FOURIER RESTRICTION ESTIMATES TO MIXED
HOMOGENEOUS SURFACES
E. FERREYRA AND M. URCIUOLO
FAMAF - CIEM (Universidad Nacional de Córdoba - Conicet).
Medina Allende s/n, Ciudad Universitaria,
5000 Córdoba.
EMail: eferrey@mate.uncor.edu
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
urciuolo@gmail.com
Received:
17 September, 2008
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Accepted:
13 February, 2009
Contents
Communicated by:
L. Pick
2000 AMS Sub. Class.:
Primary 42B10, 26D10.
Key words:
Restriction theorems, Fourier transform.
Abstract:
Let a, b be real numbers such that 2 ≤ a < b, and let ϕ : R2 → R a mixed
homogeneous function. We consider polynomial functions ϕ and also functions
of the type ϕ (x1 , x2 ) = A |x1 |a + B |x2 |b . Let Σ = {(x,
ϕ (x)) : x ∈ B}
with the Lebesgue induced measure. For f ∈ S R3 and x ∈ B, let
(Rf ) (x, ϕ (x)) = fb(x, ϕ (x)) , where fb denotes the usual Fourier transform.
For a large class of functions ϕ and for 1 ≤ p < 34 we characterize, up
to
endpoints, the pairs (p, q) such that R is a bounded operator from Lp R3 on
Lq (Σ) . We also give some sharp Lp → L2 estimates.
Acknowledgements:
Research partially supported by Secyt-UNC, Agencia Nacional de Promoción
Científica y Tecnológica.
The authors wish to thank Professor Fulvio Ricci for fruitful conversations about
this subject.
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Contents
1
Introduction
3
2
Preliminaries
5
3
The Cases ϕ (x1 , x2 ) = A |x1 |a + B |x2 |b
8
4
The Polynomial Cases
4.1 Sharp Lp − L2 Estimates . . . . . . . . . . . . . . . . . . . . . . .
11
20
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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1.
Introduction
Let a, b be real numbers such that 2 ≤ a < b, let ϕ : R2 → R be a mixed
homogeneous function of degree one with respect to the non isotropic dilations
1
1
r · (x1 , x2 ) = r a x1 , r b x2 , i.e.
(1.1)
1
1
ϕ r a x1 , r b x2 = rϕ (x1 , x2 ) ,
r > 0.
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
We also suppose ϕ to be smooth enough. We denote by B the closed unit ball of
R2 , by
Σ = {(x, ϕ (x)) : x ∈ B}
and by σ the induced Lebesgue measure. For f ∈ S (R3 ) , let Rf : Σ → C be
defined by
(1.2)
(Rf ) (x, ϕ (x)) = fb(x, ϕ (x)) ,
x ∈ B,
where fb denotes the usual Fourier transform of f. We denote by E the type set
associated to R, given by
1 1
E=
,
∈ [0, 1] × [0, 1] : kRkLp (R3 ),Lq (Σ) < ∞ .
p q
Our aim in this paper is to obtain as much information as possible about the set E,
for certain surfaces Σ of the type above described.
In the general n-dimensional case, the Lp (Rn+1 )−Lq (Σ) boundedness properties
of the restriction operator R have been studied by different authors. A very interesting survey about recent progress in this research area can be found in [11]. The
Lp (Rn+1 ) − L2 (Σ) restriction theorems for the sphere were proved by E. Stein in
vol. 10, iss. 2, art. 35, 2009
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n+4
1967, for 3n+4
< p1 ≤ 1; for 2n+4
< p1 ≤ 1 by P. Tomas in [12] and then in the same
4n+4
n+4
year by Stein for 2n+4
≤ p1 ≤ 1. The last argument has been used in several related
contexts by R. Strichartz in [9] and by A. Greenleaf in [6]. This method provides a
general tool to obtain, from suitable estimates for σ
b, Lp (Rn+1 ) − L2 (Σ) estimates
for R. Moreover, a general theorem, due to Stein, holds for smooth enough hypersurfaces with never
Gaussian curvature ([8], pp.386). There it is shown
vanishing
1 1
n+4
1
that in this case, p , q ∈ E if 2n+4
≤ p1 ≤ 1 and − n+2
+ n+2
≤ 1q ≤ 1, also that
n p
n
this last relation is the best possible and that no restriction theorem of any kind can
n+2
n+2
n+4
hold for f ∈ Lp (Rn+1 ) when p1 ≤ 2n+2
([8, pp.388]). The cases 2n+2
< p1 < 2n+4
are not completely solved. The best results for surfaces with non vanishing curvature like the paraboloid and the sphere are due to T. Tao [10]. Restriction theorems
for the Fourier transform to homogeneous polynomial
surfaces in R3 are obtained in
[4]. Also, in [1] the authors obtain sharp Lp Rn+l − L2 (Σ) estimates for certain
homogeneous surfaces Σ of codimension l in Rn+l .
In Section 2 we give some preliminary results.
In Section 3 we consider ϕ (x1 , x2 ) = A |x1 |a + B |x2 |b , A 6= 0, B 6= 0. We describe completely, up to endpoints, the pairs p1 , 1q ∈ E with p1 > 34 . A fundamental
tool we use is Theorem 2.1 of [2].
In Section 4 we deal with polynomial functions
Under certain hypothesis about
ϕ.
3
1
1 1
ϕ we can prove that if 4 < p ≤ 1 and the pair p , q satisfies some sharp conditions,
4
1 1
then p , q ∈ E. Finally we obtain some L 3 − Lq estimates and also some sharp
Lp − L2 estimates.
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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2.
Preliminaries
We take ϕ to be a mixed homogeneous and smooth enough function that satisfies
(1.1). If V is a measurable set in R2 , we denote ΣV = {(x, ϕ (x)) : x ∈ V } and σ V
as the associated surface measure. Also, for f ∈ S (R3 ) , we define RV f : ΣV → C
by
RV f (x, ϕ (x)) = fb(x, ϕ (x))
x∈V;
we note that RB = R, σ B = σ and ΣB = Σ.
For x = (x1 , x2 ) letting kxk = |x1 |a + |x2 |b , we define
2 1
A0 = x ∈ R : ≤ kxk ≤ 1
2
and for j ∈ N,
S
Thus B ⊆
Aj . A standard homogeneity argument (see, e.g. [5]) gives, for
j∈N∪{0}
From this we obtain the following remarks.
1
Remark 1. If p1 , 1q ∈ E then 1q ≥ − a+b+ab
+
a+b p
1
− a+b+ab
a+b p
(2.2)
then
1 1
,
p q
∈ E.
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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Aj = 2−j · A0 .
1 ≤ p, q ≤ ∞,
a+b 1
a+b+ab
1 a+b+ab (2.1) RAj Lp (R3 ),Lq (ΣAj ) = 2−j ab ( q − a+b + p a+b ) RA0 Lp (R3 ),Lq (ΣA0 ) .
Remark 2. If
Fourier Restriction
Estimates
+
a+b+ab
a+b
1
q
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a+b+ab
.
a+b
< ≤ 1 and
A
R 0 p 3 q A < ∞,
L (R ),L (Σ 0 )
Close
We will use a theorem due to Strichartz (see [9]), whose proofrelies on the Stein
complex interpolation theorem, which gives Lp (R3 ) − L2 ΣV estimates for the
V . In [4] we obtained inforoperator RV depending on the behavior at infinity of σc
mation about the size of the constants. There we found the following:
Remark 3. If V is a measurable set in R2 of positive measure and if
c
−τ
σ V (ξ) ≤ A (1 + |ξ3 |)
for some τ > 0 and for all ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , then there exists a positive constant
cτ such that
V
1
R p 3 2 V ≤ cτ A 2(1+τ )
L (R ),L (Σ )
for p = 2+2τ
.
2+τ
In [2] the authors obtain a result (Theorem 2.1, p.155) from which they also obtain
the following consequence
Remark 4 ([2, Corollary 2.2]). Let I, J be two real intervals, and let
M = {(x1 , x2 , ψ (x1 , x2 )) : (x1 , x2 ) ∈ I × J} ,
2
∂ ψ
where ψ : I × J → R is a smooth function such that either ∂x2 (x1 , x2 ) ≥ c > 0 or
1
2
∂ ψ
∂x2 (x1 , x2 ) ≥ c > 0, uniformly on I × J. If M has the Lebesgue surface measure,
2
1
1
= 3 1 − p and 34 < p1 ≤ 1 then there exists a positive constant c such that
q
(2.3)
for f ∈ S(R3 ).
b f |M Lq (M )
≤ c kf kLp (R3 )
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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Following the proof of Theorem 2.1 in [2] we can
2check that if in the last remark
−k −k+1 , k ∈ N in the case that ∂∂xψ2 (x1 , x2 ) ≥ c > 0 uniformly
we take J = 2 , 2
1
on I × J with c independent of k, or I = 2−k , 2−k+1 , k ∈ N in the other case, then
we can replace (2.3) by
1
1
b (2.4)
≤ c0 2−k( p + q −1) kf kLp (R3 )
f |M Lq (M )
with c0 independent of k.
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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3.
The Cases ϕ (x1 , x2 ) = A |x1 |a + B |x2 |b
In this cases we characterize, up to endpoints, the pairs p1 , 1q ∈ E with 34 < p1 ≤ 1.
We also obtain some border segments. If either A = 0 or B = 0, ϕ becomes
homogeneous and these cases are treated in [4]. For the remainder situation we
obtain the following
Theorem 3.1. Let a, b, A, B ∈ R with 2 ≤ a ≤ b, A 6= 0, B 6= 0, let ϕ (x1 , x2 ) =
A |x1 |a + B |x2 |b and let E be the type set associated to ϕ. If 43 < p1 ≤ 1 and
1
1 1
a+b+ab
1
− a+b+ab
+
<
≤
1
then
,
∈ E.
a+b p
a+b
q
p q
1
Proof. Suppose 34 < p1 ≤ 1 and − a+b+ab
+ a+b+ab
< 1q ≤ 1. By Remark 2 it
a+b p
a+b
is enough to prove
0 (2.2).
1 Now,
A0 is contained in the union of the rectangles Q =
1
[−1, 1] × 2 , 1 , Q = 2 , 1 × [−1,
to the x1 and
1] , and its symmetrics with respectS
x2 axes. Now we will study RQ Lp (R3 ),Lq (ΣQ ) . We decompose Q =
Qk with
k∈N
−k+1
−k −k+1 1
−k
Qk = −2
, −2
∪ 2 ,2
× ,1 .
2
Now, as in Theorem 1, (3.2), in [3] we have
a−2
d
−1
Q
k
σ (ξ) ≤ A2k 2 (1 + |ξ3 |)
and then Remark 3 implies
Q R k 4
(3.1)
L 3 (R3 ),L2
(ΣQk )
≤ c2k
a−2
8
.
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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2
∂ ϕ
Also, since ∂x2 (x1 , x2 ) ≥ c > 0 uniformly on Qk , from (2.4) we obtain
2
Q R k p 3 q Q ≤ c0 2−k( p1 + 1q −1)
L (R ),L (Σ k )
for 1q = 3 1 − p1 and 43 < p1 ≤ 1. Applying the Riesz interpolation theorem and
then performing the sum on k ∈ N we obtain
Q
R p 3 q Q < ∞,
L (R ),L (Σ )
for
2+3a
2+a
1−
1
p
<
1
q
3
4
≤ 1 and
1
p
<
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
≤ 1. In a similar way we get that
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Q0 R Lp (R3 ),Lq (ΣQ
1
for 2+3b
1
−
<
2+b
p
is analogous. Thus
1
q
3
4
≤ 1 and
<
1
p
0
)
< ∞,
≤ 1. The study for the symmetric rectangles
A
R 0 p 3 q A < ∞
L (R ),L (Σ 0 )
for
3
4
<
1
p
1
≤ 1 and − a+b+ab
+
a+b p
a+b+ab
a+b
i) If
1
q
≤ 1 and the theorem follows.
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Remark 5.
b+2
8
<
Contents
<
1
q
ii) The point
≤ 1 then
3 1
,
4 q
a+b+2ab 1
,
2a+2b+2ab 2
∈ E.
∈ E.
Close
From (3.1) and the Hölder inequality we obtain that
Q k( a−2
− 2−q
R k 4
8
2q )
≤
c2
Q
3
q
L 3 (R ),L (Σ k )
for
1
2
≤
1
q
≤ 1. Then if
a+2
8
< 1q ≤ 1 we perform the sum over k ∈ N to get
Q
R 4
< ∞,
L 3 (R3 ),Lq (ΣQ )
for these q’s. Analogously, if b+2
<
8
Q0 R 1
q
thus since a ≤ b, if
b+2
8
<
1
q
E. Ferreyra and M. Urciuolo
≤ 1 we get
4
L 3 (R3 ),Lq (ΣQ0 )
≤ 1,
A
R 0 4
L 3 (R3 ),Lq
Fourier Restriction
Estimates
vol. 10, iss. 2, art. 35, 2009
< ∞,
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(ΣA0 )
< ∞,
and i) follows from Remark 2.
Assertion ii) follows from Remark 3, since from Lemma 3 in [3] we have that
1
1
|b
σ (ξ)| ≤ c (1 + |ξ3 |)− a − b .
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4.
The Polynomial Cases
In this section we deal with mixed homogeneous polynomial functions ϕ satisfying
(1.1). The following result is sharp (up to the endpoints) for 34 < p1 ≤ 1, as a
consequence of Remark 1.
Theorem 4.1. Let ϕ be a mixed homogeneous polynomial function satisfying (1.1).
Suppose that the gaussian curvature of Σ does not vanish identically and that at
each point of ΣB−{0} with vanishing curvature, at least one principal curvature is
1
different from zero. If (a, b) 6= (2, 4) , 43 < p1 ≤ 1 and − a+b+ab
+ a+b+ab
< 1q ≤ 1
a+b p
a+b
then p1 , 1q ∈ E.
Proof. We first study the operator RA0 . Let (x01 , x02 ) ∈ A0 . If Hessϕ (x01 , x02 ) 6= 0
there exists a neighborhood U of (x01 , x02 ) such that Hessϕ (x1 , x2 ) 6= 0 for (x1 , x2 ) ∈
U. From the proposition in [8, pp. 386], it follows that
U
R p 3 q U < ∞
(4.1)
L (R ),L (Σ )
for 1q = 2 1 − p1 and 34 ≤ p1 ≤ 1. Suppose now that Hessϕ (x01 , x02 ) = 0 and that
∂2ϕ
∂x21
(x01 , x02 ) 6= 0 or
∂2ϕ
∂x22
(x01 , x02 ) 6= 0. Then there exists a neighborhood V =
2
2
I × J of (x01 , x02 ) such that either ∂∂xϕ2 (x1 , x2 ) ≥ c > 0 or ∂∂xϕ2 (x1 , x2 ) ≥ c > 0
1
2
uniformly on V. So from Remark 4 we obtain that
V
R p 3 q V < ∞
(4.2)
L (R ),L (Σ )
for 1q = 3 1 − p1 and 34 < p1 ≤ 1. From (4.1), (4.2) and Hölder´s inequality, it
follows that
A
R 0 p 3 q A < ∞
(4.3)
L (R ),L (Σ 0 )
either
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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1
p
for ≥ 3 1 −
and 34 < p1 ≤ 1. So, if a+b+ab
≥ 3, the theorem follows from
a+b
Remark 2. The only cases left are (a, b) = (3, 4) , (a, b) = (3, 5) , (a, b) = (4, 5)
and (a, b) = (2, b) , b > 2. If (a, b) = (3, 4) and ϕ has a monomial of the form
ai,j xi y j , with aij 6= 0, then 3i + 4j = 1 so 4i + 3j = 12 and so either (i, j) = (0, 4) or
(i, j) = (3, 0). So ϕ (x1 , x2 ) = a3,0 x31 + a0,4 x42 . The hypothesis about the derivatives
of ϕ imply that a3,0 6= 0 and a0,4 6= 0 and the theorem follows using Theorem
3.1 in each quadrant. The cases (a, b) = (3, 5) , or (a, b) = (4, 5) are completely
analogous.
Now we deal with the cases (a, b) = (2, b) , b > 2. We note that
1
q
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
b
ϕ (x1 , x2 ) = Ax21 + Bx1 x22 + Cxb2
(4.4)
where B = 0 for b odd. The hypothesis about ϕ implies A 6= 0. For b odd,
ϕ (x1 , x2 ) = Ax21 + Cxb2 and since C 6= 0 (on the contrary Hessϕ (x1 , x2 ) ≡ 0), the
theorem follows using Theorem 3.1 as before. Now we consider b even and ϕ given
by (4.4). If B = 0 the theorem follows as above, so we suppose B 6= 0.
(4.5)
Hessϕ (x1 , x2 )
b
−2
2
=−
x2
4
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b
B 2 b2 + 8ACb − 8ACb2 x22 − 2(b − 2)ABbx1 .
Full Screen
So if Hessϕ (x01 , x02 ) = 0 then either x02 = 0 or
B 2 b2 + 8ACb − 8ACb
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2
b
0 2
− 2(b − 2)ABbx01 = 0.
In the first case we have b > 4. We take a neighborhood W1 = I × −2−k0 , 2−k0 ⊂
A0 , k0 ∈ N, of the point (x01 , 0) such that Hessϕ vanishes, on W1 , only along the
x1 axes. For k ∈ N, k > k0 , we take Uk = I × Jk where Jk = [−2−k+1 , −2−k ] ∪
x2
Close
[2−k , 2−k+1 ] . So W1 = ∪Uk . For (x1 , x2 ) ∈ Uk , it follows from (4.5) that
b
|Hessϕ (x1 , x2 )| ≥ c2−k( 2 −2) ,
so for ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 ,
b−4
d
Uk (ξ) ≤ c2k 4 (1 + |ξ |)−1
σ
3
and from Remark 3 we get
U R k 4
(4.6)
L 3 (R3 ),L2
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
(
Σ Uk
)
≤ c2k
b−4
16
vol. 10, iss. 2, art. 35, 2009
.
2
Also, since ∂∂xϕ2 (x1 , x2 ) ≥ c > 0 uniformly on Uk , as in (2.4) we obtain
Title Page
1
U R k p 3 q U ≤ c2−k(2− p2 )
L (R ),L (Σ k )
(4.7)
3
4
1
p
for < ≤ 1 and
we obtain
(4.8)
for
1
qt
1
q
= 3 1−
1
p
. From (4.6), (4.7) and the Riesz Thorin theorem
= t 12 + (1 − t) 3 1 −
1
p
and
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U −(1−t)(2− p2 ))
R k p 3 q U ≤ c2k(t b−4
16
L t (R ),L t (Σ k )
Contents
1
pt
= t 34 + (1 − t) p1 .
A simple computation shows that if p1 = 34 then the exponent in (4.8) is negative
8
for t < t0 = 4+b
and that
2 + 3b
1
−
< 0,
qt0
4 (2 + b)
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so for
1
p
>
3
4
and t < t0 , both near enough, the exponent is still negative and
1
2 + 3b
1
−
1−
< 0,
qt
2+b
pt
thus
W
R 1 p 3 q W < ∞
L (R ),L (Σ 1 )
(4.9)
for
3
4
<
1
p
near enough and
1
q
=
2+3b
2+b
1−
B 2 b2 + 8ACb − 8ACb2
1
p
x02
Hessϕ (x1 , x02 )
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
. Finally, if
2b
vol. 10, iss. 2, art. 35, 2009
− 2(b − 2)ABbx01 = 0
−k−1
x01 |
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−k
then we study the order of
for 2
≤ |x1 −
≤ 2 , k ∈ N.
(x0 ) 2b −2 b
2
B 2 b2 + 8ACb − 8ACb2 x02 2 − 2(b − 2)ABbx1 (4.10) 4
(x0 ) 2b −2
= 2
(b − 2)ABb x1 − x01 ≥ c2−k .
2
We take the following neighborhood of (x01 , x02 ) , W2 = ∪k∈N Vk , with
1
1
1
0
−k−1
0
−k
r 2 x1 , r b x2 : 2
Vk =
≤ x1 − x1 ≤ 2 , ≤ r ≤ 2 .
2
From the homogeneity of ϕ and (4.10) we obtain
1
1
2 0 2
b
Hessϕ r x1 , r x2 = r1− b Hessϕ x1 , x02 ≥ c2−k ,
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then from Proposition 6 in [8, p. 344], we get for ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3
k
c
−1
V
k
σ (ξ) ≤ c2 2 (1 + |ξ3 |) ,
so from Remark 3
V R k 4
L 3 (R3 ),L2
k
(ΣVk )
≤ c2 8
and by Hölder’s inequality, for q < 2 we have
V k( 18 − 2−q
R k 4
2q ) .
≤
c2
V
L 3 (R3 ),Lq (Σ k )
This exponent is negative for 1q > 58 and so we sum on k to obtain
W
R 2 4
(4.11)
<∞
L 3 (R3 ),Lq (ΣW2 )
for
5
8
<
1
q
≤ 1. Since b ≥ 6,
5
8
≤
2+3b
4(2+b)
and then from (4.1), (4.9) and (4.11), we get
A
R 0 p 3 q A < ∞,
L (R ),L (Σ 0 )
1
for 34 < p1 near enough and 1q > 2+3b
1
−
and the theorem follows from standard
2+b
p
considerations involving Hölder’s inequality, the Riesz Thorin theorem and from
Remark 2.
Remark 6. In the case (a, b) = (2, b) , b > 2, we have (4.11). In a similar way we
get, from (4.6) and Hölder’s inequality,
W
R 1 4
<∞
L 3 (R3 ),Lq (ΣW1 )
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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for
b+4
16
<
1
q
≤ 1. So
kRkL 43 (R3 ),Lq (Σ) < ∞
5
2+3b
for max 8 , b+4
,
< 1q ≤ 1. We observe that if b = 6 then 58 = b+4
= 2+3b
,
16 8+4b
16
8+4b
1
thus from Remark 1 we see that, in this case, this condition for q is sharp, up to the
end point.
Now we will show some examples of functions ϕ not satisfying the hypothesis of
the previous theorem, for which we obtain that the portion of the type set E in the
region 34 < p1 ≤ 1 is smaller than the region
1 1
3
1
a + b + ab
1
1
Ea,b =
,
: < ≤ 1,
1−
< ≤1
p q
4
p
a+b
p
q
stated in Theorem 4.1.
We consider ϕ (x1 , x2 ) = x21 , which is a mixed homogeneous function satisfying
(1.1) for any b > 2. In this case ϕx1 x1 ≡ 2 but Hessϕ ≡ 0. From Remark 2.8 in [4]
1
1
and Remark 4 we obtain that the corresponding type set is the region q ≥ 3 1 − p ,
3
4
< p1 ≤ 1 which is smaller than the region Ea,b .
We consider now a mixed homogeneous function ϕ satisfying (1.1), of the form
Fourier Restriction
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E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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ϕ (x1 , x2 ) = xl2 P (x1 , x2 ) ,
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with P (x1 , 0) 6= 0 for x1 6= 0. Since a < b it can be checked that l ≥ 2 and that for
l > 2, ϕx1 x1 (x1 , 0) = ϕx2 x2 (x1 , 0) = 0. Moreover
(4.13)
Hessϕ = x22l−2 Px1 x1 l (l − 1) P + 2lx2 Px2 + x22 Px2 x2
− (lPx1 + x2 Px1 x2 )2 ,
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(4.12)
which vanishes at (x1 , 0) . A computation shows that the second factor is different
from zero at a point of the form (x1 , 0) . So Hessϕ does not vanish identically.
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Proposition
4.2. Let ϕ be a mixed
function satisfying (1.1) and (4.12).
homogeneous
1
1
1 1
If p , q ∈ E then q ≥ (l + 1) 1 − p .
h −1 i
Proof. Let f ε = χKε the characteristic function of the set Kε = 0, 31 × 0, ε 3 ×
h −l i
ε
1 1
0, 3M , with M =
max
P (x1 , x2 ) . If p , q ∈ E then
(x1 ,x2 )∈[0,1]×[0,1]
kRfε kLq (Σ) ≤ c kfε kLp (R3 ) = cε
(4.14)
− 1+l
p
.
By the other side,
kRfε kLq (Σ) ≥
Wε
1q
q
b
fε (x1 , x2 , ϕ (x1 , x2 )) dx1 dx2
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1
,
1
× [0, ε] . Now, for (x1 , x2 ) ∈ Wε and (y1 , y2 , y3 ) ∈ Kε ,
2
|x1 y1 + x2 y2 + ϕ (x1 , x2 ) y3 | ≤ 1
so
b
f
(x
,
x
,
ϕ
(x
,
x
))
ε 1 2
1
2 Z
−i(x
y
+x
y
+ϕ(x
,x
)y
)
1 2 3
dy1 dy2 dy3 = e 11 22
Z Kε
≥
cos (x1 y1 + x2 y2 + ϕ (x1 , x2 ) y3 ) dy1 dy2 dy3 ≥ cε−1−l .
Kε
Thus
(4.15)
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
Z
where Wε =
Fourier Restriction
Estimates
1
kRfε kLq (Σ) ≥ cε−1−l+ q .
The proposition follows from (4.14) and (4.15).
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We note that in the case that (a + b) l > ab (for example ϕ (x1 , x2 ) = x42 (x21 + x42 ))
the portion of the type set corresponding to 43 < p1 ≤ 1 will be smaller than the region
Ea,b .
Also, ϕ (x1 , x2 ) = x22 (x1 + x22 ) is an example where a = 2, b = 4, Hessϕ (x1 , x2 )
= −4x22 and if x2 = 0 and x1 6= 0, ϕx2 x2 (x1 , x2 ) = 2x1 6= 0. Again, since
12 = (a + b) l > ab = 8, we get that the portion of the type set corresponding
to 34 < p1 ≤ 1 will be smaller than the region Ea,b .
Proposition 4.3. Let ϕ be a mixed homogeneous
function satisfying (1.1) and (4.12)
b
3
1
1
1
with l ≥ 2 . If 4 ≤ p ≤ 1 and q > (l + 1) 1 − p , then
A
R 0 p 3 q A ≤ c.
L (R ),L (Σ 0 )
Proof. Let (x01 , x02 ) ∈ A0 , if Hessϕ (x01 , x02 ) 6= 0, as in the proof of Theorem 4.1
we find a neighborhood U of (x01 , x02 ) such that (4.1) holds. If Hessϕ (x01 , x02 ) = 0,
by (4.13), either x02 = 0 or the polynomial Q given by Px1 x1 (l(l − 1)P + 2lx2 Px2
+x22 Px2 x2 ) − (lPx1 + x2 Px1 x2 )2 vanishes at (x01 , x02 ) . In the first case, using the fact
that P (x1 , 0) 6= 0 for x1 6= 0, we get that
Px1 x1 l (l − 1) P − l2 Px21 x01 , 0 6= 0.
We take a neighborhood W1 of the point (x01 , 0) and Uk as in the proof of Theorem
4.1. So for (x1 , x2 ) ∈ Uk ,
−k(2l−2)
|Hessϕ (x1 , x2 )| ≥ c2
and so
2k(l−1)
d
.
σ Uk (ξ1 , ξ2 , ξ3 ) ≤
1 + |ξ3 |
Fourier Restriction
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E. Ferreyra and M. Urciuolo
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By the other side,
d
σ Uk (ξ1 , ξ2 , ξ3 ) ≤ 2−k
so for 0 ≤ τ ≤ 1,
d
σ Uk (ξ1 , ξ2 , ξ3 ) ≤
2k(τ l−1)
(1 + |ξ3 |)τ
Fourier Restriction
Estimates
and by Remark 3
U k(τ l−1)
R k p 3 2 U ≤ cτ 2 2(1+τ )
L (R ),L (Σ k )
for p =
2(1+τ )
2+τ
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
and so Hölder’s inequality implies, for 1 ≤ q < 2,
U 2−q
τ l−1
R k p 3 q U ≤ cτ 2k( 2(1+τ ) − 2q )
k
L (R ),L (Σ )
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and a computation shows that this exponent is negative for
Thus
W
R 1 p 3 q W < ∞
(4.16)
L (R ),L (Σ 1 )
1
q
> (l + 1) 1 −
1
p
.
for 43 ≤ p1 ≤ 1 and (l + 1) 1 − p1 < 1q ≤ 1. Now we suppose Q (x01 , x02 ) = 0. We
observe that
deg Q ≤ 2 deg P − 2 ≤ 2 (b − l) − 2 ≤ 2l − 2
Hessϕ (x1 , x02 )
x01
and so
vanishes at
with order at most 2l − 2. Then defining W2
and Vk as in the proof of Theorem 4.1, we have
Hessϕ x1 , x02 ≥ 2−k(2l−2)
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and as in the previous case we obtain
W
R 2 p 3 q W < ∞
(4.17)
L (R ),L (Σ 2 )
for 43 ≤ p1 ≤ 1 and
(4.17) and (4.1).
1
q
> (l + 1) 1 − p1 . The proposition follows from (4.16),
From Proposition 4.3 and Remark 2 we obtain the following result, sharp up to
the end points, for 34 ≤ p1 ≤ 1.
Theorem 4.4. Let ϕ be a mixed homogeneous function satisfying (1.1)
and (4.12)
3
b
a+b+ab
1
1
with l ≥ 2 . If m = max l + 1, a+b , 4 ≤ p ≤ 1 and q > m 1 − p1 , then
1 1
∈ E.
,
p q
Fourier Restriction
Estimates
E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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4.1.
Sharp Lp − L2 Estimates
In [4] we obtain sharp Lp −L2 estimates for the restriction of the Fourier transform to
homogeneous polynomial surfaces in R3 . The principal tools we used there were two
Littlewood Paley decompositions. Adapting this proof to the setting of non isotropic
dilations we obtain the following results.
Lemma 4.5. Let
then
1 1
,
p 2
a+b+2ab
2a+2b+2ab
≤ p1 ≤ 1. If
A
R 0 p 3 2 A < ∞
L (R ),L (Σ 0 )
∈ E.
Proof. From (2.1), the lemma follows from a process analogous to the proof of
Lemma 4.3 in [4].
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Theorem 4.6.
i) If ϕ is a mixed homogeneous polynomial
function satisfying the hypothesis of
a+b+2ab 1
Theorem 4.1 then 2a+2b+2ab
, 2 ∈ E.
ii) Let
1
p0
= max
a+b+2ab 2l+1
,
2a+2b+2ab 2l+2
. If ϕ is a mixed homogeneous polynomial
function satisfying the hypothesis of Theorem 4.4 then p10 , 12 ∈ E.
Proof. i) If a+b+ab
≥ 3, i) follows from (4.3) and Lemma 4.5. The cases (a, b) =
a+b
(3, 4) , (a, b) = (3, 5) and (a, b) = (4, 5) are solved in Remark 5, part ii). The cases
(a, b) = (2, b) with b odd or B = 0 are also included in Remark 5, part ii). For the
remainder cases (2, b), we observe that, if b > 6, from the proof of Theorem 4.1 we
obtain
A
R 0 p 3 2 A < ∞,
(4.18)
L (R ),L (Σ 0 )
for
1
p
=
a+b+2ab
,
2a+2b+2ab
so i) follows from Lemma 4.5. For b = 6, as before we get
W
R 1 p 3 2 W < ∞,
L (R ),L (Σ 1 )
Fourier Restriction
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and
V R k p 3 2 V < ∞
L (R ),L (Σ k )
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a+b+2ab
for k ∈ N, p1 = 2a+2b+2ab
. In a similar way to Lemma 4.3 of [4], we use a unidimensional Littlewood Paley decomposition to obtain
W
R 2 p 3 2 W < ∞
L (R ),L (Σ 2 )
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and then we have (4.18) for
1
p
=
a+b+2ab
.
2a+2b+2ab
So i) follows from Lemma 4.5.
ii) From the proof of Proposition 4.3, we use
a uni-dimensional
Littlewood Paley
a+b+2ab
decomposition to obtain (4.18) for p1 = max 2a+2b+2ab
, 2l+1
,
and
ii) follows from
2l+2
Lemma 4.5.
Remark 7. In [7] the authors obtain sharp estimates for the Fourier transform of
measures σ associated to surfaces Σ like ours, when ϕ is a polynomial function
satisfiyng (1.1) and the condition that ϕ and Hessϕ do not vanish simultaneously on
B−{(0, 0)} . In these cases, part i) of the above theorem follows from Remark 3. We
observe that our hypotheses are less restrictive, for example ϕ (x1 , x2 ) = x41 x22 + x10
2
satisfies the hypothesis of part i) of the above theorem but ϕ and Hessϕ vanish at
any (x1 , x2 ) with x2 = 0.
Fourier Restriction
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E. Ferreyra and M. Urciuolo
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References
[1] L. DE CARLY AND A. IOSEVICH, Some sharp restriction theorems for homogeneous manifolds, The Journal of Fourier Analysis and Applications, 4(1)
(1998), 105–128.
[2] S.W. DRURY AND K. GUO, Some remarks on the restriction of the Fourier
transform to surfaces, Math. Proc. Camb. Phil. Soc., 113 (1993), 153–159.
[3] E. FERREYRA, T. GODOY AND M. URCIUOLO, Lp − Lq estimates for
convolution operators with n-dimensional singular measures, The Journal of
Fourier Analysis and Applications, 3(4) (1997), 475–484.
[4] E. FERREYRA, T. GODOY AND M. URCIUOLO, Restriction theorems for
the Fourier transform to homogeneous polynomial surfaces in R3 , Studia
Math., 160(3) (2004), 249–265.
[5] E. FERREYRA AND M. URCIUOLO, Restriction theorems for anisotropically
homogeneous hypersurfaces of Rn+1 , Georgian Math. Journal, 15(4) (2008),
643–651.
[6] A. GREENLEAF, Principal curvature in harmonic analysis, Indiana U. Math.
J., 30 (1981), 519–537.
[7] A. IOSEVICH AND E. SAWYER, Oscilatory integrals and maximal averages
over homogeneous surfaces, Duke Math. J., 82(1) (1996), 103–141.
[8] E.M. STEIN, Harmonic Analysis, Real - Variable Methods, Orthogonality, and
Oscillatory Integrals, Princeton University Press, Princeton New Jersey (1993).
[9] R.S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces
and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705–713.
Fourier Restriction
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E. Ferreyra and M. Urciuolo
vol. 10, iss. 2, art. 35, 2009
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[10] T. TAO, A sharp bilinear restriction estimate on paraboloids. GAFA, Geom. and
Funct. Anal., 13 (2003), 1359–1384.
[11] T. TAO, Some recent progress on the restriction conjecture. Fourier analysis
and convexity, Appl. Numer. Harmon. Anal., Birkhaüser Boston, Boston MA
(2004), 217–243.
[12] P. TOMAS, A restriction theorem for the Fourier transform, Bull. Amer. Math.
Soc., 81 (1975), 477–478.
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