SlideShare a Scribd company logo
1 of 27
Download to read offline
Scattering theory analogues of several
classical estimates in Fourier analysis
Vjekoslav Kovaˇc (University of Zagreb)
Joint work with Jelena Rupˇci´c (University of Zagreb)
6th Croatian Mathematical Congress
Zagreb, June 15, 2016
Classical Fourier analysis
The trigonometric/Fourier series
a(t) =
+∞
n=−∞
Ane2πint
(at least formally)
Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1)
Convergence? In which sense? Under which conditions?
Classical Fourier analysis
The trigonometric/Fourier series
a(t) =
+∞
n=−∞
Ane2πint
(at least formally)
Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1)
Convergence? In which sense? Under which conditions?
The Fourier transform
f (ξ) :=
+∞
−∞
f (x)e−2πixξ
dx (at least formally)
Suppose that f is locally integrable and supported in [0, +∞)
d
dx a(x, ξ) = f (x)e−2πixξ
, a(0, ξ) = 0
=⇒ a(x, ξ) =
x
0 f (y)e−2πiyξdy = f 1[0,x](ξ), “a(+∞, ξ) = f (ξ)”
Nonlinear Fourier analysis
The SU(1,1) trigonometric product
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
An Bne2πint
Bne−2πint An
An > 0, Bn ∈ C, A2
n − |Bn|2 = 1
Nonlinear Fourier analysis
The SU(1,1) trigonometric product
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
An Bne2πint
Bne−2πint An
An > 0, Bn ∈ C, A2
n − |Bn|2 = 1
The SU(1,1) Fourier transform / the Dirac scattering transform
d
dx
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
=
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
0 f (x)e−2πixξ
f (x)e2πixξ 0
a(0, ξ) b(0, ξ)
b(0, ξ) a(0, ξ)
=
1 0
0 1
, “ f (ξ) =
a(+∞, ξ) b(+∞, ξ)
b(+∞, ξ) a(+∞, ξ)
”
Suppose that f is locally integrable and supported in [0, +∞)
=⇒ a(·, ξ) and b(·, ξ) exist as absolutely continuous solutions
Nonlinear Fourier analysis
SU(1, 1) :=
A B
B A
: A, B ∈ C, |A|2
− |B|2
= 1
aN(t) bN(t)
bN(t) aN(t)
SU(1,1)
=
N
n=0
An Bne2πint
Bne−2πint An
SU(1,1)
Nonlinear Fourier analysis
SU(1, 1) :=
A B
B A
: A, B ∈ C, |A|2
− |B|2
= 1
aN(t) bN(t)
bN(t) aN(t)
SU(1,1)
=
N
n=0
An Bne2πint
Bne−2πint An
SU(1,1)
su(1, 1) =
A B
B A
: A, B ∈ C, A ∈ iR
d
dx
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
SU(1,1)
=
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
SU(1,1)
0 f (x)e−2πixξ
f (x)e2πixξ 0
su(1,1)
This is not the linear Fourier transform on SU(1,1)!
Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
Integral equations:
a(x, ξ) = 1 +
x
0
f (y)e2πiyξ
b(y, ξ)dy
b(x, ξ) =
x
0
f (y)e−2πiyξ
a(y, ξ)dy
Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
Integral equations:
a(x, ξ) = 1 +
x
0
f (y)e2πiyξ
b(y, ξ)dy
b(x, ξ) =
x
0
f (y)e−2πiyξ
a(y, ξ)dy
Picard’s iteration gives certain multilinear series expansions
Born’s approximation: b(x, ξ) ≈ f 1[0,x](ξ) when f L1
(R) 1
We care about the long term behavior and cannot linearize!
Motivation
Eigenproblem for the Dirac operator:
L :=
d
dx −¯f
f − d
dx
, i.e. L
ϕ
ψ
=
ϕ − ¯f ψ
f ϕ − ψ
Motivation
Eigenproblem for the Dirac operator:
L :=
d
dx −¯f
f − d
dx
, i.e. L
ϕ
ψ
=
ϕ − ¯f ψ
f ϕ − ψ
L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem:
L
ϕ(·, ξ)
ψ(·, ξ)
= −πiξ
ϕ(·, ξ)
ψ(·, ξ)
Motivation
Eigenproblem for the Dirac operator:
L :=
d
dx −¯f
f − d
dx
, i.e. L
ϕ
ψ
=
ϕ − ¯f ψ
f ϕ − ψ
L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem:
L
ϕ(·, ξ)
ψ(·, ξ)
= −πiξ
ϕ(·, ξ)
ψ(·, ξ)
i.e. ∂x ϕ(x, ξ) + πiξϕ(x, ξ) = f (x)ψ(x, ξ)
∂x ψ(x, ξ) − πiξψ(x, ξ) = f (x)ϕ(x, ξ)
i.e. ∂x ϕ(x, ξ)eπixξ
a(x,ξ)
= f (x)e2πixξ
ψ(x, ξ)e−πixξ
b(x,ξ)
∂x ψ(x, ξ)e−πixξ
b(x,ξ)
= f (x)e−2πixξ
ϕ(x, ξ)eπixξ
a(x,ξ)
Carleson’s theorem
Classical/linear — Carleson (1966)
(An) ∈ 2
(Z) =⇒ lim
N→+∞
N
n=−N
Ane2πint
exists for a.e. t ∈ T
f ∈ L2
(R) =⇒ lim
R→+∞
+R
−R
f (x)e−2πixξ
dx exists for a.e. ξ ∈ R
Carleson’s theorem
Classical/linear — Carleson (1966)
(An) ∈ 2
(Z) =⇒ lim
N→+∞
N
n=−N
Ane2πint
exists for a.e. t ∈ T
f ∈ L2
(R) =⇒ lim
R→+∞
+R
−R
f (x)e−2πixξ
dx exists for a.e. ξ ∈ R
Nonlinear analogues — Open question
+∞
n=0
|Bn|2
< ∞
?
=⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
f ∈ L2
(R)
?
=⇒ lim
x→+∞
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
exists for a.e. ξ ∈ R
Even finiteness of supx∈[0,+∞) |a(x, ξ)| for a.e. ξ ∈ R is open
Muscalu, Tao, Thiele (2002): the Cantor group “toy-model”
Hausdorff-Young inequalities
Classical/linear — Young (1913), Hausdorff (1923)
1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp
(R)
≤ f Lp
(R)
Babenko (1961), Beckner (1975)
1 < p < 2 =⇒ f Lp
(R)
≤ p1/2p
p
1/2p
<1
f Lp
(R)
Hausdorff-Young inequalities
Classical/linear — Young (1913), Hausdorff (1923)
1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp
(R)
≤ f Lp
(R)
Babenko (1961), Beckner (1975)
1 < p < 2 =⇒ f Lp
(R)
≤ p1/2p
p
1/2p
<1
f Lp
(R)
Nonlinear analogues — Christ and Kiselev (2001)
1 ≤ p ≤ 2 =⇒ (log |a(+∞, ·)|2
)1/2
Lp
(R)
≤ Cp f Lp
(R)
p = 1 trivial by Gronwall’s lemma
p = 2 an identity (with C2 = 1) by the contour integration
Open question: Does Cp stay bounded as p ↑ 2?
K. (2010): confirmed in the Cantor group “toy-model”
Lacunary trigonometric series
1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj
Norm convergence — Zygmund (1920s)
N
j=1
Aj e2πimj t
Lp
t (T)
∼p,q
N
j=1
|Aj |2
1/2
, 0 < p < ∞
∞
j=1
Aj e2πimj t
converges in Lp
⇐⇒ (Aj ) ∈ 2
(N)
Lacunary trigonometric series
1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj
Norm convergence — Zygmund (1920s)
N
j=1
Aj e2πimj t
Lp
t (T)
∼p,q
N
j=1
|Aj |2
1/2
, 0 < p < ∞
∞
j=1
Aj e2πimj t
converges in Lp
⇐⇒ (Aj ) ∈ 2
(N)
Convergence a.e. — Kolmogorov (1924)
(Aj ) ∈ 2
(N) =⇒
∞
j=1
Aj e2πimj t
converges for a.e. t ∈ T
Converse of convergence a.e. — Zygmund (1930)
∞
j=1
Aj e2πimj t
conv. on a set of measure > 0 =⇒ (Aj ) ∈ 2
(N)
Lacunary SU(1,1) trigonometric products
ρ: SU(1, 1) × SU(1, 1) → [0, +∞)
ρ(G1, G2) := log 1 + G−1
1 G2 − I op
ρ is a complete metric on SU(1, 1)
Lacunary SU(1,1) trigonometric products
ρ: SU(1, 1) × SU(1, 1) → [0, +∞)
ρ(G1, G2) := log 1 + G−1
1 G2 − I op
ρ is a complete metric on SU(1, 1)
dp(g1, g2) :=



ρ(g1(t), g2(t)) Lp
t (T) for 1 ≤ p < ∞
ρ(g1(t), g2(t)) p
Lp
t (T)
for 0 < p < 1
Lp
(T, SU(1, 1)) := g : T → SU(1, 1) : dp(I, g) < +∞
dp is a complete metric on Lp
(T, SU(1, 1))
Lacunary SU(1,1) trigonometric products
1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
Aj Bj e2πimj t
Bj e−2πimj t Aj
Lacunary SU(1,1) trigonometric products
1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
Aj Bj e2πimj t
Bj e−2πimj t Aj
K. and Rupˇci´c (2016)
Assume q ≥ 2 and take 0 < p < ∞
lim
N→+∞
aN bN
bN aN
exists in dp ⇐⇒
∞
j=1
|Bj |2
< +∞
Recall ∞
j=1 |Bj |2 < +∞ ⇐⇒ ∞
j=1 log Aj < +∞
⇐⇒ ∞
j=1(A2
j + |Bj |2) < +∞
Lacunary SU(1,1) trigonometric products
K. and Rupˇci´c (2016)
Assume q ≥ 2
∞
j=1
|Bj |2
< +∞ =⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
Lacunary SU(1,1) trigonometric products
K. and Rupˇci´c (2016)
Assume q ≥ 2
∞
j=1
|Bj |2
< +∞ =⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
K. and Rupˇci´c (2016)
Assume q ≥ 3
lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists on a set of measure > 0
=⇒
∞
j=1
|Bj |2
< +∞
References
Introductory literature:
Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City
Graduate Summer School, unpublished lecture notes, 2003,
available at arXiv:1201.5129 [math.CA]
Tao, An introduction to the nonlinear Fourier transform,
unpublished note, 2002
Muscalu, Tao, Thiele, several papers, 2001–2007
Ablowitz, Kaup, Newell, Segur, The inverse scattering
transform — Fourier analysis for nonlinear problems,
Stud. Appl. Math. 53 (1974), 249–315
References
Introductory literature:
Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City
Graduate Summer School, unpublished lecture notes, 2003,
available at arXiv:1201.5129 [math.CA]
Tao, An introduction to the nonlinear Fourier transform,
unpublished note, 2002
Muscalu, Tao, Thiele, several papers, 2001–2007
Ablowitz, Kaup, Newell, Segur, The inverse scattering
transform — Fourier analysis for nonlinear problems,
Stud. Appl. Math. 53 (1974), 249–315
Thank you for your attention!

More Related Content

What's hot

Density theorems for Euclidean point configurations
Density theorems for Euclidean point configurationsDensity theorems for Euclidean point configurations
Density theorems for Euclidean point configurationsVjekoslavKovac1
 
Paraproducts with general dilations
Paraproducts with general dilationsParaproducts with general dilations
Paraproducts with general dilationsVjekoslavKovac1
 
On Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular IntegralsOn Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
 
Bellman functions and Lp estimates for paraproducts
Bellman functions and Lp estimates for paraproductsBellman functions and Lp estimates for paraproducts
Bellman functions and Lp estimates for paraproductsVjekoslavKovac1
 
A Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cubeA Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
 
Estimates for a class of non-standard bilinear multipliers
Estimates for a class of non-standard bilinear multipliersEstimates for a class of non-standard bilinear multipliers
Estimates for a class of non-standard bilinear multipliersVjekoslavKovac1
 
Density theorems for anisotropic point configurations
Density theorems for anisotropic point configurationsDensity theorems for anisotropic point configurations
Density theorems for anisotropic point configurationsVjekoslavKovac1
 
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsA T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
 
On maximal and variational Fourier restriction
On maximal and variational Fourier restrictionOn maximal and variational Fourier restriction
On maximal and variational Fourier restrictionVjekoslavKovac1
 
Trilinear embedding for divergence-form operators
Trilinear embedding for divergence-form operatorsTrilinear embedding for divergence-form operators
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
 
Multilinear Twisted Paraproducts
Multilinear Twisted ParaproductsMultilinear Twisted Paraproducts
Multilinear Twisted ParaproductsVjekoslavKovac1
 
Quantitative norm convergence of some ergodic averages
Quantitative norm convergence of some ergodic averagesQuantitative norm convergence of some ergodic averages
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
 
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
 
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
 
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Rene Kotze
 

What's hot (20)

Density theorems for Euclidean point configurations
Density theorems for Euclidean point configurationsDensity theorems for Euclidean point configurations
Density theorems for Euclidean point configurations
 
Paraproducts with general dilations
Paraproducts with general dilationsParaproducts with general dilations
Paraproducts with general dilations
 
On Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular IntegralsOn Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular Integrals
 
Bellman functions and Lp estimates for paraproducts
Bellman functions and Lp estimates for paraproductsBellman functions and Lp estimates for paraproducts
Bellman functions and Lp estimates for paraproducts
 
A Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cubeA Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cube
 
Estimates for a class of non-standard bilinear multipliers
Estimates for a class of non-standard bilinear multipliersEstimates for a class of non-standard bilinear multipliers
Estimates for a class of non-standard bilinear multipliers
 
Density theorems for anisotropic point configurations
Density theorems for anisotropic point configurationsDensity theorems for anisotropic point configurations
Density theorems for anisotropic point configurations
 
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsA T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
 
On maximal and variational Fourier restriction
On maximal and variational Fourier restrictionOn maximal and variational Fourier restriction
On maximal and variational Fourier restriction
 
Trilinear embedding for divergence-form operators
Trilinear embedding for divergence-form operatorsTrilinear embedding for divergence-form operators
Trilinear embedding for divergence-form operators
 
Multilinear Twisted Paraproducts
Multilinear Twisted ParaproductsMultilinear Twisted Paraproducts
Multilinear Twisted Paraproducts
 
Quantitative norm convergence of some ergodic averages
Quantitative norm convergence of some ergodic averagesQuantitative norm convergence of some ergodic averages
Quantitative norm convergence of some ergodic averages
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
 Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli... Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
 
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
 
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
 
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
 

Similar to Scattering theory analogues of several classical estimates in Fourier analysis

An Efficient Boundary Integral Method for Stiff Fluid Interface Problems
An Efficient Boundary Integral Method for Stiff Fluid Interface ProblemsAn Efficient Boundary Integral Method for Stiff Fluid Interface Problems
An Efficient Boundary Integral Method for Stiff Fluid Interface ProblemsAlex (Oleksiy) Varfolomiyev
 
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihoodDeep Learning JP
 
Phase diagram at finite T & Mu in strong coupling limit of lattice QCD
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDPhase diagram at finite T & Mu in strong coupling limit of lattice QCD
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDBenjamin Jaedon Choi
 
Introducing Zap Q-Learning
Introducing Zap Q-Learning   Introducing Zap Q-Learning
Introducing Zap Q-Learning Sean Meyn
 
University of manchester mathematical formula tables
University of manchester mathematical formula tablesUniversity of manchester mathematical formula tables
University of manchester mathematical formula tablesGaurav Vasani
 
dhirota_hone_corrected
dhirota_hone_correcteddhirota_hone_corrected
dhirota_hone_correctedAndy Hone
 
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...Edmundo José Huertas Cejudo
 
A Fibonacci-like universe expansion on time-scale
A Fibonacci-like universe expansion on time-scaleA Fibonacci-like universe expansion on time-scale
A Fibonacci-like universe expansion on time-scaleOctavianPostavaru
 
Mathematical formula tables
Mathematical formula tablesMathematical formula tables
Mathematical formula tablesSaravana Selvan
 
Geometric and viscosity solutions for the Cauchy problem of first order
Geometric and viscosity solutions for the Cauchy problem of first orderGeometric and viscosity solutions for the Cauchy problem of first order
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
 
Non-Hermitian noncommutative models in quantum optics and their superiorities
Non-Hermitian noncommutative models in quantum optics and their superioritiesNon-Hermitian noncommutative models in quantum optics and their superiorities
Non-Hermitian noncommutative models in quantum optics and their superioritiesSanjib Dey
 
Complexity of exact solutions of many body systems: nonequilibrium steady sta...
Complexity of exact solutions of many body systems: nonequilibrium steady sta...Complexity of exact solutions of many body systems: nonequilibrium steady sta...
Complexity of exact solutions of many body systems: nonequilibrium steady sta...Lake Como School of Advanced Studies
 
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...OctavianPostavaru
 
Signals and Systems Formula Sheet
Signals and Systems Formula SheetSignals and Systems Formula Sheet
Signals and Systems Formula SheetHaris Hassan
 
ch9.pdf
ch9.pdfch9.pdf
ch9.pdfKavS14
 
On the Jensen-Shannon symmetrization of distances relying on abstract means
On the Jensen-Shannon symmetrization of distances relying on abstract meansOn the Jensen-Shannon symmetrization of distances relying on abstract means
On the Jensen-Shannon symmetrization of distances relying on abstract meansFrank Nielsen
 

Similar to Scattering theory analogues of several classical estimates in Fourier analysis (20)

2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...
2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...
2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...
 
An Efficient Boundary Integral Method for Stiff Fluid Interface Problems
An Efficient Boundary Integral Method for Stiff Fluid Interface ProblemsAn Efficient Boundary Integral Method for Stiff Fluid Interface Problems
An Efficient Boundary Integral Method for Stiff Fluid Interface Problems
 
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood
【DL輪読会】Unbiased Gradient Estimation for Marginal Log-likelihood
 
Phase diagram at finite T & Mu in strong coupling limit of lattice QCD
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDPhase diagram at finite T & Mu in strong coupling limit of lattice QCD
Phase diagram at finite T & Mu in strong coupling limit of lattice QCD
 
Introducing Zap Q-Learning
Introducing Zap Q-Learning   Introducing Zap Q-Learning
Introducing Zap Q-Learning
 
University of manchester mathematical formula tables
University of manchester mathematical formula tablesUniversity of manchester mathematical formula tables
University of manchester mathematical formula tables
 
dhirota_hone_corrected
dhirota_hone_correcteddhirota_hone_corrected
dhirota_hone_corrected
 
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...
Zeros of orthogonal polynomials generated by a Geronimus perturbation of meas...
 
A Fibonacci-like universe expansion on time-scale
A Fibonacci-like universe expansion on time-scaleA Fibonacci-like universe expansion on time-scale
A Fibonacci-like universe expansion on time-scale
 
Mathematical formula tables
Mathematical formula tablesMathematical formula tables
Mathematical formula tables
 
Maths04
Maths04Maths04
Maths04
 
Geometric and viscosity solutions for the Cauchy problem of first order
Geometric and viscosity solutions for the Cauchy problem of first orderGeometric and viscosity solutions for the Cauchy problem of first order
Geometric and viscosity solutions for the Cauchy problem of first order
 
Non-Hermitian noncommutative models in quantum optics and their superiorities
Non-Hermitian noncommutative models in quantum optics and their superioritiesNon-Hermitian noncommutative models in quantum optics and their superiorities
Non-Hermitian noncommutative models in quantum optics and their superiorities
 
Complexity of exact solutions of many body systems: nonequilibrium steady sta...
Complexity of exact solutions of many body systems: nonequilibrium steady sta...Complexity of exact solutions of many body systems: nonequilibrium steady sta...
Complexity of exact solutions of many body systems: nonequilibrium steady sta...
 
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...
Generalized fractional-order hybrid of block-pulse functions and Bernoulli po...
 
Signals and Systems Formula Sheet
Signals and Systems Formula SheetSignals and Systems Formula Sheet
Signals and Systems Formula Sheet
 
ch9.pdf
ch9.pdfch9.pdf
ch9.pdf
 
On the Jensen-Shannon symmetrization of distances relying on abstract means
On the Jensen-Shannon symmetrization of distances relying on abstract meansOn the Jensen-Shannon symmetrization of distances relying on abstract means
On the Jensen-Shannon symmetrization of distances relying on abstract means
 
Improper integral
Improper integralImproper integral
Improper integral
 
Imc2016 day2-solutions
Imc2016 day2-solutionsImc2016 day2-solutions
Imc2016 day2-solutions
 

Recently uploaded

Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest Management
Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest ManagementPests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest Management
Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest ManagementPirithiRaju
 
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...Thane Heins
 
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...Sharon Liu
 
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...dkNET
 
Description of cultivating Duckweed Syllabus.pdf
Description of cultivating Duckweed Syllabus.pdfDescription of cultivating Duckweed Syllabus.pdf
Description of cultivating Duckweed Syllabus.pdfHaim R. Branisteanu
 
Project report on Fasciola hepatica.docx
Project report on Fasciola hepatica.docxProject report on Fasciola hepatica.docx
Project report on Fasciola hepatica.docxpriyanshimanchanda4
 
Advance pharmacology presentation..............
Advance pharmacology presentation..............Advance pharmacology presentation..............
Advance pharmacology presentation..............SIMRAN VERMA
 
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...Thane Heins
 
Phagocytosis Pinocytosis detail presentation
Phagocytosis Pinocytosis detail presentationPhagocytosis Pinocytosis detail presentation
Phagocytosis Pinocytosis detail presentationdhaduknevil1
 
Science9 Quarter 3:Latitude and altitude.pptx
Science9 Quarter 3:Latitude and altitude.pptxScience9 Quarter 3:Latitude and altitude.pptx
Science9 Quarter 3:Latitude and altitude.pptxteleganne21
 
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptx
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptxAKSHITA A R ECOLOGICAL NICHE and Gauss lawpptx
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptxharichikku1713
 
Skin: Structure and function of the skin
Skin: Structure and function of the skinSkin: Structure and function of the skin
Skin: Structure and function of the skinheenarahangdale01
 
Introduction to Green chemistry ppt.pptx
Introduction to Green chemistry ppt.pptxIntroduction to Green chemistry ppt.pptx
Introduction to Green chemistry ppt.pptxMuskan219429
 
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...ORAU
 
Cultivating various strains of Duckweed Syllabus.pdf
Cultivating various strains of Duckweed Syllabus.pdfCultivating various strains of Duckweed Syllabus.pdf
Cultivating various strains of Duckweed Syllabus.pdfHaim R. Branisteanu
 
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.kapgateprachi@gmail.com
 
Development of a Questionnaire for Identifying Personal Values in Driving
Development of a Questionnaire for Identifying Personal Values in DrivingDevelopment of a Questionnaire for Identifying Personal Values in Driving
Development of a Questionnaire for Identifying Personal Values in Drivingstudiotelon
 
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...klada0003
 
Advances in Oncolytic Virotherapy - Creative Biolabs
Advances in Oncolytic Virotherapy - Creative BiolabsAdvances in Oncolytic Virotherapy - Creative Biolabs
Advances in Oncolytic Virotherapy - Creative BiolabsCreative-Biolabs
 
Solid waste management_13_409_U1_2024.pptx
Solid waste management_13_409_U1_2024.pptxSolid waste management_13_409_U1_2024.pptx
Solid waste management_13_409_U1_2024.pptxkrishuchavda31032003
 

Recently uploaded (20)

Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest Management
Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest ManagementPests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest Management
Pests of Maize_Dr.UPR_Identification, Binomics, Integrated Pest Management
 
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
 
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...
20240315 ACMJ Diagrams Set 2.docx . With light, motor, coloured light, and se...
 
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...
dkNET Webinar "The Multi-Omic Response to Exercise Training Across Rat Tissue...
 
Description of cultivating Duckweed Syllabus.pdf
Description of cultivating Duckweed Syllabus.pdfDescription of cultivating Duckweed Syllabus.pdf
Description of cultivating Duckweed Syllabus.pdf
 
Project report on Fasciola hepatica.docx
Project report on Fasciola hepatica.docxProject report on Fasciola hepatica.docx
Project report on Fasciola hepatica.docx
 
Advance pharmacology presentation..............
Advance pharmacology presentation..............Advance pharmacology presentation..............
Advance pharmacology presentation..............
 
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
AI Published & MIT Validated Perpetual Motion Machine Breakthroughs (2 New EV...
 
Phagocytosis Pinocytosis detail presentation
Phagocytosis Pinocytosis detail presentationPhagocytosis Pinocytosis detail presentation
Phagocytosis Pinocytosis detail presentation
 
Science9 Quarter 3:Latitude and altitude.pptx
Science9 Quarter 3:Latitude and altitude.pptxScience9 Quarter 3:Latitude and altitude.pptx
Science9 Quarter 3:Latitude and altitude.pptx
 
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptx
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptxAKSHITA A R ECOLOGICAL NICHE and Gauss lawpptx
AKSHITA A R ECOLOGICAL NICHE and Gauss lawpptx
 
Skin: Structure and function of the skin
Skin: Structure and function of the skinSkin: Structure and function of the skin
Skin: Structure and function of the skin
 
Introduction to Green chemistry ppt.pptx
Introduction to Green chemistry ppt.pptxIntroduction to Green chemistry ppt.pptx
Introduction to Green chemistry ppt.pptx
 
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...
Non equilibrium Molecular Simulations of Polymers under Flow Saving Energy th...
 
Cultivating various strains of Duckweed Syllabus.pdf
Cultivating various strains of Duckweed Syllabus.pdfCultivating various strains of Duckweed Syllabus.pdf
Cultivating various strains of Duckweed Syllabus.pdf
 
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.
SHAMPOO : OVERVIEW OF SHAMPOO AND IT'S TYPES.
 
Development of a Questionnaire for Identifying Personal Values in Driving
Development of a Questionnaire for Identifying Personal Values in DrivingDevelopment of a Questionnaire for Identifying Personal Values in Driving
Development of a Questionnaire for Identifying Personal Values in Driving
 
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...
1David Andress - The Oxford Handbook of the French Revolution-Oxford Universi...
 
Advances in Oncolytic Virotherapy - Creative Biolabs
Advances in Oncolytic Virotherapy - Creative BiolabsAdvances in Oncolytic Virotherapy - Creative Biolabs
Advances in Oncolytic Virotherapy - Creative Biolabs
 
Solid waste management_13_409_U1_2024.pptx
Solid waste management_13_409_U1_2024.pptxSolid waste management_13_409_U1_2024.pptx
Solid waste management_13_409_U1_2024.pptx
 

Scattering theory analogues of several classical estimates in Fourier analysis

  • 1. Scattering theory analogues of several classical estimates in Fourier analysis Vjekoslav Kovaˇc (University of Zagreb) Joint work with Jelena Rupˇci´c (University of Zagreb) 6th Croatian Mathematical Congress Zagreb, June 15, 2016
  • 2. Classical Fourier analysis The trigonometric/Fourier series a(t) = +∞ n=−∞ Ane2πint (at least formally) Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1) Convergence? In which sense? Under which conditions?
  • 3. Classical Fourier analysis The trigonometric/Fourier series a(t) = +∞ n=−∞ Ane2πint (at least formally) Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1) Convergence? In which sense? Under which conditions? The Fourier transform f (ξ) := +∞ −∞ f (x)e−2πixξ dx (at least formally) Suppose that f is locally integrable and supported in [0, +∞) d dx a(x, ξ) = f (x)e−2πixξ , a(0, ξ) = 0 =⇒ a(x, ξ) = x 0 f (y)e−2πiyξdy = f 1[0,x](ξ), “a(+∞, ξ) = f (ξ)”
  • 4. Nonlinear Fourier analysis The SU(1,1) trigonometric product aN(t) bN(t) bN(t) aN(t) = N n=0 An Bne2πint Bne−2πint An An > 0, Bn ∈ C, A2 n − |Bn|2 = 1
  • 5. Nonlinear Fourier analysis The SU(1,1) trigonometric product aN(t) bN(t) bN(t) aN(t) = N n=0 An Bne2πint Bne−2πint An An > 0, Bn ∈ C, A2 n − |Bn|2 = 1 The SU(1,1) Fourier transform / the Dirac scattering transform d dx a(x, ξ) b(x, ξ) b(x, ξ) a(x, ξ) = a(x, ξ) b(x, ξ) b(x, ξ) a(x, ξ) 0 f (x)e−2πixξ f (x)e2πixξ 0 a(0, ξ) b(0, ξ) b(0, ξ) a(0, ξ) = 1 0 0 1 , “ f (ξ) = a(+∞, ξ) b(+∞, ξ) b(+∞, ξ) a(+∞, ξ) ” Suppose that f is locally integrable and supported in [0, +∞) =⇒ a(·, ξ) and b(·, ξ) exist as absolutely continuous solutions
  • 6. Nonlinear Fourier analysis SU(1, 1) := A B B A : A, B ∈ C, |A|2 − |B|2 = 1 aN(t) bN(t) bN(t) aN(t) SU(1,1) = N n=0 An Bne2πint Bne−2πint An SU(1,1)
  • 7. Nonlinear Fourier analysis SU(1, 1) := A B B A : A, B ∈ C, |A|2 − |B|2 = 1 aN(t) bN(t) bN(t) aN(t) SU(1,1) = N n=0 An Bne2πint Bne−2πint An SU(1,1) su(1, 1) = A B B A : A, B ∈ C, A ∈ iR d dx a(x, ξ) b(x, ξ) b(x, ξ) a(x, ξ) SU(1,1) = a(x, ξ) b(x, ξ) b(x, ξ) a(x, ξ) SU(1,1) 0 f (x)e−2πixξ f (x)e2πixξ 0 su(1,1) This is not the linear Fourier transform on SU(1,1)!
  • 8. Nonlinear Fourier analysis In the scalar form: ∂x a(x, ξ) = f (x)e2πixξ b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ a(x, ξ) a(0, ξ) = 1, b(0, ξ) = 0
  • 9. Nonlinear Fourier analysis In the scalar form: ∂x a(x, ξ) = f (x)e2πixξ b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ a(x, ξ) a(0, ξ) = 1, b(0, ξ) = 0 Integral equations: a(x, ξ) = 1 + x 0 f (y)e2πiyξ b(y, ξ)dy b(x, ξ) = x 0 f (y)e−2πiyξ a(y, ξ)dy
  • 10. Nonlinear Fourier analysis In the scalar form: ∂x a(x, ξ) = f (x)e2πixξ b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ a(x, ξ) a(0, ξ) = 1, b(0, ξ) = 0 Integral equations: a(x, ξ) = 1 + x 0 f (y)e2πiyξ b(y, ξ)dy b(x, ξ) = x 0 f (y)e−2πiyξ a(y, ξ)dy Picard’s iteration gives certain multilinear series expansions Born’s approximation: b(x, ξ) ≈ f 1[0,x](ξ) when f L1 (R) 1 We care about the long term behavior and cannot linearize!
  • 11. Motivation Eigenproblem for the Dirac operator: L := d dx −¯f f − d dx , i.e. L ϕ ψ = ϕ − ¯f ψ f ϕ − ψ
  • 12. Motivation Eigenproblem for the Dirac operator: L := d dx −¯f f − d dx , i.e. L ϕ ψ = ϕ − ¯f ψ f ϕ − ψ L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem: L ϕ(·, ξ) ψ(·, ξ) = −πiξ ϕ(·, ξ) ψ(·, ξ)
  • 13. Motivation Eigenproblem for the Dirac operator: L := d dx −¯f f − d dx , i.e. L ϕ ψ = ϕ − ¯f ψ f ϕ − ψ L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem: L ϕ(·, ξ) ψ(·, ξ) = −πiξ ϕ(·, ξ) ψ(·, ξ) i.e. ∂x ϕ(x, ξ) + πiξϕ(x, ξ) = f (x)ψ(x, ξ) ∂x ψ(x, ξ) − πiξψ(x, ξ) = f (x)ϕ(x, ξ) i.e. ∂x ϕ(x, ξ)eπixξ a(x,ξ) = f (x)e2πixξ ψ(x, ξ)e−πixξ b(x,ξ) ∂x ψ(x, ξ)e−πixξ b(x,ξ) = f (x)e−2πixξ ϕ(x, ξ)eπixξ a(x,ξ)
  • 14. Carleson’s theorem Classical/linear — Carleson (1966) (An) ∈ 2 (Z) =⇒ lim N→+∞ N n=−N Ane2πint exists for a.e. t ∈ T f ∈ L2 (R) =⇒ lim R→+∞ +R −R f (x)e−2πixξ dx exists for a.e. ξ ∈ R
  • 15. Carleson’s theorem Classical/linear — Carleson (1966) (An) ∈ 2 (Z) =⇒ lim N→+∞ N n=−N Ane2πint exists for a.e. t ∈ T f ∈ L2 (R) =⇒ lim R→+∞ +R −R f (x)e−2πixξ dx exists for a.e. ξ ∈ R Nonlinear analogues — Open question +∞ n=0 |Bn|2 < ∞ ? =⇒ lim N→+∞ aN(t) bN(t) bN(t) aN(t) exists for a.e. t ∈ T f ∈ L2 (R) ? =⇒ lim x→+∞ a(x, ξ) b(x, ξ) b(x, ξ) a(x, ξ) exists for a.e. ξ ∈ R Even finiteness of supx∈[0,+∞) |a(x, ξ)| for a.e. ξ ∈ R is open Muscalu, Tao, Thiele (2002): the Cantor group “toy-model”
  • 16. Hausdorff-Young inequalities Classical/linear — Young (1913), Hausdorff (1923) 1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp (R) ≤ f Lp (R) Babenko (1961), Beckner (1975) 1 < p < 2 =⇒ f Lp (R) ≤ p1/2p p 1/2p <1 f Lp (R)
  • 17. Hausdorff-Young inequalities Classical/linear — Young (1913), Hausdorff (1923) 1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp (R) ≤ f Lp (R) Babenko (1961), Beckner (1975) 1 < p < 2 =⇒ f Lp (R) ≤ p1/2p p 1/2p <1 f Lp (R) Nonlinear analogues — Christ and Kiselev (2001) 1 ≤ p ≤ 2 =⇒ (log |a(+∞, ·)|2 )1/2 Lp (R) ≤ Cp f Lp (R) p = 1 trivial by Gronwall’s lemma p = 2 an identity (with C2 = 1) by the contour integration Open question: Does Cp stay bounded as p ↑ 2? K. (2010): confirmed in the Cantor group “toy-model”
  • 18. Lacunary trigonometric series 1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj Norm convergence — Zygmund (1920s) N j=1 Aj e2πimj t Lp t (T) ∼p,q N j=1 |Aj |2 1/2 , 0 < p < ∞ ∞ j=1 Aj e2πimj t converges in Lp ⇐⇒ (Aj ) ∈ 2 (N)
  • 19. Lacunary trigonometric series 1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj Norm convergence — Zygmund (1920s) N j=1 Aj e2πimj t Lp t (T) ∼p,q N j=1 |Aj |2 1/2 , 0 < p < ∞ ∞ j=1 Aj e2πimj t converges in Lp ⇐⇒ (Aj ) ∈ 2 (N) Convergence a.e. — Kolmogorov (1924) (Aj ) ∈ 2 (N) =⇒ ∞ j=1 Aj e2πimj t converges for a.e. t ∈ T Converse of convergence a.e. — Zygmund (1930) ∞ j=1 Aj e2πimj t conv. on a set of measure > 0 =⇒ (Aj ) ∈ 2 (N)
  • 20. Lacunary SU(1,1) trigonometric products ρ: SU(1, 1) × SU(1, 1) → [0, +∞) ρ(G1, G2) := log 1 + G−1 1 G2 − I op ρ is a complete metric on SU(1, 1)
  • 21. Lacunary SU(1,1) trigonometric products ρ: SU(1, 1) × SU(1, 1) → [0, +∞) ρ(G1, G2) := log 1 + G−1 1 G2 − I op ρ is a complete metric on SU(1, 1) dp(g1, g2) :=    ρ(g1(t), g2(t)) Lp t (T) for 1 ≤ p < ∞ ρ(g1(t), g2(t)) p Lp t (T) for 0 < p < 1 Lp (T, SU(1, 1)) := g : T → SU(1, 1) : dp(I, g) < +∞ dp is a complete metric on Lp (T, SU(1, 1))
  • 22. Lacunary SU(1,1) trigonometric products 1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj aN(t) bN(t) bN(t) aN(t) = N n=0 Aj Bj e2πimj t Bj e−2πimj t Aj
  • 23. Lacunary SU(1,1) trigonometric products 1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj aN(t) bN(t) bN(t) aN(t) = N n=0 Aj Bj e2πimj t Bj e−2πimj t Aj K. and Rupˇci´c (2016) Assume q ≥ 2 and take 0 < p < ∞ lim N→+∞ aN bN bN aN exists in dp ⇐⇒ ∞ j=1 |Bj |2 < +∞ Recall ∞ j=1 |Bj |2 < +∞ ⇐⇒ ∞ j=1 log Aj < +∞ ⇐⇒ ∞ j=1(A2 j + |Bj |2) < +∞
  • 24. Lacunary SU(1,1) trigonometric products K. and Rupˇci´c (2016) Assume q ≥ 2 ∞ j=1 |Bj |2 < +∞ =⇒ lim N→+∞ aN(t) bN(t) bN(t) aN(t) exists for a.e. t ∈ T
  • 25. Lacunary SU(1,1) trigonometric products K. and Rupˇci´c (2016) Assume q ≥ 2 ∞ j=1 |Bj |2 < +∞ =⇒ lim N→+∞ aN(t) bN(t) bN(t) aN(t) exists for a.e. t ∈ T K. and Rupˇci´c (2016) Assume q ≥ 3 lim N→+∞ aN(t) bN(t) bN(t) aN(t) exists on a set of measure > 0 =⇒ ∞ j=1 |Bj |2 < +∞
  • 26. References Introductory literature: Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes, 2003, available at arXiv:1201.5129 [math.CA] Tao, An introduction to the nonlinear Fourier transform, unpublished note, 2002 Muscalu, Tao, Thiele, several papers, 2001–2007 Ablowitz, Kaup, Newell, Segur, The inverse scattering transform — Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249–315
  • 27. References Introductory literature: Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes, 2003, available at arXiv:1201.5129 [math.CA] Tao, An introduction to the nonlinear Fourier transform, unpublished note, 2002 Muscalu, Tao, Thiele, several papers, 2001–2007 Ablowitz, Kaup, Newell, Segur, The inverse scattering transform — Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249–315 Thank you for your attention!