On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Gibbs flow transport for Bayesian inferenceJeremyHeng10
Minisymposium on "Selected topics in computation and dynamics: machine learning and multiscale methods" at SciCADE 2019, Innsbruck, July 2019.
https://scicade2019.uibk.ac.at/
Slides are based on the article in https://arxiv.org/abs/1509.08787
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
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1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
"The Metropolis adjusted Langevin Algorithm
for log-concave probability measures in high
dimensions", talk by Andreas Elberle at the BigMC seminar, 9th June 2011, Paris
Scalable inference for a full multivariate stochastic volatilitySYRTO Project
Scalable inference for a full multivariate stochastic volatility
P. Dellaportas, A. Plataniotis and M. Titsias UCL(London), AUEB(Athens), AUEB(Athens)
Final SYRTO Conference - Université Paris1 Panthéon-Sorbonne
February 19, 2016
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
My talk entitled "Numerical Smoothing and Hierarchical Approximations for Efficient Option Pricing and Density Estimation", that I gave at the "International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022. The talk is related to our recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://arxiv.org/abs/2111.01874) and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation" (link: https://arxiv.org/abs/2003.05708). In these two works, we introduce the numerical smoothing technique that improves the regularity of observables when approximating expectations (or the related integration problems). We provide a smoothness analysis and we show how this technique leads to better performance for the different methods that we used (i) adaptive sparse grids, (ii) Quasi-Monte Carlo, and (iii) multilevel Monte Carlo. Our applications are option pricing and density estimation. Our approach is generic and can be applied to solve a broad class of problems, particularly for approximating distribution functions, financial Greeks computation, and risk estimation.
Similar to Estimates for a class of non-standard bilinear multipliers (20)
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
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Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
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Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
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THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
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IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
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introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
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This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
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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
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
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
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
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
