Sets of -restriction and -spectral synthesis
In this paper we investigate the restriction problem. More precisely, we give sufficient conditions for the failure of a set in to have the -restriction property. We also extend the concept of spectral synthesis to for sets of -restriction when . We use our results to show that there are -values for which the unit sphere is a set of -spectral synthesis in when .
Key words and phrases:-spectral synthesis, restriction problem, set of -restriction, span of translates
2010 Mathematics Subject Classification:Primary: 42B10; Secondary: 43A15
Throughout this paper will denote the real numbers and will denote the integers. Let and let be a positive integer. Indicate by the usual Lebesgue space and denote by the usual Banach space norm on . If , then will denote the closure of in . Also will always represent the conjugate index of , that is . For the Fourier transform of is defined by
The Fourier transform can be extended to a unitary operator on and by the Hausdorff-Young inequality, can be extended to a continuous operator from to , when . Let be a closed subset of . If , then is continuous on . Consequently, the restriction of to , which we denote by , is a well-defined function on . For and . So if is a set of positive Lebesgue measure we can restrict to . The interesting question is can be restricted to when has Lebesgue measure zero? This question is the heart of the restriction problem, which we will now describe.
For , let be the set of continuous functions on and let be the set of functions in with compact support. Let be the usual Banach space formed with respect to the induced measure on . The norm on will be denoted by . Recall that the norm on is indicated by . Let and let denote the space of Schwartz functions on . The operator given by
is known as the restriction operator associated with . If can be extended to a continuous operator from , then we shall say that has property . Observe that if and has property , then it also has property . We shall say that is a set of -restriction if has property . Note that any closed set in is a set of -restriction. Furthermore, if is not a set of -restriction, then does not have property for any . The best known result concerning the restriction of to is the Stein-Tomas theorem: has property if and only if , where is a smooth compact hypersurface in with nonzero Gaussian curvature. In general though it is an extremely difficult problem to determine if has property . A more comprehensive treatment of the restriction problem, along with its history, can be found in [3, 11], and [4, Chapter 5.4] and the references therein.
In this paper we will only be concerned with the case where has property , that is is a set of -restriction. Set
and if is a set of -restriction define
This paper was inspired by the paper  where these spaces were investigated for the case when is the unit sphere in . Our first main result is:
Let be a smooth compact submanifold of codimension in . If there exists such that vanishes on , then is not a set of -restriction for .
Thus does not have property when and . If is a hypersurface, then the lower bound for becomes . We are able to improve this lower bound for hypersurfaces with the constant relative nullity condition, which we now define. Let be an open set in and let be a smooth hypersurface in . If the Hessian matrix
of has constant rank on , where , then we say that has constant relative nullity . A smooth hypersurface of is said to have constant relative nullity if every localization of has constant relative nullity . If , then is a hyperplane. It is known that hyperplanes are sets of -restriction only if . Thus we will only consider hypersurfaces with . Note that for .
Let and let be a smooth compact hypersurface in with constant relative nullity , for . If , then is not a set of -restriction.
Let and let . The translate of by , which we write as , is the function , where . For , let be the closed subspace of spanned by and its translates. The zero set of is defined by
In Section 3 we will see that if is a set of -restriction, then .
We will now briefly review the concept of spectral synthesis in . Suppose is a closed ideal in and define the zero set of by
Let be a closed set in , then is a closed ideal in with zero set . In fact, is the largest closed ideal in whose zero set is . Now let
and is the smallest closed ideal in with zero set . The set is known as a set of spectral synthesis if . A more detailed account of spectral synthesis can be found in [9, Chapter 7]. Extending the concept of spectral synthesis to for falls short since the analog to , is not well-defined for closed sets of Lebesgue measure zero in . However, for sets of -restriction is well-defined, which allows us to extend the idea of spectral synthesis to for sets of -restriction. We shall say that a set of -restriction is a set of -spectral synthesis if
We can now state:
Let and let be a smooth compact hypersurface in with constant relative nullity . If is a set of -restriction for some that satisfies one of the following:
then is a set of -spectral synthesis.
It is known that is a set of spectral synthesis in , but is not a set of spectral synthesis in for [9, Chapter 7.3]. We will use Theorem 1.3 to show that there are -values where is a set of -spectral synthesis for .
This paper is organized as follows: In Section 2 we give some background and results that will be needed for this paper. In Section 3 we will prove Theorems 1.1 and 1.2 by linking them to the problem of determining when is dense in for with on . In Section 4 we prove Theorem 1.3, and use the theorem to show that there are -values for which the unit sphere is a set of -spectral synthesis in for .
In this section we will give some results that will be used in the sequel. The convolution of two measurable functions and on is defined by
Let . Each defines a bounded linear functional on via
Sometimes we will write in place of . For closed subspaces in ,
will denote the annihilator of in . The following characterization of when is a translation-invariant subspace of will be needed later.
Let be a translation-invariant subspace of . Then if and only if for all .
Observe that for and
It follows from the translation invariance of that for all if and only if . ∎
The space is a -module since whenever and . The following proposition will not be used in the paper, but we record it here for its independent interest.
If is a set of -restriction, then is a -submodule of .
A modification of the proof of [9, Theorem 7.1.2] will show that a closed translation-invariant subspace of is translation invariant if and only if it is a -submodule of . The proposition now follows since is a closed translation-invariant subspace of . ∎
It is well known that the Fourier transform is an isomorphism on the Schwartz space , with inverse Fourier transform given by
for . A continuous linear functional on is known as a temperate distribution. A nice property of temperate distributions is that the Fourier transform can be extended to them. In fact, the Fourier transform defines an isomorphism on the temperate distributions. Indeed, if is a tempered distribution, then is the tempered distribution given by
for . The inverse Fourier transform of a temperate distribution is defined by
where . Since elements of are temperate distributions, we can define the Fourier transform for in the distributional sense when . For the rest of this paper, distribution will mean temperate distribution.
We shall write to indicate the support of , where depending on the context, is a function, measure, or distribution.
We conclude this section with a result that will be needed later.
If is a compact subset of , then there exists an for which .
Let be an open ball containing and let . The Whitney extension theorem produces a smooth function such that on and at . For the purpose of this proof only, will mean the function defined above instead of the translate of . For each there exists an open ball for which is positive on . Now choose a countable subcover of . Let
is a smooth function on . Let be an open ball satisfying Deonte by the smooth function obtained by multiplying by a smooth function that equals one on and zero on . Set where that is zero on and positive on , where is the closure of . Thus and can be expressed as for some . The proof of the proposition is now complete since . ∎
Let be a compact set in with induced measure . Suppose has the -restriction property. This is equivalent to the existence of a constant that depends on and and satisfies
for all . Condition (3.1) is equivalent to the dual condition
for all smooth functions on , and where is the inverse Fourier transform of the measure . Recall that the inverse Fourier transform of a finite Borel measure is
where . Setting on we see from (3.2) that . We record this as:
Let and let be a compact set in with induced measure . If is a set of -restriction, then .
Let and let be a compact subset of with induced measure . If is a set of -restriction, then .
Let and let be a compact subset of . If there exists an for which and , then is not a set of -restriction.
Assume is a set of -restriction. Then by Proposition 3.2 . Since for all which contradicts our hypothesis . ∎
3.1. Proof of Theorem 1.1
3.2. Proof of Theorem 1.2
We will prove Theorem 1.2 by proving a more general theorem. We start with a definition. Let be a closed subset of . We shall say that is -thin if the only distribution that satisfies and is .
Let and let be a compact subset of . If is -thin, then is not a set of -restriction.
4. -spectral synthesis
We start with a definition. Let be a -dimensional submanifold in with induced Lebesgue measure . We shall say that has the -approximate property if for each distribution with and , we can find a sequence of measures on , absolutely continuous with respect to , such that as . Our results on sets of -spectral synthesis are an immediate consequence of previous work by Guo on sets that have the -approximate property [5, 7]. In fact, it is stated in  that the -approximate property is a variation of the spectral synthesis property. Sets with the -restriction property allows us to make this statement more transparent. Specifically, Theorem 4.1 will show that -spectral synthesis follows from the -approximate property for submanifolds with the -restriction property.
4.1. Proof of Theorem 1.3
Suppose is a -dimensional submanifold of and let be the induced Lebesgue measure on . Also assume that is a set of -restriction for some . Let denote the closed subspace of generated by
Let and let be a compact, smooth -dimensional submanifold of and assume that has the -restriction property. Let be the induced measure on . If has the -approximate property, then is a set of -spectral synthesis.
Since is a set of -restriction, by Lemma 3.1. We begin by showing showing . Let . We can assume that , where for some smooth function on . Let , using the argument from Proposition 3.2 we obtain that , which implies that by Proposition 2.1.
Now let . Since and has the -approximate property, there exists a sequence of measures , where is smooth on , such that . Thus , which implies . Clearly, . Therefore, is a set of -spectral synthesis. ∎
4.2. -spectral synthesis and the unit sphere
We mentioned in the Introduction that for is not a set of spectral synthesis. Using Theorem 1.3 we will be able to show that there are -values for which is a set of -spectral synthesis. By [5, Theorem 1], has the -approximate property when and ; and has the -approximate property for when . It follows from the Stein-Tomas theorem that is a set of -restriction for . Consequently, Theorem 4.1 yields that is a set of -spectral synthesis for and for is a set of -spectral synthesis for . The upper bound in this inequality is probably not sharp, in fact it would not surprise us if it is . However, the lower bound is sharp. Indeed, using [6, Lemma 2.3(ii)] a distribution can be constructed with that satisfies for ,
Since there exists with and . Thus, for is not a set of -spectral synthesis for .
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