On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
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
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
allows us to extend the definition of to
Moreover, since obviously
from the Riez-Thorin theorem we obtain
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
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
Remark: Since the restriction property is obvious, taking Moreover, we can take any Indeed
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 ,
Remark. The statement of the above theorem still holds if Indeed,
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
In this case, and we already know that these conditions about and are also necessary.
Back to the Stein conjecture, in the paper  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
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 .
Another problem related with the restriction conjecture is the following. Fix and following  we use to denote the statement that is bounded on where and is the Bochner Riesz multiplier
The Bochner-Riesz conjecture says that holds for every and for every In  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
We denote by We try to obtain information about the type set
for some and for every
The set of pairs for which the equality holds is called the homogeneity line.
If does not vanish identically we know that the inequalities
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
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
If for we obtain
(i) If then
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
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 .
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 ).
 Ferreyra E., Urciuolo M. Restriction theorems for anisotropically homogeneous hypersurfaces of To appear in Georgian Mathematical Journal. [ Links ]
 Ferreyra E., Godoy T., Urciuolo M. Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in Studia Math. 160, 249-265, 2004. [ Links ]
 Tao T. The Bochner-Riesz conjecture implies the Restriction conjecture Duke Math. J. 96, 363-376, 1999. [ Links ]
 Tao T. Some recent progress on the restriction conjecture Fourier Analysis and convexity, p.217-243, Appl. Numer. Harmon. Anal., (2004). [ Links ]
 Wolff T. Thomas Wolff's Lectures in Harmonic Analysis AMS, University lecture series, vol 29, 2003. [ Links ]
Universidad Nacional de Córdoba,
Medina Allende s/n, Ciudad Universitaria
Córdoba 5000, Argentina
Recibido: 10 de abril de 2008
Aceptado: 2 de julio de 2008