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.
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
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
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.
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
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...Francesco Tudisco
We consider the p-Laplacian on discrete graphs, a nonlinear operator that generalizes the standard graph Laplacian (obtained for p=2). We consider a set of variational eigenvalues of this operator and analyze the nodal domain count of the corresponding eigenfunctions. In particular, we show that the famous Courant’s nodal domain theorem for the linear Laplacian carries over almost unchanged to the nonlinear case. Moreover, we use the nodal domains to prove a higher-order Cheeger inequality that relates the k-way graph cut to the k-th variational eigenvalue of the p-Laplacian
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
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.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
1. Boundedness of the Twisted
Paraproduct
Vjekoslav Kovaˇc, UCLA
Indiana University, Bloomington
January 13, 2011
Mathematical Institute, University of Bonn
February 3, 2011
2. 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(x) := 2kϕ(2kx), ψk(x) := 2kψ(2kx)
3. 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
Littlewood-Paley projections in the 1st and the 2nd
variable respectively
(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
4. Twisted paraproduct – Estimates
We are interested in
strong-type estimates
T(F, G) Lpq/(p+q)(R2) p,q F Lp(R2) G Lq(R2)
weak-type estimates
α |T(F, G)| > α
(p+q)/pq
p,q F Lp(R2) G Lq(R2)
Theorem (K., 2010)
Operators Td and Tc satisfy the strong bound if
1 < p, q < ∞, 1
p + 1
q > 1
2.
Additionally, operators Td and Tc satisfy the weak bound
when p = 1, 1 ≤ q < ∞ or q = 1, 1 ≤ p < ∞.
The weak estimate fails for p = ∞ or q = ∞.
5. Range of exponents
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
6. 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 – direct proof,
K. (2010)
the rest of the shaded region
– conditional proof,
Bernicot (2010)
dashed segments –
counterexamples,
K. (2010)
points D, E – the direct
proof simplifies
Continuous version Tc
transition using the
Jones-Seeger-Wright square
function
7. Some motivation
1D bilinear Hilbert transform
Tα,β(f , g)(x) := p.v.
R
f (x − αt) g(x − βt)
dt
t
Bounds proved by Lacey and Thiele (1997–1999).
2D bilinear Hilbert transform
TA,B(F, G)(x, y) :=
p.v.
R2
F (x, y) − A(s, t) G (x, y) − B(s, t) K(s, t) ds dt
|∂α ˆK(ξ, η)| α (ξ2 + η2)−|α|/2
Introduced by Demeter and Thiele (2008). They proved bounds in
the range 2 < p, q < ∞, 1
p + 1
q > 1
2, and for almost all cases
depending on matrices A and B.
8. Some motivation
Essentially the only unresolved case was (denoted “Case 6”):
T(F, G)(x, y) := p.v.
R2
F(x − s, y) G(x, y − t) K(s, t) ds dt
Using “cone decomposition” it reduces to bilinear multipliers with
symbols
ˆK(ξ, η) =
k∈Z
ˆϕ(2−k
ξ) ˆψ(2−k
η)
which are twisted paraproducts.
9. Bounding the dyadic version
ϕd
I := |I|−1/21I the Haar scaling function
ψd
I := |I|−1/2(1Ileft
− 1Iright
) the Haar wavelet
C = dyadic squares in R2
Td(F, G)(x, y) =
I×J∈C R2
F(u, y)G(x, v)ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y)dudv
Λd(F, G, H) =
R2
Td(F, G)(x, y)H(x, y) dx dy
=
I×J∈C R4
F(u, y)G(x, v)H(x, y) ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y) du dx dv dy
10. Introducing trees
A finite convex tree is a collection T of dyadic squares such that
There exists QT ∈ T , called the root of T , satisfying Q ⊆ QT
for every Q ∈ T .
Whenever Q1 ⊆ Q2 ⊆ Q3 and Q1, Q3 ∈ T , then also Q2 ∈ T .
A leaf of T is a square that is not contained in T , but its parent is.
L(T ) = the family of leaves of T
Squares in L(T ) partition QT .
11. Local forms
For any finite convex tree T we define:
ΛT (F, G, H) :=
I×J∈T R4
F(u, y)G(x, v)H(x, y)
ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y) du dx dv dy
Θ
(2)
T (F1, F2, F3, F4) :=
I×J∈T R4
F1(u, v)F2(x, v)F3(u, y)F4(x, y)
ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y) du dv dx dy
Θ
(1)
T (F1, F2, F3, F4) :=
I×J∈T R4
F1(u, v)F2(x, v)F3(u, y)F4(x, y)
ψd
I (u)ψd
I (x) ϕd
Jleft
(v)ϕd
Jleft
(y) + ϕd
Jright
(v)ϕd
Jright
(y) du dv dx dy
Observe that ΛT (F, G, H) = Θ
(2)
T (1, G, F, H).
12. A notion from additive combinatorics
For Q = I × J ∈ C we define:
Gowers box inner-product and norm
[F1, F2, F3, F4] (Q) :=
1
|Q|2
I I J J
F1(u, v)F2(x, v)F3(u, y)F4(x, y) du dx dv dy
F (Q) := [F, F, F, F]
1/4
(Q)
The box Cauchy-Schwarz inequality:
[F1, F2, F3, F4] (Q) ≤ F1 (Q) F2 (Q) F3 (Q) F4 (Q)
A special case of Brascamp-Lieb inequalities:
F (Q) ≤ 1
|Q| Q |F|2 1/2
13. Local forms
For any finite collection of squares F we define:
ΞF (F1, F2, F3, F4) :=
Q∈F
|Q| F1, F2, F3, F4 (Q)
=
I×J∈F R4
F1(u, v)F2(x, v)F3(u, y)F4(x, y)
ϕd
I (u)ϕd
I (x)ϕd
J (v)ϕd
J (y) du dv dx dy
A single-scale estimate:
ΞL(T )(F1, F2, F3, F4) ≤ |QT |
4
j=1
max
Q∈L(T )
Fj (Q)
14. Telescoping identity over trees
Lemma (Telescoping identity)
For any finite convex tree T with root QT we have
Θ
(1)
T (F1, F2, F3, F4) + Θ
(2)
T (F1, F2, F3, F4)
= ΞL(T )(F1, F2, F3, F4) − Ξ{QT }(F1, F2, F3, F4)
15. Telescoping identity over trees
Lemma (Telescoping identity)
For any finite convex tree T with root QT we have
Θ
(1)
T (F1, F2, F3, F4) + Θ
(2)
T (F1, F2, F3, F4)
= ΞL(T )(F1, F2, F3, F4) − Ξ{QT }(F1, F2, F3, F4)
Proof, part 1.
When T consists of only one square, the identity reduces to
j∈{left,right}
ψd
I (u)ψd
I (x)ϕd
Jj
(v)ϕd
Jj
(y) + ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y)
=
i,j∈{left,right}
ϕd
Ii
(u)ϕd
Ii
(x)ϕd
Jj
(v)ϕd
Jj
(y) − ϕd
I (u)ϕd
I (x)ϕd
J (v)ϕd
J (y)
16. Telescoping identity over trees
Lemma (Telescoping identity)
For any finite convex tree T with root QT we have
Θ
(1)
T (F1, F2, F3, F4) + Θ
(2)
T (F1, F2, F3, F4)
= ΞL(T )(F1, F2, F3, F4) − Ξ{QT }(F1, F2, F3, F4)
Proof, part 2.
For a general finite convex tree T the following sum
Q∈T Q is a child of Q
Ξ{Q}
− Ξ{Q}
telescopes into the right hand side.
17. Reduction inequalities
Lemma (Reduction inequalities)
Θ
(1)
T (F1, F2, F3, F4) ≤ Θ
(1)
T (F1, F1, F3, F3)
1
2 Θ
(1)
T (F2, F2, F4, F4)
1
2
Θ
(2)
T (F1, F2, F3, F4) ≤ Θ
(2)
T (F1, F2, F1, F2)
1
2 Θ
(2)
T (F3, F4, F3, F4)
1
2
18. Reduction inequalities
Lemma (Reduction inequalities)
Θ
(1)
T (F1, F2, F3, F4) ≤ Θ
(1)
T (F1, F1, F3, F3)
1
2 Θ
(1)
T (F2, F2, F4, F4)
1
2
Θ
(2)
T (F1, F2, F3, F4) ≤ Θ
(2)
T (F1, F2, F1, F2)
1
2 Θ
(2)
T (F3, F4, F3, F4)
1
2
Proof.
Rewrite Θ
(2)
T (F1, F2, F3, F4) as
I×J∈T R2 R
F1(u, v)F2(x, v)ψd
J (v)dv
·
R
F3(u, y)F4(x, y)ψd
J (y)dy ϕd
I (u)ϕd
I (x) du dx
and apply the Cauchy-Schwarz inequality, first over (u, x) ∈ I × I,
and then over I × J ∈ T .
19. Single tree estimate
For any finite convex tree T we have:
Proposition (Single tree estimate)
Θ
(2)
T (F1, F2, F3, F4) |QT |
4
j=1
max
Q∈L(T )
Fj (Q)
ΛT (F, G, H)
|QT | max
Q∈L(T )
F (Q) max
Q∈L(T )
G (Q) max
Q∈L(T )
H (Q)
20. Proof of the single tree estimate
Θ
(2)
T (F1, F2, F3, F4)
ww ''
Θ
(2)
T (F1, F2, F1, F2)
Θ
(2)
T (F3, F4, F3, F4)
Θ
(1)
T (F1, F2, F1, F2)
~~
Θ
(1)
T (F3, F4, F3, F4)
~~
Θ
(1)
T (F1, F1, F1, F1) Θ
(1)
T (F2, F2, F2, F2) Θ
(1)
T (F3, F3, F3, F3) Θ
(1)
T (F4, F4, F4, F4)
21. Proof of the single tree estimate
How do we control Θ
(1)
T (Fj , Fj , Fj , Fj ) ?
Θ
(1)
T (Fj , Fj , Fj , Fj )
≥0
+ Θ
(2)
T (Fj , Fj , Fj , Fj )
≥0
+ Ξ{QT }(Fj , Fj , Fj , Fj )
≥0 if Fj ≥0
= ΞL(T )(Fj , Fj , Fj , Fj )
22. Estimates in the local L2
range
Proposition
|Λd(F, G, H)| p,q,r F Lp G Lq H Lr
for 1
p + 1
q + 1
r = 1, 2 p, q, r ∞
23. Estimates in the local L2
range
Proposition
|Λd(F, G, H)| p,q,r F Lp G Lq H Lr
for 1
p + 1
q + 1
r = 1, 2 p, q, r ∞
Proof, part 1.
PF
k := Q : 2k ≤ supQ ⊇Q F (Q ) 2k+1
MF
k = maximal squares in PF
k
PG
k , MG
k , PH
k , MH
k defined analogously
Pk1,k2,k3 := PF
k1
∩ PG
k2
∩ PH
k3
Mk1,k2,k3 = maximal squares in Pk1,k2,k3
24. Estimates in the local L2
range
Proposition
|Λd(F, G, H)| p,q,r F Lp G Lq H Lr
for 1
p + 1
q + 1
r = 1, 2 p, q, r ∞
Proof, part 2.
For each Q ∈ Mk1,k2,k3
TQ := {Q ∈ Pk1,k2,k3 : Q ⊆ Q}
is a convex tree with root Q. We apply the single tree estimate to
each of the trees TQ.
It remains to show
k1,k2,k3∈Z
2k1+k2+k3
Q∈Mk1,k2,k3
|Q| p,q,r F Lp G Lq H Lr
25. Estimates in the local L2
range
Proposition
|Λd(F, G, H)| p,q,r F Lp G Lq H Lr
for 1
p + 1
q + 1
r = 1, 2 p, q, r ∞
Proof, part 3.
i.e.
k1,k2,k3∈Z
2k1+k2+k3
min
Q∈MF
k1
|Q|,
Q∈MG
k2
|Q|,
Q∈MH
k3
|Q|
p,q,r F Lp G Lq H Lr
i.e.
k∈Z
2pk
Q∈MF
k
|Q| p F p
Lp ,
k∈Z
2qk
Q∈MG
k
|Q| q G q
Lq ,
k∈Z
2rk
Q∈MH
k
|Q| r H r
Lr
26. Estimates in the local L2
range
Proposition
|Λd(F, G, H)| p,q,r F Lp G Lq H Lr
for 1
p + 1
q + 1
r = 1, 2 p, q, r ∞
Proof, part 4.
M2F := sup
Q∈C
1
|Q| Q
|F|2
1/2
1Q
For each Q ∈ MF
k we have 1
|Q| Q |F|2 1/2
≥ F (Q) ≥ 2k and
so by disjointness
Q∈MF
k
|Q| ≤ |{M2F ≥ 2k
}|
k∈Z
2pk
|{M2F ≥ 2k
}| ∼p M2F p
Lp p F p
Lp
27. Extending the range of exponents
Proposition (Bernicot, 2010)
If for some (p, q), 1 p, q ∞
Td(F, G) Lpq/(p+q) p,q F Lp G Lq
then
Td(F, G) Lp/(p+1),∞ p,q F Lp G L1
Td(F, G) Lq/(q+1),∞ p,q F L1 G Lq
28. Extending the range of exponents
Proposition (Bernicot, 2010)
If for some (p, q), 1 p, q ∞
Td(F, G) Lpq/(p+q) p,q F Lp G Lq
then
Td(F, G) Lp/(p+1),∞ p,q F Lp G L1
Td(F, G) Lq/(q+1),∞ p,q F L1 G Lq
Proof.
Perform one-dimensional Calder´on-Zygmund decomposition in
each fiber F(·, y) or G(x, ·).
Even simpler in the dyadic setting: there is no contribution of “the
bad part” outside of “the bad set”.
29. Bounding the continuous version
Let us assume that ψk = φk+1 − φk for some φ Schwartz,
R φ = 1. The general case is obtained by composing with a
bounded Fourier multiplier in the second variable.
Proposition (Jones, Seeger, Wright, 2008)
If ϕ is Schwartz and R ϕ = 1, then the square function
SJSWf :=
k∈Z
Pϕk
f − Ekf
2 1/2
satisfies SJSWf Lp(R) p f Lp(R)
for 1 p ∞.
31. Bounding the continuous version
Proposition
Tc(F, G) − Td(F, G) Lpq/(p+q) p,q F Lp G Lq
Proof, part 1.
Taux(F, G) :=
k∈Z
(E
(1)
k F)(P
(2)
ψk
G)
Tc(F, G) − Taux(F, G)
≤
k∈Z
P(1)
ϕk
F − E
(1)
k F
2 1/2
S
(1)
JSWF
k∈Z
P
(2)
ψk
G
2 1/2
S(2)G
32. Bounding the continuous version
Proposition
Tc(F, G) − Td(F, G) Lpq/(p+q) p,q F Lp G Lq
Proof, part 2.
Taux(F, G) = FG −
k∈Z
(∆
(1)
k F)(P
(2)
φk+1
G)
Td(F, G) = FG −
k∈Z
(∆
(1)
k F)(E
(2)
k+1G)
Taux(F, G) − Td(F, G)
≤
k∈Z
∆
(1)
k F
2 1/2
S
(1)
d F
k∈Z
P
(2)
φk
G − E
(2)
k G
2 1/2
S
(2)
JSWG
33. Endpoint counterexample
We disprove
L∞
(R2
) × Lq
(R2
) → Lq,∞
(R2
)
for 1 ≤ q ∞.
G(x, y) := 1[0,2−n)(x)
n
k=1
Rk(y)
Rk := k-th Rademacher function =
J⊆[0,1), |J|=2−k+1
(1Jleft
− 1Jright
)
G Lq ∼q 2−n/qn1/2 by Khintchine’s inequality
(∆
(2)
k G)(x, y) = 1[0,2−n)(x)Rk+1(y) for k = 0, 1, . . . , n − 1
34. Endpoint counterexample
We choose F so that there is no cancellation between different
scales in Td(F, G).
(E
(1)
k F)(x, y) = Rk+1(y) for x ∈ [0, 2−n), k = 0, 1, . . . , n − 1
F(x, y) :=
2Rj (y) − Rj+1(y), for x ∈ [2−j , 2−j+1), j = 1, . . . , n−1
Rn(y), for x ∈ [0, 2−n+1)
Then Td(F, G) = n 1[0,2−n)×[0,1)
Td(F, G) Lq,∞
F L∞ G Lq
q
2−n/qn
2−n/qn1/2
= n1/2
35. A byproduct of the proof
Theorem (K., 2010)
Θ(2)
(F1, F2, F3, F4) :=
I×J∈C R4
F1(u, v)F2(x, v)F3(u, y)F4(x, y)
ϕd
I (u)ϕd
I (x)ψd
J (v)ψd
J (y) du dv dx dy
Θ(2)
(F1, F2, F3, F4) p1,p2,p3,p4
4
j=1
Fj L
pj ,
for 1
p1
+ 1
p2
+ 1
p3
+ 1
p4
= 1, 2 p1, p2, p3, p4 ∞
Higher-dimensional generalizations are straightforward.
36. Subsequent research
The next step in the Demeter-Thiele program:
More singular 2D bilinear Hilbert transform
p.v.
R
F(x − t, y) G(x, y − t)
dt
t
A reason for caution:
Bi-parameter bilinear Hilbert transform
p.v.
R2
F(x − s, y − t) G(x + s, y + t)
ds
s
dt
t
satisfies no Lp estimates – Muscalu, Pipher, Tao, Thiele (2004)
