PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 1–24 DOI: 10.2298/PIM1206001J

MICRO-LOCAL ANALYSIS IN SOME SPACES OF ULTRADISTRIBUTIONS

Karoline Johansson, Stevan Pilipović, Nenad Teofanov and Joachim Toft

Communicated by Gradimir Milovanović

Abstract. We extend some results from [14 ] and [19 ], concerning wave-front sets of Fourier–Lebesgue and modulation space types, to a broader class of spaces of ultradistributions. We relate these wave-front sets one to another and to the usual wave-front sets of ultradistributions.

Furthermore, we give a description of discrete wave-front sets by introducing the notion of discretely regular points, and prove that these wave-front sets coincide with corresponding wave-front sets in [19 ]. Some of these investigations are based on the properties of the Gabor frames.

1. Introduction

Wave-front sets with respect to Fourier–Lebesgue and modulation spaces were introduced in [19] and studied further in [18, 20, 21]. Among other properties, it was proved that wave-front sets of Fourier–Lebesgue and modulation spaces coincide, and that the usual wave-front sets with respect to smoothness (cf. [13, Sections 8.1–8.3]) can be obtained as wave-front sets of sequences of Fourier–Lebesgue or modulation spaces. Discrete versions of wave-front sets of Fourier–Lebesgue and modulation spaces, related to those wave-front sets in [24], were introduced and studied in [14]. In particular, it was proved that these wave-front sets agree with corresponding wave-front sets in [19].

In this paper we put questions from [14,19] in a broader context, where we allow the involved distributions to be Beurling or Roumieu type ultradistributions. This is done by relaxing a polynomial type conditions on the involved weight functions, into a subexponential type condition. An important benefit is that the families of Fourier–Lebesgue and modulation spaces in [14, 19] are significantly enlarged, since growth/decay properties of these weights are crucial concerning growth and regularity limitations on the involved distributions. For example, in this extended situation, the modulation spaces might contain positive functions growing subexponentially. We refer to [11, 27] for a review of these facts.

2010 Mathematics Subject Classification: Primary 35A18, 35S30, 42B05, 35H10.

Key words and phrases: Wave-front sets, weighted Fourier–Lebesgue spaces, ultradistributions.

In this section we set the stage with a brief overview of basic notions. Then, in Section 2 we introduce wave-front sets of a Beurling type ultradistribution with respect to weighted Fourier–Lebesgue space and show that they satisfy appropriate micro-local properties, Theorem 2.1. Then, in subsection 2.1 we show that the most common wave-front sets of ultradistributions given in [12,17,22] can be described within our approach, Proposition 2.1.

An important part of our investigations is to establish identification properties between wave-front sets of Fourier–Lebesgue and modulation space types. This is the subject of Theorem 3.1 in Section 3. Although we follow the framework of [19], we note that several new problems appear when dealing with ultradistributions. For example, several properties of the wave-front sets depend on properties of the shorttime Fourier transform in the framework of ultradistributions, cf. Proposition 3.2.

Finally, in Section 4 we introduce discrete versions of wave-front sets of ultradistributions and prove their invariance properties (cf. Theorem 4.2). The main ideas of our approach can be traced back to [14], see also [24, 25]. In order to be self-contained, we could not avoid certain repetitions of [14]. However, here we provide additional explanations of the construction and introduce the notion of discretely regular points. We believe that the results in form of series, established when introducing discrete wave-front sets, might be useful for numerical analysis of micro-local properties of functions and ultradistributions. For example, we use Gabor frames for the description of discrete wave-front sets and note that the Gabor frame coefficients give information on micro-local properties of the signal in such way. (See [7, 8] for numerical treatment of Gabor frame theory.)

Since compactly supported smooth functions are used in the process of microlocalization we are limited to the use of weights with almost exponential growth at infinity described by the Beurling–Domar condition. We refer to subsection 1.1 for the notions and to [10] for a discussion on the role of weights in time-frequency analysis.

Our investigation can therefore be considered as the starting point in the study of analytic wave-front sets and pseudodifferential operators with ultrapolynomial symbols, also known as symbol-global type operators. This will be done in a separate paper, [15].

1.1. Basic notions and notation. In this subsection we collect some notation and notions which will be used in the sequel.

We put N = {0, 1, 2, …}, ⟨x⟩ = (1 + |x|^{2})^{1/2} , for x ∈ R^{d}, and A ≲ B to indicate A ⩽ cB for a suitable constant c > 0. The symbol B_{1} ↪ B_{2} denotes the continuous and dense embedding of the topological vector space B_{1} into B_{2}. The scalar product in L^{2} is denoted by (·, ·)_{L^{2}} = (·, ·). Translation and modulation operators are given by T_{x}f(t) = f(t − x) and M_{ξ}f(t) = e^{i⟨ξ,t⟩}f(t).

1.1.1. Weights. In general, a weight function is a nonnegative function in L_{loc}^{∞}.

Definition 1.1. Let ω, v be nonnegative functions. Then (1) v is called submultiplicative if v(x + y) ⩽ v(x)v(y), for each x, y ∈ R^{d};

(2) ω is called v-moderate if ω(x + y) ≲ v(x)ω(y), for each x, y ∈ R^{d}. For a given submultiplicative weight v the set of all v-moderate weights will be denoted by ℳ_{v}.

If v is even and ω ∈ ℳ_{v}, then 1/v ≲ ω ≲ v, ω ≠ 0 everywhere and 1/ω ∈ ℳ_{v}. In the sequel v will always stand for an even submultiplicative function. Sub-

multiplicativity implies that v is dominated by an exponential function, i.e.,

v ⩽ Ce^{k|·|} for some C, k > 0.

Let s > 1. By ℳ_{{s}} (R^{d}) we denote the set of all weights which are moderate with respect to a weight v which satisfies v ⩽ Ce^{k|·|^{1/s}} for some positive constants C and k. The weight v satisfy the Beurling–Domar non-quasi-analyticity condition which takes the form ∑ _{n=0}^{∞} \frac{1}{n^{2}} logv(nx) < ∞, x ∈ R^{d}. We refer to [10] for more facts about such weights.

1.1.2. Test function spaces and their duals. Next we introduce spaces of test functions and their duals in the context of spaces of ultradistributions. These test function spaces correspond to the spaces C_{0}^{∞} , 𝒮 and C^{∞} in the distribution theory in [12, 26]. We start by giving the definition of Gelfand-Shilov type spaces.

Definition 1.2. Let s > 1 and A > 0. We denote by S_{A}^{s}(R^{d}) the space of all functions φ ∈ C^{∞} (R^{d}) such that the norm

∥φ∥_{s,A} = sup_{α,β∈N_{0}^{d}} sup_{x∈R^{d}} ( \frac{A^{|α+β|}}{α!^{s}β!^{s}} ⟨x⟩^{|α|} |φ^{(β)} (x)|)

is finite. Then the spaces S^{(s)} (R^{d}) and S^{{s}} (R^{d}) are given by

S^{(s)} (R^{d}) = ⋂_{A>0} S_{A}^{s}(R^{d}) S^{{s}} (R^{d}) = ⋃_{A>0} S_{A}^{s}(R^{d}).

The topologies for S^{(s)} (R^{d}) and S^{{s}} (R^{d}) are given by the projective and inductive limit, respectively, i.e.,

S^{(s)} (R^{d}) = proj lim_{A→∞} S_{A}^{s}(R^{d}), S^{{s}} (R^{d}) = ind lim_{A→0} S_{A}^{s}(R^{d}).

We note that S_{A}(R^{d}) is a Banach space, for every A > 0, and its dual is denoted by (S_{A})′(R^{d}). Then the Gelfand-Shilov type distribution spaces (S^{(s)} )′(R^{d}) and (S^{{s}} )′(R^{d}) are defined as

(S^{(s)} )′(R^{d}) = ⋃_{A>0} (S_{A}^{s})′(R^{d}), (S^{{s}} )′(R^{d}) = ⋂_{A>0} (S_{A}^{s})′(R^{d}).

These spaces are the strong dual spaces of S^{(s)} (R^{d}) and S^{{s}} (R^{d}), and are called the spaces of tempered ultradistributions of Beurling type and Roumieu type, respectively. If s > t, then

S^{(t)}(R^{d}) ↪ S^{{t}}(R^{d}) ↪ S^{(s)} (R^{d}) ↪ S^{{s}} (R^{d})
↪ (S^{{s}} )′(R^{d}) ↪ (S^{(s)} )′(R^{d}) ↪ (S^{{t}})′(R^{d}) ↪ (S^{(t)})′(R^{d}).

In order to perform (micro-)local analysis we use the following test function spaces on open sets, cf. [16].

Definition 1.3. Let X be an open set in R^{d}. For a given compact set K ⊂ X, s > 1 and A > 0 we denote by E_{A,K}^{s} (X) the space of all φ ∈ C^{∞} (X) such that the norm

(1.1)  ∥φ∥_{s,A,K} = sup_{β∈N_{0}^{n}} sup_{x∈K} \frac{A^{|β|}}{β!^{s}} |φ^{(β)} (x)|

is finite.

The space of functions φ ∈ C^{∞} (X) such that (1.1) holds and suppφ ⊆ K is denoted by D_{A}^{s}(K).

Let (K_{n})_{n} be a sequence of compact sets such that K_{n} ⊂⊂ K_{n+1} and ⋃ K_{n} =X . Then

E^{(s)} (X) = proj lim_{n→∞} (proj lim_{A→∞} E_{A,K_{n}}^{s} )(X),
E^{{s}} (X) = proj lim_{n→∞} (ind lim_{A→0} E_{A,K_{n}}^{s} )(X),
D^{(s)} (X) = ind lim_{n→∞} (proj lim_{A→∞} D_{A}^{s}(K_{n})),
D^{{s}} (X) = ind lim_{n→∞} (ind lim_{A→0} D_{A}^{s}(K_{n})).

Obviously, D^{(s)} (X) (D^{{s}} (X) resp.) are subspaces of E^{(s)} (X) (of E^{{s}} (X) resp.) whose elements are compactly supported. We also remark that a usual notation for the space D^{{s}} (X) is G_{0}^{s}(X) (cf. [22]).

Remark 1.1. Let ∗ denote (s) or {s}. Then D^{∗}, S^{∗} and E^{∗} correspond to C_{0}^{∞} , 𝒮 and C^{∞} , respectively, and D^{∗} ⊆ C_{0}^{∞} , S^{∗} ⊆ 𝒮 and E^{∗} ⊆ C^{∞} .

The spaces of linear functionals over D^{(s)} (X) and D^{{s}} (X), denoted by (D^{(s)} )′(X) and (D^{{s}} )′(X) respectively, are called the spaces of ultradistributions of Beurling and Roumieu type respectively, while the spaces of linear functionals over E^{(s)} (X) and E^{{s}} (X), denoted by (E^{(s)} )′(X) and (E^{{s}} )′(X), respectively are called the spaces of ultradistributions of compact support of Beurling and Roumieu type respectively, [16]. We have

(E^{{s}} )′(X) ⊆ (E^{(s)} )′(X), (E^{(s)} )′(X) ⊆ (E^{(s)} )′(R^{d}) and (E^{{s}} )′(X) ⊆ (E^{{s}} )′(R^{d}).

Clearly,

(E^{{s}} )′(R^{d}) ⊆ (S^{{s}} )′(R^{d}) ⊆ (D^{{s}} )′(R^{d})
(E^{(s)} )′(R^{d}) ⊆ (S^{(s)} )′(R^{d}) ⊆ (D^{(s)} )′(R^{d}).

Any ultra-distribution with compact support can be viewed as an element of (S^{(1)})′(R^{d}). More generally, by similar arguments as in the distribution theory in [12], it follows that E^{∗} are exactly those elements in S^{∗} or D^{∗} with compact support.

1.1.3. Fourier–Lebesgue spaces. The Fourier transform ℱ is a linear and continuous mapping on 𝒮′(R^{d}) which takes the form

(ℱ f)(ξ) = f̂(ξ) ≡ (2π)^{−d/2} ∫_{R^{d}} f(x)e^{−i⟨x,ξ⟩} dx

when f ∈ L^{1}(R^{d}). It is a homeomorphism on (S^{{s}} )′(R^{d}) (on (S^{(s)} )′(R^{d}) resp.) which restricts to a homeomorphism on S^{{s}} (R^{d}) (on S^{(s)} (R^{d}) resp.) and to a unitary operator on L^{2}(R^{d}).

Let q ∈ [1, ∞], s > 1 and ω ∈ ℳ_{{s}} (R^{d}). The (weighted) Fourier–Lebesgue space ℱ L_{(ω)}^{q} (R^{d}) is the inverse Fourier image of L_{(ω)}^{q} (R^{d}), i.e., ℱ L_{(ω)}^{q} (R^{d}) consists

of all f ∈ (S^{(s)} )′(R^{d}) such that ∥f∥_{ℱL_{(ω)}^{q}} ≡ ∥f̂· ω∥_{L^{q}} is finite. If ω = 1, then the notation ℱ L^{q} is used instead of ℱ L_{(ω)}^{q} . We note that if ω(ξ) = ⟨ξ⟩^{s}, then ℱ L_{(ω)}^{q} is the Fourier image of the Bessel potential space H_{s}^{p}.

Remark 1.2. Whenever it is convenient, we permit an x dependence for the weight ω in the definition of Fourier–Lebesgue spaces. More precisely, for each ω ∈ ℳ_{{s}} (R^{2d}) we let ℱ L_{(ω)}^{q} be the set of all ultradistributions f such that

∥f∥_{ℱL_{(ω)}^{q}} ≡ ∥f̂ω(x, ·)∥_{L^{q}}

is finite. Since ω is v-moderate, it follows that different choices of x give rise to equivalent norms. Therefore the condition ∥f∥_{ℱL_{(ω)}^{q}} < ∞ is independent of x, and

it follows that ℱ L_{(ω)}^{q} (R^{d}) is independent of x although ∥ · ∥_{ℱL_{(ω)}^{q}} might depend on x.

2. Wave-front sets of Fourier–Lebesgue type in spaces of Beurling type ultradistributions

In this section we introduce wave-front sets of Fourier–Lebesgue type in spaces of ultradistributions of Beurling type.

Let s > 1, q ∈ [1, ∞], and Γ ⊆ R^{d} ∖ 0 be an open cone. If f ∈ (S^{(s)} )′(R^{d}) and ω ∈ ℳ_{{s}} (R^{2d}) we define

(2.1)  |f|_{ℱL_{(ω)}^{q,Γ}} = |f|_{ℱL_{(ω),x}^{q,Γ}} ≡ ( ∫_{Γ} |f̂(ξ)ω(x, ξ)|^{q}dξ) ^{1/q}

(with obvious interpretation when q = ∞). We note that | · |_{ℱL_{(ω),x}^{q,Γ}} defines a semi-

norm on (S^{(s)} )′(R^{d}) which might attain the value +∞. Since ω is v-moderate it follows that different x ∈ R^{d} gives rise to equivalent seminorms |f|_{ℱL_{(ω),x}^{q,Γ}} . Fur-

thermore, if Γ = R^{d} ∖ 0, f ∈ ℱ L_{(ω)}^{q} (R^{d}) and q < ∞, then |f|_{ℱL_{(ω),x}^{q,Γ}} agrees with

the Fourier–Lebesgue norm ∥f∥_{ℱL_{(ω),x}^{q}} of f.

For the sake of notational convenience we set

(2.2)  B = ℱ L_{(ω)}^{q} = ℱ L_{(ω)}^{q} (R^{d}), and | · |_{B(Γ)} = | · |_{ℱL_{(ω),x}^{q,Γ}} .

We let Θ_{B}(f) = Θ_{ℱL_{(ω)}^{q}} (f) be the set of all ξ ∈ R^{d} ∖ 0 such that |f|_{B(Γ)} < ∞, for an open conical neighbourhood Γ = Γ_{ξ} of ξ. We also let Σ_{B}(f) be the complement of Θ_{B}(f) in R^{d} ∖ 0. Then Θ_{B}(f) and Σ_{B}(f) are open, respectively closed, subsets in R^{d} ∖ 0, which are independent of the choice of x ∈ R^{d} in (2.1).

Definition 2.1. Let s > 1, q ∈ [1, ∞], B be as in (2.2), and let X be an open subset of R^{d}. If ω ∈ ℳ_{{s}} (R^{2d}), then the wave-front set of f ∈ (D^{(s)} )′(X), WF_{B}(f) ≡ WF_{ℱL_{(ω)}^{q}} (f) with respect to B consists of all pairs (x_{0}, ξ_{0}) in X×(R^{d}∖0)

such that ξ_{0} ∈ Σ_{B}(φf) holds for each φ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0.

We note that WF_{B}(f) is a closed set in R^{d} × (R^{d} ∖ 0), since it is obvious that its complement is open. We also note that if x ∈ R^{d} is fixed and ω_{0}(ξ) = ω(x, ξ), then WF_{B}(f) = WF_{ℱL_{(ω_{0} )}^{q}} (f), since Σ_{B} is independent of x.

The following theorem shows that wave-front sets with respect to ℱ L_{(ω)}^{q} satisfy appropriate micro-local properties. It also shows that such wave-front sets are decreasing with respect to the parameter q, and increasing with respect to the weight ω.

Theorem 2.1. Let s > 1, q, r ∈ [1, ∞], X be an open set in R^{d} and ω, ϑ ∈

ℳ_{{s}} (R^{2d}) be such that r ⩽ q, and ω(x, ξ) ≲ ϑ(x, ξ). Also let B be as in (2.2) and put B_{0} = ℱ L_{(ϑ)}^{r} (R^{d}). If f ∈ (D^{(s)} )′(X) and φ ∈ D^{(s)} (X), then

WF_{B}(φ f) ⊆ WF_{B_{0}} (f).

Proof. By the definitions it is sufficient to prove Σ_{B}(φf) ⊆ Σ_{B_{0}} (f) when φ ∈ D^{(s)} (X), ϑ = ω, and f ∈ (E^{(s)} )′(R^{d}), since the statement only involves local assertions. For the same reasons we may assume that ω(x, ξ) = ω(ξ) is independent of x. Finally, we prove the assertion for r ∈ [1, ∞). The case r = ∞ follows by similar arguments and is left to the reader.

Choose open cones Γ_{1} and Γ_{2} in R^{d} such that \overline{Γ_{2}} ⊆ Γ_{1}. We will use the fact that if f ∈ (E^{(s)} )′(R^{d}), then |f̂(ξ)ω(ξ)| ≲ e^{N_{0} |ξ|^{1/s}} for some N_{0} > 0 and prove that for every N > 0, there exist C_{N} > 0 such that

(2.3) |φf|_{B(Γ_{2} )} ⩽ C_{N} ( |f|_{B_{0} (Γ_{1} )} + sup_{ξ∈R^{d}} (|f̂(ξ)ω(ξ)|e^{−N|ξ|^{1/s}} )) when \overline{Γ}_{2} ⊆ Γ_{1}.

Since ω ∈ ℳ_{{s}} (R^{2d}) by letting F (ξ) = |f̂(ξ)ω(ξ)| and ψ(ξ) = |φ̂(ξ)v(ξ)| we get

|φf|_{B(Γ_{2} )} = ( ∫_{Γ_{2}} |ℱ (φf)(ξ)ω(ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ_{2}} ( ∫_{R^{d}} ψ(ξ − η)F (η) dη) ^{q}dξ) ^{1/q} ≲ J_{1} + J_{2},

where

J_{1} = ( ∫_{Γ_{2}} ( ∫_{Γ_{1}} ψ(ξ − η)F (η) dη) ^{q}dξ) ^{1/q},

J_{2} = ( ∫_{Γ_{2}} ( ∫_{∁Γ_{1}} ψ(ξ − η)F (η) dη) ^{q}dξ) ^{1/q}.

Let q_{0} be chosen such that 1/r_{0}+1/r = 1+1/q, and let χ_{Γ_{1}} be the characteristic function of Γ_{1}. Then Young’s inequality gives

J_{1} ⩽ ( ∫_{R^{d}} ( ∫_{Γ_{1}} ψ(ξ − η)F (η) dη) ^{q}dξ) ^{1/q}
= ∥ψ ∗ (χ_{Γ_{1}} F)∥_{L^{q}} ⩽ ∥ψ∥_{L^{r_{0}}} ∥χ_{Γ_{1}} F ∥_{L^{r}} = C_{ψ} |f|_{B_{0} (Γ_{1} )} ,

where C_{ψ} = ∥ψ∥_{L^{q_{0}}} < ∞. If φ ∈ D^{(s)} (X), then for every N > 0 there exist C_{N} > 0 such that

(2.4)  ψ(ξ) = |φ̂(ξ)v(ξ)| ⩽ C_{N} e^{−(N+k)|ξ|^{1/s}} e^{k|ξ|^{1/s}} ⩽ C_{N} e^{−N|ξ|^{1/s}} .
In order to estimate J_{2}, we note that \overline{Γ_{2}} ⊆ Γ_{1} implies that
(2.5) |ξ − η|^{1/s} > 2c max(|ξ|^{1/s} , |η|^{1/s} ) ⩾ c(|ξ|^{1/s} + |η|^{1/s} ),  ξ ∈ Γ_{2}, eta ∉ Γ_{1}

holds for some constant c > 0, since this is true when 1 = |ξ| ⩾ |η|. A combination of (2.4) and (2.5) implies that for every N_{1} > 0 we have

ψ(ξ − η) ≲ Ce^{−2N_{1} (|ξ|^{1/s} +|η|^{1/s} )} .

This gives

J_{2} ≲ ( ∫_{Γ_{2}} ( ∫_{∁Γ_{1}} e^{−2N_{1} (|ξ|^{1/s} +|η|^{1/s} )} F (η) dη) ^{r}dξ) ^{1/r}

≲ ( ∫_{Γ_{2}} ( ∫_{∁Γ_{1}} e^{−2N_{1} (|ξ|^{1/s} +|η|^{1/s} )} e^{N_{1} |η|^{1/s}} (e^{−N_{1} |η|^{1/s}} F (η)) dη) ^{r}dξ) ^{1/r}

≲ sup_{η∈R^{d}} |e^{−N_{1} |η|^{1/s}} F (η))|,

which proves (2.3) and the result follows. □

Next we modify the definitions from [19] concerning wave-front sets with respect to sequences of spaces. Let ω_{j} ∈ ℳ_{{s}} (R^{2d}) and q_{j} ∈ [1, ∞] when j belongs to some index set J, and let B be the array of spaces, given by

(2.6)  (B_{j}) ≡ (B_{j})_{j∈J} , where B_{j} = ℱ L_{(ω_{j} )}^{q_{j}} = ℱ L_{(ω_{j} )}^{q_{j}} (R^{d}), j ∈ J.

If s > 1, f ∈ (D^{(s)} )′(R^{d}), and (B_{j}) is given by (2.6), then we let Θ_{(B_{j} )}^{sup}(f) be the set of all ξ ∈ R^{d} ∖ 0 such that for some Γ = Γ_{ξ} and each j ∈ J it holds |f|_{B_{j} (Γ)} < ∞. We also let Θ_{(B_{j} )}^{inf}(f) be the set of all ξ ∈ R^{d} ∖ 0 such that for some Γ = Γ_{ξ} and some j ∈ J it holds |f|_{B_{j} (Γ)} < ∞. Finally we let Σ_{(B_{j} )}^{sup}(f) and Σ_{(B_{j} )}^{inf}(f) be the complements in R^{d} ∖ 0 of Θ_{(B_{j} )}^{sup}(f) and Θ_{(B_{j} )}^{inf}(f) respectively.

Definition 2.2. Let J be an index set, q_{j} ∈ [1, ∞], ω_{j} ∈ ℳ_{{s}} (R^{2d}) when

j ∈ J, (B_{j}) be as in (2.6), and let X be an open subset of R^{d}.

(1) The wave-front set of f ∈ (D^{(s)} )′(X), of sup-type with respect to (B_{j}), WF_{(B_{j} )}^{sup}(f), consists of all pairs (x_{0}, ξ_{0}) in X × (R^{d} ∖ 0) such that ξ_{0} ∈ Σ_{(B_{j} )}^{sup}(φf) holds for each φ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0;

(2) The wave-front set of f ∈ (D^{(s)} )′(X), of inf-type with respect to (B_{j}), WF_{(B_{j} )}^{inf}(f) consists of all pairs (x_{0}, ξ_{0}) in X × (R^{d} ∖ 0) such that ξ_{0} ∈ Σ_{(B_{j} )}^{inf}(φf) holds for each φ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0.

Remark 2.1. We recall that if f ∈ 𝒟′(R^{d}), and ω_{j} (x, ξ) = ⟨ξ⟩^{j} for j ∈ J = N, then it follows that WF_{(B_{j} )}^{sup}(f) in Definition 2.2 is equal to the standard wave front set WF(f) in Chapter VIII in [12].

2.1. Comparisons to other types of wave-front sets. Let ω ∈ ℳ_{{s}} (R^{2d}) be moderated with respect to a weight of polynomial growth at infinity and let f ∈ D′(X). Then WF_{ℱL_{(ω)}^{q}} (f) in Definition 2.1 is the same as the wave-front set introduced in [19, Definition 3.1]. Therefore, the information on regularity in the background of wave-front sets of Fourier–Lebesgue type in Definition 2.1 might be compared to the information obtained from the classical wave-front sets, cf. Example 4.9 in [19].

Next we compare the wave-front sets introduced in Definition 2.1 to the wavefront sets in spaces of ultradistributions given in [12, 17, 22].

Let s > 1 and let X be an open subset of R^{d}. The ultradistribution f ∈ (D^{(s)} )′(X) (f ∈ (D^{{s}} )′(X)) is (s)-micro-regular ({s}-micro-regular) at (x_{0}, ξ_{0}) if there exists φ ∈ D^{(s)} (X) (φ ∈ D^{{s}} (X)) such that φ(x) = 1 in a neighborhood of x_{0} and an open cone Γ which contains ξ_{0} such that

(2.7)  |ℱ (φf)(ξ)| ≲ e^{−N|ξ|^{1/s}} , ξ ∈ Γ,

for each N > 0 (for some N > 0). The (s)-wave-front set ({s}-wave-front set) of f, WF_{(s)} (f) (WF_{{s}} (f)) is defined as the complement in X × R^{d} ∖ 0 of the set of all (x_{0}, ξ_{0}) where f is (s)-micro-regular ({s}-micro-regular), cf. [22, Definition 1.7.1].

The {s}-wave-front set WF_{{s}} (f) can be found in [17] and agrees with certain wave-front set WF_{L}(f) introduced in [12, Chapter 8.4].

Remark 2.2. Let s > 1, X ⊆ R^{d} be open, f ∈ (D^{{s}} )′(X), φ ∈ E^{{s}} (X) and

φ_{0} ∈ D^{(s)} (X) be such that φ(x) = 1 in a neighborhood suppφ_{0}. Also let Γ_{0}, Γ be open cones such that \overline{Γ_{0}} ⊆ Γ. If (2.7) holds for some N > 0, then it follows by straightforward computations, using similar arguments as in the proof of Theorem 2.1 that (2.7) is still true for some N > 0 after φ has been replaced by φ_{0}. Hence it follows that the following conditions are equivalent:

(1) (x_{0}, ξ_{0}) ∉ WF_{{s}} (f);

(2) for some φ ∈ D^{{s}} (X), such that φ(x) = 1 in a neighborhood of x_{0}, a conical neighborhood Γ of ξ and for some N > 0, it follows that (2.7) holds; (3) for some φ ∈ D^{(s)} (X), such that φ(x) = 1 in a neighborhood of x_{0}, a conical neighborhood Γ of ξ and for some N > 0, it follows that (2.7) holds.

Consequently we may always choose φ in D^{(s)} (X) in the definition of WF_{{s}} (f), when f ∈ (D^{{s}} )′(X).

Proposition 2.1. Let s > 1, and let B_{j} be the same as in (2.6) with q_{j} ∈ [1, ∞] and ω_{j} (ξ) ≡ e^{j|ξ|^{1/s}} . Then the following is true:

(1) if f ∈ (D^{{s}} )′(R^{d}), then

WF_{(B_{j} )}^{inf}(f) = ⋂_{j>0} WF_{B_{j}} (f) = WF_{{s}} (f) ⊆ WF_{(s)} (f);
(2) if f ∈ (D^{(s)} )′(R^{d}), then

WF_{(s)} (f) = ⋃_{j>0} WF_{B_{j}} (f) ⊆ WF_{(B_{j} )}^{sup}(f).

Proof. Let Γ ⊆ R^{d} ∖ 0 be a cone, ε > 0 and let r, q ∈ [1, ∞] be such that r ⩽ q. Then Hölder’s inequality implies that |f|_{ℱL_{(ω_{j} )}^{r} (Γ)} ⩽ C|f|_{ℱL_{(ω_{j+ε} )}^{q} (Γ)} , for a constant C > 0 which only depends on ε > 0 and d. A combination of this fact and (2.3) then shows that if C_{j} = ℱ L_{(ω_{j} )}^{∞} (R^{d}), then

⋂_{j>0} WF_{B_{j}} (f) = ⋂_{j>0} WF_{C_{j}}(f),  ⋃_{j>0} WF_{B_{j}} (f) = ⋃_{j>0} WF_{C_{j}}(f)
WF_{(B_{j} )}^{inf}(f) = WF_{(C_{j} )}^{inf}(f),  WF_{(B_{j} )}^{sup}(f) = WF_{(C_{j} )}^{sup}(f).

Hence we may assume that q_{j} = ∞, for every j. The result is now a straightforward consequence of the definitions. The proof is complete. □

3. Wave-front sets with respect to modulation spaces

In this section we define wave-front sets with respect to modulation spaces, and show that they coincide with wave-front sets of Fourier–Lebesgue types.

3.1. Modulation spaces. In this subsection we consider properties of modulation spaces which will be used in microlocal analysis of ultradistributions.

Let s > 1 and let ϕ ∈ S^{(s)} (R^{d}) be fixed. Then the short-time Fourier transform (STFT) of f ∈ S^{(s)} (R^{d}) with respect to the window ϕ is given by

V_{ϕ}f(x, ξ) = (2π)^{−d/2} ∫_{R^{d}} f(y) \overline{ϕ(y − x)} e^{−i⟨ξ,y⟩} dy.

The map (f, ϕ) ↦ V_{ϕ}f from S^{(s)} (R^{d})×S^{(s)} (R^{d}) to S^{(s)} (R^{2d}) extends uniquely to a continuous mapping from (S^{(s)} )′(R^{d}) × (S^{(s)} )′(R^{d}) to (S^{(s)} )′(R^{2d}) by duality.

Moreover, if ϕ ∈ S^{(s)} (R^{d}) ∖ 0 fixed and f ∈ (S^{(s)} )′(R^{d}), then

(3.1)  f ∈ S^{(s)} (R^{d}) ⟺ V_{ϕ}f ∈ S^{(s)} (R^{2d}).

We refer to [11, 27] for the proofs, as well as more details on STFT in the context of Gelfand–Shilov spaces.

Now we recall the definition of modulation spaces. Let s > 1, ω ∈ ℳ_{{s}} (R^{2d}), p, q ∈ [1, ∞], and the window ϕ ∈ S^{(s)} (R^{d})∖0 be fixed. Then the modulation space M_{(ω)}^{p,q}(R^{d}) is the set of all ultra-distributions f ∈ (S^{(s)} )′(R^{d}) such that

∥f∥_{M_{(ω)}^{p,q}} = ∥f∥_{M_{(ω)}^{p,q,ϕ}} ≡ ∥V_{ϕ}f ω∥_{L_{1}^{p,q}} < ∞.

Here ∥ · ∥_{L_{1}^{p,q}} is the norm given by

∥F ∥_{L_{1}^{p,q}} ≡ ( ∫_{R^{d}} ( ∫_{R^{d}} |F (x, ξ)|^{p}dx) ^{q/p}dξ) ^{1/q},

when F ∈ L_{loc}^{1}(R^{2d}) (with obvious interpretation when p = ∞ or q = ∞). Furthermore, the modulation space W_{(ω)}^{p,q}(R^{d}) consists of all f ∈ (S^{(s)} )′(R^{d}) such that

∥f∥_{W_{(ω)}^{p,q}} = ∥f∥_{W_{(ω)}^{p,q,ϕ}} ≡ ∥V_{ϕ}f ω∥_{L_{2}^{p,q}} < ∞,

where ∥ · ∥_{L_{2}^{p,q}} is the norm given by

∥F ∥_{L_{2}^{p,q}} ≡ ( ∫_{R^{d}} ( ∫_{R^{d}} |F (x, ξ)|^{q}dξ) ^{p/q}dx) ^{1/p},

when F ∈ L_{loc}^{1}(R^{2d}).

If ω = 1, then the notations M^{p,q} and W^{p,q} are used instead of M_{(ω)}^{p,q} and W_{(ω)}^{p,q} respectively. Moreover we set M_{(ω)}^{p} = W_{(ω)}^{p} = M_{(ω)}^{p,p} and M^{p} = W^{p} = M^{p,p}. We note that M^{p,q} are modulation spaces of classical forms, while W^{p,q} are classical forms of Wiener amalgam spaces. (See [3] concerning the terminology.)

If s > 1, p, q ∈ [1, ∞] and ω ∈ ℳ_{{s}} (R^{2d}), then one can show that the spaces ℱ L_{(ω)}^{q} (R^{d}), M_{(ω)}^{p,q}(R^{d}) and W_{(ω)}^{p,q}(R^{d}) are locally the same, in the sense that

ℱ L_{(ω)}^{q} (R^{d}) ∩ (E^{(s)} )′(R^{d}) = M_{(ω)}^{p,q}(R^{d}) ∩ (E^{(s)} )′(R^{d}) = W_{(ω)}^{p,q}(R^{d}) ∩ (E^{(s)} )′(R^{d}).

This follows by similar arguments as in [23] (and replacing the space of polynomially moderated weights 𝒫(R^{2d}) with ℳ_{{s}} (R^{2d})). Later on we extend these properties in the context of wave-front sets and recover the equalities above.

The proof of the next proposition concerning topological questions of modulation spaces, and properties of the adjoint of the short-time Fourier transform V_{ϕ}^{∗}F, can be found in [1]. Here we recall that ⟨V_{ϕ}^{∗}F, f⟩ ≡ ⟨F, V_{ϕ}f⟩, f ∈ S^{(s)} (R^{d}), when s > 1, ω ∈ ℳ_{{s}} (R^{2d}), ϕ ∈ S^{(s)} ∖ 0 and F (x, ξ) ∈ L_{(ω)}^{p,q} (R^{2d}).

Proposition 3.1. Let s > 1, ω ∈ ℳ_{{s}} (R^{2d}), p, q ∈ [1, ∞], and ϕ, ϕ_{1} ∈

S^{(s)} (R^{d}), with (ϕ, ϕ_{1})_{L^{2}} ≠ 0. Then the following is true:

(1) the operator V_{ϕ}^{∗} from S^{(s)} (R^{2d}) to S^{(s)} (R^{d}) extends uniquely to a continuous operator from L_{(ω)}^{p,q} (R^{2d}) to M_{(ω)}^{p,q}(R^{d}), and

∥V_{ϕ}^{∗}F ∥_{M_{(ω)}^{p,q}} ⩽ C∥V_{ϕ_{1}} ϕ∥_{L_{(v)}^{1}} ∥F ∥_{L_{(ω)}^{p,q}} ;

(2) M_{(ω)}^{p,q}(R^{d}) is a Banach space whose definition is independent on the choice of window ϕ ∈ S^{(s)} ∖ 0; (3) the set of windows can be extended from S^{(s)} (R^{d}) ∖ 0 to M_{(v)}^{1}(R^{d}) ∖ 0.

3.2. Wave-front sets with respect to modulation spaces. Next we define wave-front sets with respect to modulation spaces and show that they agree with corresponding wave-front sets of Fourier–Lebesgue types. More precisely, we prove that [19, Theorem 6.1] holds if the weights of polynomial growth are replaced by more general submultiplicative weights.

Let s > 1, ϕ ∈ S^{(s)} (R^{d}) ∖ 0, ω ∈ ℳ_{{s}} , Γ ⊆ R^{d} ∖ 0 be an open cone and let p, q ∈ [1, ∞]. For any f ∈ (S^{(s)} )′(R^{d}) we set

(3.2) |f|_{B(Γ)} = |f|_{B(ϕ,Γ)} ≡ ( ∫_{Γ} ( ∫_{R^{d}} |V_{ϕ}f(x, ξ)ω(x, ξ)|^{p}dx) ^{q/p}dξ) ^{1/q}

when B = M_{(ω)}^{p,q} = M_{(ω)}^{p,q}(R^{d}).

We note that |f|_{B(Γ)} = ∥f∥_{M_{(ω)}^{p,q}} when Γ = R^{d} ∖ 0 and ϕ ∈ S^{(s)} (R^{d}), and that |f|_{B(ϕ,Γ)} might attain +∞.

We also set

(3.3) |f|_{B(Γ)} = |f|_{B(ϕ,Γ)} ≡ ( ∫_{R^{d}} ( ∫_{Γ} |V_{ϕ}f(x, ξ)ω(x, ξ)|^{q}dξ) ^{p/q}dx) ^{1/p}

when B = W_{(ω)}^{p,q} = W_{(ω)}^{p,q}(R^{d})

and note that similar properties hold for this semi-norm compared to (3.2).

Let ω ∈ ℳ_{{s}} (R^{2d}), p, q ∈ [1, ∞], f ∈ (S^{(s)} )′(R^{d}), and let B = M_{(ω)}^{p,q} or B = W_{(ω)}^{p,q}. Then Θ_{B}(f), Σ_{B}(f) and the wave-front set WF_{B}(f) of f with respect to the modulation space B are defined in the same way as in Section 2, after replacing the semi-norms of Fourier–Lebesgue types in (2.1) with the semi-norms in (3.2) or (3.3) respectively.

We need the following proposition when proving that the wave-front sets of Fourier–Lebesgue and modulation space types are the same. The first part is an extension of [1, Proposition 4.2].

Proposition 3.2. Let s > 1. Then the following is true:

(1) if f ∈ (E^{(s)} )′(R^{d}) and ϕ ∈ S^{(s)} (R^{d}), then

(3.4)  |V_{ϕ}f(x, ξ)| ≲ e^{−h|x|^{1/s}} e^{ε|ξ|^{1/s}} , for some h > 0 and ε > 0;

(2) if f ∈ (E^{{s}} )′(R^{d}) and ϕ ∈ S^{(s)} (R^{d}), then (3.4) holds for every h > 0 and ε > 0;

(3) if f ∈ (D^{(s)} )′(R^{d}) and ϕ ∈ D^{(s)} (R^{d}) ∖ 0, then f ∈ (E^{(s)} )′(R^{d}), if and only if suppV_{ϕ}f ⊆ K × R^{d} for some compact set K, and then

(3.5)  |V_{ϕ}f(x, ξ)| ≲ e^{ε|ξ|^{1/s}} , for some ε > 0.

(4) if f ∈ (D^{(s)} )′(R^{d}) and ϕ ∈ D^{(s)} (R^{d}) ∖ 0, then f ∈ (E^{{s}} )′(R^{d}), if and only if suppV_{ϕ}f ⊆ K × R^{d} for some compact set K and (3.5) holds for every ε > 0.

Proof. We only prove (1) and (3). The other statements follow by similar arguments and are left for the reader.

In order to prove (1) we assume that f ∈ (E^{(s)} )′(R^{d}) and ϕ ∈ S^{(s)} (R^{d}). Also let ψ ∈ D^{(s)} (R^{d}) be such that ψ = 1 in suppf. Then for some ε, h > 0 we have

|V_{ψ} ϕ(x, ξ)| ≲ e^{−h|x|^{1/s} −2ε|ξ|^{1/s}} , and |f̂(ξ)| ≲ e^{ε|ξ|^{1/s}} .

By straightforward calculations, it follows that

|V_{ϕ}f(x, ξ)| = |(V_{ϕ}(ψf))(x, ξ)| ≲ (|V_{ψ} ϕ(x, ·)| ∗ |f̂|)(ξ)
= ∫ |V_{ψ} ϕ(x, ξ − η)||f̂(η)| dη ≲ ∫ e^{−h|x|^{1/s} −2ε|ξ−η|^{1/s}} e^{ε|η|^{1/s}} dη
⩽ e^{−h|x|^{1/s}} ∫ e^{−2ε|η|^{1/s} +2ε|ξ|^{1/s} +ε|η|^{1/s}} dη ≲ e^{−h|x|^{1/s} +2ε|ξ|^{1/s}} ,

and (1) follows.

Next we prove (3). First assume that ϕ ∈ D^{(s)} (R^{d}) ∖ 0 and f ∈ (E^{(s)} )′(R^{d}). Since both ϕ and f have compact support, it follows that supp(V_{ϕ}f) ⊆ K × R^{d}. Furthermore, |V_{ϕ}f(x, ξ)| ≲ e^{ε(|x|^{1/s} +|ξ|^{1/s} )} , for some ε > 0, in view of [1]. Since V_{ϕ}f(x, ξ) has compact support in the x-variable, it follows that

|V_{ϕ}f(x, ξ)| ≲ e^{ε|ξ|^{1/s}} .

In order to prove the reverse direction we assume that suppV_{ϕ}f ⊆ K × R^{d}, for a compact set K. Assume that suppϕ ⊆ K and choose φ ∈ D^{(s)} (R^{d}) such that suppφ ∩ 2K = ∅. Then (f, φ) = (∥ϕ∥_{L^{2}} )^{−2} (V_{ϕ}f, V_{ϕ}φ) = 0, which implies that f has compact support. Here the first equality is Moyal’s identity (cf. [9]). This implies that f has compact support and the condition f ∈ (D^{(s)} )′(R^{d}) now gives f ∈ (E^{(s)} )′(R^{d}). □

Theorem 3.1. Let s > 1, p, q ∈ [1, ∞], ω ∈ ℳ_{{s}} (R^{2d}), B = ℱ L_{(ω)}^{q} (R^{d}), and

let C = M_{(ω)}^{p,q}(R^{d}) or C = W_{(ω)}^{p,q}(R^{d}). If f ∈ (D^{(s)} )′(R^{d}), then

(3.6)  WF_{B}(f) = WF_{C}(f).

In particular, WF_{C}(f) is independent of p and ϕ ∈ S^{(s)} (R^{d}) ∖ 0 in (3.2) and (3.3).

In the proof of Theorem 3.1, the main part concerns proving that the wave-front sets of modulation types are independent of the choice of window ϕ ∈ S^{(s)} (R^{d})∖0.

Proof. We only consider the case C = M_{(ω)}^{p,q}. The case C = W_{(ω)}^{p,q} follows by similar arguments and is left for the reader. We may assume that f ∈ (E^{(s)} )′(R^{d}) and that ω(x, ξ) = ω(ξ) since the statements only concern local assertions.

In order to prove that WF_{C}(f) is independent of ϕ ∈ S^{(s)} (R^{d}) ∖ 0, we assume that ϕ, ϕ_{1} ∈ S^{(s)} (R^{d}) ∖ 0 and let | · |_{C_{1} (Γ)} be the semi-norm in (3.2) after ϕ has been replaced by ϕ_{1}. Let Γ_{1} and Γ_{2} be open cones in R^{d} such that \overline{Γ_{2}} ⊆ Γ_{1}. The asserted independence of ϕ follows if we prove that

(3.7)  |f|_{C(Γ_{2} )} ⩽ C(|f|_{C_{1} (Γ_{1} )} + 1),

for some positive constant C. Let

Ω_{1} = {(x, ξ) ; ξ ∈ Γ_{1}} ⊆ R^{2d} and Ω_{2} = ∁Ω_{1} ⊆ R^{2d},

with characteristic functions χ_{1} and χ_{2} respectively, and set

F_{k}(x, ξ) = |V_{ϕ_{1}} f(x, ξ)| ω(ξ) χ_{k}(x, ξ), k = 1, 2,

and G = |V_{ϕ}ϕ_{1}(x, ξ)v(ξ)|. Since ω is v-moderate, it follows from [9, Lemma 11.3.3] that

|V_{ϕ}f(x, ξ) ω(x, ξ)| ≲ ((F_{1} + F_{2}) ∗ G)(x, ξ),

which implies that |f|_{C(Γ_{2} )} ≲ J_{1} + J_{2}, where

J_{k} = ( ∫_{Γ_{2}} ( ∫_{R^{d}} |(F_{k} ∗ G)(x, ξ)|^{p}dx) ^{q/p}dξ) ^{1/q}, k = 1, 2.

By Young’s inequality J_{1} ⩽ ∥F_{1} ∗ G∥_{L_{1}^{p,q}} ⩽ ∥G∥_{L^{1}} ∥F_{1}∥_{L_{1}^{p,q}} = C|f|_{C_{1} (Γ_{1} )} , where C = ∥G∥_{L^{1}} = ∥V_{ϕ}ϕ_{1}(x, ξ)v(ξ)∥_{L^{1}} < ∞, in view of (3.1).

Next we consider J_{2}. For ξ ∈ Γ_{2} fixed and integrating over η ∈ ∁Γ_{1}, it follows from (2.5), Propositon 3.2 and (3.1) that for some N, k, ε > 0 and every h > 0 we have

|(F_{2} ∗ G)(x, ξ)| ≲ ∫∫_{R^{2d}} e^{−N|y|^{1/s}} e^{ε|η|^{1/s}} e^{−h(|x−y|^{1/s} +|ξ−η|^{1/s} )} v(ξ − η) dy dη
≲ ∫∫_{R^{2d}} e^{−N|y|^{1/s}} e^{ε|η|^{1/s}} e^{−h|x−y|^{1/s} −hc(|ξ|^{1/s} +|η|^{1/s} )} e^{k(|ξ|^{1/s} +|η|^{1/s} )} dy dη
≲ e^{−N_{1} |x|^{1/s}} e^{(k−hc)|ξ|^{1/s}} ∫∫_{R^{2d}} e^{−N_{1} |y|^{1/s}} e^{(k+ε−hc)|η|^{1/s}} dy dη,
≲ e^{−N_{1} |x|^{1/s}} e^{(k−hc)|ξ|^{1/s}} < ∞,

for some N_{1} > 0, provided h is chosen large enough. Therefore

J_{2} = ( ∫_{Γ_{2}} ( ∫_{R^{d}} |(F_{2} ∗ G)(x, ξ)|^{p}dx) ^{q/p}dξ) ^{1/q}

≲ ( ∫_{Γ_{2}} ( ∫_{R^{d}} ( e^{−N_{1} |x|^{1/s}} e^{(k−hc)|ξ|^{1/s}} )^{p}dx) ^{q/p}dξ) ^{1/q} < ∞.

This proves that (3.7), and hence WF_{C}(f) is independent of ϕ ∈ S^{(s)} (R^{d}) ∖ 0.

In order to prove (3.6) we assume from now on that ϕ in (3.2) is real-valued and has compact support. Let p_{0} ∈ [1, ∞] be such that p_{0} ⩽ p and set C_{0} = M_{(ω)}^{p_{0} ,q}. The result follows if we prove

Θ_{C_{0}}(f) ⊆ Θ_{B}(f) ⊆ Θ_{C}(f) when p_{0} = 1, p = ∞, (3.8)

Θ_{C}(f) ⊆ Θ_{C_{0}}(f). (3.9)

The proof of the first inclusion in (3.8) follows from the estimates

|f|_{B(Γ)} ≲ ( ∫_{Γ} |f̂(ξ)ω(ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ} |ℱ ( f ∫_{R^{d}} ϕ(· − x) dx) (ξ)ω(ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ} ( ∫_{R^{d}} |ℱ (fϕ(· − x))(ξ)ω(ξ)| dx) ^{q}dξ) ^{1/q}

= ( ∫_{Γ} ( ∫_{R^{d}} |V_{ϕ}f(x, ξ)ω(ξ)| dx) ^{q}dξ) ^{1/q} = C|f|_{C_{0} (Γ)} ,

for a positive constant C.

Next we prove the second inclusion in (3.8). We have

|f|_{C(Γ_{2} )} = ( ∫_{Γ_{2}} sup_{x∈R^{d}} |V_{ϕ}f(x, ξ)ω(x, ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ_{2}} sup_{x∈R^{d}} |(|f̂| ∗ |ℱ (ϕ(· − x))|)(ξ)ω(ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ_{2}} |(|f̂| ∗ |ϕ̂|)(ξ)ω(ξ)|^{q}dξ) ^{1/q}

≲ ( ∫_{Γ_{2}} ((|f̂· ω| ∗ |ϕ̂ · v|)(ξ))^{q}dξ) ^{1/q},

where ϕ ∈ D^{(s)} (X) is chosen such that ϕ = 1 in suppf. The second inclusion in (3.8) now follows by straightforward computations, using similar arguments as in the proof of (2.3). The details are left for the reader.

It remains to prove (3.9). Let K ⊆ R^{d} be a compact set chosen such that V_{ϕ}f(x, ξ) = 0 outside K ×R^{d}, and let p_{1} ∈ [1, ∞] be chosen such that 1/p_{1}+1/p_{0} = 1 + 1/p. By Hölder’s inequality we get

|f|_{C_{0} (Γ)} = ( ∫_{Γ} ( ∫_{R^{d}} |V_{ϕ}f(x, ξ)ω(x, ξ)|^{p_{0}} dx) ^{q/p_{0}} dξ) ^{1/q}

⩽ C_{K} ( ∫_{Γ} ( ∫_{R^{d}} |V_{ϕ}f(x, ξ)ω(x, ξ)|^{p}dx) ^{q/p}dξ) ^{1/q} = C_{K} |f|_{C(Γ)} .

This gives (3.9), and the proof is complete. □

Remark 3.1. Let s > 1, p, q ∈ [1, ∞], and ω ∈ ℳ_{{s}} (R^{2d}). If f ∈ (E^{(s)} )′(R^{d}),

then it follows from the definition of the wave-front sets that then

f ∈ B ⟺ WF_{B}(f) = ∅,

when B is equal to ℱ L_{(ω)}^{q} , M_{(ω)}^{p,q} or W_{(ω)}^{p,q}. In particular

ℱ L_{(ω)}^{q} ∩ (E^{(s)} )′(R^{d}) = M_{(ω)}^{p,q} ∩ (E^{(s)} )′(R^{d}) = W_{(ω)}^{p,q} ∩ (E^{(s)} )′(R^{d}),

by Theorem 3.1.

In particular, we recover Corollary 6.2 in [19], Theorem 2.1 and Remark 4.6 in [23].

Remark 3.2. In some situations we may relax the condition on the window function ϕ ∈ S^{(s)} (R^{d}) into ϕ ∈ S^{{s}} (R^{d}). In fact, let s > 1, ϕ ∈ S^{{s}} (R^{d}) ∖ 0 and let f ∈ (S^{{s}} )′(R^{d}). Then V_{ϕ}f makes sense as an element in (S^{{s}} )′(R^{2d}) ∩ C^{∞} (R^{2d}). Furthermore, if B = M_{(ω)}^{p,q}(R^{d}), then analogous versions of the sets Θ_{B}(f), Σ_{B}(f) and WF_{B}(f) can be defined by replacing the condition φ ∈ D^{(s)} (R^{d}) with φ ∈ D^{{s}} (R^{d}). The investigations in this section then show that Theorem 3.1 still holds after the assumptions on f and ϕ were changed in this way.

4. Discrete versions of wave-front sets

The main goal of this section is to introduce discrete wave-front sets with respect to Fourier–Lebesgue and modulation spaces, and to relate them with the corresponding wave-front sets of continuous types, given in the previous sections. To that aim, in the first part we introduce discrete analogues of Fourier–Lebesgue norms and relate them to corresponding continuous ones, given in (2.1). Finally we define discrete versions of wave-front sets and prove that they agree one to another and to the corresponding continuous ones.

4.1. Discrete semi-norms in Fourier–Lebesgue spaces. In this subsection we introduce discrete analogues of the semi-norms in (2.1), and show that these semi-norms are finite if and only if the corresponding nondiscrete semi-norms are finite. The techniques used here are similar to those in [14].

Assume that q ∈ [1, ∞], s > 1, ω ∈ ℳ_{{s}} (R^{d}), B = ℱL_{(ω)}^{q} (R^{d}), and H ⊆ R^{d} is a discrete set. Then we set

|f|_{B(H)}^{(D)} ≡ ( ∑_{ξ_{l} ∈H} |f̂(ξ_{l})ω(ξ_{l})|^{q}) ^{1/q},  f̂ ∈ C(R^{d}) ∩ (S^{(s)} )′(R^{d})

with obvious modifications when q = ∞. As in the continuous case, we may allow weight functions in ℳ_{{s}} (R^{2d}), i.e., ω = ω(x, ξ). However, again we note that the condition |f|_{B(H)}^{(D)} < ∞ is independent of x ∈ R^{d}. For the proof of the main result of this part, we need two lemmas.

We recall that by a lattice Λ we mean the set

Λ = {a_{1}e_{1} + ⋯+ a_{d}e_{d}; a_{1}, …, a_{d} ∈ Z},

where e_{1}, …, e_{d} is a basis in R^{d}.

The following Lemma was proved for distributions, cf. [14, 24, 25].

Lemma 4.1. Let s > 1, f ∈ (E^{(s)} )′(R^{d}), Γ and Γ_{0} be open cones in R^{d} ∖ 0 such that \overline{Γ_{0}} ⊆ Γ, q ∈ [1, ∞], and let Λ ⊆ R^{d} be a lattice. If |f|_{B(Γ)} is finite, then |f|_{B(Γ_{0} ∩Λ)}^{(D)} is finite.

Proof. We only prove the result for q < ∞, leaving the small modifications in the case q = ∞ for the reader. Assume that |f|_{B(Γ)} < ∞, and let H = Γ_{0} ∩ Λ. Also let φ ∈ D^{(s)} (R^{d}) be such that φ = 1 in suppf. Then

(|f|_{B(Γ_{0} ∩Λ)}^{(D)} )^{q} = ∑_{ξ_{l} ∈H} |ℱ (φf)(ξ_{l})ω(ξ_{l})|^{q}

= (2π)^{−qd/2} ∑_{ξ_{l} ∈H} |∫ φ̂(ξ_{l} − η)f̂(η)ω(ξ_{l}) dη| ^{q} ≲ (S_{1} + S_{2}),

where

S_{1} = ∑_{ξ_{l} ∈H} ( ∫_{Γ} ψ(ξ_{l} − η)F (η) dη) ^{q} , S_{2} = ∑_{ξ_{l} ∈H} ( ∫_{∁Γ} ψ(ξ_{l} − η)F (η) dη) ^{q} .

Here we set F (ξ) = |f̂(ξ)ω(ξ)| and ψ(ξ) = |φ̂(ξ)v(ξ)| as in the proof of Theorem 2.1. We need to estimate S_{1} and S_{2}. By Hölder’s inequality we get

S_{1} = ∑_{ξ_{l} ∈H} ( ∫_{Γ} ψ(ξ_{l} − η)F (η) dη) ^{q} = ∑_{ξ_{l} ∈H} ( ∫_{Γ} ψ(ξ_{l} − η)^{1/q′}(ψ(ξ_{l} − η)^{1/q}F (η)) dη) ^{q}

⩽ ∥ψ∥_{L^{1}}^{q/q′} ∑_{ξ_{l} ∈H} ∫_{Γ} ψ(ξ_{l} − η)F (η)^{q}dη ⩽ C′∫_{Γ} F (η)^{q}dη = C′|f|_{B(Γ)}^{q} ,

where

C′ = ∥ψ∥_{L^{1}}^{q/q′} sup_{η∈R^{d}} ∑_{ξ_{l} ∈H} ψ(ξ_{l} − η)

is finite by (2.4). This proves that S_{1} is finite.

It remains to prove that S_{2} is finite. We observe that

|ξ_{l} − η|^{1/s} ⩾ 2c max(|ξ_{l}|^{1/s} , |η|^{1/s} ) ⩾ c(|ξ_{l}|^{1/s} + |η|^{1/s} )

when ξ_{l} ∈ H and η ∈ ∁Γ, for some c > 0. Since f ∈ (E^{(s)} )′(R^{d}), it follows that |F | ≲ e^{N_{0} |·|^{1/s}} for a positive constant N_{0}. Furthermore, since φ ∈ D^{(s)} (R^{d}), it follows that for every N ⩾ 0 we have ψ ≲ e^{−N|·|^{1/s}} . This gives

S_{2} ≲ ∑_{ξ_{l} ∈H} ( ∫_{∁Γ} e^{−N|ξ_{l} −η|^{1/s}} e^{N_{0} |η|^{1/s}} dη) ^{q}

≲ ∑_{ξ_{l} ∈H} e^{−qNc|ξ_{l} |^{1/s}} ( ∫ e^{−(Nc−N_{0} )|η|^{1/s}} dη) ^{q} ,

where we have used the fact that ω is v-moderate. The result now follows, since the right-hand side is finite when N > N_{0}/c. The proof is complete. □

Next we prove a converse of Lemma 4.1, in the case when the lattice Λ is dense enough. Let e_{1}, …, e_{d} in R^{d} be a basis for Λ, i.e., for some x_{0} ∈ Λ we have

Λ = {x_{0} + t_{1}e_{1} + ⋯+ t_{d}e_{d}; t_{1}, …, t_{d} ∈ Z}.

A parallelepiped D, spanned by e_{1}, …, e_{d} for Λ and with corners in Λ, is called a Λ-parallelepiped. This means that for some x_{0} ∈ Λ and for some basis e_{1}, …, e_{d} for Λ we have D = {x_{0} + t_{1}e_{1} + ⋯+ t_{d}e_{d}; t_{1}, …, t_{d} ∈ [0, 1]}.

We let A(Λ) be the set of all Λ-parallelepipeds. For future references we note that if D_{1}, D_{2} ∈ A(Λ), then their volumes |D_{1}| and |D_{2}| agree, and for convenience we let ∥Λ∥ denote the common value, i.e., ∥Λ∥ = |D_{1}| = |D_{2}|.

Let Λ_{1} and Λ_{2} be lattices in R^{d} with bases e_{1}, …, e_{d} and ε_{1}, …, ε_{d} respectively. Then the pair (Λ_{1}, Λ_{2}) is called admissible lattice pair, if for some 0 < c ⩽ 2π we have ⟨e_{j} , ε_{j} ⟩ = c and ⟨e_{j} , ε_{k}⟩ = 0 when j ≠ k. If in addition c < 2π, then (Λ_{1}, Λ_{2}) is called a strongly admissible lattice pair. If instead c = 2π, then the pair (Λ_{1}, Λ_{2}) is called a weakly admissible lattice pair.

Here we note that if the lattice pair (Λ_{1}, Λ_{2}) is weakly admissible, then every choice of ϕ in (7.2.2) in [12] gives rise to a Fourier series expansion of a D_{1}-periodic distribution f, where D_{1} ∈ A(Λ_{1}). Hence, there is an ambiguity concerning the choice of ϕ for expressing f in a Fourier series. On the other hand, if (Λ_{1}, Λ_{2}) is strongly admissible and the restriction of f to the open set D_{1} has compact support, then the Fourier coefficients are defined in a canonical way. (See also (4.2) below.)

Lemma 4.2. Let s > 1, (Λ_{1}, Λ_{2}) be an admissible lattice pair, D_{1} ∈ A(Λ_{1}), and let f ∈ (E^{(s)} )′(R^{d}) be such that an open neighbourhood of its support is contained in D_{1}. Also let Γ and Γ_{0} be open cones in R^{d} such that \overline{Γ_{0}} ⊆ Γ. If |f|_{B(Γ∩Λ_{2} )}^{(D)} is finite, then |f|_{B(Γ_{0} )} is finite.

Proof. Since D_{1} contains an open neighbourhood of the support of f, we may modify Λ_{1} (and therefore D_{1}) such that the lattice pair (Λ_{1}, Λ_{2}) is strongly admissible, and such that the hypothesis still holds. From now on we therefore assume that (Λ_{1}, Λ_{2}) is strongly admissible.

We use similar arguments as in the proof of Lemma 4.1. Again we prove the result only for q < ∞. The small modifications to the case q = ∞ are left for the reader.

Assume that |f|_{B(Γ∩Λ_{2} )}^{(D)} < ∞, and let φ ∈ D^{(s)} (D_{1}^{∘}) be equal to one in the support of f, where D_{1}^{∘} denotes the interior of the set D_{1}. By expanding f = φf into a Fourier series on D_{1} we get f̂(ξ) = C ∑ _{ξ_{l} ∈Λ_{2}} φ̂(ξ−ξ_{l})f̂(ξ_{l}), where the positive constant C only depends on Λ_{2}. (Cf. e.g. (4.1) below.) We have

(|f|_{B(Γ_{0} )} )^{q} = ∫_{Γ_{0}} |f̂(ξ)ω(ξ)|^{q}dξ = C^{q} ∫_{Γ_{0}} | ∑_{ξ_{l} ∈Λ_{2}} φ̂(ξ − ξ_{l})f̂(ξ_{l})ω(ξ)|^{q}dξ ≲ S_{1} + S_{2},

where

S_{1} = ∫_{Γ_{0}} | ∑_{ξ_{l} ∈H_{1}} φ̂(ξ − ξ_{l})f̂(ξ_{l})ω(ξ)|^{q}dξ, S_{2} = ∫_{Γ_{0}} | ∑_{ξ_{l} ∈H_{2}} φ̂(ξ − ξ_{l})f̂(ξ_{l})ω(ξ)|^{q}dξ,

H_{1} = Γ ∩ Λ_{2} and H_{2} = ∁Γ ∩ Λ_{2}.

We have to estimate S_{1} and S_{2}. Let ω be moderate with respect to the weight v(·) = e^{k|·|^{1/s}} . By Minkowski’s inequality we get

S_{1} ⩽ C ∫_{Γ_{0}} ( ∑_{ξ_{l} ∈H_{1}} |φ̂(ξ − ξ_{l})v(ξ − ξ_{l})||f̂(ξ_{l})ω(ξ_{l})|) ^{q}dξ

⩽ C′∫_{Γ_{0}} ( ∑_{ξ_{l} ∈H_{1}} |φ̂(ξ − ξ_{l})v(ξ − ξ_{l})||f̂(ξ_{l})ω(ξ_{l})|^{q}) dξ ⩽ C′′ ∑_{ξ_{l} ∈H_{1}} |f̂(ξ_{l})ω(ξ_{l})|^{q},

where C′ = C sup_{ξ} ∥φ̂(ξ − ξ_{l})v(ξ − ξ_{l})∥_{l^{1} (Λ_{2} )}^{q/q′} < ∞, and C′′ = C′∥φ∥_{ℱL_{(v)}^{1}} < ∞.

This proves that S_{1} is finite when |f|_{B(Γ∩Λ_{2} )}^{(D)} < ∞.

It remains to prove that S_{2} is finite. We recall that

|ξ − ξ_{l}|^{1/s} ⩾ 2c max(|ξ|^{1/s} , |ξ_{l}|^{1/s} ) ⩾ c(|ξ|^{1/s} + |ξ_{l}|^{1/s} ) when ξ ∈ Γ_{0} and ξ_{l} ∈ H_{2},

and use the same arguments as in the proof of Lemma 4.1 to obtain

S_{2} ≲ ∫_{Γ_{0}} ( ∑_{ξ_{l} ∈H_{2}} e^{−N|ξ−ξ_{l} |^{1/s}} e^{N_{0} |ξ_{l} |^{1/s}} ) ^{q}dξ

≲ ∫_{Γ_{0}} e^{−qNc|ξ|^{1/s}} ( ∑_{ξ_{l} ∈H_{2}} e^{−(Nc−N_{0} )|ξ_{l} |^{1/s}} ) ^{q}dξ.

The result now follows, since the right-hand side is finite when N > N_{0}/c. The proof is complete. □

Theorem 4.1. Let s > 1, (Λ_{1}, Λ_{2}) be an admissible lattice pair, D_{1} ∈ A(Λ_{1}),

and let f ∈ (E^{(s)} )′(R^{d}) be such that an open neighbourhood of its support is contained in D_{1}. Also let Γ and Γ_{0} be open cones in R^{d} such that \overline{Γ_{0}} ⊆ Γ. If |f|_{B(Γ∩Λ_{2} )}^{(D)}

is finite, then |φf|_{B(Γ_{0} ∩Λ_{2} )}^{(D)} is finite for every φ ∈ D^{(s)} (X).

For the proof we recall that |φf|_{B(Γ_{0} )} is finite when f ∈ (E^{(s)} )′(R^{d}), φ ∈ D^{(s)} (X), and |f|_{B(Γ)} is finite. This follows from the proof of Theorem 2.1.

Proof. Let Γ_{1}, Γ_{2} be open cones such that \overline{Γ_{j}} ⊆ Γ_{j+1} for j = 0, 1, \overline{Γ_{2}} ⊆ Γ,

and assume that |f|_{B(Γ∩Λ_{2} )}^{(D)} < ∞. Then Lemma 4.2 shows that |f|_{B(Γ_{2} )} is finite.

Hence, Theorem 2.1 implies that |φf|_{B(Γ_{1} )} < ∞. This gives |φf|_{B(Γ_{0} ∩Λ_{2} )}^{(D)} < ∞, in view of Lemma 4.1. The proof is complete. □

4.2. Gabor pairs. In this subsection we recall in Definition 4.1 the notion of Gabor pairs, which are later on used in the definition of discrete version of wavefront sets with respect to modulation spaces. We refer to [14] for an explanation that conditions in Definition 4.1 are quite general.

By Definition 4.1 it follows that our analysis can be applied to the most general classes of non-quasianalytic ultradistributions, and it also points out the role of Beurling–Domar weights in definitions of ℱ L_{(ω)}^{q} (R^{d}) and M_{(ω)}^{p,q}(R^{d}), cf. [2,10,11]. On the other hand, a larger class of quasianalytic ultradistributions can not be treated by the technique given here, since the corresponding test function spaces do not contain smooth functions of compact support.

Assume that e_{1}, …, e_{d} is a basis for the lattice Λ_{1}, and that (Λ_{1}, Λ_{2}) is a weakly admissible lattice pair. If f ∈ L_{loc}^{2} is periodic with respect to Λ_{1}, and D is the parallelepiped, spanned by {e_{1}, …, e_{d}}, then we may make Fourier expansion of f as

(4.1)  f(x) = ∑_{ξ_{l} ∈Γ_{2}} c_{l}e^{i⟨x,ξ_{l} ⟩} ,  x ∈ R^{d}

(with convergence in L_{loc}^{2}), where the coefficients c_{l} are given by

(4.2)  c_{l} = ∫_{Δ} f(y)e^{−i⟨y,ξ_{l} ⟩} dy.

Here y = y_{1}e_{1} + ⋯+ y_{d}e_{d}, dy = dy_{1} ⋯ dy_{d}, and Δ = [0, 1]^{d}. For nonperiodic functions and distributions we instead make Gabor expansions. Because of the support properties of the involved Gabor atoms and their duals, we are usually forced to change the assumption on the involved lattice pairs. More precisely, instead of assuming that (Λ_{1}, Λ_{2}) should be a weakly admissible lattice pair, we assume from now on that (Λ_{1}, Λ_{2}) is a strongly admissible lattice pair, with Λ_{1} = {x_{j} }_{j∈J} and Λ_{2} = {ξ_{l}}_{l∈J} . Also let s > 1 and

(4.3) ϕ, ψ ∈ D^{(s)} (R^{d}), ϕ_{j,l} (x) = ϕ(x − x_{j})e^{i⟨x,ξ_{l} ⟩} , ψ_{j,l} (x) = ψ(x − x_{j})e^{i⟨x,ξ_{l} ⟩}

be such that {ϕ_{j,l} }_{j,l∈J} and {ψ_{j,l} }_{j,l∈J} are dual Gabor frames (see [6, 9] for the definition and basic properties of Gabor frames and their duals). If f ∈ (S^{(s)} )′(R^{d}) then

(4.4)  f = ∑_{j,l∈J} c_{j,l} ϕ_{j,l} ,

where

(4.5)  c_{j,l} = C_{ϕ,ψ} · (f, ψ_{j,l} )

and the constant C_{ϕ,ψ} depends on the frames only. Here (·, ·) denotes the unique extension of the L^{2}-form on S^{(s)} (R^{d}) × S^{(s)} (R^{d}) into (S^{(s)} )′(R^{d}) × (S^{(s)} )′(R^{d}).

Note that the convergence is in (S^{(s)} )′(R^{d}) due to Proposition 3.2.

Definition 4.1. Assume that ε ∈ (0, 1], {x_{j} }_{j∈J} = Λ_{1} ⊆ R^{d} and {ξ_{l}}_{l∈J} = Λ_{2} ⊆ R^{d} are lattices and let Λ_{1}(ε) = εΛ_{1}. Also let ϕ, ψ ∈ C_{0}^{∞} (R^{d}) be nonnegative, and set

ϕ^{ε} = ϕ(·/ε), ϕ_{j,l}^{ε} = ϕ^{ε}(· − εx_{j})e^{i⟨·,ξ_{l} ⟩},
ψ^{ε} = ψ(·/ε), ψ_{j,l}^{ε} = ψ^{ε}(· − εx_{j})e^{i⟨·,ξ_{l} ⟩}

when εx_{j} ∈ Λ_{1}(ε) (i.e., x_{j} ∈ Λ_{1}) and ξ_{l} ∈ Λ_{2}. Then the pair

(4.6)  ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} )

is called a Gabor pair with respect to the lattices Λ_{1} and Λ_{2} if for each ε ∈ (0, 1], the sets {ϕ_{j,l}^{ε} }_{j,l∈J} and {ψ_{j,l}^{ε} }_{j,l∈J} are dual Gabor frames.

By Definition 4.1 and Chapters 5–13 in [9] it follows that if f ∈ (S^{(s)} )′(R^{d}) and if ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ) is a Gabor pair, then

(4.4)′  f = ∑_{j,l∈J} c_{j,l} (ε)ϕ_{j,l}^{ε}

in (S^{(s)} )′(R^{d}), for every ε ∈ (0, 1], where

(4.5)′  c_{j,l} (ε) = (f, ψ_{j,l}^{ε} ).

We remark that if the pair in (4.6) is a Gabor pair, then it follows from the investigations in [9] that the lattice pair (Λ_{1}, Λ_{2}) in Definition 4.1 is strongly admissible.

The following proposition explains that any pair of dual Gabor frames satisfying a mild additional condition, generates a Gabor pair. The proof can be found in [14].

Proposition 4.1. Let ϕ, ψ ∈ C_{0}^{∞} (R^{d}) be nonnegative functions and let ϕ_{j,l}

and ψ_{j,l} be given by (4.3). Also, let Λ_{1} and Λ_{2} be the same as in Definition 4.1. If {ϕ_{j,l} }_{j,l∈J} and {ψ_{j,l} }_{j,l∈J} are dual Gabor frames such that

(4.7)  ∑_{x_{j} ∈Λ_{1}} ϕ(· − x_{j})ψ(· − x_{j}) = ∥Λ_{1}∥^{−1} ,

holds, then (4.6) is a Gabor pair.

Remark 4.1. If ϕ = ψ, then (4.7) describes the tight frame property of the corresponding Gabor frame, cf. [9, Theorem 6.4.1].

Remark 4.2. Let p, q ∈ [1, ∞], ω ∈ ℳ_{{s}} (R^{2d}), and f ∈ (E^{(s)} )′(R^{d}). If

({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ) is a Gabor pair such that (4.4) and (4.5) hold, then it follows that f ∈ M_{(ω)}^{p,q}(R^{d}) if and only if

∥f∥_{[ε]} ≡ ( ∑_{l∈J} ( ∑_{j∈J} |c_{j,l} (ε)ω(εx_{j} , ξ_{j})|^{p})^{q/p} )^{1/q}

is finite for every ε ∈ (0, 1]. Furthermore, for every ε ∈ (0, 1], the norm f ↦ ∥f∥_{[ε]} is equivalent to the modulation space norm (1.3) (cf. [2, 4, 5, 9].)

4.3. Discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We start with two definitions.

Definition 4.2. Let s > 1, q ∈ [1, ∞], f ∈ (S^{(s)} )′(R^{d}), X be an open subset

of R^{d}, (x_{0}, ξ_{0}) ∈ X × (R^{d} ∖ 0), ω ∈ ℳ_{{s}} (R^{d}) and B = ℱ L_{(ω)}^{q} . The point (x_{0}, ξ_{0}) is called discretely regular with respect to B if

|φf|_{B(Γ∩Λ_{2} )}^{(D)} < ∞,

for some choice of strongly admissible lattice pair (Λ_{1}, Λ_{2}) such that x_{0} ∉ Λ_{1}, an open conical neighborhood Γ of ξ_{0}, and some choice of φ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0.

For the definition of discrete wave-front sets of modulation spaces, we consider Gabor pairs ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ), and let J_{x_{0}} (ε) = J_{x_{0}} (ε, ϕ, ψ) = J_{x_{0}} (ε, ϕ, ψ, Λ_{1}) be the set of all j ∈ J such that x_{0} ∈ suppϕ_{j,l}^{ε} or x_{0} ∈ suppψ_{j,l}^{ε} .

Definition 4.3. Let s > 1, p, q ∈ [1, ∞], f ∈ (S^{(s)} )′(R^{d}), X be an open

subset of R^{d}, (x_{0}, ξ_{0}) ∈ X × (R^{d} ∖ 0), ω ∈ ℳ_{{s}} (R^{2d}) and C = M_{(ω)}^{p,q}(R^{d}). Also let Λ_{1}, Λ_{2} ⊆ R^{d} be lattices such that x_{0} ∉ Λ_{1}, ϕ, ψ ∈ D^{(s)} (R^{d}) be such that ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ) is a Gabor pair (with respect to Λ_{1} and Λ_{2}), and let c_{j,l} (ε) be the same as in (4.5)′. The point (x_{0}, ξ_{0}) is called discretely regular with respect to C if

( ∑_{ξ_{l} ∈Γ∩Λ_{2}} ( ∑_{j∈J_{x_{0}} (ε)} |c_{j,l} (ε)ω(ξ_{l})|^{p})^{q/p} )^{1/q} < ∞,

for some ε ∈ (0, 1], an open conical neighborhood Γ of ξ_{0}, and for some φ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0.

The discrete wave-front set of f with respect to C (B), denoted DF_{C}(f) (DF_{B}(f)), consists of all (x_{0}, ξ_{0}) ∈ X × (R^{d} ∖ 0), where f is not discretely regular with respect to C (B).

Roughly speaking, (x_{0}, ξ_{0}) ∈ DF_{C}(f) means that f is not locally in C, in the direction ξ_{0}. The following result shows that our wave-front sets coincide.

Theorem 4.2. Let s > 1, X ⊆ R^{d} be open and let f ∈ (D^{(s)} )′(X). Then

(4.8)  WF_{B}(f) = WF_{C}(f) = DF_{B}(f) = DF_{C}(f).

Proof. By Theorem 3.1 and Lemmas 4.1 and 4.2, it follows that the first two equalities in (4.8) hold. The result therefore follows if we prove that DF_{B}(f) = DF_{C}(f).

First assume that (x_{0}, ξ_{0}) ∉ DF_{B}(f), and choose φ ∈ D^{(s)} (X), an open neighbourhood X_{0} ⊂ \overline{X_{0}} ⊂ X of x_{0} and conical neighbourhoods Γ, Γ_{0} of ξ_{0} such that

• \overline{Γ_{0}} ⊆ Γ, φ(x) = 1 when x ∈ X_{0},
• |φ f|_{B(H)}^{(D)} < ∞, when H = Λ_{2} ∩ Γ.

Now let ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ) be a Gabor pair and choose ε ∈ (0, 1] such that suppϕ_{j,l}^{ε} and suppψ_{j,l}^{ε} are contained in X_{0} when x_{0} ∈ suppϕ_{j,l}^{ε} and x_{0} ∈ suppψ_{j,l}^{ε} . Since

c_{j,l} (ε) = C(f, ψ_{j,l}^{ε} )_{L^{2} (R^{d})} = ℱ (f ψ(·/ε − x_{j}))(ξ_{l}),

it follows from these support properties that if H_{0} = Λ_{2} ∩ Γ_{0}, then

(4.9)  ( ∑_{ξ_{l} ∈H_{0}} |ℱ (f ψ(·/ε − x_{j}))(ξ_{l})ω(ξ_{l})|^{q}) ^{1/q} = |f ψ(·/ε − x_{j})|_{B(H_{0} )}^{(D)}
= |f φψ(·/ε − x_{j})|_{B(H_{0} )}^{(D)} ,

when j ∈ J_{x_{0}} (ε). Hence, by combining Theorem 4.1 with the facts that J_{x_{0}} (ε) is finite and |φ f|_{B(H)}^{(D)} < ∞, it follows that the expressions in (4.9) are finite and

( ∑_{ξ_{l} ∈H_{0}} ( ∑_{j∈J_{x_{0}} (ε)} |ℱ (f ψ(·/ε − x_{j}))(ξ_{l})ω(ξ_{l})|^{p}) ^{q/p}) ^{1/q} < ∞.

This implies that (x_{0}, ξ_{0}) ∉ DF_{C}(f), and we have proved that DF_{C}(f) ⊆ DF_{B}(f).

In order to prove the reverse inclusion we assume that (x_{0}, ξ_{0}) ∉ DF_{C}(f), and we choose ε ∈ (0, 1], Gabor pair ({ϕ_{j,l} }_{j,l∈J} , {ψ_{j,l} }_{j,l∈J} ) and conical neighbourhoods Γ, Γ_{0} of ξ_{0} such that \overline{Γ_{0}} ⊆ Γ and

(4.10)  ( ∑_{ξ_{l} ∈H} ( ∑_{j∈J_{x_{0}} (ε)} |ℱ (f ψ(·/ε − x_{j}))(ξ_{l})ω(ξ_{l})|^{p}) ^{q/p}) ^{1/q} < ∞,

when H = Λ_{2} ∩ Γ. Also choose φ, κ ∈ D^{(s)} (X) such that φ(x_{0}) ≠ 0 and

κ(x) ∑_{j∈J_{x_{0}} (ε)} ψ(x/ε − x_{j}) = 1, when x ∈ suppφ.

Since J_{x_{0}} (ε) is finite, Hölder’s inequality gives

|φ f|_{B(H_{0} )}^{(D)} = | ∑_{j∈J_{x_{0}} (ε)} (φκ) (f ψ(·/ε − x_{j}))|_{B(H_{0} )}^{(D)}

⩽ ( ∑_{ξ_{l} ∈H_{0}} ( ∑_{j∈J_{x_{0}} (ε)} |ℱ ((φκ)f ψ(·/ε − x_{j}))(ξ_{l})ω(ξ_{l})|) ^{q}) ^{1/q}

≲ ( ∑_{ξ_{l} ∈H_{0}} ( ∑_{j∈J_{x_{0}} (ε)} |ℱ ((φκ)f ψ(·/ε − x_{j}))(ξ_{l})ω(ξ_{l})|^{p}) ^{q/p}) ^{1/q},

where H_{0} = Λ_{2} ∩ Γ_{0}. By Theorem 4.1 and (4.10) it now follows that the righthand side in the last estimates is finite. Hence, |φ f|_{B(H_{0} )}^{(D)} < ∞, which shows that (x_{0}, ξ_{0}) ∉ DF_{B}(f), and we have proved that DF_{B}(f) ⊆ DF_{C}(f). The proof is complete. □

We may define discrete versions, DF_{(B_{j} )}^{inf} (f) and DF_{(B_{j} )}^{inf} (f), of the wave-front sets WF_{(B_{j} )}^{inf} (f) and WF_{(B_{j} )}^{inf} (f) of sequence types, as it is done in [14]. Then it follows that Theorem 4.2 can be extended to involve such wave-front sets. Hence Proposition 2.1 is still true, after WF_{(B_{j} )}^{inf} (f) and WF_{(B_{j} )}^{inf} (f) are replaced by DF_{(B_{j} )}^{inf} (f) and DF_{(B_{j} )}^{inf} (f), respectively. In particular we obtain the following discrete interpretation of the wave-front set WF_{{s}} (f).

Corollary 4.1. Let q ∈ [1, ∞], s > 1, and let ω_{k}(ξ) ≡ e^{k|ξ|^{1/s}} , ξ ∈ R^{d}, k > 0.

If f ∈ (D^{(s)} )′(R^{d}), then

⋂_{k>0} DF_{ℱL_{(ω_{k} )}^{q}} (f) = WF_{{s}} (f).

We remark that a discrete analogue of WF_{{s}} (f) also can be introduced in a similar way as in [24, 25]. Let us denote this set by WF_{s,T} (f), and refer to it as toroidal s-wave-front set. It can be proved that

WF_{s,T} (f) = T^{d} × Z^{d} ∩ WF_{{s}} (f),

where T^{d} is the torus in R^{d}.

A significant difference between the toroidal wave-front sets and our discrete wave-front sets lies in the fact that WF_{s,T} (f) only informs about the rational directions for the propagation of singularities of f at a certain point, while DF(f) = WF(f) takes care of all directional for f to that point, we refer to [14] for an example.

Acknowledgement. This work is supported by project no. 174024, MESTD of Republic of Serbia.

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Department of Mathematics and Informatics (Received 28 02 2012) University of Novi Sad Novi Sad Serbia

 stevan.pilipovic@dmi.uns.ac.rs
   nenad.teofanov@dmi.uns.ac.rs

Department of Computer Science, Mathematics and Physics Linnæus University Växjö Sweden

 karoline.johansson@lnu.se
       joachim.toft@lnu.se