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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dez. 2008
Restriction of the Fourier transform
Marta Urciuolo
Abstract. This paper contains a brief survey about the state of progress on the restriction of the Fourier transform and its connection with other conjectures. It contains also a description of recent related results that we have obtained.
If the integral defining
![∫ ^ -ix.ξ f (ξ) = e f (x)dx](/img/revistas/ruma/v49n2/2a051x.png)
is absolutely convergent for every and defines a continuos function on
For more general functions the extension of the definition of
requires density arguments. In particular if
the identity of Plancherel
![∥ ∥ ∥∥ ^f∥∥ = ∥f ∥ , 2 2](/img/revistas/ruma/v49n2/2a057x.png)
allows us to extend the definition of to
Moreover, since obviously
![∥ ∥ ∥∥f^∥∥ ≤ ∥f∥ , ∞ 1](/img/revistas/ruma/v49n2/2a0510x.png)
from the Riez-Thorin theorem we obtain
![∥ ∥ ∥∥f^∥∥ ≤ ∥f∥ , p′ p](/img/revistas/ruma/v49n2/2a0511x.png)
for
and
the Hölder conjugate of
So we can extend the notion of
to these
Suppose that is a given smooth submanifold of
and that
is its induced Lebesgue measure .
If we say that the
restriction property is valid for
if there exists
so that the inequality
![( ∫ ) || ||q 1∕q |f^(ξ)| dμ ≤ Ap,q (Σ0 )∥f ∥Lp(Rn ) Σ0](/img/revistas/ruma/v49n2/2a0526x.png)
holds for each whenever
is an open subset of
with compact closure in
Because
is dense
we can, in this case, define
on
for each
The determination of optimal ranges for the exponents and
are difficult problems which have not yet been completely solved.
In paragraph 2 we describe some known results about certain submanifolds with this property, and we also describe the connection with the Kakeya and the Bochner Riesz conjectures.
In Paragraph 3 we state the results that we have obtained for hypersurfaces given as the graph of certain homogeneous polynomial functions.
From now on we will suppose that is a compact submanifold of
and we wil study the restrition operator
where
![∫ гf (ξ) = ^f |Σ (ξ) = e-ix.ξf (x )dx ∀ξ ∈ Σ.](/img/revistas/ruma/v49n2/2a0543x.png)
Remark: Since the
restriction property is obvious, taking
Moreover, we can take any
Indeed
![[∫ ||∫ ||q ]1∕q ∥гf ∥ q = | e-ixξf (x)dx| dμ (ξ) L (Σ) Σ | Rn |](/img/revistas/ruma/v49n2/2a0548x.png)
![[∫ ( ∫ )q ]1∕q ≤ |f(x)|dx dμ (ξ) = ∥f ∥L1(Rn )μ(Σ )1∕q. Σ Rn](/img/revistas/ruma/v49n2/2a0549x.png)
As usual, for we define
by
Theorem (P. A. Tomas, E. Stein, 1975) Let be the unit sphere of
let
and
There exists
such that, for
,
![∥гf ∥Lq (Sn-1) ≤ Ap,q ∥f∥Lp (Rn).](/img/revistas/ruma/v49n2/2a0559x.png)
Remark. The statement of the above theorem still holds if
Indeed,
![∥ гf ∥Lq(Sn-1) ≤ ∥гf ∥ ( n--1)p′ L n+1 (Sn-1)](/img/revistas/ruma/v49n2/2a0562x.png)
The proof of the theorem extends naturally to submanifold of
of dimension
with never vanishing Gaussian curvature.
In general, it can be proved that the condition is necessary. It is not known if the condition about
is also necessary.
We have the following result. If is a compact submanifold of
and for some
is a bounded operator, then
.
In the case of the sphere, studying it can be checked that if
then
in other words,
is a necessary condition for
to have an
restriction property. This result can also be proved for submanifols with never vanishing Gaussian curvature.
For these submanifols then, everything is done, except in the sector . The Stein conjecture says that for submanifolds of codimension one in
with never vanishing Gaussian curvature we should be able to obain the statement of the theorem in that sector. For
this result has already been proved.
Theorem. (Fefferman 1970) Let be a curve in
with never vanishing curvature and let
be a subarc of
If
and
then there exists
such that, for
![∥гf ∥Lq(γ ) ≤ Ap,q (γ0)∥f ∥Lp(R2). 0](/img/revistas/ruma/v49n2/2a0591x.png)
In this case, and we already know that these conditions about
and
are also necessary.
Back to the Stein conjecture, in the paper [4] there is a very interesting survey about the recent improvements that different authors have obtained , for the cases of the sphere and the paraboloid.
The restriction conjecture is related with the Kakeya conjecture, that is stated as follows The Hausdorff dimension of a Kakeya set in is n. Up to these days, it is only known that this last conjecture is true for
but it is still an open problem for greater dimensions.
Definition. A Kakeya set, or a Besicovitch set is a compact set which contains a unitary segment in each direction, i.e
![n- 1 n ∀e ∈ S ∃x ∈ R : x + te ∈ E,](/img/revistas/ruma/v49n2/2a0598x.png)
![[ ] 1- 1- ∀t ∈ - 2, 2 .](/img/revistas/ruma/v49n2/2a0599x.png)
An old (from about 1920) and well known result due to Besicovitch asserts that for there exist Kakeya sets in
with measure zero.
We define now the concept of Hausdorff dimension. For and
we set
where the infimum is taken over the countable coverings of
by discs
with
We define It is easy to check that there exists
called the Hausdorff dimension of
such that
for
and
for
Fefferman y Bourgain proved that if the restriction conjecture holds for the sphere , with
then the Kakeya conjecture also holds. A very nice approach to these subjects can be found in [5].
Another problem related with the restriction conjecture is the following. Fix
and
following [3] we use
to denote the statement that
is bounded on
where
and
is the Bochner Riesz multiplier
![( ) ^Sδf (ξ) = 1 - |ξ |2 δ ^f (ξ ). +](/img/revistas/ruma/v49n2/2a05125x.png)
The Bochner-Riesz conjecture says that holds for every
and for every
In [3] the author proves that the Bochner Riesz conjecture implies the restriction conjecture.
We (jointly with Elida Ferreyra and Tomás Godoy) study hypersufaces de
given as a compact subset of the graph of a homogeneous polynomial function
of degree
![{ ( )} Σ = x1,x2,φ (x1,x2) : x21 + x22 ≤ 1 .](/img/revistas/ruma/v49n2/2a05133x.png)
We denote by We try to obtain information about the type set
![{ ( ) } E = 1, 1 ∈ Q : ∥гf ∥ q ≤ c∥f ∥ p 3 p q L (Σ) L (R )](/img/revistas/ruma/v49n2/2a05135x.png)
for some and for every
3.1. Necessary conditions. A simple homogeneity argument shows that if then
![( ) ( ) 1-≥ - m- + 1 1-+ m- + 1 . q 2 p 2](/img/revistas/ruma/v49n2/2a05139x.png)
The set of pairs for which the equality holds is called the homogeneity line.
If does not vanish identically we know that the inequalities
![1 2 1 2 --≥ - --+ 2 and --> -- q p p 3](/img/revistas/ruma/v49n2/2a05142x.png)
are neccesary conditions for a pair The first inequality is the same than the corresponding to homogeneity degree
. Trying to obtain as much information as we could about
, (a sharp result would be to obtain that
is the set given by
and
) we found some difficulties that suggested the existence of another line with greater slope than the slope of the homogeneity line, providing a better necessary condition. Indeed, if
does not vanish identically on
but if it vanishes in some point
it vanishes on a finite union of lines through the origin. If
is one of such lines, the vanishing order
of
in any point of
is independent of the point on
plays a fundamental role. We define
and we obtain that if
then
![( ) ( ) 1- ^m- 1- m^ q ≥ - 2 + 1 p + 2 + 1 .](/img/revistas/ruma/v49n2/2a05161x.png)
We remark that in some cases, For example, if
and its vanishing order on the
axis is
. In this case the line corresponding to
has bigger slope than the slope of the homogeneity line, and so we obtain a better necessary condition.
3.2. Sufficient Conditions. If possibly after a linear change of coordinates that leaves
invariant, we have
and it is easy to see that in this case the set
is the type set corresponding to the curve
in
. We obtained then the following result
Let be a homogeneous polynomial function of degree
such that
Then for
![{( ) } E ∘ = 1-, 1 ∈ Q : 1-> - m-+--1 + m + 1 p q q p](/img/revistas/ruma/v49n2/2a05178x.png)
and for
![{( ) } ∘ 1 1 3 1 3 E = --,-- ∈ (-,1] × [0,1] :--> - -+ 3 . p q 4 q p](/img/revistas/ruma/v49n2/2a05180x.png)
If for
we obtain
(i) If then
![E∘ =](/img/revistas/ruma/v49n2/2a05184x.png)
![{( ) } 1 1 1 ( m ) 1 m p-,q- ∈ Q :-q > - 2-+ 1 p + 2- + 1 ,](/img/revistas/ruma/v49n2/2a05185x.png)
(ii) if
![( ( ] ) E ∘ ∩ 3,1 × [0,1] = 4](/img/revistas/ruma/v49n2/2a05187x.png)
![{ ( ) } 1-1- 1- (m- ) 1- m- p,q ∈ Q :q > - 2 + 1 p + 2 + 1](/img/revistas/ruma/v49n2/2a05188x.png)
![( ( 3 ] ) ∩ -,1 × [0,1] 4](/img/revistas/ruma/v49n2/2a05189x.png)
and also for
In the region given by
we can not give neither a positive nor a negative answer to the question if
belongs to
Also, we don't know wether
belongs to
or not.
We did not expect to obtain positive results for since our proof basically consists in applying the Stein-Tomas theorem to the restriction of the Fourier transform to the shells
![{( - j- 1 2 2 -j)} Σj = x1,x2,φ (x1,x2) : 2 ≤ x 1 + x 2 ≤ 2 ,](/img/revistas/ruma/v49n2/2a05199x.png)
that have non vanishing curvature, and then scaling.
If does not vanish identically on
but if it vanishes in some point
we obtain the same results than before, with
replaced by
Finally, in every case we obtain a sharp estimate.
The techniques that we use were:
- Asymptotic developments and Van der Corput lemmas for oscillatory integrals.
- Real and complex interpolation.
- Littlewood Paley theory.
These results are in the paper [2].
Lately, with E. Ferreyra, we studied the cases of anisotropically homogeneous surfaces. For and
the unit ball of
we consider
of the form
and we studied the restriction of the Fourier transform to the surface
given by
We obtained a poligonal region contained in the type set
In some cases this result is sharp (see [1]).
[1] Ferreyra E., Urciuolo M. Restriction theorems for anisotropically homogeneous hypersurfaces of To appear in Georgian Mathematical Journal. [ Links ]
[2] Ferreyra E., Godoy T., Urciuolo M. Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in Studia Math. 160, 249-265, 2004. [ Links ]
[3] Tao T. The Bochner-Riesz conjecture implies the Restriction conjecture Duke Math. J. 96, 363-376, 1999. [ Links ]
[4] Tao T. Some recent progress on the restriction conjecture Fourier Analysis and convexity, p.217-243, Appl. Numer. Harmon. Anal., (2004). [ Links ]
[5] Wolff T. Thomas Wolff's Lectures in Harmonic Analysis AMS, University lecture series, vol 29, 2003. [ Links ]
Marta Urciuolo
CIEM-FaMAF,
Universidad Nacional de Córdoba,
Medina Allende s/n, Ciudad Universitaria
Córdoba 5000, Argentina
urciuolo@mate.uncor.edu
Recibido: 10 de abril de 2008
Aceptado: 2 de julio de 2008