PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 1–10 DOI: 10.2298/PIM150107001M

WAVE FRONTS VIA FOURIER SERIES COEFFICIENTS

Snježana Maksimović, Stevan Pilipović, Petar Sokoloski, and Jasson Vindas

Abstract. Motivated by the product of periodic distributions, we give a new description of the wave front and the Sobolev-type wave front of a distribution f ∈ 𝒟′(ℝ^{d}) in terms of Fourier series coefficients.

1. Introduction

In this article microlocal properties of a distribution f at x_{0} ∈ ℝ^{d} are detected through the Fourier series expansion of periodizations of φf, where φ is a cut-off function near x_{0}. In contrast to [7], where weighted type wave front sets have been discussed by the use of Gabor and dual Gabor frames depending on an additional continuous parameter ε → 0, we shall show that the classical Fourier basis can be used for microlocal analysis. Our approach leads to discretized definitions of wave fronts in terms of Fourier coefficients. This is the main novelty of the paper, which also includes proofs for the equivalence between these discretized definitions and Hörmander’s approach.

The space of periodic distributions is one of the basic Schwartz spaces and has been studied in many books and papers in the second half of the last century. We refer here just to few of them [14, 17, 15, 1, 2, 8] (see [16] for applications in summability of Fourier series). In the context of our paper, we mention the recent articles [11, 12] and book [13], where Ruzhansky and Turunen have studied generalized functions on a torus 𝕋^{d}. Their interest there lies in pseudo-differential operators and microlocal analysis over 𝕋^{d} × ℤ^{d}. On the other hand, at present time, Hörmander’s notion of the wave front set attracts a lot of interest among mathematicians and there exists a vast literature related to this basic notion and its important role in the qualitative analysis of PDEs and ΨDOs. We mention the basic books of Hörmander [5, 6] as standard references for classical and Sobolev type wave front sets; the articles [9, 7] deal with weighted type wave fronts, while [3, 4] study extended wave fronts by considering local and global versions with respect to various Banach and Fréchet spaces of functions over the configuration and the frequency domains.

2010 Mathematics Subject Classification: Primary 35A18; Secondary 46F10.

Key words and phrases: wave fronts; Fourier series; multiplication of distributions.

Communicated by Gradimir Milovanović.

It is well known that the product of two distributions can be defined if their wave fronts are in a “good” position with respect to each other. This motivates us to study the product and wave fronts via spaces of periodic distributions. In Subsection 1.3 below we discuss an elementary approach to local multiplication based on Fourier series. The main results of this article are presented in Sections 2 and 3. In Theorems 2.1 and 3.1 we characterize the wave front and the Sobolev type wave front of a distribution f ∈ 𝒟′(ℝ^{d}) by estimates of the Fourier coefficients of its localizations. It should be mentioned that the toroidal wave fronts have been studied in [12, 13] through Fourier series as in this paper, but our approach is quite different and is related to the Hörmander’s wave fronts in a precise fashion. Let us note that Sobolev type wave fronts were not considered in [12, 13].

1.1. Notation. For x = (x_{1}, …, x_{d}) ∈ ℝ^{d}, we write |x| = \sqrt{x_{1}^{2} + ⋯+ x_{d}^{2}}

and ⟨x⟩ = (1 + |x|^{2})^{1/2} . Let 0 < η ⩽ 1. We will use the notation

I_{η,x} = ∏_{j=1}^{d} (x_{j} − η/2, x_{j} + η/2) and I_{η} := I_{η,0} .

Throughout the article, the word periodic always refers to functions or distributions on ℝ^{d} which are periodic of period 1 in each variable, i.e., f(x + n) = f(x), x ∈ ℝ^{d}, n ∈ ℤ^{d}. We also use the notation e_{y} for e_{y}(x) = e^{2πiy·x} , y ∈ ℝ^{d}. We will consider periodic extensions of localizations of distributions around a point x_{0} ∈ ℝ^{d}, so if a distribution g is supported by I_{η,x_{0}} , with 0 < η < 1, we shall write g_{p}(x) := ∑ _{n∈ℤ^{d}} g(x + n) for its periodic extension.

1.2. Basic spaces. The space of periodic test functions 𝒫 = 𝒫(ℝ^{d}) consists of smooth periodic functions; its topology is given via the sequence of norms ∥φ∥_{k} = sup_{x∈I_{1} ,|α|⩽k} |φ^{(α)} (x)|, k ∈ ℕ. Obviously, φ ∈ 𝒫 if and only if ∑ _{n∈ℤ^{d}} |φ_{n}|^{2}⟨n⟩^{2k} < ∞ for every k ∈ ℤ, where φ_{n} = ∫_{I_{1}} φ(x)e^{−2πin·x} dx = ⟨φ, e_{−n} ⟩, n ∈ ℤ^{d}. The dual space of 𝒫, the space of periodic distributions, is denoted by 𝒫′. One has: f = ∑ _{n∈ℤ^{d}} f_{n}e_{n} ∈ 𝒫′ if and only if ∑ _{n∈ℤ^{d}} |f_{n}|^{2}⟨n⟩^{−2k_{0}} < ∞, for some k_{0} > 0. If f = ∑ _{n∈ℤ^{d}} f_{n}e_{n} ∈ 𝒫′ and φ = ∑ _{n∈ℤ^{d}} φ_{n}e_{n} ∈ 𝒫, then their dual pairing is given by ⟨f, φ⟩ = ∑ _{n∈ℤ^{d}} f_{n}φ_{−n} .

Let ν, ω be positive functions over ℤ^{d}. We call ω a ν-moderate weight if there is C such that

(1.1)  ω(m + n) ⩽ Cω(m) ν(n), ∀m, n ∈ ℤ^{d}.

If we take ν to be a polynomial, we call ω polynomially moderate. The set of all polynomially moderate weights on ℤ^{d} will be denoted as Pol(ℤ^{d}). For ω ∈ Pol(ℤ^{d}), we define

𝒫l_{ω}^{q} = {f ∈ 𝒫′: {f_{n}ω(n)}_{n∈ℤ^{d}} ∈ l^{q}, where f_{n} = ⟨f, e_{−n} ⟩}

supplied with the norm ∥f∥_{𝒫l_{ω}^{q}} = ∥{f_{n}ω(n)}∥_{l^{q}}. We consider from now only values of q ⩾ 1. Clearly, 𝒫l_{ω_{1}}^{q_{1}} ⊆ 𝒫l_{ω_{2}}^{q_{2}} , if q_{1} ⩽ q_{2} and ω_{2} ⩽ Cω_{1}.

We will also consider the local space 𝒫l_{ω, loc}^{q} consisting of distributions f ∈ 𝒟′(ℝ^{d}) such that the periodic extensions (φf)_{p} ∈ 𝒫l_{ω}^{q}, for all x_{0} ∈ ℝ^{d} and φ ∈ 𝒟(I_{1,x_{0}} ). Its topology is defined via the family of seminorms ∥f∥_{x_{0} ,φ} = ∥(φf)_{p}∥_{𝒫l_{ω}^{q}} , where x_{0} ∈ ℝ^{d} and φ ∈ 𝒟(I_{1,x_{0}} ). For the sake of completeness, we give the following elementary proposition (its proof also shows that the definition of 𝒫l_{ω, loc}^{q} is consistent).

Proposition 1.1. 𝒫l_{ω}^{q} ⊂ 𝒫l_{ω, loc}^{q} .
Proof. Let f ∈ 𝒫l_{ω}^{q} and φ ∈ 𝒟(I_{1,x_{0}} ), then (φf)_{p} = φ_{p}f. Write f =

∑ _{n} f_{n}e_{n} and φ_{p} = ∑ _{n} φ_{n}e_{n} ∈ 𝒫. By (1.1) and the generalized Minkowski inequality, we have

∥φ_{p}f∥_{𝒫l_{ω}^{q}} ⩽ C( ∑_{n} ( ∑_{j} |φ_{j} |ν(j)|f_{n−j} |ω(n − j))^{q})^{1/q}
⩽ C∥φ_{p}∥_{𝒫l_{ν}^{1}} ∥f∥_{𝒫l_{ω}^{q}} < ∞.  □
Set ω_{s}(n) = ⟨n⟩^{s}, s ∈ ℝ. For convenience, we write 𝒫l_{s}^{q} := 𝒫l_{ω_{s}}^{q} and

𝒫l_{s, loc}^{q} := 𝒫l_{ω_{s} , loc}^{q} . We clearly have

𝒫 = ⋂_{s⩾0} 𝒫l_{s}^{q} = ⋂_{ω∈𝒫ol(ℤ^{d})} 𝒫l_{ω}^{q} and 𝒫′ = ⋃_{s⩽0} 𝒫l_{s}^{q} = ⋃_{ω∈𝒫ol(ℤ^{d})} 𝒫l_{ω}^{q}.

Moreover,

ℰ = ⋂_{s⩾0} 𝒫l_{s, loc}^{q} = ⋂_{ω∈𝒫ol(ℤ^{d})} 𝒫l_{ω, loc}^{q} and 𝒟′_{F} = ⋃_{s⩽0} 𝒫l_{s, loc}^{q} = ⋃_{ω∈𝒫ol(ℤ^{d})} 𝒫l_{ω, loc}^{q} ,

where ℰ is the space of all smooth functions and 𝒟′_{F} is the space of distributions of finite order on ℝ^{d}.

1.3. Multiplication. In this section we make some comments about the multiplication of distributions. Assume that the indices q, q_{1}, q_{2} ∈ [1, ∞] are such that \frac{1}{q_{1}} + \frac{1}{q_{2}} = \frac{1}{q} + 1. We fix two weight functions ω, ν ∈ Pol(ℤ^{d}) and we assume ω is ν-moderate (cf. 1.1).

We start with products in the spaces of type 𝒫l_{ω}^{q}. Here we define the product via Fourier coefficients. Indeed, let f_{1} = ∑ _{n∈ℤ^{d}} f_{1,n} e_{n} ∈ 𝒫l_{ω}^{q_{1}} and f_{2} = ∑ _{n∈ℤ^{d}} f_{2,n} e_{n} ∈ 𝒫l_{ν}^{q_{2}}. We define their product as f := f_{1}f_{2} := ∑ _{n∈ℤ^{d}} f_{n}e_{n}, where

f_{n} = ∑_{j∈ℤ^{d}} f_{1,n−j} f_{2,j} , n ∈ ℤ^{d}.

(One can easily see that the formula for the Fourier coefficients f_{n} is just the convolution formula on the integer lattice.) We will check in Proposition 1.2 that f ∈ 𝒫l_{ω}^{q}.

The previous definition allows us to introduce multiplication in the local versions of these spaces. In fact, let now f_{1} ∈ 𝒫l_{ω, loc}^{q_{1}} and f_{2} ∈ 𝒫l_{ν, loc}^{q_{2}} . To define their product f = f_{1}f_{2}, we proceed locally. Let x_{0} ∈ ℝ^{d} and 0 < η < 1. Let ϕ ∈ 𝒟(I_{1,x_{0}} ) be such that ϕ(x) = 1 for x ∈ I_{η,x_{0}} . We define f_{I_{η,x_{0}}} ∈ 𝒟′(I_{η,x_{0}} ) as the restriction of (ϕf_{1})_{p} (ϕf_{2})_{p} to I_{η,x_{0}} . Note that different choices of ϕ lead to different Fourier coefficients but, by Proposition 1.1, we have f_{I_{η,x_{0}}} = f_{I_{η′,x′_{0}}} on I_{η,x_{0}} ∩ I_{η′,x′_{0}} . The {f_{I_{η,x_{0}}} } thus gives rise to a distribution f ∈ 𝒫l_{ω, loc}^{q} and we define the product of f_{1}f_{2} := f.

Proposition 1.2. The mappings

(1.2)  𝒫l_{ω}^{q_{1}} × 𝒫l_{ν}^{q_{2}} ∋ (f_{1}, f_{2}) ↦ f_{1}f_{2} ∈ 𝒫l_{ω}^{q},
(1.3)  𝒫l_{ω, loc}^{q_{1}} × 𝒫l_{ν, loc}^{q_{2}} ∋ (f_{1}, f_{2}) ↦ f_{1}f_{2} ∈ 𝒫l_{ω, loc}^{q}

are continuous.

Proof. The continuity of (1.3) follows at once from that of (1.2). For (1.2), Young’s inequality and (1.1) yield

∥f_{1}f_{2}∥_{𝒫l_{ω}^{q}} ⩽ C∥f_{1}∥_{𝒫l_{ω}^{q_{1}}} ∥f_{2}∥_{𝒫l_{ν}^{q_{2}}} .  □

In particular, we have

Corollary 1.1. Let s, s_{1}, s_{2} ∈ ℝ be such that s_{1} + s_{2} ⩾ 0, s ⩽ min{s_{1}, s_{2}}. Then the two mappings

𝒫l_{s_{1}}^{q_{1}} × 𝒫l_{s_{2}}^{q_{2}} ∋ (f_{1}, f_{2}) ↦ f_{1}f_{2} ∈ 𝒫l_{s}^{q},
𝒫l_{s_{1} , loc}^{q_{1}} × 𝒫l_{s_{2} , loc}^{q_{2}} ∋ (f_{1}, f_{2}) ↦ f_{1}f_{2} ∈ 𝒫l_{s, loc}^{q}

are continuous.

Proof. We may assume s_{1} ⩾ 0 and s = s_{2}. It is obvious that s_{1} ⩾ |s_{2}| has to hold in order to have s_{1} + s_{2} ⩾ 0. The result then follows from Proposition1.2 upon setting ω(n) = ⟨n⟩^{s_{2}} and ν(n) = ⟨n⟩^{s_{1}} because (1.1) holds for them. □

Concerning the local products from Corollary 1.1, exactly the same method from the proof of Theorem 3.1 below applies to show that the local space 𝒫l_{s, loc}^{2} coincides with the local Sobolev space H_{loc}^{s}(ℝ^{d}). Therefore, the multiplicative product for the local spaces in Corollary 1.1 agrees with the one defined by Hörmander in [6, Sect. 8.2]. Moreover, it is also worth mentioning that our results from the next sections imply that one can go beyond local products and in fact define the multiplicative product by microlocalization as in [6, Sect. 8.3]. We leave the formulation of such definitions to the reader. Theorem 3.1 below shows that the microlocal version of our multiplication also agrees with Hörmander’s one.

2. Wave front

Our goal in this section is to describe the wave front of f ∈ 𝒟′(ℝ^{d}) via the Fourier series coefficients of the periodic extension of an appropriate localization of f around x_{0} ∈ ℝ^{d}, as explained in the previous section. Recall (x_{0}, ξ_{0}) ∉ W F (f) if there exist ψ ∈ 𝒟(ℝ^{d}) with ψ ≡ 1 in a neighborhood of x_{0} and an open cone Γ ⊂ ℝ^{d} containing ξ_{0} such that

(2.1)  (∀N > 0)(∃C_{N} > 0)(∀ξ ∈ Γ)(|f̂ψ (ξ)| ⩽ C_{N} ⟨ξ⟩^{−N} ).

The following theorem tells that we can discretize (2.1).

Theorem 2.1. Let f ∈ 𝒟′(ℝ^{d}) and (x_{0}, ξ_{0}) ∈ ℝ^{n} × (ℝ^{d} ∖ {0}). The following conditions are equivalent:

(i) There exist ϕ ∈ 𝒟(I_{ε,x_{0}} ), with ε ∈ (0, 1) and ϕ ≡ 1 in a neighborhood of x_{0}, and an open cone Γ containing ξ_{0} such that

(∀N ∈ ℕ)(∃C_{N} > 0)(∀n ∈ Γ ∩ ℤ^{d})(|f̂ϕ (n)| ⩽ C_{N} ⟨n⟩^{−N} ).

(ii) (x_{0}, ξ_{0}) ∉ W F (f).

Proof. It is well known that, by shrinking the conic neighborhood of ξ_{0}, one may choose ψ in (2.1) with arbitrarily small support around x_{0}. Thus, (ii) implies (i). So, it is enough to show that (i) implies (ii). Assume (i). We divide the proof in two steps. We first prove that there are ε′ and an open cone ξ_{0} ∈ Γ_{1} such that

(2.2) (∀B bounded set in 𝒟(I_{ε′,x_{0}} ))(∀N > 0)(∃C′_{N} > 0)

(∀n ∈ Γ_{1} ∩ ℤ^{d})( sup_{φ∈B} |f̂φ (n)| ⩽ \frac{C′_{N}}{⟨n⟩^{N}}) .

We choose ε′ in such a way that ϕ(x) = 1 for every x ∈ I_{ε′,x_{0}} . For the cone, we select Γ_{1} an open cone with ξ_{0} ∈ Γ_{1} and \overline{Γ}_{1} ⊂ Γ ∪ \{0\}. Let us show that (2.2) holds with these choices. Let 0 < c < 1 be a constant smaller than the distance between ∂Γ and the intersection of \overline{Γ}_{1} with the unit sphere. Clearly, {y ∈ ℝ^{d} : (∃ξ ∈ Γ_{1})(|ξ − y| ⩽ c|ξ|)} ⊂ Γ. Let B ⊂ 𝒟(I_{ε′,x_{0}} ) be a bounded set. We have that ϕφ = φ, ∀φ ∈ B. Moreover, note that f̂φ (n) are precisely the Fourier coefficients of the periodic distribution (fϕ)_{p}(φ)_{p}. Therefore, for φ ∈ B and n ∈ Γ_{1} ∩ ℤ^{d},

|f̂φ (n)| = | ∑_{j∈ℤ^{d}} f̂ϕ (n − j)φ̂(j)| ⩽ ( ∑_{|j|⩽c|n|} + ∑_{|j|>c|n|} ) |f̂ϕ (n − j)φ̂(j)|

=: I_{1}(n) + I_{2}(n)

Further on,

I_{1}(n) = ∑_{|n−j|⩽c|n|} |f̂ϕ (j)∥φ̂(n − j)| ⩽ C sup_{|n−j|⩽c|n|} |f̂ϕ (j)|,

where C only depends on B. Since |n − j| ⩽ c|n| implies |j| ⩾ (1 − c)|n|,

(2.3) sup_{φ∈B, n∈Γ_{1}} ⟨n⟩^{N} I_{1}(n) ⩽ C sup_{n∈Γ_{1}} ⟨n⟩^{N} sup_{|n−j|⩽c|n|} |f̂ϕ (j)|

⩽ C sup_{j∈Γ_{ξ_{0}}} (1 − c)^{−N} ⟨j⟩^{N} |f̂ϕ (j)| = C(1 − c)^{−N} C_{N} .

For the estimate of I_{2} we use that |n − j| ⩽ (1 + c^{−1} )|j| if |j| ⩾ c|n|. Moreover, by the Paley–Wiener theorem, there are M, D > 0 such that

|f̂ϕ (n − j)| ⩽ D⟨n − j⟩^{M} , n, j ∈ ℤ^{d}.

Due to the boundedness of B ⊂ 𝒟(ℝ^{d}),

sup_{φ∈B} ∑_{j∈ℤ^{d}} ⟨j⟩^{M+N} |φ̂(j)| =: K_{N} < ∞.

Thus, for the second term, we have

(2.4)  sup_{n∈Γ_{1}} ⟨n⟩^{N} I_{2}(n) ⩽ D sup_{n∈Γ_{1}} ⟨n⟩^{N} ∑_{|j|⩾c|n|} ⟨n − j⟩^{M} |φ̂(j)|
⩽ Dc^{−N} (1 + c^{−1} )^{M} K_{N} , ∀φ ∈ B.

Combining (2.3) and (2.4), we get (2.2).

We now deduce that (x_{0}, ξ_{0}) ∉ W F (f) with the aid of (2.2). Let ψ ∈ 𝒟(I_{ε′,x_{0}} ) be equal to 1 in a neighborhood of x_{0}. Then, the set B = {φ_{t} := ψe_{−t} : t ∈ [0, 1)^{d}} is a bounded subset of 𝒟(I_{ε′,x_{0}} ). So,

sup_{t∈[0,1)^{d}} |f̂ψ (n + t)| = sup_{t∈[0,1)^{d}} |f̂φ_{t} (n)| ⩽ \frac{C′_{N}}{⟨n⟩^{N}}, ∀n ∈ Γ_{1} ∩ ℤ^{d},

that is,

sup_{ξ∈(Γ_{1} ∩ℤ^{d})+[0,1)^{d}} ⟨ξ⟩^{N} |f̂ψ (ξ)| ⩽ (1 + 4d)^{N/2} C′_{N} .

Select now an open conic neighborhood Γ_{2} of ξ_{0} such that \overline{Γ}_{2} ⊂ Γ_{1} ∪ \{0\} and find c′ such that {y ∈ ℝ^{d} : (∃ξ ∈ Γ_{2})(|ξ − y| ⩽ c′|ξ|)} ⊂ Γ_{1}. The latter condition implies that Γ_{2} ∩ {ξ ∈ ℝ^{d} : |ξ|c′ ⩾ 1} ⊂ (Γ_{1} ∩ ℤ^{d}) + [0, 1)^{d} and hence

sup_{ξ∈Γ_{2}}⟨ξ⟩^{N} |f̂ψ (ξ)| ⩽ max{C′′_{N}, (1 + 4d)^{N/2} C′_{N} } = C_{N} < ∞,

where C′′_{N} = sup_{ξ∈Γ_{2} , |ξ|<1/c′} ⟨ξ⟩^{N} |f̂ψ (ξ)|. This shows that (x_{0}, ξ_{0}) ∉ W F (f), as claimed. □

We mention that Theorem 2.1 also follows from the relation between discrete and Hörmander’s wave front sets proved in Theorem 7.4 in [12], once one observes that the notion is local and so it does not depend of a particular parametrization.

3. Sobolev wave front

In this section we deal with wave fronts of Sobolev type. We slightly reformulate Hörmander’s definition [6].

Definition 3.1. Let f ∈ 𝒟′(ℝ^{d}), (x_{0}, ξ_{0}) ∈ ℝ^{d} × (ℝ^{d} ∖ {0}), and s ∈ ℝ. We

say that f is Sobolev microlocally regular at (x_{0}, ξ_{0}) of order s, that is (x_{0}, ξ_{0}) ∉ W F_{s}(f), if there exist an open cone Γ around ξ_{0} and ψ ∈ 𝒟(ℝ^{d}) with ψ ≡ 1 in a neighborhood of x_{0} such that ∫_{Γ} |ψ̂f (ξ)|^{2}⟨ξ⟩^{2s} dξ < ∞ .

We shall now refine Theorem 2.1.

Theorem 3.1. Let f ∈ 𝒟′(ℝ^{d}). The following two conditions are equivalent:

(i) There exist an open cone Γ around ξ_{0}, ϕ ∈ 𝒟(I_{η,x_{0}} ), η ∈ (0, 1), with ϕ ≡ 1 in a neighborhood of x_{0}, such that

(3.1)  ∑_{n∈Γ∩ℤ^{d}} |a_{n}|^{2}⟨n⟩^{2s} < ∞, where (fϕ)_{p} = ∑_{n∈ℤ^{d}} a_{n}e_{n}.

(ii) (x_{0}, ξ_{0}) ∉ W F_{s}(f).

Proof. (i) ⇒ (ii). Assume (3.1). Choose an open cone Γ_{1} so that \overline{Γ_{1}} ⊂ Γ∪\{0\} and ξ_{0} ∈ Γ_{1}. Find 0 < ε < η such that ϕ(x) = 1 for all x ∈ I_{ε,x_{0}} . We first prove that: For every bounded set B ⊂ 𝒟(I_{ε,x_{0}} )

(3.2)  sup_{φ∈B} ∑_{n∈Γ_{1} ∩ℤ^{d}} |f̂φ (n)|^{2}⟨n⟩^{2s} < ∞.

Fix a bounded subset B ⊂ 𝒟(I_{ε,x_{0}} ). In view of the choice of ε, we have that fφ = fφϕ and so f̂φ (n) = ∑ _{j∈ℤ^{d}} a_{j} φ̂(n − j), for every φ ∈ B. We fix a constant 0 < c < 1 that is smaller than the distance between ∂Γ and the intersection of \overline{Γ}_{1} with the unit sphere, and also smaller than the distance between ∂Γ_{1} and the intersection of \overline{ℝ^{d} ∖ Γ} with the unit sphere. One has that ξ ∈ Γ_{1} and y ∉ Γ imply |ξ − y| > c max{|ξ|, |y|}. We keep φ ∈ B. By Peetre’s inequality, we have

( ∑_{n∈Γ_{1} ∩ℤ^{d}} |f̂φ (n)|^{2}⟨n⟩^{2s} )^{1/2} ⩽ C( ∑_{n∈Γ_{1} ∩ℤ^{d}} ( ∑_{j∈ℤ^{d}} |a_{j} |⟨j⟩^{s} |φ̂(n − j)|⟨n − j⟩^{|s|} )^{2} )^{1/2}
⩽ C(I_{1}(φ) + I_{2}(φ)),

where

I_{1}(φ) = ( ∑_{n∈Γ_{1} ∩ℤ^{d}} ( ∑_{j∈Γ∩ℤ^{d}} |a_{j} |⟨j⟩^{s} |φ̂(n − j)|⟨n − j⟩^{|s|} )^{2} )^{1/2} ,

I_{2}(φ) = ( ∑_{n∈Γ_{1} ∩ℤ^{d}} ( ∑_{j∉Γ∩ℤ^{d}} |a_{j} |⟨j⟩^{s} |φ̂(n − j)|⟨n − j⟩^{|s|} )^{2} )^{1/2} .

By Young’s inequality and the fact that B is a bounded set,

sup_{φ∈B} I_{1}(φ) ⩽ ( ∑_{n∈Γ∩ℤ^{d}} |a_{n}|^{2}⟨n⟩^{2s})^{1/2} sup_{φ∈B} ∑_{n∈ℤ^{d}} |φ(n)|⟨n⟩^{|s|} < ∞.

We now estimate I_{2}(φ). Since fϕ is compactly supported, ⟨j⟩^{s}|a_{j} | = ⟨j⟩^{s}|f̂ϕ (j)| ⩽ D⟨j⟩^{k}, ∀n ∈ ℤ^{d}, for some D > 0 and k > 0. The fact that B is bounded yields the existence of C′ > 0 such that |φ̂(j)| ⩽ C′⟨j⟩^{−k−|s|−3(d+1)/2} . Because of the choice of Γ_{1}, we have

sup_{φ∈B}(I_{2}(φ))^{2} ⩽ (DC′)^{2} ∑_{n∈Γ_{1} ∩ℤ^{d}} ( ∑_{j∉Γ∩ℤ^{d}} ⟨j⟩^{k}⟨n − j⟩^{−k−3(d+1)/2} )^{2}

⩽ (DC′)^{2}c^{−2k−3(d+1)} ∑_{n∈Γ_{1} ∩ℤ^{d}} ⟨n⟩^{−d−1} ( ∑_{j∉Γ∩ℤ^{d}} ⟨j⟩^{−d−1} )^{2}.

Thus (3.2) has been established. We now deduce (ii) from (3.2). Once again we shrink the conic neighborhood of ξ_{0}. So let Γ_{2} be an open cone such that \overline{Γ_{2}} ⊂ Γ_{1} ∪ \{0\} and ξ_{0} ∈ Γ_{2}. Let ψ ∈ 𝒟(I_{ε,x_{0}} ) be equal to 1 in a neighborhood of x_{0}. Find r > 0 such that Γ_{2} ∩ {ξ ∈ ℝ^{d} : |ξ| ⩾ r} ⊂ (Γ_{1} ∩ ℤ^{d}) + [0, 1)^{d}. For each n ∈ Γ_{1} ∩ ℤ^{d}, write Λ_{n} = n + [0, 1]^{d}. Then, by (3.2) and Peetre’s inequality,

∫\begin{matrix}ξ∈Γ_{2} \\ |ξ|⩾r \end{matrix} |ψ̂f (ξ)|^{2}⟨ξ⟩^{2s} dξ⩽ C ∑_{n∈Γ_{1} ∩ℤ^{d}} ⟨n⟩^{2s} ∫_{Λ_{n}} |ψ̂f (ξ)|^{2}dξ

= C ∫_{[0,1]^{d}} ∑_{n∈Γ_{1} ∩ℤ^{d}} ⟨n⟩^{2s} |ψ̂f (n + t)|^{2}dt

⩽ C sup_{t∈[0,1]^{d}} ∑_{n∈Γ_{1} ∩ℤ^{d}} ⟨n⟩^{2s} |{ \atop e_{−t} ψf}(n)|^{2} < ∞.

Therefore, (x_{0}, ξ_{0}) ∉ W F (f).

(ii) ⇒ (i). A variant of the argument employed above, but with integrals instead of sums, applies to show that (x_{0}, ξ_{0}) ∉ W F_{s}(f) implies the following property: There exist an open cone Γ and ε ∈ (0, 1) such that for every bounded set B ⊂ 𝒟(I_{ε,x_{0}} )

(3.3)  sup_{ψ∈B} ∫_{Γ} |f̂ψ (ξ)|^{2}⟨ξ⟩^{2s} dξ < ∞.

We leave such details to the reader. So, assume that (3.3) holds. Let Γ_{1} be an open cone containing ξ_{0} such that \overline{Γ}_{1} ⊂ Γ ∪ \{0\}. Then, there exists some r > 0 such that (Γ_{1} + [0, 1]^{d}) ∩ {ξ ∈ ℝ^{d} : |ξ| ⩾ r} ⊂ Γ. Let ϕ ∈ 𝒟(I_{ε,x_{0}} ) such that ϕ ≡ 1 in a neighborhood of x_{0}. Consider a measurable function t : Γ → [0, 1]^{d}. Taking the bounded set

B = {ψ_{j,t} ∈ 𝒟(I_{ε,x_{0}} ) : ψ_{j,t} (x) = x_{j} e^{−2πix·t(ξ)} ϕ(x), ξ ∈ Γ, j = 1, …, d}

in (3.3), we obtain that there is a constant C > 0 such that

(3.4)  ∫_{Γ} |∇(ϕ̂f )(ξ + t(ξ))|^{2}⟨ξ⟩^{2s} dξ < C.

The constant C is actually independent of t. For each n ∈ Γ_{1} ∩ ℤ^{d}, let Λ_{n} be the unit cube n + [0, 1]^{d} = ∏ _{j=1}^{d} [n_{j} , n_{j} + 1]. Note that Λ_{n} ⊂ Γ if |n| ⩾ r. Then

( ∑_{n∈Γ_{1} ∩ℤ^{d}} |ϕ̂f (n)|^{2}⟨n⟩^{2s} )^{1/2} = ( ∑_{n∈Γ_{1} ∩ℤ^{d}} ∫_{Λ_{n}} |ϕ̂f (n)|^{2}⟨n⟩^{2s}dξ)^{1/2} ⩽ I_{1}^{1/2} + I_{2}^{1/2} ,

where I_{1} := ∑ _{n∈Γ_{1} ∩ℤ^{d}} ∫_{Λ_{n}} |ϕ̂f (n) − ϕ̂f (ξ)|^{2}⟨n⟩^{2s} dξ and

I_{2} := ∑_{n∈Γ_{1} ∩ℤ^{d}} ∫_{Λ_{n}} |ϕ̂f (ξ)|^{2}⟨n⟩^{2s} dξ

⩽ ∑_{|n|⩽r} ∫_{Λ_{n}} |ϕ̂f (ξ)|^{2}⟨n⟩^{2s} dξ + C′∫_{Γ} |ϕ̂f (ξ)|^{2}⟨ξ⟩^{2s}dξ < ∞.

It remains to show that I_{1} is finite. Given θ > 0, define t_{θ} : Γ → [0, 1]^{d} as

t_{θ}(ξ) = \{ \begin{matrix}θ(n − ξ) & if ξ ∈ Λ_{n} and |ξ| ⩾ r, \\ 0 & otherwise. \end{matrix}

We now make use of (3.4). Since

|ϕ̂f (ξ) − ϕ̂f (n)|^{2} ⩽ |n − ξ| ∫_{0}^{1} |∇(ϕ̂f )(ξ + θ(n − ξ))|^{2}dθ,

we have

I_{1} ⩽ ∑_{\begin{matrix}|n|⩽r \\ n∈Γ_{1} ∩ℤ^{d} \end{matrix}} ∫_{Λ_{n}} |ϕ̂f (n) − ϕ̂f (ξ)|^{2}⟨n⟩^{2s} dξ

+ C′ sup_{θ∈[0,1]} ∫_{Γ} |∇(ϕ̂f )(ξ + t_{θ}(ξ))|^{2}⟨ξ⟩^{2s} dξ < ∞.

This completes the proof. □

Acknowledgement. This work is supported by the Project 174024 of the Serbian Ministry of Science.

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Faculty of Electrical Engineering (Received 26 11 2014) University of Banja Luka Bosnia and Herzegovina

snjezana.maksimovic@etfbl.net

Department of Mathematics and Informatics University of Novi Sad Novi Sad, Serbia

pilipovic@dmi.uns.ac.rs

Faculty of Mathematics and Natural Sciences St. Cyril and Methodius University, Skopje, Macedonia

petar@pmf.ukim.mk

Department of Mathematics Ghent University Ghent, Belgium

jvindas@cage.ugent.be