Micro-local analysis in Fourier Lebesgue and modulation spaces.

Transcription

1 Micro-local analysis in Fourier Lebesgue and modulation spaces. Stevan Pilipović University of Novi Sad Nagoya, September 30, 2009 (Novi Sad) Nagoya, September 30, / 52

2 Introduction We introduce wave-front sets of (weighted) Fourier Lebesgue spaces, a familly of wave-front sets which contains the wave-front sets of Sobolev type, introduced by Hörmander as well as the classical wave-front sets with respect to smoothness as special cases. Roughly speaking, the wave-front set WF F L q (f ) of a distribution f on R d with respect to the (weighted) Fourier Lebesgue space F L q (Rd ) = { f S (R d ) ; f (ξ)ω(ξ) L q (R d ) } (1) with exponent q [1, ] and weight function ω L loc (Rd ), consists of all pairs (x 0, ξ 0 ) in R d (R d \ 0) such that no localizations of the distribution at x 0 belongs to F L q (Rd ) in the direction ξ 0. (Novi Sad) Nagoya, September 30, / 52

3 Preliminaries ω is called v-moderate if ω(x + y) Cω(x)v(y) (2) P(R d ) is the set of all polynomially moderated functions on R d. If ω(x, ξ) P(R 2d ) is constant with respect to the x-variable (ξ-variable), then we sometimes write ω(ξ) (ω(x)) instead if ω(x, ξ). In this case we consider ω as an element in P(R 2d ) or in P(R d ) depending on the situation. P 0 (R d ) is the set of all ω P(R d ) C (R d ) such that α ω/ω L for all multi-indices α. It follows that for each ω P(R d ), there is an element ω 0 P 0 (R d ) such that for some constant C. C 1 ω 0 ω Cω 0, (3) (Novi Sad) Nagoya, September 30, / 52

4 Preliminaries For each non-negative real number ρ, we also let P ρ (R 2d ) be the set of all ω(x, ξ) in P 0 (R 2d ) such that ξ ρ β α x β ξ ω(x, ξ) ω(x, ξ) L (R 2d ). The short-time Fourier transform of f S (R d ) with respect to ϕ is defined by (V ϕ f )(x, ξ) = F (f ϕ( x))(ξ). We note that the left-hand side makes sense, since it is the partial Fourier transform of the tempered distribution with respect to the y-variable. F (x, y) = (f ϕ)(y, y x) (Novi Sad) Nagoya, September 30, / 52

5 Modulation and Fourier-Lebesgue spaces Assume that ω P(R 2d ), p, q [1, ], and that χ S (R d ) \ 0. Then the modulation space M p,q (Rd ) is the set of all f S (R d ) such that f M p,q = f M p,q,χ ( ( ) q/p ) 1/q (4) V χ f (x, ξ)ω(x, ξ) p dx dξ < (with obvious interpretation when p = or q = ). The (weighted) Fourier-Lebesgue space F L q (Rd ) is the Banach space which consists of all f S (R d ) such that f F L q = f F L q,x f ω(x, ) L q <. (5) (Novi Sad) Nagoya, September 30, / 52

6 Modulation and Fourier-Lebesgue spaces Theorem Assume that p, q, p j, q j [1, ] for j = 1, 2, and that ω, ω 1, ω 2, v P(R 2d ) are such that ω is v-moderate. Then the following is true: (1) if χ M(v) 1 (Rd ) \ 0, then f M p,q (Rd ) if and only if (4) holds, i. e. M p,q (Rd ) is independent of the choice of χ. Moreover, M p,q is a Banach space under the norm in (4), and different choices of χ give rise to equivalent norms; (2) if p 1 p 2, q 1 q 2 and ω 2 ω 1, then S (R d ) M p 1,q 1 (ω 1 ) (Rd ) M p 2,q 2 (ω 2 ) (Rd ) S (R d ); (Novi Sad) Nagoya, September 30, / 52

7 Theorem (3) the sesqui-linear form (, ) on S extends to a continuous map from M p,q (Rd ) M p,q (1/ω) (Rd ) to C. On the other hand, if a = sup (a, b), where the supremum is taken over all b M p,q (1/ω) (Rd ) such that b M p,q 1, then and M p,q are (1/ω) equivalent norms; (4) if p, q <, then S (R d ) is dense in M p,q (Rd ). The dual space of M p,q (Rd ) can be identified with M p,q (1/ω) (Rd ), through the form (, ) L 2. Moreover, S (R d ) is weakly dense in M (Rd ). (Novi Sad) Nagoya, September 30, / 52

8 Modulation and Fourier-Lebesgue spaces Locally, the spaces F L q that and Mp,q F L q E = M p,q E p,q (and W ) are the same, in the sense (= W p,q E ). We extend these properties in context of the new type of wave-front sets, based on modulation and Fourier-Lebesgue spaces. (Novi Sad) Nagoya, September 30, / 52

9 Modulation and Fourier-Lebesgue spaces Feichtinger and Gröchenig introduced and established the theory on coorbit spaces, which in particular involve modulation spaces. If ω P(R 4d ) and χ S (R 2d ), then an example of a coorbit space which is not a modulation space is M (R 2d ) which consists of all a S (R 2d ) such that ( a M e sup H,a,ω (ξ, z) ) dz ξ R d is finite, where H,a,ω (ξ, z) = sup V χ a(x, ζ, ξ, z)ω(x, ζ, ξ, z). x,ζ R d We note that the definition of M is similar to the definition of the modulation spaces M,1 (R2d ) and M, (R 2d ), since a M,1 = H,a,ω L 1, and a M, = H,a,ω L. (Novi Sad) Nagoya, September 30, / 52

10 Pseudodifferential operators Assume that a S (R 2d ), and that t R is fixed. Then the pseudo-differential operator a t (x, D), defined by the formula (a t (x, D)f )(x) = (Op t (a)f )(x) = (2π) m a((1 t)x + ty, ξ)f (y)e i x y,ξ dydξ. (6) is a linear and continuous operator on S (R d ). For general a S (R 2d ), the pseudo-differential operator a t (x, D) is defined as the continuous operator from S (R d ) to S (R d ) with distribution kernel (Novi Sad) Nagoya, September 30, / 52

11 Pseudodifferential operators K t,a (x, y) = (2π) m/2 (F 1 2 a)((1 t)x + ty, y x), (7) Here F 2 F is the partial Fourier transform of F (x, y) S (R 2d ) with respect to the y-variable. This definition makes sense, since the mappings F 2 and F (x, y) F ((1 t)x + ty, y x) are homeomorphisms on S (R 2d ). Note that this definition of a t (x, D) agrees with the Kohn-Nirenberg representation a(x, D) when t = 0. (Novi Sad) Nagoya, September 30, / 52

12 Pseudodifferential operators Any linear and continuous operator T from S (R d ) to S (R d ) has a distribution kernel K in S (R 2d ) in view of kernel theorem of Schwartz. By Fourier s inversion formula we may then find a unique a S (R 2d ) such that (7) is fulfilled with K = K t,a. Consequently, for every fixed t R, there is a one to one correspondence between linear and continuous operators from S (R d ) to S (R d ), and Op t (S (R 2d )), the set of all a t (x, D) such that a S (R 2d ). (Novi Sad) Nagoya, September 30, / 52

13 Pseudodifferential operators In particular, if a S (R 2d ) and s, t R, then there is a unique b S (R 2d ) such that a s (x, D) = b t (x, D). By straight-forward applications of Fourier s inversion formula, it follows that a s (x, D) = b t (x, D) b(x, ξ) = e i(t s) Dx,D ξ a(x, ξ) (8) (Novi Sad) Nagoya, September 30, / 52

14 Pseudodifferential operators Assume that ρ, r R are fixed. Then S r ρ,0 (R2d ) is the set of all a C (R 2d ) such that for each pairs of multi-indices α and β, there is a constant C α,β such that We usually assume that 0 < ρ 1. α x β ζ a(x, ζ) C α,β ζ r ρ β (Novi Sad) Nagoya, September 30, / 52

15 Pseudodifferential operators More generally, let g = g ρ be the Riemannian metric on R 2d, defined by the formula ( gρ )(y,η) (x, ζ) = x 2 + η 2ρ ζ 2 and assume that ω 0 P ρ (R 2d ). Then we recall that S(ω 0, g ρ ) consists of all a C (R 2d ) such that α x β ζ a(x, ζ) C α,βω 0 (x, ζ) ζ ρ β. We note that S(ω 0, g ρ ) = S r ρ,0 (R2d ) when ω 0 (x, ζ) = ζ r. (Novi Sad) Nagoya, September 30, / 52

16 Pseudodifferential operators Theorem Assume that p, q, p j, q j [1, ] for j = 1, 2, satisfy 1/p 1 1/p 2 = 1/q 1 1/q 2 = 1 1/p 1/q, q p 2, q 2 p. Also assume that ω P(R 2d R 2d ) and ω 1, ω 2 P(R 2d ) satisfy ω 2 (x, ζ + ξ) Cω(x, ζ, ξ, z) (9) ω 1 (x + z, ζ) for some constant C. If a M p,q (R2d ), then a(x, D) from S (R d ) to S (R d ) extends uniquely to a continuous mapping from M p 1,q 1 (ω 1 ) (Rd ) to M p 2,q 2 (ω 2 ) (Rd ). (Novi Sad) Nagoya, September 30, / 52

17 Pseudodifferential operators when ρ R and s R 4. Definition ω s,ρ (x, ζ, ξ, z) = ω(x, ζ, ξ, z) x s 4 ξ s 3 ζ ρs 2 z s 1, (10) Assume that s R 4 is such that s 2 0, ρ R, ω P(R 4d ), and that ω s,ρ is given by (10). Then the symbol class s,ρ (R2d ) is the set of all a S (R 2d ) which satisfy α ζ a M,1 (ω u(s,α),ρ ) (R2d ), u(s, α) = (s 1, α, s 3, s 4 ),, for each multi-indices α such that α 2s 2. (Novi Sad) Nagoya, September 30, / 52

18 Pseudodifferential operators Assume that ω P ρ and ω 0 (x, ζ) = ω(x, ζ, 0, 0). Then it follows from the embedding properties between modulation spaces and Besov spaces, and identification properties of modulation spaces with different weights that (Novi Sad) Nagoya, September 30, / 52

19 Pseudodifferential operators a S(ω0 1, g ρ) ω 0 a Sρ,0 0 x s 4 a s 1,s 2,s 3 0 s,ρ (11) α x β ξ a(x, ζ) Cω 0(x, ζ) 1 ζ ρ β. this connects the s,ρ operators. spaces to classical theory of pseudo-differential (Novi Sad) Nagoya, September 30, / 52

20 Pseudodifferential operators a S(ω 1, g ρ ), where ω P(R 2d ) and ρ (0, 1]. Assume that ω P(R 2d ), ρ (0, 1] and a S(ω 1, g ρ ). For each open cone Γ R d \ 0, open set U R d and real number R > 0, let Ω U,Γ,R { (x, ξ) ; x U, ξ Γ, ξ > R }. Also let Ξ U,Γ,R,ρ be the set of all c S 0 ρ,0 (R2d ) such that c = 1 on Ω U,Γ,R. (Novi Sad) Nagoya, September 30, / 52

21 Pseudodifferential operators The pair (x 0, ξ 0 ) R d (R d \ 0) is called non-characteristic for a S(ω 1, g ρ ) (with respect to ω P ρ (R 2d )), if there is a conical neighbourhood Γ of ξ 0, a neighbourhood U of x 0, a real number R > 0, and elements b S(ω, g ρ ), c Ξ U,Γ,R,ρ and h S ρ ρ,0 (R2d ) and such that b(x, ξ)a(x, ξ) = c(x, ξ) + h(x, ξ), (x, ξ) R 2d. The pair (x 0, ξ 0 ) R d (R d \ 0) is called characteristic for a (with respect to ω P ρ (R 2d )), if it is not non-characteristic for a with respect to ω P ρ (R 2d ). The set of characteristic pairs for a is denoted by Char(a) = Char (a). (Novi Sad) Nagoya, September 30, / 52

22 Wave front sets Assume that ϕ S (R d ) \ 0, ω P(R 2d ), Γ R d \ 0 is an open cone and p, q [1, ] are fixed. For any f S (R d ), let ( ( q/p ) 1/q V ϕ f (x, ξ)ω(x, ξ) dx) p dξ f M p,q,γ = f M p,q,γ,ϕ Γ and f F L q,γ,x ( Γ f (ξ)ω(x, ξ) q dξ) 1/q (12) (Novi Sad) Nagoya, September 30, / 52

23 Wave front sets We note that M p,q,γ and F L q,γ define semi-norms on S which,x might attain the value +. Since ω is v-moderate for some v P(R 2d ), it follows that different x R d gives rise to equivalent semi-norms f F L q,γ. Furthermore, if Γ = R d \ 0, then f,x M p,q,γ and f F L q,γ agree,x with the modulation space norm f M p,q respectively of f. f F L q,γ,x and Fourier Lebesgue norm (Novi Sad) Nagoya, September 30, / 52

24 Wave front sets We let Θ M p,q (f ) = Θ M p,q,ϕ(f ) and Θ F L q (f ) be the sets of all ξ R d \ 0 < respectively, for some such that f M p,q,γ,ϕ < and f F L q,γ,x Γ = Γ ξ. Here recall that Γ ξ is an open cone in R d \ 0 which contains ξ. We also let Σ M p,q,ϕ (f ) and Σ F L q (f ) be the complements of Θ M p,q,ϕ and Θ F L q (f ) respectively in R d \ 0. Then Θ M p,q,ϕ(f ) and Σ M p,q,ϕ open respectively closed subsets with respect to the standard topology in R d \ 0, and Θ F L q (f ) and Σ F L q x R d in (12). We have now the following result. (f ) are independent of the choice of (f ) (f ) are (Novi Sad) Nagoya, September 30, / 52

25 Wave front sets Theorem Assume that p, q [1, ], ϕ C0 (Rd ) \ 0, χ S (R d ), and that ω P(R 2d ). Also assume that f E (R d ). Then and Θ M p,q,ϕ(f ) = Θ F L q (f ), Σ M p,q,ϕ(f ) = Σ F L q (f ), (13) Σ M p,q,ϕ (χf ) Σ M p,q,ϕ(f ), Σ F L q (χf ) Σ F L q (f ). (14) (Novi Sad) Nagoya, September 30, / 52

26 Wave front sets Let ϕ S (R d ) \ 0 be fixed, and assume that ω P(R 2d ) and p, q [1, ]. Then the Wiener amalgam related space W p,q (Rd ) consists of all f S (R d ) such that f W p,q = f W p,q,ϕ ( ( F (f τ x ϕ)(ξ)ω(x, ξ) q dξ) p/q dx ) 1/p <. We note that W p,q = F Mq,p (eω) when ω(x, ξ) = ω( ξ, x). Hence, most of the properties for modulation spaces carry over to spaces of the form W p,q. (Novi Sad) Nagoya, September 30, / 52

27 Wave front sets For any open cone Γ in R d, ϕ C0 (Rd ) and f D (R d ), we may now, in a way similar as for modulation spaces and Fourier-Lebesgue spaces, define ( ( ) p/q ) 1/p f W p,q,γ = f W p,q,γ,ϕ V ϕ f (x, ξ)ω(x, ξ) q dξ dx (with obvious interpretation when p = or q = ). We let Θ W p,q(f ) = Θ W p,q,ϕ(f ) be the sets of all ξ R d \ 0 such that f W p,q,γ,ϕ < for some Γ = Γ ξ. We also let Σ W p,q,ϕ complements of Θ W p,q,ϕ Γ (f ) be the (f ) in R d \ 0. It is now straight-forward to check that the formulas in (13) and (14) hold when M p,q is replaced by W p,q. (Novi Sad) Nagoya, September 30, / 52

28 Wave front sets Corollary Assume that p, q [1, ], f D (R d ), ϕ C 0 (Rd ) \ 0, x 0, y 0 R d, ξ 0 R d \ 0 and that ω P(R 2d ). Also let ω 0 (ξ) = ω(y 0, ξ). Then the following conditions are equivalent: 1 there exists an open cone Γ = Γ ξ0 and χ C0 (Rd ) such that χ(x 0 ) 0, and χf M p,q,γ,ϕ < (i. e. ξ 0 Θ M p,q(χf )); 2 there exists an open cone Γ = Γ ξ0 and χ C0 (Rd ) such that χ(x 0 ) 0, and χf W p,q,γ,ϕ < (i. e. ξ 0 Θ W p,q(χf )); 3 there exists an open cone Γ = Γ ξ0 and χ C0 (Rd ) such that χ(x 0 ) 0, and χf F L q < (i. e. ξ 0 Θ F L q (χf )); 4 there exists an open cone Γ = Γ ξ0 and χ C0 (Rd ) such that χ(x 0 ) 0, and χf F L q < (i. e. ξ 0 Θ (ω 0 ) F L q (χf )). (ω 0 ) (Novi Sad) Nagoya, September 30, / 52

29 Wave front sets Definition Assume that p, q [1, ], f D (R d ), ω 0 P(R d ) and ω P(R 2d ) are such that ω 0 (ξ) = ω(y 0, ξ) for some y 0 R d. The wave-front set WF F L q (f ) WF (ω 0 ) F L q (f ) WF M p,q (f ) WF W p,q(f ) with respect to F L q (ω 0 ) (Rd ), M p,q (Rd ) or W p,q (Rd ) consists of all pairs (x 0, ξ 0 ) in R d (R d \ 0) such that (x 0, ξ 0 ) Σ F L q (χf ) = Σ (ω 0 ) F L q (χf ) = Σ M p,q,ϕ holds for each χ C 0 (Rd ) such that χ(x 0 ) 0. (χf ) = Σ W p,q,ϕ(χf ) (Novi Sad) Nagoya, September 30, / 52

30 Wave front sets We note that WF F L q (f ) in Definition 7 is independent of the choice of (ω 0 ) y 0 R d and p [1, ], since ω is v-moderate for some polynomial v, and M p 1,q 1 (ω 1 ) M p 2,q 2 (ω 2 ) when p 1 p 2, q 1 q 2 and ω 2 Cω 1. Furthermore, from these arguments it also follows that when p 1, p 2 [1, ] and WF p M 1,q(f ) = WF p (ω 1 ) M 2,q(f ) (ω 2 ) C 1 ω 2 (x, ξ) x N ω 1 (x, ξ) Cω 2 (x, ξ) x N, for some constants C and N. (Novi Sad) Nagoya, September 30, / 52

31 Wave front sets Theorem Assume that f D (R d ), q j [1, ] and ω j P(R 2d ) for j = 1, 2 satisfy q 1 q 2, and ω 2 (x, ξ) Cω 1 (x, ξ), (15) for some constant C which is independent of x, ξ R d. Then WF q F L 2 (f ) WF q (ω 2 ) F L 1 (f ). (ω 1 ) (Novi Sad) Nagoya, September 30, / 52

32 Discrete wave front sets We introduce discrete wave-front sets and prove that these wave-front sets agree with the introduced one. Assume that q [1, ], ω P(R 2d ) and that H R d is a discrete set. Then we set ( f (D) F L q = f (D) (H) F L q x, (H) f (ξ k )ω(x, ξ k ) q) 1/q {ξ k } H (with obvious modifications when q = ). As in the continuous case, we note that the condition f (D) F L q x, (H) < is independent of x R d. From now on we assume that ω is independent of x. (Novi Sad) Nagoya, September 30, / 52

33 Discrete wave front sets Lemma Assume that Γ and Γ 0 are open cones in R d such that Γ 0 Γ, and that Λ R d is a lattice. Also assume that f E (R d ) and ω P(R d ). If f F L q,γ is finite, then f (D) F L q (Γ is finite. 0 Λ) (Novi Sad) Nagoya, September 30, / 52

34 Discrete wave front sets In the next result we prove a converse to Lemma 9, in the case that 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 t d e d ; t 1,..., t d Z }. Then the parallelepiped, spanned by e 1,..., e d and with corners in Λ is called a Λ-parallelepiped. We let A(Λ) be the set of all Λ-parallelepipeds. (Novi Sad) Nagoya, September 30, / 52

35 Discrete wave front sets Assume that Λ 1 and Λ 2 are 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 an 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 Lemma Assume that q [1, ], (Λ 1, Λ 2 ) is a strongly admissible lattice pair, K A(Λ 1 ), and that f E (R d ) is such that an open neighbourhood of its support is contained in K. Also assume that Γ and Γ 0 are open cones in R d such that Γ 0 Γ. If f (D) F L q (Γ Λ 2) is finite, then f F L q,γ 0 is finite. (Novi Sad) Nagoya, September 30, / 52

36 Discrete wave front sets For non-periodic functions and distributions we instead make Gabor expansions. More precisely, let (Λ 1, Λ 2 ) be a strongly admissible lattice pair, with Λ 1 = {x j } j J and Λ 2 = {ξ k } k J. Also let ϕ, ψ C 0 (R d ), ϕ j,k (x) = ϕ(x x j )e i x,ξ k and ψ j,k (x) = ψ(x x j )e i x,ξ k (16) be such that {ϕ j,k } j,k J and {ψ j,k } j,k J are dual Gabor frames. (Novi Sad) Nagoya, September 30, / 52

37 Discrete wave front sets If f S (R d ), then where f = c j,k ϕ j,k, (17) j,k J c j,k = (f, ψ j,k ) L 2 (R d ) (18) and the constant C ϕ,ψ depends on the frames only. Here the series converges in S (R d ). (Novi Sad) Nagoya, September 30, / 52

38 Discrete wave front sets By replacing the lattices Λ 1 and Λ 2 here above with ελ 1 and Λ 2 /ε, and ϕ and ψ with ϕ ε = ϕ( /ε) and ψ ε = ψ( /ε) respectively, we still have f = c j,k (ε)ϕ ε j,k, (17) j,k J c j,k (ε) = C ϕ,ψ (ε)(f, ψ ε j,k ) L 2 (R d ), (18) ϕ ε j,k = ϕ j,k( /ε), ψ ε j,k = ψ j,k( /ε). (19) Here the constants C ϕ,ψ (ε) depends on ϕ, ψ and ε. (Novi Sad) Nagoya, September 30, / 52

39 Discrete wave front sets In some situations it is convenient to play with the parameter ε in ελ 1, ϕ ε = ϕ( /ε) and ψ ε = ψ( /ε), but keeping Λ 2 fixed and independent of ε. A problem is then that (17) and (18) might be violated. In the following we establish sufficient conditions for this to work properly. We first introduce admissible Gabor pairs. (Novi Sad) Nagoya, September 30, / 52

40 Discrete wave front sets Definition Assume that ε (0, 1], {x j } j J = Λ 1 R d and {ξ k } k J = Λ 2 R d are lattices and let Λ 1 (ε) = ελ 1. Also let ϕ, ψ C0 (Rd ) be non-negative, and set ϕ ε = ϕ( /ε), ψ ε = ψ( /ε), ϕ ε j,k = ϕε ( εx j )e i,ξ k, ψ ε j,k = ψε ( εx j )e i,ξ k, (20) when εx j Λ 1 (ε) (i. e. x j Λ 1 ) and ξ k Λ 2. Then the pair ({ϕ ε j,k } j,k J, {ψ ε j,k } j,k J) is called an admissible Gabor pair (AGP) if for each ε (0, 1], the sets {ϕ ε j,k } j,k J and {ψ ε j,k } j,k J are dual Gabor frames. (Novi Sad) Nagoya, September 30, / 52

41 Discrete wave front sets Lemma Assume that 0 < ε 1, ϕ, ψ C0 (Rd ) are non-negative, ϕ j,k, ψ j,k, ϕ ε j,k and ψj,k ε are given by (16) and x j Λ 1 ϕ( x j )ψ( x j ) = (2π) d Λ 2, (21) holds. Also assume that {ϕ j,k } j,k J and {ψ j,k } j,k J are dual Gabor frames, and that (21) holds. Then ({ϕ ε j,k } j,k J, {ψ ε j,k } j,k J) is an admissible Gabor pair. (Novi Sad) Nagoya, September 30, / 52

42 Discrete wave front sets Assume that p, q [1, ], ω P(R 2d ), f S (R d ), ({ϕ ε j,k } j,k J, {ψ ε j,k } j,k J) is an admissible Gabor pair, and that (17) and (18) hold. Then it follows that f M p,q (Rd ), if and only if ( ( f [ε] c j,k (ε)ω(εx j, ξ j ) p) q/p) 1/q k J j J if finite. Furthermore, for every ε (0, 1], the norm f f [ε] is equivalent to the modulation space norm (4). (Novi Sad) Nagoya, September 30, / 52

43 Discrete wave front sets Definition Assume that ω P(R d ), f D (X ), x 0 X, (Λ 1, Λ 2 ) is a strongly admissible lattice pair in R d and that {ξ k } j J = Λ 2. Also assume that D (Λ 1 ) contains x 0. Then the discrete wave-front set DF F L q (f ) consists of all (x 0, ξ 0 ) R d (R d \ 0) such that for each χ C0 (D X ) with χ(x 0 ) 0 and each open conical neighbourhood Γ of ξ 0, it holds χf (D) F L q =. (Λ) (Novi Sad) Nagoya, September 30, / 52

44 Discrete wave front sets For the definition of discrete wave-front sets of modulation space type, we consider admissible Gabor pairs ({ϕ ε j,k } j,k J, {ψ ε j,k } j,k J), ε (0, 1], and let J x0 (ε) = J x0 (ε, ϕ, ψ) = J x0 (ε, ϕ, ψ, Λ 1 ) be the set of all j J such that x 0 supp ϕ ε j,k or x 0 supp ψ ε j,k (Novi Sad) Nagoya, September 30, / 52

45 Discrete wave front sets Definition Assume that ω P(R 2d ), f D (X ), x 0 X and ({ϕ ε j,k } j,k J, {ψj,k ε } j,k J), ε (0, 1], are admissible Gabor pairs with respect to the lattices Λ 1 and Λ 2 in R d. Then the discrete wave-front set DF M p,q(f ) consists of all (x 0, ξ 0 ) R d (R d \ 0) such that for each ε (0, 1] and each open conical neighbourhood Γ of ξ 0, it holds ( {ξ k } Γ Λ 2 ( j J x0 (ε) c j,k (ε)ω(ξ k ) p) q/p) 1/q =, where f = c j,k(ε) ϕ ε j,k, and c j,k(ε) = C ϕ,ψ (f, ψj,k ε ) L 2 (R d ) and the j,k J constant C ϕ,ψ depends on the frames only. (Novi Sad) Nagoya, September 30, / 52

46 Discrete wave front sets Theorem Assume that X R d is open, ω P(R 2d ), f D (X ) and p, q [1, ]. Then all the wave fronts are equal. (Novi Sad) Nagoya, September 30, / 52

47 Wave-front sets and pseudo-differential operators We consider mapping properties for pseudo-differential operators in background of wave-front sets. In particular we prove such properties for pseudo-differential operators when the symbols belong to s,ρ for appropriate s R 4, ρ R and ω P(R 4d ). In order to state the results we use the convention (ϑ 1, ϑ 2 ) (ω 1, ω 2 ) (22) when ω j, ϑ j P(R d ) for j = 1, 2 satisfy ϑ j Cω j for some constant C. If instead ω j P(R 2d ), then it follows that ω j (x, ζ 1 + ζ 2 ) Cω j (x, ζ 1 ) ζ 2 t j, (23) for some constants C > 0 and t j, j = 1, 2, independent of x, ζ 1, ζ 2 R d. Then it is necessary that t 1 and t 2 are non-negative. Here we let ω s,ρ to be as in (10) and we use the notation (ω 1, ω 2 ) ω when (9) holds for some constant C. (Novi Sad) Nagoya, September 30, / 52

48 Wave-front sets and pseudo-differential operators Theorem Assume that 0 < ρ 1, ω j, ϑ j P(R 2d ) for j = 1, 2, ω P ρ (R 4d ), (ω 1, ω 2 ) ω s,ρ and (ϑ 1, ϑ 2 ) ω s,ρ, for some s R 4 such that 0 s 1, s 2 N, s 3 > t 1 + t 2 + 2d, where t 1 and t 2 are suitably chosen. Also assume that a s,ρ (R2d ) and that f M (ϑ 1 ) (Rd ). Then WF F L q (a(x, D)f ) WF (ω 2 ) F L q (f ). (ω 1 ) We need some preparations for the proof. The first part concerns the contribution to the wave-front set of a(x, D)f at a particular point x 0, when the support of f is far away from x 0. The following proposition shows that this contribution is limited. Here we let S 0 0,0 (Rd ) be the set of all χ C (R d ) such that α χ L for all multi-indices α. (Novi Sad) Nagoya, September 30, / 52

49 Wave-front sets and pseudo-differential operators Theorem Assume that ω P ρ (R 4d ), s R 4 is such that 0 s 1 and 0 s 2 Z, a s,ρ (R2d ), χ 1 C0 (Rd ) and χ 2 S0,0 0 (Rd ) are such that supp χ 1 supp χ 2 =, and let T a be the operator from S (R d ) to S (R d ), defined by the formula (T a f )(x) = χ 1 (x)(a(x, D)(χ 2 f ))(x) = (2π) d a(x, ζ)χ 1 (x)χ 2 (y)f (y)e i x y,ζ dydζ. Then the following is true: (1) T a = a 0 (x, D), for some a 0 S (R 2d ) such that α ζ a 0(x, ζ) s 4 0 M,1 (ω s,ρ) (R2d ), (24) (Novi Sad) Nagoya, September 30, / 52

50 Theorem for all multi-indices α such that α 2s 2 ; (2) if p, q [1, ], and ω 1, ω 2 P(R 2d ) fulfill ω 2 (x, ξ) ω 1 (y, η) Cω s,ρ(x, η, ξ η, y x) = Cω(x, η, ξ η, y x) x s 4 ξ η s 3 η 2ρs 2 x y s 1, then the definition of T a extends uniquely to a continuous map from M p,q (ω 1 ) (Rd ) to M p,q (ω 2 ) (Rd ). (Novi Sad) Nagoya, September 30, / 52

51 S ρ,δ (R2d ) consists of all a C (R 2d ) such that α x β ξ a(x, ξ) C α,βω(x, ξ) ξ ρ β +δ α. (25) Theorem Assume that 0 < ρ, ω 1, ω 2 P(R 2d ) and ω 0 P ρ,δ (R 2d ) satisfy. If a S (ω 0) ρ,0 (R2d ) and q [1, ], then ω 0 (x, ξ) C ω 1(x, ξ) ω 2 (x, ξ), (26) WF F L q (a(x, D)f ) WF (ω 2 ) F L q (f ), f S (R d ). (27) (ω 1 ) (Novi Sad) Nagoya, September 30, / 52

52 Theorem Assume that 0 δ < ρ 1, ω 1, ω 2 P(R 2d ) and ω 0 P ρ,δ (R 2d ) satisfy. If a S (ω 0) ρ,δ (R2d ) and q [1, ], then C 1 ω 1(x, ξ) ω 2 (x, ξ) ω 0(x, ξ), (28) WF F L q (f ) WF (ω 1 ) F L q (a(x, D)f ) Char (ω0 )(a), f S (R d ). (29) (ω 2 ) (Novi Sad) Nagoya, September 30, / 52