Revista de la Unión Matemática Argentina - Restriction of the Fourier transform

## Indicators

• Cited by SciELO

• Similars in SciELO

## On-line version ISSN 1669-9637

### Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 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.

1. Introduction

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.

2. Some known results

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 [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

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

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.

3. Our results

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

3.1. Necessary conditions. A simple homogeneity argument shows that if then

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

and for

If for we obtain

(i) If then

(ii) if

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 [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]).

References

[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,