arXiv:1703.05557v1 [math.CA] 16 Mar 2017
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR
SMALL POTENTIALS
VJEKOSLAV KOVAČ, DIOGO OLIVEIRA E SILVA, AND JELENA RUPČIĆ
Abstract. The nonlinear Hausdorff–Young inequality follows from the work of Christ and
Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently
of the exponent. We show that the nonlinear Hausdorff–Young quotient admits an even better
upper bound than the linear one, provided that the function is sufficiently small in the L1
norm. The proof combines perturbative techniques with the sharpened version of the linear
Hausdorff–Young inequality due to Christ.
1. Introduction
In this paper, we investigate the Hausdorff–Young inequality for a nonlinear version of the
Fourier transform, and establish Theorem 1.1 below. Before stating it precisely, we briefly
discuss the linear case. Given a complex-valued integrable function f on the real line, we
normalize its Fourier transform as follows:
Z
b
f (x)e−2πixξ dx.
f (ξ) =
R
In this way, the Fourier transform is a contraction from L1 (R) to L∞ (R) and it extends to
a unitary operator on L2 (R). Standard interpolation tools can then be used to show that,
for any p ∈ [1, 2], the Fourier transform is also a contraction from Lp (R) to Lq (R), where
p
denotes the exponent conjugate to p. This is the content of the classical Hausdorff–
q = p−1
Young inequality. Its sharp version was first established by Babenko [2] in the case when the
exponent q is an even integer, and then by Beckner [3] for general exponents. It states that,
if p ∈ [1, 2], then
kfbkLq (R) ≤ Bp kf kLp (R)
(1.1)
for every f ∈ Lp (R), where the optimal constant is given by
1
Bp = p 2p q
1
− 2q
.
(1.2)
An easy computation shows that Gaussians, i.e. functions of the form
G(x) = ce−ax
2 +vx
(1.3)
with a > 0 and c, v ∈ C, turn inequality (1.1) into an equality. In other words, Gaussians
are extremizers for inequality (1.1). In the converse direction, Lieb [8] has shown that all
extremizers for inequality (1.1) are in fact Gaussians. Recently, Christ [4] further refined
Date: March 17, 2017.
2010 Mathematics Subject Classification. Primary 42A38; Secondary 34L25.
Key words and phrases. Nonlinear Fourier transform, Dirac scattering transform, Hausdorff–Young
inequality.
1
2
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
inequality (1.1) by establishing the following sharpened version: Given p ∈ (1, 2), there exists
a constant cp > 0 such that, for every nonzero function f ∈ Lp (R),
dist2p (f, G) kf kLp (R) .
(1.4)
kfbkLq (R) ≤ Bp − cp
kf k2Lp (R)
Here, the distance from f ∈ Lp (R) to the set of all Gaussians, denoted G, is naturally defined
as
distp (f, G) = inf kf − GkLp (R) .
G∈G
We now describe the nonlinear setting of the present paper. We are interested in the simplest nonlinear model of the Fourier transform, also known as the Dirac scattering transform
or the SU(1,1)-scattering transform. To describe it precisely, take a measurable, bounded,
and compactly supported function f : R → C, which will often be referred to as a potential.
Consider the initial-value problem
∂ a(x, ξ)
a(−∞, ξ)
1
f (x)e2πixξ a(x, ξ)
0
,
=
.
(1.5)
=
b(x, ξ)
b(−∞, ξ)
0
f (x)e−2πixξ
0
∂x b(x, ξ)
For each ξ ∈ R, this problem has a unique solution a(·, ξ), b(·, ξ) in the class of absolutely
continuous functions. We simply write a(ξ), b(ξ) in place of the limits a(+∞, ξ), b(+∞, ξ),
and define the nonlinear Fourier transform of the potential f to be the function
a(ξ)
2
R → C , ξ 7→
.
b(ξ)
If f vanishes outside some interval I = [α, β], then the initial condition in (1.5) translates
into a(α, ξ) = 1, b(α, ξ) = 0, while a(+∞, ξ), b(+∞, ξ) can be interpreted as a(β, ξ), b(β, ξ),
respectively. The differential equation forces
|a(ξ)|2 − |b(ξ)|2 = |a(α, ξ)|2 − |b(α, ξ)|2 = 1,
which in particular means that a certain size of the above vector is retained by the quantity
|a(ξ)| alone. Occasionally it is more convenient to add an extra column and turn the above
vector into a 2 × 2 matrix belonging to the classical Lie group SU(1, 1), see e.g. [9].
Sources of motivation for considering this precise instance of the nonlinear Fourier transform
include the eigenvalue problem for the Dirac operator, the study of completely integrable
systems and scattering theory, and the Riemann–Hilbert problem; see the expository paper
[10] for further information, and the notes [11] for several related examples in the discrete
setting. The Dirac scattering transform is the simplest case of a more general transform, the
AKNS–ZS nonlinear Fourier transform; see [1], [13] for details.
There is a strong parallel between the nonlinear and the linear Fourier transforms. It is a
straightforward exercise to verify the following analogues of the symmetry rules for the linear
Fourier transform.
• Unimodular homogeneity: If f (x) = eiθ f1 (x), where θ ∈ R, then
a(ξ) = a1 (ξ),
b(ξ) = eiθ b1 (ξ).
• Modulation symmetry: If f (x) = e2πixξ0 f1 (x), where ξ0 ∈ R, then
a(ξ) = a1 (ξ − ξ0 ), b(ξ) = b1 (ξ − ξ0 ).
• Translation symmetry: If f (x) = f1 (x − x0 ), where x0 ∈ R, then
a(ξ) = a1 (ξ),
b(ξ) = e−2πix0 ξ b1 (ξ).
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
3
• L1 -normalized dilation symmetry: If f (x) = λ−1 f1 (λ−1 x), where λ > 0, then
a(ξ) = a1 (λξ), b(ξ) = b1 (λξ).
• Conjugation symmetry: If f (x) = f1 (x), then
a(ξ) = a1 (−ξ), b(ξ) = b1 (−ξ).
• A substitute for additivity: If f (x) = f1 (x) + f2 (x), where the support of f1 lies to the
left of that of f2 , then
a(ξ) = a1 (ξ)a2 (ξ) + b1 (ξ)b2 (ξ), b(ξ) = a1 (ξ)b2 (ξ) + b1 (ξ)a2 (ξ).
We proceed to describe some nonlinear analogues of standard estimates for the linear Fourier
transform. The nonlinear Riemann–Lebesgue estimate, which follows easily from Grönwall’s
inequality, states that
(log |a(ξ)|2 ) 21 ∞ ≤ log(|a(ξ)|+|b(ξ)|) ∞ ≤ kf k 1 ,
L (R)
L (R)
L (R)
ξ
ξ
for every potential f . The nonlinear Plancherel identity, which is a well-known scattering
identity that can be established via complex contour integration (see e.g. the Appendix in
[9]), states that
(log |a(ξ)|2 ) 21 2
= kf kL2 (R) ,
L (R)
ξ
for every potential f as before. This equality can even be used to extend the definition of a, b
for functions f that are only square-integrable, but in this case the existence of the pointwise
limits a(+∞, ξ), b(+∞, ξ) for a.e. ξ ∈ R is a well-known open problem (see [6], [9]). Even if
interpolation is not available in the present nonlinear setting, the work of Christ and Kiselev
on the spectral theory of one-dimensional Schrödinger operators [5], [6] establishes a version of
the nonlinear Hausdorff–Young inequality which translates into the present context as follows:
If p ∈ [1, 2], then there exists a constant Cp < ∞, such that
(log |a(ξ)|2 ) 21 q
≤ Cp kf kLp (R) ,
(1.6)
L (R)
ξ
for every potential f . An interesting question raised in [9] is whether the constants Cp can
be chosen uniformly in p, as p → 2− . This has been confirmed in a particular toy model in
[7], but remains an open problem in its full generality. By considering truncated Gaussian
potentials
2
G(x) = e−(N x) 1[−1,1] (x)
as N → +∞ and linearizing, one can check that the constant in (1.6) is at least as large as
Beckner’s constant (1.2), i.e. Cp ≥ Bp . One may be tempted to conjecture that the optimal
constant in (1.6) is actually Cp = Bp .
While these questions are left open by the present work, we are able to provide some further
supporting evidence of their validity by considering the behaviour of the nonlinear Hausdorff–
Young ratio for sufficiently small potentials. The main result of this paper is the following
theorem.
1
−
1
p
, and Bp = p 2p q 2q . Let H, W > 0. Then there exist
Theorem 1.1. Let p ∈ (1, 2), q = p−1
δ > 0 and ε > 0, depending on p, H, W , with the following property: If I ⊂ R is an interval
of length W , and f : R → C is a measurable function such that |f | ≤ H 1I and kf kL1 (R) ≤ δ,
then
(log |a(ξ)|2 ) 12 q
(1.7)
≤ Bp − εkf k2L1 (R) kf kLp (R) .
L (R)
ξ
A few remarks may help to further orient the reader.
4
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
• The parameters H, W provide upper bounds for the height and the width of the function f , respectively. They are fixed but can be arbitrarily large.
• Inequality (1.7) implies (1.6) with an optimal constant Cp = Bp , but only within
the restricted class of potentials considered in the theorem, and this class is allowed
to depend on p. That way Theorem 1.1 does not claim uniform boundedness of the
constants in (1.6) for any particular family of functions. It rather fixes the value
of p and shows that the nonlinear Hausdorff–Young inequality beats the linear one
in the asymptotic regime when kf kL1 → 0. In particular, the constant Bp is never
attained for nonzero potentials in a sufficiently small neighborhood of the zero function
(depending on p, H, W ).
• Between the lines of the first part of the proof below (see §2), one can easily obtain
(log |a(ξ)|2 ) 12 q
≤ Bp exp(kf kL1 (R) )kf kLp (R) .
(1.8)
L (R)
ξ
Estimate (1.8) provides a cheap version of (1.7), with Bp − εkf k2L1 replaced by Bp +
O(kf kL1 ), as kf kL1 → 0, but the interest lies, of course, in obtaining the estimate
with a negative sign. However, (1.8) at least shows that the mere uniformity of the
constants Cp is trivial for potentials that are controlled in the L1 norm.
• It is easy to observe that (1.7) is invariant under L1 -normalized dilations applied to f .
That way one can trade width for height or vice versa, and conclude that δ, ε depend
only on the product HW . This fact can also be checked by tracking the dependence
of various parameters in the proof.
Let us briefly comment on the proof of Theorem 1.1, which spans over the next three
sections. The upshot is that for most choices of f one can simply estimate the error arising
from linearization, while for the remaining ones we verify that the nonlinear effect actually
improves the estimate. To implement this strategy, we set up a case distinction, depending
on whether the function f is far or close to the set of Gaussians, in an appropriate sense.
The former case is the subject of §2, where we invoke Christ’s sharpened Hausdorff–Young
inequality (1.4) in order to absorb the error terms coming from linearization. The latter case is
the subject of §3, where we use a perturbative argument to expand the functional in question
around a suitable Gaussian that provides a good approximation for f in both the Lp and the L1
senses. We are naturally led to study a certain quartic operator Q and its quadrilinear variant
Φ, which enjoy various symmetries. The operator Q can in turn be pointwise dominated by a
power of the maximally truncated Fourier transform, defined as follows:
Z
(1.9)
(F∗ f )(ξ) = sup f (x)e−2πixξ dx,
I
I
where the supremum is taken over all intervals I ⊆ R. The classical Menshov–Paley–Zygmund
inequality states that, for every p ∈ (1, 2), there exists Mp < ∞ such that, for every f ∈ Lp (R),
kF∗ f kLq (R) ≤ Mp kf kLp (R) .
(1.10)
The argument makes crucial use of estimate (1.10) in order to control the error terms, and
to verify that the second order variation about the aforementioned Gaussian has the correct
sign. Proofs of several technical lemmata are deferred to §4. Finally, in the last section §5, we
provide a counterexample to the natural question of whether an inequality
fb q
(log |a(ξ)|2 ) 12 q
≤
(1.11)
L (R)
L (R)
ξ
might hold for a general bounded compactly supported potential f . More precisely, when
q = 4, we exhibit an explicit linear combination of indicator functions of six contiguous
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
5
intervals of unit length that disproves (1.11). Because of this, we believe that Theorem 1.1
cannot be established solely by regarding (1.7) as a small perturbation of (1.1).
Notation. If x, y are real numbers, we write x = O(y) or x . y if there exists a finite absolute
constant C such that |x| ≤ C|y|. If we want to make explicit the dependence of the constant
C on some parameter α, we write x = Oα (y) or x .α y. We also write x ∨ y = max{x, y} and
x ∧ y = min{x, y}. The real and imaginary parts of a complex number z are denoted by Re(z)
and Im(z). The indicator function of a set E ⊆ R is denoted by 1E . Throughout the paper it
will be understood that all constants may depend on the admissible parameters p, H, W .
2. Far from Gaussians
Let f : R → C satisfy the assumptions of Theorem 1.1. Our first task is to make precise
what it means for f to be far away from the set of Gaussians. The following notion will be
suitable for our purposes: Assume that
where γ is shorthand notation for
distp (f, G) ≥ γkf kLp (R) ,
8B kf k 1 1
2
p
L
.
γ=
cp
(2.1)
Here, Bp denotes Beckner’s constant (1.2) and cp is the constant promised by Christ’s refinement (1.4). This precise choice of γ will become clear as the proof unfolds. In particular, note
1
that γ . δ 2 . Going back to the defining ODE (1.5), it is straightforward to check that the
functions a and b satisfy the integral equations
Z x
a(x, ξ) = 1 +
f (t)e2πitξ b(t, ξ) dt,
−∞
Z x
f (t)e−2πitξ a(t, ξ) dt.
b(x, ξ) =
−∞
Adding the two equations, the triangle inequality yields
Z x
Z x
−2πitξ
|f (t)| |b(t, ξ)| + |a(t, ξ) − 1| dt.
f (t)e
dt +
|b(x, ξ)| + |a(x, ξ) − 1| ≤ −∞
−∞
Taking the Lq norm in ξ and invoking Minkowski’s integral inequality, we obtain
Z x
|b(x, ξ)| + |a(x, ξ) − 1| q
c
≤ f 1(−∞,x] Lq (R) +
|f (t)||b(t, ξ)| + |a(t, ξ) − 1|Lq (R) dt.
L (R)
ξ
−∞
To estimate the quantity kfc1(−∞,x] kLq (R) , we further split the analysis into two cases.
ξ
Case 1. kf 1(x,+∞)kLp (R) < γ2 kf kLp (R) . In this case, for each G ∈ G we have that
γ
kf 1(−∞,x] − GkLp (R) ≥ kf − GkLp (R) − kf 1(x,+∞)kLp (R) ≥ kf kLp (R) .
2
It follows that
γ
distp (f 1(−∞,x], G) ≥ kf 1(−∞,x]kLp (R) ,
2
and Christ’s improved Hausdorff–Young inequality yields
γ 2 e p kf kLp (R) ,
c
kf 1(−∞,x] kLp (R) ≤ B
f 1(−∞,x] Lq (R) ≤ Bp − cp
2
e p is not really a constant, since γ
e p is defined as B
e p = Bp − cp ( γ )2 . Note that B
where B
2
1
depends on the L norm of f .
6
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
Case 2. kf 1(x,+∞) kLp (R) ≥ γ2 kf kLp (R) . In this case, the sharp Hausdorff–Young inequality
(1.1) yields
γ p p1
fc1(−∞,x] q
p
1
≤
B
kf
kf kLp (R) .
1
−
k
≤
B
p
p
(−∞,x] L (R)
L (R)
2
Bernoulli’s inequality can then be invoked to verify that
1 γ p e
γ p p1
≤ Bp ,
(2.2)
≤ Bp 1 −
Bp 1 −
2
p 2
provided δ is chosen to be sufficiently small. Here we also use that p < 2.
In both cases, we obtain
Z x
e
|b(t, ξ)| + |a(t, ξ) − 1| q dt.
|b(x, ξ)| + |a(x, ξ) − 1| q
p
|f
(t)|
≤
B
kf
k
+
p
L
(R)
L (R)
L (R)
ξ
ξ
−∞
Grönwall’s lemma then implies
Z
|b(x, ξ)| + |a(x, ξ) − 1| q
e
≤ Bp exp
L (R)
ξ
x
−∞
|f (t)| dt kf kLp (R) .
1
2
Letting x → +∞ and estimating (log |a|2 ) ≤ |b|, we finally have that
e p exp(kf k 1 )kf kLp (R) .
(log |a(ξ)|2 ) 12 q
≤B
L (R)
L (R)
ξ
The obtained inequality shows that, in this case, the only loss in passing from the linear to the
nonlinear setting amounts to the exponential factor, which tends to 1 as kf kL1 → 0. Recall
the choice of γ from (2.1). It remains to choose δ, ε > 0 small enough, so that (2.2) holds, and
e p exp(kf k 1 ) = Bp (1 − 2kf k 1 ) 1 + kf k 1 + O(kf k2 1 )
B
L
L
L
L
= Bp 1 − kf kL1 + O(kf k2L1 ) ≤ Bp − εkf k2L1 .
Thus we have verified the desired inequality (1.7) in the case when the function f is far
from the Gaussians. We analyse the complementary situation in the next section, where an
additional smallness condition will be imposed on δ, ε.
3. Close to Gaussians
We are now working under the assumption
distp (f, G) < γkf kLp (R) ,
1
where γ was defined in (2.1). In particular, γ . kf kL2 1 . Thus, there exists a Gaussian G ∈ G,
such that
kf − GkLp (R) < γkf kLp (R) .
(3.1)
This readily implies
kf kLp (R) ≤ 2kGkLp (R) ,
(3.2)
provided γ < 21 . Recall our working assumptions that f vanishes outside an interval I of
length W , that |f | is bounded from above by H, and that it satisfies kf kL1 ≤ δ. Under these
conditions,
kf − GkL1 (I) . kf − GkLp (I) ≤ kf − GkLp (R) < γkf kLp (R) = γkf kLp (I)
1
1
1
1
. kf kL2 1 (I) kf kLp1 (I) kf kLq ∞ (I) . δ p
− 21
kf kL1 (I) .
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
7
The penultimate step in this chain of inequalities follows from the choice of γ and log-convexity
of the Lp norms. As a consequence,
kf kL1 (I) ≤ 2kGkL1 (I) ,
provided δ is small enough. This readily implies
kf kL1 (R) ≤ 2kGkL1 (R) .
(3.3)
Inequalities (3.2) and (3.3) ensure that the Gaussian G is a good approximation for the function
f on the whole real line, both in the Lp and in the L1 senses.
We now proceed to derive the first nontrivial term in the expansion of the left-hand side of
inequality (1.7) for any q > 2. The first observation is that
log(|a(x, ξ)|2 ) = − log(1 − |r(x, ξ)|2 ),
where the reflection coefficient r =
b
a
(3.4)
satisfies Riccati’s differential equation
∂r
(x, ξ) = f (x)e−2πixξ − f (x)e2πixξ r(x, ξ)2 ,
r(−∞, ξ) = 0.
∂x
In other words, the reflection coefficient is given by the integral equation
Z x
Z x
f (t)e2πitξ r(t, ξ)2 dt.
f (t)e−2πitξ dt −
r(x, ξ) =
(3.5)
−∞
−∞
From identity (3.4) we obtain
i.e.
∂r
2 Re r(x, ξ) ∂x
(x, ξ)
∂
2
−2πixξ
log(|a(x, ξ)| ) =
=
2
Re
f
(x)e
r(x,
ξ)
,
∂x
1 − |r(x, ξ)|2
log(|a(ξ)|2 ) = 2 Re
Z
f (x1 )e−2πix1 ξ r(x1 , ξ) dx1 .
(3.6)
R
Using the integral equation (3.5) to substitute for r(x1 , ξ) on the right-hand side of (3.6),
Z Z x1
2
f (x1 )f (x2 )e−2πi(x1 −x2 )ξ dx2 dx1
log(|a(ξ)| ) = 2 Re
R
Z x1
Z −∞
f (x1 )f (x2 )e−2πi(x1 +x2 )ξ r(x2 , ξ)2 dx2 dx1 .
− 2 Re
R
−∞
The first summand on the right-hand side can be recognized as
Z
Z
f (x1 )f (x2 )e−2πi(x1 −x2 )ξ dx1 dx2 +
f (x2 )f (x1 )e2πi(x2 −x1 )ξ dx1 dx2 = |fb(ξ)|2 .
{x1 >x2 }
{x2 >x1 }
Repeating this procedure once again, i.e. substituting for r(x2 , ξ) and symmetrizing in the
variables x1 and x2 , we conclude that
log(|a(ξ)|2 ) = |fb(ξ)|2 − (Qf )(ξ) + (Ef )(ξ),
(3.7)
where Q is the quartic operator defined as
Z
f (x1 )f (x2 )f (x3 )f (x4 ) e2πi(−x1 −x2 +x3 +x4 )ξ dx1 dx2 dx3 dx4
(Qf )(ξ) = Re
{(x1 ∧x2 )>(x3 ∨x4 )}
8
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
and E is the nonlinear operator given by
Z
f (x1 )f (x2 )f (x3 )f (x4 ) e2πi(−x1 −x2 +x3 −x4 )ξ r(x4 , ξ)2 dx1 dx2 dx3 dx4
(Ef )(ξ) = 2 Re
− Re
Z
{(x1 ∧x2 )>(x3 ∨x4 )}
f (x1 )f (x2 )f (x3 )f (x4 ) e2πi(−x1 −x2 −x3 −x4 )ξ r(x3 , ξ)2 r(x4 , ξ)2 dx1 dx2 dx3 dx4 .
{(x1 ∧x2 )>(x3 ∨x4 )}
We now use the first numerical inequality established in Lemma 4.1 below. Substituting
u = |fb(ξ)|2 and v = −(Qf )(ξ) + (Ef )(ξ)
into the inequality from part (a) of Lemma 4.1, and then integrating in ξ, we conclude from
(3.7) that
q
q
(log |a(ξ)|2 ) 21 q q
≤ fbLq (R) − H(f ) + R(f ),
(3.8)
Lξ (R)
2
where the second term is given by
Z
H(f ) = (Qf )(ξ)|fb(ξ)|q−2 dξ,
R
and the remainder R(f ) is bounded by a linear combination (with coefficients depending only
on q) of integrals
Z
Z
Z
q
q
|(Ef )(ξ)| 2 dξ,
|(Ef )(ξ)||fb(ξ)|q−2 dξ,
(3.9)
|(Qf )(ξ)| 2 dξ,
R
R
and, if q > 4, also
Z
R
|(Qf )(ξ)|2 |fb(ξ)|q−4 dξ,
R
Z
R
|(Ef )(ξ)|2 |fb(ξ)|q−4 dξ.
(3.10)
Approximating the function f with the Gaussian G as discussed at the beginning of this
section, we are thus reduced to showing that H(G) is a large enough positive quantity, that
H(G) provides a good approximation for H(f ), and that the remainder term R(f ) is appropriately small. This is accomplished via the following sequence of lemmata, whose proofs are
deferred to the next section in order not to obscure the main line of reasoning. All of them
hold under the ongoing assumption that f satisfies the hypotheses of Theorem 1.1, and that
G ∈ G approximates f in the sense of (3.1).
The first lemma shows that the quantity H(G) is not too small.
Lemma 3.1.
H(G) & kGk2L1 (R) kGkqLp (R) .
The second lemma shows that H(G) provides a good approximation for H(f ).
Lemma 3.2.
|H(f ) − H(G)| . γ (q−2)∧1 kGk2L1 (R) kGkqLp (R) .
The third lemma shows smallness of the remainder term R(f ).
Lemma 3.3.
|R(f )| . δ(q−2)∧2 kf k2L1 (R) kf kqLp (R) .
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
9
We are now in a position to finish the proof of the theorem. If the parameter δ is chosen
to be small enough, then so is γ, and the three lemmata combine with bounds (3.2), (3.3) to
yield
q
q
|H(f ) − H(G)| + |R(f )| ≤ H(G).
2
4
If ε > 0 is small enough, then this inequality together with (3.2), (3.3), (3.8), Lemma 3.1, and
the sharp Hausdorff–Young inequality (1.1) imply
q
fb q − q H(G)
(log |a(ξ)|2 ) 21 q q
≤
L (R)
Lξ (R)
4
q
≤ fbLq (R) − εqBpq−1 kf k2L1 (R) kf kqLp (R)
≤ Bqp − εqBpq−1 kf k2L1 (R) kf kqLp (R) .
One last application of Bernoulli’s inequality finally yields
1
Bqp − εqBpq−1 kf k2L1 (R) q ≤ Bp − εkf k2L1 (R) ,
as desired. This completes the proof of Theorem 1.1 modulo the verification of the lemmata,
which is the content of the next section.
4. Proofs of lemmata
We start with some elementary numerical inequalities.
Lemma 4.1. Given an exponent 2 < q < ∞, the following inequalities hold.
(a) For u ≥ 0 and v ∈ R one has
q
q
q
q
u 2 + 2q vu 2 −1 + Dq |v| 2
if 2 < q ≤ 4,
2
q
q
q
q
|u + v| ≤
q
−1
−2
2
2
2
2
2
u + 2 vu
+ Dq |v| + Dq |v| u
if q > 4,
with some finite constant Dq > 0 depending only on q.
(b) For u ≥ 0 and v ∈ R one has
q−2
if 2 < q ≤ 3,
|u + v|q−2 − uq−2 ≤ Eq |v|q−2
|v|
+ |v|uq−3 if q > 3,
with some finite constant Eq > 0 depending only on q.
Proof. (a) No generality is lost in assuming that u 6= 0. We can then divide both sides of the
q
inequality by u 2 , and substituting t = uv we are left with checking that
q
q
q
if 2 < q ≤ 4,
|t| 2
q
|1 + t| 2 − 1 − t .q
2
2
|t| 2 + t if q > 4.
If Q(t) denotes the quotient of the two sides of the inequality,

q
|1+t| 2 −1− q2 t


if 2 < q ≤ 4,
q
|t| 2
q
Q(t) =
q

 |1+t| 2q −1− 2 t if q > 4,
2
|t| 2 +t
then we need to show boundedness from above of the function Q on R \ {0}. This is a simple
consequence of the continuity of Q and finiteness of the limits:
0
if 2 < q < 4,
lim Q(t) = 1.
lim Q(t) =
q(q−2)
t→±∞
t→0
if q ≥ 4,
8
10
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
v
u
turns the inequality into
q−2
|t|
if 2 < q ≤ 3,
q−2
|1 + t|
− 1 .q
|t|q−2 + |t| if q > 3.
(b) This time the substitution t =
If we again denote the quotient of the two sides by Q(t), then the inequality follows as before
from the continuity of Q and finiteness of the limits:
0
if 2 < q < 3,
lim Q(t) =
lim Q(t) = 1.
q
−
2
if q ≥ 3,
t→±∞
t→0
Proof of Lemma 3.1. Start by noting that the quotient
H(f )
kf k2L1 (R) kf kqLp (R)
(4.1)
is invariant under arbitrary scalings, modulations, translations, and L1 -normalized dilations.
Indeed, if f (x) = cf1 (x) for some c ∈ C, then H(f ) = |c|q+2 H(f1 ). Moreover, if f (x) =
e2πixξ0 f1 (x) for some ξ0 ∈ R, then H(f ) = H(f1 ). Similarly, if f (x) = f1 (x − x0 ) for some
x0 ∈ R, then H(f ) = H(f1 ). Finally, dilation invariance is easily seen from the Fourier
representation
Z
ϕ(t)e−2πitξ dt,
(Qf )(ξ) =
R
where the function ϕ is given by
Z
ϕ(t) = β
f (x1 )f (x2 )f (x3 )f (x4 ) dσ3 (x1 , x2 , x3 , x4 ).
(4.2)
{x1 +x2 −x3 −x4 =t,
(x1 ∧x2 )>(x3 ∨x4 ) or (x1 ∨x2 )<(x3 ∧x4 )}
Here we are integrating over a region in the affine hyperplane {x1 + x2 − x3 − x4 = t} ⊂ R4
with respect to the 3-dimensional Hausdorff measure σ3 . The constant β > 0 is unimportant
and it is coming from the non-orthogonal choice of coordinates. It follows that, if f (x) =
λ−1 f1 (λ−1 x), then
1 t
, (Qf )(ξ) = (Qf1 )(λξ), H(f ) = λ−1 H(f1 ).
ϕ(t) = ϕ1
λ
λ
On the other hand,
kf k2L1 (R) kf kqLp (R) = λ−1 kf1 k2L1 (R) kf1 kqLp (R) .
This shows that the expression (4.1) is invariant under L1 -normalized dilations, as claimed.
Now, going back to (1.3) and writing
G(x) = ce−ax
2 +vx
= ce
Re(v)2
4a
eiIm(v)x e−a(x−
Re(v) 2
)
2a
,
one sees that any Gaussian can be brought to standard form by an application of an appropriate
scaling, modulation, translation, and dilation. Given the symmetries of (4.1) just discussed, in
verifying the claim of the lemma we can assume that the Gaussian approximation G coincides
with the standard Gaussian, G(x) = exp(−πx2 ). In this case, we are reduced to checking that
H(G) > 0. Again writing QG as the Fourier transform of some function ϕ, then from formula
(4.2) above with f = G it immediately follows that ϕ is nonnegative and not identically zero.
q−2 is the Fourier transform of another Gaussian function ψ, given by
b
Also, |G(ξ)|
πx2
1
e− q−2 ,
ψ(x) = √
q−2
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
11
which is clearly strictly positive. It remains to invoke unitarity of the linear Fourier transform
and observe that
b q−2 2
H(G) = QG, |G|
= hϕ, ψiL2 (R) > 0.
L (R)
This concludes the proof of the lemma.
The proofs of the two remaining lemmata rely on the observation that Qf can be pointwise
controlled by the maximally truncated Fourier transform F∗ f , defined in (1.9). Indeed,
Z
2
Z +∞
−2πiyξ
f (y)e
dy |f (x3 )||f (x4 )| dx3 dx4
sup |(Qf )(ξ)| ≤
=
x
R2 x∈R
2
kf kL1 (R) (F∗ f )2 (ξ).
(4.3)
In a similar way, Ef can be controlled pointwise by F∗ f . To see this, start by noting that
the reflection coefficient satisfies |r(x, ξ)| ≤ 1. It then follows from the integral equation (3.5)
that also |r(x, ξ)| ≤ 2kf kL1 . This time we get
Z
2
Z +∞
f (y)e−2πiyξ dy |f (x3 )||f (x4 )||r(x4 , ξ)|2 dx3 dx4
sup |(Ef )(ξ)| ≤ 3
≤
x
R2 x∈R
12kf k4L1 (R) (F∗ f )2 (ξ).
(4.4)
Proof of Lemma 3.2. We start by rewriting H(f ) − H(G) as
Z
Z
q−2
q−2
q−2
b
b
b
(Qf )(ξ) − (QG)(ξ) |G(ξ)|
dξ.
(Qf )(ξ) |f (ξ)|
− |G(ξ)|
dξ +
(4.5)
R
R
Set h = f − G. Part (b) of Lemma 4.1 allows us to bound the first integral in (4.5) by a
multiple of
Z
Z
q−3
q−2
b
b
|(Qf )(ξ)||b
h(ξ)||G(ξ)|
dξ.
|(Qf )(ξ)||h(ξ)|
dξ and, if q > 3, also
R
R
To bound these integrals, start by observing that estimates (4.3) and (1.10) imply
Z
Z
q
q
2
|(Qf )(ξ)| dξ ≤ kf kL1 (R) (F∗ f )q (ξ) dξ . kf kqL1 (R) kf kqLp (R) .
R
R
Hölder’s inequality and the Hausdorff–Young inequality then imply
Z
Z
2 Z
1− 2
q
q
q
q−2
b
|(Qf )(ξ)||h(ξ)|
dξ ≤
|(Qf )(ξ)| 2 dξ
|b
h(ξ)|q dξ
R
R
R
. kf k2L1 (R) kf k2Lp (R) khkLq−2
p
(R)
. γ q−2 kGk2L1 (R) kGkqLp (R) ,
where the last inequality is a consequence of (3.1), (3.2), and (3.3). In particular, this term is
acceptable. In a similar way,
Z
q
2
q−3
b
|(Qf )(ξ)||b
h(ξ)||G(ξ)|
dξ . kf k2L1 (R) kf k2Lp (R) khkLp (R) kGkLq−3
p
(R) . γkGkL1 (R) kGkLp (R)
R
12
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
is likewise acceptable for q > 3. Now we focus on the second integral from (4.5). It is useful
to introduce the quadrilinear operator Φ by
Φ(f1 , f2 , f3 , f4 )(ξ)
Z
= Re
f1 (x1 )f2 (x2 )f3 (x3 )f4 (x4 ) e2πi(−x1 −x2 +x3 +x4 )ξ dx1 dx2 dx3 dx4 ,
{(x1 ∧x2 )>(x3 ∨x4 )}
so that Q(f ) = Φ(f, f, f, f ). It can be estimated either by
|Φ(f1 , f2 , f3 , f4 )(ξ)|
Z
Z
|f1 (x1 )||f2 (x2 )| sup ≤
R2
x∈R
x
−2πiyξ
f3 (y)e
Z
dy sup x∈R
−∞
= kf1 kL1 (R) kf2 kL1 (R) (F∗ f3 )(ξ)(F∗ f4 )(ξ),
x
−∞
f4 (y)e−2πiyξ dy dx1 dx2
or (similarly as Q) by
|Φ(f1 , f2 , f3 , f4 )(ξ)| ≤ kf3 kL1 (R) kf4 kL1 (R) (F∗ f1 )(ξ)(F∗ f2 )(ξ).
(4.6)
(4.7)
Multilinearity implies
q
2
Qf − QG = Φ(G, G, G, h) + Φ(G, G, h, f ) + Φ(G, h, f, f ) + Φ(h, f, f, f ),
so the L norm of this function is controlled using (4.6), (4.7), and Hölder’s inequality by
kQf − QGk
q
L 2 (R)
≤ kGk2L1 (R) kF∗ GkLq (R) kF∗ hkLq (R) + kGk2L1 (R) kF∗ hkLq (R) kF∗ f kLq (R)
+ kf k2L1 (R) kF∗ GkLq (R) kF∗ hkLq (R) + kf k2L1 (R) kF∗ hkLq (R) kF∗ f kLq (R) .
Invoking the Menshov–Paley–Zygmund inequality, (3.1), (3.2), and (3.3) then gives
kQf − QGk
q
L 2 (R)
. γkGk2L1 (R) kGk2Lp (R) .
Finally, yet another application of Hölder’s inequality combined with the linear Hausdorff–
Young inequality for G bounds the second integral in (4.5) by a constant times
γkGk2L1 (R) kGkqLp (R) .
As noted before, this concludes the proof of the lemma.
Proof of Lemma 3.3. The proof parallels that of the previous lemma, and so we shall be brief.
Using Hölder’s inequality, the pointwise estimates (4.3) and (4.4), and that fact that the
maximally truncated Fourier transform F∗ satisfies the estimate (1.10), we conclude that the
five integrals in (3.9) and (3.10) are respectively controlled by
q
kQf k 2 q
L 2 (R)
q
. kf k2q
kF∗ f kqLq (R) . kf k2q
kf kqLp (R) ,
L1 (R)
L1 (R)
kEf k 2 q
L 2 (R)
kEf k
q
L 2 (R)
kQf k2 q
2q
q
kfbkLq−2
q
(R) . kf kL1 (R) kf kLp (R)
L 2 (R)
kEf k2 q
L 2 (R)
≤ kf kqL1 (R) kF∗ f kqLq (R) . kf kqL1 (R) kf kqLp (R) ,
2
q
q
q
kfbkLq−4
q
(R) . kf kL1 (R) kf kLp (R)
2q
q
kfbkLq−4
q
(R) . kf kL1 (R) kf kLp (R)
Since kf kL1 (R) ≤ δ, the result follows.
q
4
kf kLq−2
p
(R) = kf kL1 (R) kf kLp (R) ,
4
q
4
q
q
4
kf kLq−4
p
(R) = kf kL1 (R) kf kLp (R) ,
q
8
kf kLq−4
p
(R) = kf kL1 (R) kf kLp (R) .
A SHARP NONLINEAR HAUSDORFF–YOUNG INEQUALITY FOR SMALL POTENTIALS
13
5
7
4
3
2
2
1
1
-3
-3
-2
-2
-1
-1
0
1
2
3
-3
-2
-1
0
1.5
0.15
0.5
0.05
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
1
1
1
, 100
, respectively.
Figure 1. Graphs of (log |a|2 ) 2 for α = 1, 21 , 10
5. A counterexample
Now we disprove inequality (1.11). For computational simplicity, take q = 4. Define:
F = −1[0,1) + 81[1,2) + 71[2,3) − 61[3,4) + 51[4,5) − 31[5,6)
(5.1)
and consider the potential f = αF for a variable parameter α > 0. Graphs of the function
1
ξ 7→ (log |a(ξ)|2 ) 2 for several values of α are depicted in Figure 1. Let us remark that
1
the graph looks almost identically as the one of |fb|, due to the effects of
already for α = 100
linearization.
1
provides a direct
Straightforward numerical computation of the L4 norms for α = 100
counterexample to (1.11):
(log |a(ξ)|2 ) 21 4
fb 4
= 0.12075670 . . . .
=
0.12075839
.
.
.
,
L (R)
L (R)
ξ
The above decimal numbers were evaluated using the Mathematica [12] command NIntegrate
with options WorkingPrecision → 1000 and AccuracyGoal → 14. The step-function 5.1 was
discovered by numerical evaluation of the 6-linear generalization of the form H(F ) for indicator
functions of arbitrary intervals [kj , kj + 1), where kj ∈ {0, 1, . . . , 5} for each j ∈ {0, 1, . . . , 5},
and subsequent numerical optimization of the coefficients.
Acknowledgements
V.K. was supported in part by the Croatian Science Foundation under the project 3526.
V.K. and J.R. were partially supported by the bilateral DAAD-MZO grant Multilinear singular integrals and applications. D.O.S. was partially supported by the Hausdorff Center for
Mathematics and DFG grant CRC1060. This work was started during a pleasant visit of the
second author to the University of Zagreb, whose hospitality is greatly appreciated.
14
V. KOVAČ, D. OLIVEIRA E SILVA, AND J. RUPČIĆ
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Vjekoslav Kovač, Department of Mathematics, Faculty of Science, University of Zagreb,
Bijenička cesta 30, 10000 Zagreb, Croatia
E-mail address: [email protected]
Diogo Oliveira e Silva, Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn,
Germany
E-mail address: [email protected]
Jelena Rupčić, Faculty of Transport and Traffic Sciences, University of Zagreb, Vukelićeva
4, 10000 Zagreb, Croatia
E-mail address: [email protected]