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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 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  1 n f ∈ L (R ), the integral defining

 ∫ ^ -ix.ξ f (ξ) = e f (x)dx

is absolutely convergent for every ξ ∈ Rn and defines a continuos function on Rn.

For more general functions f the extension of the definition of ^f requires density arguments. In particular if f ∈ L1 (Rn ) ∩ L2 (Rn) , the identity of Plancherel

∥ ∥ ∥∥ ^f∥∥ = ∥f ∥ , 2 2

allows us to extend the definition of f^ to L2 (Rn ).

Moreover, since obviously

∥ ∥ ∥∥f^∥∥ ≤ ∥f∥ , ∞ 1

from the Riez-Thorin theorem we obtain

∥ ∥ ∥∥f^∥∥ ≤ ∥f∥ , p′ p

for  1 n p n f ∈ L (R ) ∩ L (R ), 1 ≤ p ≤ 2, and  ′ p the Hölder conjugate of p. So we can extend the notion of f^ to these  p n L (R ).

Suppose that Σ is a given smooth submanifold of Rn and that μ is its induced Lebesgue measure .

If 1 ≤ p ≤ ∞, we say that the Lp restriction property is valid for Σ if there exists q = q(p), 1 ≤ q ≤ ∞, so that the inequality

( ∫ ) || ||q 1∕q |f^(ξ)| dμ ≤ Ap,q (Σ0 )∥f ∥Lp(Rn ) Σ0

holds for each f ∈ S (Rn ) whenever Σ0 is an open subset of Σ with compact closure in Σ. Because S (Rn ) is dense Lp(Rn ) we can, in this case, define  ^ f on Σ (a.e. with respect to μ), for each f ∈ Lp (Rn ).

The determination of optimal ranges for the exponents p and q 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 Σ given as the graph of certain homogeneous polynomial functions.

2. Some known results

From now on we will suppose that Σ is a compact submanifold of Rn and we wil study the restrition operator г : f → ^f |Σ , where

 ∫ гf (ξ) = ^f |Σ (ξ) = e-ix.ξf (x )dx ∀ξ ∈ Σ.

Remark: Since |гf | ≤ ∥f ∥L1(Rn ), the  1 L restriction property is obvious, taking q = ∞. Moreover, we can take any 1 ≤ q ≤ ∞. Indeed

 [∫ ||∫ ||q ]1∕q ∥гf ∥ q = | e-ixξf (x)dx| dμ (ξ) L (Σ) Σ | Rn |
 [∫ ( ∫ )q ]1∕q ≤ |f(x)|dx dμ (ξ) = ∥f ∥L1(Rn )μ(Σ )1∕q. Σ Rn

As usual, for 1 ≤ p ≤ ∞, we define  ′ p by 1 -1 p + p′ = 1.

Theorem (P. A. Tomas, E. Stein, 1975) Let Sn- 1  be the unit sphere of Rn,  let 1 ≤ p ≤ 22nn++23  and  ( ) q = nn-+11-p′. There exists  A(p,q)  such that, for f ∈ S (Rn ) , 

∥гf ∥Lq (Sn-1) ≤ Ap,q ∥f∥Lp (Rn).

Remark. The statement of the above theorem still holds if 1 ≤ p ≤ 2n+2, n+3  q ≤ (n--1)p ′. n+1 Indeed,

∥ гf ∥Lq(Sn-1) ≤ ∥гf ∥ ( n--1)p′ L n+1 (Sn-1)

The proof of the theorem extends naturally to submanifold Σ of  n R of dimension n - 1, with never vanishing Gaussian curvature.

In general, it can be proved that the condition  ( ) q ≤ nn-+11- p′ is necessary. It is not known if the condition about p is also necessary.

We have the following result. If Σ  is a compact submanifold of Rn and for some 1 ≤ p,q ≤ ∞,  г : Lp (Rn ) → Lq (Σ )  is a bounded operator, then  p′ n ^μ ∈ L (R ) .

In the case of the sphere, studying ^μ, it can be checked that if  ′ ^μ ∈ Lp (Rn) , then 1p > n+21n , in other words, 1p > n+21n- is a necessary condition for г to have an Lp 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 n+21n-< 1p < 2nn++32 . The Stein conjecture says that for submanifolds of codimension one in Rn, n > 2, with never vanishing Gaussian curvature we should be able to obain the statement of the theorem in that sector. For n = 2 this result has already been proved.

Theorem. (Fefferman 1970) Let γ  be a curve in R2  with never vanishing curvature and let γ0  be a subarc of γ.  If 34 < 1p ≤ 1  and -1 + 1 ≥ 1 3q p  then there exists A (γ ) p,q 0  such that, for f ∈ S (R2 ),  

∥гf ∥Lq(γ ) ≤ Ap,q (γ0)∥f ∥Lp(R2). 0

In this case, n+1-= 3, 2n 4 and we already know that these conditions about p and q 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  n R  is n. Up to these days, it is only known that this last conjecture is true for n = 2, but it is still an open problem for greater dimensions.

Definition. A Kakeya set, or a Besicovitch set is a compact set E ⊂ Rn, which contains a unitary segment in each direction, i.e

 n- 1 n ∀e ∈ S ∃x ∈ R : x + te ∈ E,
 [ ] 1- 1- ∀t ∈ - 2, 2 .

An old (from about 1920) and well known result due to Besicovitch asserts that for n ≥ 2, there exist Kakeya sets in  n R with measure zero.

We define now the concept of Hausdorff dimension. For α > 0 and E ⊂ Rn we set H ɛ (E) = inf ∞∑ (r )α, α j=1 j where the infimum is taken over the countable coverings of E by discs D (x ,r ) j j with rj < ɛ.

We define  ɛ H α(E) = lɛi→m0 H α(E ). It is easy to check that there exists α0, called the Hausdorff dimension of E such that H α(E) = 0 for α > α 0 and H (E ) = ∞ α for α < α . 0

Fefferman y Bourgain proved that if the restriction conjecture holds for the sphere  n-1 S , with n > 2 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 n ≥ 2, 1 ≤ p ≤ ∞ and α > 0, following [3] we use BR (p,α ) to denote the statement that  δ(p)+α S is bounded on  p L , where  ( | | ) δ(p) = max n ||1p - 12|| - 12,0 and Sδ is the Bochner Riesz multiplier

 ( ) ^Sδf (ξ) = 1 - |ξ |2 δ ^f (ξ ). +

The Bochner-Riesz conjecture says that BR (p,ɛ)  holds for every 1 ≤ p ≤ ∞ and for every ɛ > 0. 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 Σ de R3 given as a compact subset of the graph of a homogeneous polynomial function φ of degree m ≥ 2,

 { ( )} Σ = x1,x2,φ (x1,x2) : x21 + x22 ≤ 1 .

We denote by Q = [0,1] × [0,1 ]. 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 )

for some c > 0 and for every  3 f ∈ S (R ).

3.1. Necessary conditions. A simple homogeneity argument shows that if ( ) 1, 1 ∈ E p q then

 ( ) ( ) 1-≥ - m- + 1 1-+ m- + 1 . q 2 p 2

The set of pairs (1 1) p, q for which the equality holds is called the homogeneity line.

If det φ′′ does not vanish identically we know that the inequalities

1 2 1 2 --≥ - --+ 2 and --> -- q p p 3

are neccesary conditions for a pair ( ) 1 1 p, q ∈ E. The first inequality is the same than the corresponding to homogeneity degree 2 . Trying to obtain as much information as we could about E , (a sharp result would be to obtain that E is the set given by  ( ) ( ) 1≥ - m-+ 1 1 + m-+ 1 q 2 p 2 and 1 > 2 p 3 ) 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 detφ ′′ does not vanish identically on R2 \{0 }, but if it vanishes in some point x0 ⁄= 0, it vanishes on a finite union of lines through the origin. If Lj, 1 ≤ j ≤ k, is one of such lines, the vanishing order α j of det φ′′(x) in any point of L j (α j is independent of the point on Lj) plays a fundamental role. We define ^m = max {m, α1,...,αk } and we obtain that if ( ) 1p, 1q ∈ E then

 ( ) ( ) 1- ^m- 1- m^ q ≥ - 2 + 1 p + 2 + 1 .

We remark that in some cases, αj > m. For example, if  7 φ (x1,x2) = x2 (x1 + x2),  ′′ 12 det φ (x1, x2) = - 49x 2 , and its vanishing order on the x1 axis is 12 . In this case the line corresponding to ^m has bigger slope than the slope of the homogeneity line, and so we obtain a better necessary condition.

3.2. Sufficient Conditions. If  ′′ det φ ≡ 0, possibly after a linear change of coordinates that leaves E invariant, we have  m φ (x1, x2) = x2 , and it is easy to see that in this case the set E is the type set corresponding to the curve (t,tm ) in R2 . We obtained then the following result

Let φ : R2 → R  be a homogeneous polynomial function of degree m ≥ 2  such that det φ′′(x ) ≡ 0.  Then for m ≥ 3

 {( ) } E ∘ = 1-, 1 ∈ Q : 1-> - m-+--1 + m + 1 p q q p

and for m = 2

 {( ) } ∘ 1 1 3 1 3 E = --,-- ∈ (-,1] × [0,1] :--> - -+ 3 . p q 4 q p

If det φ′′(x ,x ) ⁄= 0 1 2 for (x ,x ) ∈ R2\ {0} , 1 2 we obtain

(i) If m ≥ 6, then

E∘ =
{( ) } 1 1 1 ( m ) 1 m p-,q- ∈ Q :-q > - 2-+ 1 p + 2- + 1 ,

(ii) if m < 6

 ( ( ] ) E ∘ ∩ 3,1 × [0,1] = 4
{ ( ) } 1-1- 1- (m- ) 1- m- p,q ∈ Q :q > - 2 + 1 p + 2 + 1
 ( ( 3 ] ) ∩ -,1 × [0,1] 4

and also ( ) 3 1 4,q ∈ E  for ^m+2- 1 8 < q ≤ 1.

In the region given by  ( ) ( ) 1q ≥ - m2 + 1 1p + m2-+ 1 , 23 < 1p < 34, we can not give neither a positive nor a negative answer to the question if ( 1 1) p,q belongs to E. Also, we don't know wether (3, m+2-) 4 8 belongs to E or not.

We did not expect to obtain positive results for 1 3 p < 4 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 ,

that have non vanishing curvature, and then scaling.

If  ′′ det φ does not vanish identically on  2 R \{0} , but if it vanishes in some point x0 ⁄= 0, we obtain the same results than before, with m replaced by m^.

Finally, in every case we obtain a sharp Lp (R3) → L2 (Σ ) 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 β1,...,βn > 1, and B the unit ball of  n R , we consider φ : Rn → R of the form φ (x1,...,xn) = Σni=1|xi|βi and we studied the restriction of the Fourier transform to the surface S given by S = {(x,φ (x)) : x ∈ B }. We obtained a poligonal region contained in the type set E. In some cases this result is sharp (see [1]).

References

[1]    Ferreyra E., Urciuolo M. Restriction theorems for anisotropically homogeneous hypersurfaces of Rn+1. 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 R3. 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