Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

0.18 MB·

by  Antonio Córdoba

RADIAL MULTIPLIERS AND RESTRICTION TO SURFACES OF
THE FOURIER TRANSFORM IN MIXED-NORM SPACES
6
1 ANTONIOCO´RDOBA1 ANDERICLATORRE1
0
2
n Abstract. In this article we revisit some classical conjectures in harmonic
a analysisinthesettingofmixednormspacesLpradL2ang(Rn). Weproducesharp
J bounds for the restriction of the Fourier transform to compact hypersurfaces
of revolution in the mixed norm setting and study an extension of the disc
9
multiplier. Wealsopresentsomeresultsforthediscreterestrictionconjecture
1
andstateanintriguingopenproblem.

]
A
C
1. Introduction
.
h
t Thewell-knownrestrictionconjecture,firstproposedbyE.M.Stein,assertsthat
a
the restriction of the Fourier transformof a givenintegrable function f to the unit
m
[ sphere, fˆ|Sn−1, yields a bounded operator from Lp(Rn), n ≥ 2, to Lq Sn−1 so
long as
(cid:0) (cid:1)
2 2n 1 n+1 1
1 p< , 1 .
v ≤ n+1 q ≥ n 1 − p
0 − (cid:18) (cid:19)
This conjecture has been fully provedonly in dimension n=2 by C. Fefferman [7]
7
8 (see also [4] for an alternative geometrical proof). In higher dimensions, the best
3 known result is the particular case q = 2 and 1 p 2(n+1), which proof was
0 ≤ ≤ n+3
obtained independently by P. Tomas and E. M. Stein [13].
.
1 The periodic analogue,i.e. for Fourierseries,wasobservedby A. Zygmund[16],
0 but also in two dimensions. It asserts that for any trigonometric polynomial
6
1 P (x)= a e2πiν·x, ν Z2,
ν
: ∈
v |νX|=R
i the following inequality holds:
X
r P . P ,
a k kL4(Q) k kL2(Q)
uniformly on R>0 and where Q is any unit square in the plane.
The alternative proof given in [4] allows us to connect both the periodic and
the nonperiodic restriction theorems, explaining the reason for the apparently dif-
ferent numerologies of the corresponding (p,q) exponent ranges. It also raises an
interesting question about the location of lattice points in small arcs of circles [3].
The first result in this paper goes further in that direction: Given ξ a finite
j
{ }
set of points in the circle ξ =R of the plane, let us consider
{k k }
1
M :=sup# ξk, ξk ξj R2 .
k − k≤
j
n o
We have:
1- The authors are partially supported by the grant MTM2014-56350-P from the Ministerio
deCienciaeInnovacio´n (Spain).
1
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 2
Theorem 1. The following inequality holds
sup ˆ ake2πiξk·x 4dµ(x) 14 .M12 |ak|2 21 , (1.1)
µ(Q)=1(cid:20) Q(cid:12)X (cid:12) (cid:21) (cid:16)X (cid:17)
where the suprermum is(cid:12)(cid:12)taken over all(cid:12)(cid:12)unit squares of R2 and µ corresponds to the
Lebesgue measure.
The corresponding result in higher dimensions (n 3) is an interesting open
≥
problem:
Conjecture 2. Let {ξj} ⊂ SRn−1 and M := supj# ξk, kξk−ξjk≤R21 , is it
true that n o
n−1
sup ˆ ake2πiξk·x n2−n1 dµ(x) 2n .M12 |ak|2 21 . (1.2)
µ(Q)=1(cid:20) Q(cid:12)X (cid:12) (cid:21) (cid:16)X (cid:17)
Although there are(cid:12) many interes(cid:12)ting publications by several authors throwing
(cid:12) (cid:12)
somelightontherestrictionconjecture,itsproofremainsopenindimensionn 3.
≥
One of the more remarkable improvements was B. Barcelo’s thesis [12]. He proved
that Fefferman’s result also holds for the cone in R3. Another interesting result
was given by L. Vega in his Ph.D. thesis [14], where he obtained the best result
in the Stein-Tomas restriction inequality when the space Lp(Rn) is replaced by
Lp L2 (Rn).
rad ang
Here we shall consider the restriction of the Fourier transform to other surfaces
of revolution in these mixed norm spaces. Several special cases have already been
treated [8, 9] but we present a more general and unified proof for “all” compact
surfaces of revolution:
Γ= (g(z),θ,z) Rn+1, θ Sn−1, a z b, 0 g C1(a,b) .
∈ ∈ ≤ ≤ ≤ ∈
That is, in (cid:8)Rn+1, n 2, we consider cylindrical coordinates (r,θ,z) w(cid:9)here the
first components (r,θ≥) correspond to the standard polar coordinates in Rn; 0 <
r < , θ Sn−1, and z R denotes the zenithal coordinate. In this coordinate
syste∞m, the∈Lp L2 L2 ∈Rn+1 norm is given by
rad zen ang
∞ (cid:0) ∞(cid:1) p2 p1
rn−1 f(r,θ,z)2dθdz dr .
ˆ0 (cid:18)ˆ−∞ˆSn−1| | (cid:19) !
We can state our result.
Theorem 3. Let Γ be a compact surface of revolution, then the restriction of the
Fourier transform to Γ is a bounded operator from Lp L2 L2 Rn+1 to L2(Γ),
rad zen ang
i.e. there eists a finite constant C such that
p
(cid:0) (cid:1)
∞ 2 12
g(z)n−1 1+g′(z)2 fˆ(g(z),θ,z) dθdz .
ˆ ˆ
(cid:18) −∞ Sn−1 q (cid:12) (cid:12) (cid:19)
(cid:12) .C(cid:12) f , (1.3)
(cid:12) (cid:12)pk kLpradL2zenL2ang(Rn+1)
so long as 1 p< 2n .
≤ n+1
AcentralpointinthisareaisC.Fefferman’sobservationthatthediscmultiplier
in Rn for n 2, given by the formula
≥
T f(ξ)=χ (ξ)fˆ(ξ),
0 B(0,1)
d
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 3
is bounded on Lp(Rn) only in the trivial case p=2. However, it was later proved
(see ref [6] and [10]) that T is bounded on the mixed norm spaces Lp L2 (Rn)
0 rad ang
if and only if 2n < p < 2n . Here we extend that result to a more general class
n+1 n−1
of radial multipliers.
Theorem 4. Let T be a Fourier multiplier defined by
m
(T f)ˆ(ξ):=m(ξ )fˆ(ξ), (1.4)
m
| |
for all rapidly decreasing smooth functions f, where m satisfies the following hypo-
thesis:
(1) Supp(m) [a,b] R+, and m is differentiable in the interior (a,b).
⊂ ⊂
(2) b m′(x) dx< .
a | | ∞
T is th´en bounded in Lp L2 (Rn) so long as 2n <p< 2n .
m rad ang n+1 n−1
Finally, let us observe that Theorem 4 admits different extensions taking into
account Littlewood-Paley theory. Some vector valued and weighted inequalities
are satisfied by T and the so called universal Kakeya maximal function acting on
0
radial functions.
2. Restriction in the discrete setting
Proof of Theorem 1. Firstletusobservethat,byaneasyargument,wecanassume
M =1 without loss of generality. Next we take a smooth cut-off ϕ sot that
1
ϕ 1 on B 0, ,
≡ 2
(cid:18) (cid:19)
ϕ 0 when x 1,
≡ k k≥
ϕ C∞ R2 .
∈ 0
We can then write (cid:0) (cid:1)
f(ξ) = a ϕ(ξ+ξ )e2πiξ·q
k k
k
X
= a ϕ (ξ)e2πiξ·q,
k k
k
X
where q is a point in R2. We have
fˆ(x)= akϕˆ(x q)e2πiξk·(x−q).
−
k
X
Note that the L4 norm of fˆmajorizes the left hand side of (1.1),
4 4
ˆ fˆ(x) dx ≥ ˆ ake2πiξk·(x−q)ϕˆ(x−q) dx
(cid:12) (cid:12) x−q∈Q0(cid:12)X (cid:12)
(cid:12) (cid:12) (cid:12) 4 (cid:12)
(cid:12) (cid:12) & ˆ a(cid:12)ke2πiξk·x dx, (cid:12)
Q
(cid:12)X (cid:12)
(cid:12) (cid:12)
where Q = 1,1 2 and Q=q+(cid:12)Q . (cid:12)
0 −2 2 0
(cid:2) (cid:3)
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 4
On the other hand, we have
4
fˆ(x) dx = f f(ξ)2dξ
ˆ ˆ | ∗ |
(cid:12) (cid:12)
(cid:12) (cid:12) 2
(cid:12) (cid:12)
= a a ϕ ϕ (ξ)eiξ·q dξ.
ˆ (cid:12) k j k∗ j (cid:12)
(cid:12)k,j (cid:12)
(cid:12)X (cid:12)
(cid:12) (cid:12)
Furthermore,becausethesupportsofϕ(cid:12)k andϕj haveafiniteov(cid:12)erlapping,uniformly
(cid:12) (cid:12)
on the radius R.
4 2
fˆ(x) dx. a 2 ,
ˆ | k|
q.e.d. (cid:12)(cid:12) (cid:12)(cid:12) (cid:16)X (cid:17) (cid:3)
(cid:12) (cid:12)
Using similar arguments we can obtain the following analogous result: In R2
let us consider the parabola γ(t) = t,t2 and a set of real numbers ξ so that
j
{ }
t t 1, then
j+1 j
| − |≥ (cid:0) (cid:1)
1
sup a e2πiγ(tj)·x . a 2 2 .
(cid:13) j (cid:13) | j|
µ(Q)=1(cid:13)(cid:13)Xj (cid:13)(cid:13)L4(Q) (cid:16)X (cid:17)
(cid:13) (cid:13)
An interesting open ques(cid:13)tion is to decide(cid:13)if the L4 norm could be replaced by an
(cid:13) (cid:13)
Lp norm (p > 4) in the inequality above. It is known that p = 6 fails, but for
4<p<6 it is, as far as we know, an interesting open problem [2].
3. The restriction conjecture in mixed norm spaces
Recall that in Rn+1 we establish cylindrical coordinates (r,θ,z), where (r,θ)
correspondstotheusualsphericalcoordinatesinRn andz Rdenotesthezenithal
∈
component. We will also use the notation (ρ,φ,ζ) to refer to the same coordinate
system.
The Lp L2 L2 Rn+1 norm is therefore given by
rad zen ang
(cid:0) ∞(cid:1) ∞ p2 p1
f = rn−1 f(r,θ,z)2dθdz dr . (3.1)
k kLp,2,2 ˆ0 (cid:18)ˆ−∞ˆSn−1| | (cid:19) !
Let g be a continuous positive function supported on a compact interval I of
the real line that is almost everywhere differentiable, and consider the surface of
revolution in Rn+1 given by
Γ:= (g(z),θ,z) Rn+1, θ Sn−1, <z < . (3.2)
∈ ∈ −∞ ∞
We are interested in the restriction to Γ of the Fourier transform of functions in
(cid:8) (cid:9)
the Schwartz class Rn+1 . The restriction inequality
S
(cid:0) (cid:1)fˆ C f
L2(Γ) ≤ pk kLp,2,2(Rn+1)
(cid:13) (cid:13)
for 1 p< 2n is, by dual(cid:13)ity(cid:13), equivalent to the extension estimate:
≤ n+1 (cid:13) (cid:13)
fdΓ C f
Lq,2,2(Rn+1) ≤ qk kL2(Γ)
(cid:13) (cid:13)
for q > 2n . (cid:13)d(cid:13)
n−1 (cid:13) (cid:13)
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 5
To compute fdΓ let us recall
d dΓ = g(z)n−1 1+(g′(z))2dzdθ
= G (z)dzqdθ,
1
so that
∞
fdΓ(ρ,φ,ζ)= G (z)f(z,θ)e−izζe−i(ρg(z))θ·φdθdz. (3.3)
ˆ ˆ 1
−∞ Sn−1
Next we usde the spherical harmonic expansion
f(z,θ)= a (z)Yj(θ),
k,j k
k,j
X
whereforeachk, Yj isanorthonormalbasisofthesphericalharmonics
k
j=1,...,d(k)
degree k. We thennobotain:
∞ 1
fdΓ(ρ,φ,ζ)= 2πikYj(φ)ρ−n−22 g(z)n2 1+(g′(z))2 2
k ˆ ·
Xk,j −∞ (cid:16) (cid:17)
d ·ak,j(z)Jk+n−22 (ρg(z))e−izζdz,
whereJ denotes Bessel’sfunction oforderν (see ref. [15]). Denoting by G (z):=
ν 2
1
g(z)n2 1+(g′(z))2 2, the Fourier transform fdΓ becomes
(cid:16) (cid:17)
∞
k,j 2πikYkj(φ)ρ−n−22 ˆ−∞G2(z)ak,j(z)dJk+n−22 (ρg(z))e−izζdz. (3.4)
X
Taking into account the orthogonality of the elements of the basis Yj to-
k
getherwithPlancherel’sTheoreminthez-variable,weobtainthattheminxedonorm
q
fdΓ is up to a constant equal to
Lq,2,2
(cid:13) (cid:13)
(cid:13)d(cid:13) q
(cid:13) ∞ (cid:13) ∞ 2
ˆ ρ−qn−22+n−1 ˆ |g(ζ)|n 1+(g′(ζ))2 |ak,j(ζ)|2|Jνk(ρg(ζ))|2dζ dρ,
0 Xk,j −∞ (cid:12) (cid:12)
 (cid:12)(cid:12) (cid:12)(cid:12) (3.5)
where ν =k+ n−2. On the other hand we have
k 2
2
∞
f 2 = a (z)Yj(θ) g(z)n−1 1+g′(z)2dθdz
ˆΓ| | ˆ−∞ˆSn−1(cid:12)(cid:12)j,k k,j k (cid:12)(cid:12) q
(cid:12)X (cid:12)
∞ (cid:12) (cid:12)
= ˆ |ak(cid:12)(cid:12),j(z)|2g(z)n−1 (cid:12)(cid:12)1+g′(z)2dz. (3.6)
j,k −∞ q
X
Therefore our theorem will be a consecuence of the following fact:
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 6
Lemma 5. Given any sequence of positive indices ν with ν n−2 for all j
{ j} j ≥ 2
and Schwartz functions a , the following inequality holds:
j
q
∞ ∞ 2
ˆ ρ−qn−22+n−1 ˆ |g(z)|n 1+(g′(z))2 |aj(z)|2 Jνj(ρg(z)) 2dz dρ
0 Xj −∞ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) q 
∞ 1 2
. g(z)n−1 1+(g′(z))2 2 a (z)2dz , (3.7)
 ˆ | | | j | 
Xj −∞ (cid:16) (cid:17)
for q > 2n .  
n−1
Remark 6. Taking into account the hypothesis about g we will look for estimates
depending uponA=sup g(x) andB =sup g′(x),where I is the compact
x∈I| | x∈I| |
support of g. It is also easy to see that we can reduce ourselves to consider the
sums overthe family of indices ν ∞ suchthatν n−2. Thereforeit is enough
{ j}j=1 j ≥ 2
to show
q
∞ ∞ 2
ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj (ρg(z)) 2dz dρ
0 j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
∞ 2
. b (z)2dz (3.8)
 ˆ | j | 
j −∞
X
for a family of smooth functions b and indexesν n−2. 
{ j}j j ≥ 2
In order to show (3.8) we will need a sharp control of the decay of Bessel func-
tions; namely the following estimates:
Lemma 7. The following estimates hold for ν 1.
≥
(1) J (r) 1 , when r 2ν.
ν ≤ r1/2 ≥
(2) J (r) 1, when r 1ν.
ν ≤ ν ≤ 2
(3) J ν+ρν1/3 1 , when 0 ρ 3ν2/3.
ν ≤ ρ1/4ν1/3 ≤ ≤ 2
(4) J ν ρν1/3 1 , when 1 ρ 3ν2/3.
ν(cid:0) − (cid:1)≤ ρν1/3 ≤ ≤ 2
(5) J (r) rν, as r 0.
ν(cid:0) ≤ (cid:1) →
Theseassymptoticsfollowbythe stationaryphasemethodasitis shownin[15],
[1] and [5].
Proof of Lemma 5 . To prove 3.8 we shall first decompose the ρ-integration in dy-
adic parts: [0, )=[0,1) ∞ [2n,2n+1).
∞ ∪n=0
S
q
1 ∞ 2
ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj (ρg(z)) 2dz dρ
0 j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
2M ∞ 2
+ ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj(ρg(z)) 2dz dρ, (3.9)
M M j −∞
X X (cid:12) (cid:12)
 (cid:12) (cid:12) 
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 7
where M =2m, m=0,1,...
For the lower integrand, we have the following splitting:
1 1 1
ρ−qn−22+n−1[...]q2 dρ = A ρ−qn−22+n−1[...]q2 dρ+ ρ−qn−22+n−1[...]q2 dρ
ˆ ˆ ˆ
0 0 1
A
= I +II.
In order to bound I we invoke Minkowski’s inequality and property 5. of Lemma
7.
q
∞ 1 q 2q 2
I . ˆ−∞Xj ˆ0A nρ−(n−2)+q2(n−1)|bj(z)|2(cid:12)Jνj(ρz)(cid:12)2o2 dρ! dz .
 (cid:12) (cid:12) q 
∞ 1 2q 2
≤ ˆ−∞ j |bj(z)|2A2νj ˆ0A ρ−qn−22+(n−1)+qνjdρ! dz ,
X
 
where A= g . Since the sum is taken over all ν n−2, the inner integrand is
k k∞ j ≥ 2
well defined and we can bound
q
∞ 2
I .Aqn−21−n ˆ |bj(z)|2dz . (3.10)
j −∞
X
 
The second part is similarly bounded
q
∞ 2
II . 1+Aqn−21−n  ˆ |bj(z)|2dz . (3.11)
(cid:16) (cid:17) Xj −∞
 
Then Lemma 5 will be a consequence of the following claim:
Claim 8. For all q >4, the following inequality holds true
q
2M ∞ 2
ρ b (z)2 J (ρg(z)) 2dz dρ
ˆ  ˆ | j | νj 
M j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
∞ 2
.M4−2q ˆ |bj(z)|2dz . (3.12)
−∞ j
X
 
Indeed, if q >4 we need only to note that
q
2M ∞ 2
ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj(ρg(z)) 2dz dρ
M j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
2M ∞ 2
.M(n−2)(−q2+1)ˆ ρ ˆ |bj(z)|2 Jνj (ρg(z)) 2dz dρ,
M j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12) 
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 8
invoke our claim and sum over all dyadic intervals in (3.9):
q
2m+1 ∞ 2
ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj(ρg(z)) 2dz dρ
m 2m j −∞
X X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
∞ 2
. 2m(n−2)(−q2+1)+m4−2q ˆ |bj(z)|2dz . (3.13)
m −∞ j
X X
 
It is then a simple matter to check that the exponent is negative for q > 2n .
n−1
Iftheexponentq ishoweversmaller, 2n <q 4,weneedtouseanextratrick.
n−1 ≤
Note that equation (3.12) implies
q1 q1
2M ∞ 2 ∞ 2
ˆ  ˆ |bj(z)|2 Jνj(ρg(z)) 2dz dρ.M1−q21 ˆ |bj(z)|2dz ,
M j −∞ −∞ j
X (cid:12) (cid:12) X
 (cid:12) (cid:12)   
for all q >4. Then using Ho¨lder’s inequality and the previous inequality,
1
q
2M ∞ 2
b (z)2 J (ρg(z)) 2dz dρ
ˆ  ˆ | j | νj 
M j −∞
X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
2M ∞ q21 q1
.M1−qq1 ˆ  ˆ |bj(z)|2 Jνj (ρg(z)) 2dz dρ .
M j −∞
 X (cid:12)(cid:12) (cid:12)(cid:12)  
Therefore, summing over all intervals, we obtain
q
2m+1 ∞ 2
ˆ ρ−qn−22+n−1 ˆ |bj(z)|2 Jνj(ρg(z)) 2dz dρ
m 2m j −∞
X X (cid:12) (cid:12)
 (cid:12) (cid:12)  q
∞ 2
. 2m{−qn−22+n−1+1−2q}ˆ |bj(z)|2dz ,
m −∞ j
X X
 
where the exponent qn−1 +n is negative for all q > 2n .
− 2 n−1
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 9
To prove Claim 8 let us split each dyadic integrand in (3.9) in three parts cor-
responding to the differnt ranges of control of Bessel functions.
q
2M ∞ 2
ρ b (z)2 J (ρg(z)) 2dz dρ
ˆ ˆ | j | νj 
M −∞νXj∈I0 (cid:12) (cid:12)
 (cid:12) (cid:12)  q
2M ∞ 2
+ ρ b (z)2 J (ρg(z)) 2dz dρ
ˆ ˆ | j | νj 
M −∞νXj∈Ic (cid:12) (cid:12)
 (cid:12) (cid:12)  q
2
2M ∞
+ ρ b (z)2 J (ρg(z)) 2dz dρ
ˆ ˆ | j | νj 
M −∞νjX∈I∞ (cid:12) (cid:12)
 (cid:12) (cid:12) 
= I0 +Ic +I∞ ,
M M M
M
X(cid:0) (cid:1)
where I0 =[0,Mg(z)/2), Ic =[Mg(z)/2,4Mg(z)), and I∞ =[4Mg(z), ).
M M M ∞
Recall that if 2k < r, J (r) r−1/2; in I0 we have 2ν < Mg(z) < ρg(z),
| k | ≤ M j
hence
q
2M ∞ 2
IM0 ≤A−q2 ˆ ρ1−q2 ˆ |bj(z)|2dz dρ
M −∞νXj∈I0
 q 
∞ 2
≤A−2q M4−2q ˆ |bj(z)|2dz . (3.14)
−∞Xνj
 
Similarly, I∞ is also easily bounded as if k > 2r, J (r) k−1, and in I∞,
M | k | ≤ M
k >4Mg(z)>2ρg(z). Furthermore,sinceρg(z)>1,(ρg(z))−2 <(ρg(z))−1 and,
in I∞, we have J (ρg(z))2 (ρg(z))−1. This shows that again
M | k | ≤
q
∞ 2
IM∞ ≤A−q2M4−2q ˆ |bj(z)|2dz . (3.15)
−∞Xνj
 
Finally, we need to work a little bit harder than in the previous cases to obtain
a suitable estimate for Ic . First of all note that Minkowski’s inequality yields
M
q
∞ 2M q2 2q 2
IMc ≤ˆ ˆ ρ |bj(z)|2 Jνj (ρg(z)) 2 dρ dz . (3.16)
 −∞ M νXj∈Ic (cid:12) (cid:12)  
  (cid:12) (cid:12)  
In Ic we want touse estimate (3) of Lemma 7, we thus need to split the inner
M
integralso that ρg(z) ν +αν in the according range of α. Consider the family
j j
∼
of sets
Gα = M +αM31g(z)−32 ,M +(α+1)M31g(z)−23 ,
2 2
(cid:20) (cid:21)
RADIAL MULTIPLIERS AND RESTRICTION IN MIXED NORM SPACES 10
2
for α = 0,1,2,..., (Mg(z))3 , so that Gα [M,2M] and in each interval
⊇
1 h i
ρg(z) ν +αν3, and split (3.16) in the foSllowing way
∼ j j
q
2q 2q 2
∞
IMc .ˆ  ˆ ρ |bj(z)|2 Jνj (ρg(z)) 2 dρ dz ,
 −∞Xα Gα νXj∈Ic (cid:12) (cid:12)  
  (cid:12) (cid:12)  
Let us alsodefine  
A = b (z)2.
β j
| |
νjX∈Gβ
We can then invoke Lemma 7 and rearragne the sums to bound Ic by
M
q
∞ 1 q2 q2 2
ˆ−∞Xα ˆGαβX≤αAβ(|α−β|+1)1/2M23g(z)−34 ρdρ dz
   
  ∞ 1 q2  q2 q2
+ˆ−∞Xα ˆGαβX≥αAβ(|α−β|+1)2M23g(z)−34 ρdρ dz .
   
Note that the second sum is easier to control than the first. We shall, therefore,
focus on the first term, IMc,1,. Since the intervals Gα have length M31g(z)−32,
q
IMc,1 . M4−3qA2(q−31) ˆ−∞∞Xα βX≥αAβ(|α−β1|+1)2q22q dz2 .
   
Furthermore, using Young’sinequality, since q > 4, taking 2/q =1/s 1/2 we
−
obtain
1 q2 2qs
A . As
 β(α β +1)2 γ!
Xα βX≥α | − | Xγ
  q
2
. A .
γ
!
γ
X
We have thus ahowed that the central integrand Ic can also be bounded in the
M
desired way;
q
2
∞
IMc,1 .A2(q−31)M4−3q ˆ |ak|2dz . (3.17)
−∞k∈Ic
XM
 
q.e.d. (cid:3)