2 Still no Lp estimates are known Note: the symbol is singular along the 3-plane ξ1 + η2 = 0 Probably not the right way of looking at the operator
- 33. Part 1 — Open problem #1 |||| |||| Triangular Hilbert transform A singular integral approach to bilinear ergodic averages (suggested by C. Demeter and C. Thiele): 1 N N−1 k=0 f (Sk ω)g(Tk ω), ω ∈ Ω S, T : Ω → Ω are commuting measure preserving transformations L2 norm convergence as N → ∞ was shown by J.-P. Conze and E. Lesigne (1984) a.e. convergence as N → ∞ is still an open problem
- 34. Part 1 — Open problem #2 |||| |||| Trilinear Hilbert transform
- 35. Part 1 — Open problem #2 |||| |||| Trilinear Hilbert transform T(f , g, h)(x) = p.v. R f (x−t)g(x+t)h(x+2t) dt t
- 36. Part 1 — Open problem #2 |||| |||| Trilinear Hilbert transform T(f , g, h)(x) = p.v. R f (x−t)g(x+t)h(x+2t) dt t T(f , g, h)(x) = R3 πi sgn(−ξ + η + 2ζ)e2πix(ξ+η+ζ) f (ξ)g(η)h(ζ)dξdηdζ
- 37. Part 1 — Open problem #2 |||| |||| Trilinear Hilbert transform T(f , g, h)(x) = p.v. R f (x−t)g(x+t)h(x+2t) dt t T(f , g, h)(x) = R3 πi sgn(−ξ + η + 2ζ)e2πix(ξ+η+ζ) f (ξ)g(η)h(ζ)dξdηdζ Note: the symbol is singular along the 2-plane −ξ + η + 2ζ = 0 A complete mystery! Only some negative results are known: C. Demeter (2008)
- 38. Part 2 — Entangled structure | |||| |||| k-linear operator (k+1)-linear form Object of study: Multilinear singular integral forms with functions that partially share variables
- 39. Part 2 — Entangled structure | |||| |||| k-linear operator (k+1)-linear form Object of study: Multilinear singular integral forms with functions that partially share variables Schematically: Λ(F1, F2, . . .) = Rn F1(x1, x2) F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn K = singular kernel F1, F2, . . . = functions on R2
- 40. Part 2 — Generalized modulation invariance || |||| |||| An alternative viewpoint: generalized modulation invariances
- 41. Part 2 — Generalized modulation invariance || |||| |||| An alternative viewpoint: generalized modulation invariances Rn F1(x1, x2) F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn
- 42. Part 2 — Generalized modulation invariance || |||| |||| An alternative viewpoint: generalized modulation invariances Rn F1(x1, x2) F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn = Rn e2πiax1 F1(x1, x2) e−2πiax1 F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn
- 43. Part 2 — Generalized modulation invariance || |||| |||| An alternative viewpoint: generalized modulation invariances Rn F1(x1, x2) F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn = Rn e2πiax1 F1(x1, x2) e−2πiax1 F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn = Rn ϕ(x1)F1(x1, x2) 1 ϕ(x1) F2(x1, x3) . . . K(x1, . . . , xn) dx1dx2dx3 . . . dxn
- 44. Part 2 — Estimates ||| |||| |||| Goal: Lp estimates |Λ(F1, F2, . . . , Fk)| F1 Lp1 F2 Lp2 . . . Fk Lpk in a nonempty open subrange of 1 p1 + 1 p2 + . . . + 1 pk = 1
- 45. Part 2 — Estimates ||| |||| |||| Goal: Lp estimates |Λ(F1, F2, . . . , Fk)| F1 Lp1 F2 Lp2 . . . Fk Lpk in a nonempty open subrange of 1 p1 + 1 p2 + . . . + 1 pk = 1 Desired results: characterizations of Lp boundedness T(1)-type theorems
- 46. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| |||| T(F, G)(x, y) = R2 F(x − s, y) G(x, y − t) K(s, t) ds dt
- 47. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| |||| T(F, G)(x, y) = R2 F(x − s, y) G(x, y − t) K(s, t) ds dt Substitute u = x − s, v = y − t: Λ(F, G, H) = T(F, G), H = R4 F(u, y)G(x, v)H(x, y)K(x − u, y − v) dudvdxdy
- 48. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| |||| T(F, G)(x, y) = R2 F(x − s, y) G(x, y − t) K(s, t) ds dt Substitute u = x − s, v = y − t: Λ(F, G, H) = T(F, G), H = R4 F(u, y)G(x, v)H(x, y)K(x − u, y − v) dudvdxdy Non-translation-invariant generalization: Λ(F, G, H) = R4 F(u, y)G(x, v)H(x, y)K(u, v, x, y) dudvdxdy
- 49. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| |||| Λ(F, G, H) = R4 F(u, y)G(x, v)H(x, y)K(u, v, x, y) dudvdxdy Graph associated with its structure: x ◦ H G ◦ F y v ◦ ◦ u
- 50. Part 2 — Back to examples, triangular HT |||| |||| |||| T(F, G)(x, y) := p.v. R F(x + t, y)G(x, y + t) dt t
- 51. Part 2 — Back to examples, triangular HT |||| |||| |||| T(F, G)(x, y) := p.v. R F(x + t, y)G(x, y + t) dt t Substitute: z = −x − y − t, F1(x, y) = H(x, y), F2(y, z) = F(−y −z, y), F3(z, x) = G(x, −x−z) Λ(F, G, H) = T(F, G), H = R3 F1(x, y) F2(y, z) F3(z, x) −1 x + y + z dxdydz We do not know how to proceed in this example
- 52. Part 2 — Back to examples, triangular HT |||| |||| |||| Λ(F1, F2, F3) = R3 F1(x, y) F2(y, z) F3(z, x) −1 x + y + z dxdydz Associated graph: x ◦ F3F1 y ◦ F2 ◦ z Note: this graph is not bipartite
- 53. Part 2 — A manageable modification | |||| |||| |||| Quadrilinear variant: Λ(F1, F2, F3, F4) = R4 F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy
- 54. Part 2 — A manageable modification | |||| |||| |||| Quadrilinear variant: Λ(F1, F2, F3, F4) = R4 F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy Associated graph: x ◦ F3 F4 ◦ F2 y v ◦ F1 ◦ u
- 55. Part 2 — A manageable modification | |||| |||| |||| Quadrilinear variant: Λ(F1, F2, F3, F4) = R4 F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy Associated graph: x ◦ F3 F4 ◦ F2 y v ◦ F1 ◦ u x ◦ F3 F2 ◦ F4 y u ◦ F1 ◦ v Note: this graph is bipartite
- 56. Part 3 — Dyadic model operators || |||| |||| |||| Scope of our techniques
- 57. Part 3 — Dyadic model operators || |||| |||| |||| Scope of our techniques We specialize to: bipartite graphs multilinear Calder´on-Zygmund kernels K “perfect” dyadic models
- 58. Part 3 — Perfect dyadic conditions ||| |||| |||| |||| m, n = positive integers D := (x, . . . , x m , y, . . . , y n ) : x, y ∈ R the “diagonal” in Rm+n
- 59. Part 3 — Perfect dyadic conditions ||| |||| |||| |||| m, n = positive integers D := (x, . . . , x m , y, . . . , y n ) : x, y ∈ R the “diagonal” in Rm+n Perfect dyadic Calder´on-Zygmund kernel K : Rm+n → C, Auscher, Hofmann, Muscalu, Tao, Thiele (2002): |K(x1, . . . , xm, y1, . . . , yn)| i1<i2 |xi1 − xi2 | + j1<j2 |yj1 − yj2 | 2−m−n K is constant on (m+n)-dimensional dyadic cubes disjoint from D K is bounded and compactly supported
- 60. Part 3 — Bipartite structure |||| |||| |||| |||| E ⊆ {1, . . . , m}×{1, . . . , n} G = simple bipartite undirected graph on {x1, . . . , xm} and {y1, . . . , yn} xi —yj ⇔ (i, j) ∈ E
- 61. Part 3 — Bipartite structure |||| |||| |||| |||| E ⊆ {1, . . . , m}×{1, . . . , n} G = simple bipartite undirected graph on {x1, . . . , xm} and {y1, . . . , yn} xi —yj ⇔ (i, j) ∈ E |E|-linear singular form: Λ (Fi,j )(i,j)∈E := Rm+n K(x1, . . . , xm, y1, . . . , yn) (i,j)∈E Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn Assume: there are no isolated vertices in G avoids degeneracy
- 62. Part 3 — Adjoints |||| |||| |||| |||| There are |E| mutually adjoint (|E|−1)-linear operators Tu,v , (u, v) ∈ E: Λ (Fi,j )(i,j)∈E = R2 Tu,v (Fi,j )(i,j)=(u,v) Fu,v
- 63. Part 3 — Adjoints |||| |||| |||| |||| There are |E| mutually adjoint (|E|−1)-linear operators Tu,v , (u, v) ∈ E: Λ (Fi,j )(i,j)∈E = R2 Tu,v (Fi,j )(i,j)=(u,v) Fu,v Explicitly: Tu,v (Fi,j )(i,j)∈E{(u,v)} (xu, yv ) = Rm+n−2 K(x1, . . . , xm, y1, . . . , yn) (i,j)∈E{(u,v)} Fi,j (xi , yj ) i=u dxi j=v dyj
- 64. Part 3 — A T(1)-type theorem | |||| |||| |||| |||| Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013) (a) For m, n ≥ 2 and a graph G there exist positive integers di,j such that (i,j)∈E 1 di,j > 1 and the following holds. If |Λ(1Q, . . . , 1Q)| |Q|, Q dyadic square, Tu,v (1R2 , . . . , 1R2 ) BMO(R2) 1, (u, v) ∈ E, then Λ (Fi,j )(i,j)∈E (i,j)∈E Fi,j L pi,j (R2) for exponents pi,j s.t. (i,j)∈E 1 pi,j = 1, di,j < pi,j ≤ ∞.
- 65. Part 3 — A T(1)-type theorem | |||| |||| |||| |||| Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013) (a) For m, n ≥ 2 and a graph G there exist positive integers di,j such that (i,j)∈E 1 di,j > 1 and the following holds. If |Λ(1Q, . . . , 1Q)| |Q|, Q dyadic square, Tu,v (1R2 , . . . , 1R2 ) BMO(R2) 1, (u, v) ∈ E, then Λ (Fi,j )(i,j)∈E (i,j)∈E Fi,j L pi,j (R2) for exponents pi,j s.t. (i,j)∈E 1 pi,j = 1, di,j < pi,j ≤ ∞. (b) Conversely, the estimate for some choice of exponents implies the conditions.
- 66. Part 3 — A T(1)-type theorem, reformulation|| |||| |||| |||| |||| Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013) For m, n ≥ 2 and a graph G there exist positive integers di,j such that (i,j)∈E 1 di,j > 1 and the following holds. If Tu,v (1Q, . . . , 1Q) L1 (Q) |Q|, Q dyadic square, (u, v) ∈ E, then Λ (Fi,j )(i,j)∈E (i,j)∈E Fi,j L pi,j (R2) for exponents pi,j s.t. (i,j)∈E 1 pi,j = 1, di,j < pi,j ≤ ∞.
- 67. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions
- 68. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof:
- 69. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof: decomposition into paraproducts
- 70. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof: decomposition into paraproducts a stopping time argument for reducing global estimates to local estimates
- 71. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof: decomposition into paraproducts a stopping time argument for reducing global estimates to local estimates cancellative paraproducts with ∞ coefficients “most” cases of graphs G di,j related to sizes of connected components of G stuctural induction + Bellman function technique exceptional cases of graphs G
- 72. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof: decomposition into paraproducts a stopping time argument for reducing global estimates to local estimates cancellative paraproducts with ∞ coefficients “most” cases of graphs G di,j related to sizes of connected components of G stuctural induction + Bellman function technique exceptional cases of graphs G non-cancellative paraproducts with BMO coefficients reduction to cancellative paraproducts
- 73. Part 3 — Proof outline ||| |||| |||| |||| |||| The only nonstandard part — sufficiency of the testing conditions Scheme of the proof: decomposition into paraproducts a stopping time argument for reducing global estimates to local estimates cancellative paraproducts with ∞ coefficients “most” cases of graphs G di,j related to sizes of connected components of G stuctural induction + Bellman function technique exceptional cases of graphs G non-cancellative paraproducts with BMO coefficients reduction to cancellative paraproducts counterexample for m = 1 or n = 1
- 74. Part 3 — Multilinear Bellman functions |||| |||| |||| |||| |||| Bellman functions in harmonic analysis Invented by Burkholder (1980s) Developed by Nazarov, Treil, Volberg, etc. (1990s) We only keep the “induction on scales” idea
- 75. Part 3 — Multilinear Bellman functions |||| |||| |||| |||| |||| Bellman functions in harmonic analysis Invented by Burkholder (1980s) Developed by Nazarov, Treil, Volberg, etc. (1990s) We only keep the “induction on scales” idea A broad class of interesting dyadic objects can be reduced to bounding expressions of the form ΛT (F1, . . . , F ) = Q∈T |Q| AQ(F1, . . . , F ) T = a finite convex tree of dyadic squares AQ(F1, . . . , F ) = some “scale-invariant” quantity depending on F1, . . . , F and Q ∈ T
- 76. Part 3 — Calculus of finite differences |||| |||| |||| |||| |||| B = BQ(F1, . . . , F ) First order difference of B: B = BQ(F1, . . . , F ) BI×J := 1 4BIleft×Jleft + 1 4BIleft×Jright + 1 4BIright×Jleft + 1 4BIright×Jright − BI×J
- 77. Part 3 — Calculus of finite differences |||| |||| |||| |||| |||| B = BQ(F1, . . . , F ) First order difference of B: B = BQ(F1, . . . , F ) BI×J := 1 4BIleft×Jleft + 1 4BIleft×Jright + 1 4BIright×Jleft + 1 4BIright×Jright − BI×J Suppose: |A| ≤ B, i.e. |AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F ) for all Q ∈ T and nonnegative bounded measurable F1, . . . , F
- 78. Part 3 — Calculus of finite differences |||| |||| |||| |||| |||| |AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F ) |Q| |AQ(F1, . . . , F )| ≤ Q is a child of Q |Q| BQ (F1, . . . , F ) − |Q| BQ(F1, . . . , F )
- 79. Part 3 — Calculus of finite differences |||| |||| |||| |||| |||| |AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F ) |Q| |AQ(F1, . . . , F )| ≤ Q is a child of Q |Q| BQ (F1, . . . , F ) − |Q| BQ(F1, . . . , F ) |ΛT (F1, . . . , F )| ≤ Q∈L(T ) |Q| BQ(F1, . . . , F ) − |QT | BQT (F1, . . . , F ) B = a Bellman function for ΛT
- 80. Part 4 — Ordinary paraproduct | |||| |||| |||| |||| |||| Dyadic version Td(f , g) := k∈Z (Ekf )(∆kg) Ekf := |I|=2−k 1 |I| I f 1I , ∆kg := Ek+1g − Ekg
- 81. Part 4 — Ordinary paraproduct | |||| |||| |||| |||| |||| Dyadic version Td(f , g) := k∈Z (Ekf )(∆kg) Ekf := |I|=2−k 1 |I| I f 1I , ∆kg := Ek+1g − Ekg Continuous version Tc(f , g) := k∈Z (Pϕk f )(Pψk g) Pϕk f := f ∗ ϕk, Pψk g := g ∗ ψk ϕ, ψ Schwartz, supp( ˆψ) ⊆ {ξ ∈ R : 1 2 ≤|ξ| ≤ 2} ϕk(t) := 2kϕ(2kt), ψk(t) := 2kψ(2kt)
- 82. Part 4 — Twisted paraproduct || |||| |||| |||| |||| |||| Dyadic version Td(F, G) := k∈Z (E (1) k F)(∆ (2) k G) E (1) k martingale averages in the 1st variable ∆ (2) k martingale differences in the 2nd variable
- 83. Part 4 — Twisted paraproduct || |||| |||| |||| |||| |||| Dyadic version Td(F, G) := k∈Z (E (1) k F)(∆ (2) k G) E (1) k martingale averages in the 1st variable ∆ (2) k martingale differences in the 2nd variable Continuous version Tc(F, G) := k∈Z (P(1) ϕk F)(P (2) ψk G) P (1) ϕk , P (2) ψk L-P projections in the 1st and the 2nd variable (P (1) ϕk F)(x, y) := R F(x−t, y)ϕk(t)dt (P (2) ψk G)(x, y) := R G(x, y −t)ψk(t)dt
- 84. Part 4 — Twisted paraproduct || |||| |||| |||| |||| |||| Dyadic version Td(F, G) := k∈Z (E (1) k F)(∆ (2) k G) Continuous version Tc(F, G) := k∈Z (P(1) ϕk F)(P (2) ψk G) Bilinear multipliers from our theorems reduce to these using cone decomposition of the symbol: m = j m[j] from the Fourier series m[j] (ξ1, η2) = k∈Z ϕ [j] k (ξ1) ψ [j] k (η2)
- 85. Part 4 — Twisted paraproduct, estimates ||| |||| |||| |||| |||| |||| B( ), _ 2 1 C( )_ 2 1 , _ 2 1 1 2 _,1 4 _ )(E D( )_ 2 1 , 0 0,1 2 _ )(A _ 4 1 _1 q p 1_ 1 0 10 the shaded region – the strong estimate two solid sides of the square – the weak estimate two dashed sides of the square – no estimates the white region – unresolved
- 86. Part 4 — Proof outline |||| |||| |||| |||| |||| |||| B( ), _ 2 1 C( )_ 2 1 , _ 2 1 1 2 _,1 4 _ )(E D( )_ 2 1 , 0 0,1 2 _ )(A _ 4 1 _1 q p 1_ 1 0 10 Dyadic version Td ABC – a very special case of the technique in Part 3 the rest of the shaded region – conditional proof, F. Bernicot (2010) dashed segments – counterexamples D, E – an alternative purely Bellman function proof Continuous version Tc transition using the Jones-Seeger-Wright square function
- 87. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| |||| Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1 The general case is then obtained by composing with a bounded Fourier multiplier in the second variable
- 88. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| |||| Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1 The general case is then obtained by composing with a bounded Fourier multiplier in the second variable A. Calder´on (1960s), R. L. Jones, A. Seeger and J. Wright (2008) If ϕ is Schwartz and R ϕ = 1, then the square function SF := k∈Z Pϕk F − EkF 2 1/2 satisfies SF Lp (R) p F Lp (R) for 1 < p < ∞.
- 89. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| |||| Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1 The general case is then obtained by composing with a bounded Fourier multiplier in the second variable A. Calder´on (1960s), R. L. Jones, A. Seeger and J. Wright (2008) If ϕ is Schwartz and R ϕ = 1, then the square function SF := k∈Z Pϕk F − EkF 2 1/2 satisfies SF Lp (R) p F Lp (R) for 1 < p < ∞. Proposition Tc(F, G) − Td(F, G) Lpq/(p+q) p,q F Lp G Lq
- 90. Part 4 — “Entangled” + cont. kernel | |||| |||| |||| |||| |||| |||| General bipartite graphs G How to obtain boundedness of Λ (Fi,j )(i,j)∈E := Rm+n K(x1, . . . , xm, y1, . . . , yn) (i,j)∈E Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn at least for some continuous singular kernels K?
- 91. Part 4 — “Entangled” + cont. kernel | |||| |||| |||| |||| |||| |||| General bipartite graphs G How to obtain boundedness of Λ (Fi,j )(i,j)∈E := Rm+n K(x1, . . . , xm, y1, . . . , yn) (i,j)∈E Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn at least for some continuous singular kernels K? We can average “entangled” dyadic operators from Part 3 over translated, dilated, and rotated dyadic grids Partial results: One can recover some very special kernels K Possibly all sufficiently smooth translation-invariant kernels This is still far from a complete T(1)-type theorem
- 92. Currently open problems || |||| |||| |||| |||| |||| |||| Further directions:
- 93. Currently open problems || |||| |||| |||| |||| |||| |||| Further directions: Translating the results to the case of more general continuous C-Z kernels K Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type theorem
- 94. Currently open problems || |||| |||| |||| |||| |||| |||| Further directions: Translating the results to the case of more general continuous C-Z kernels K Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type theorem Forms corresponding to non-bipartite graphs (such as odd cycles, recall a triangle)
- 95. Currently open problems || |||| |||| |||| |||| |||| |||| Further directions: Translating the results to the case of more general continuous C-Z kernels K Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type theorem Forms corresponding to non-bipartite graphs (such as odd cycles, recall a triangle) More singular kernels K, like K(x, y, z) = 1 x+y+z
- 96. Thank you! ||| |||| |||| |||| |||| |||| |||| Thank you!