Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform

Abstract

An asymptotic series of general symbol of pseudo-differential operator is obtained by using the theory of fractional Fourier transform.

1. Introduction

Namias [1] introduced fractional Fourier transform which is a generalization of Fourier transform. Fractional Fourier transform is the most important tool, which is frequently used in signal processing and other branches of mathematical sciences and engineering. The fractional Fourier transform can be considered as a rotation by an angle 𝛼 in time-frequency plane and is also called rotational Fourier transform or angular Fourier transform. The fractional Fourier transform [2, 3], with angle 𝛼 of a function 𝑢(𝑥), is defined by 𝑢_𝛼(𝑥)=𝐹_𝛼𝑢(𝜉)=_ℝ𝐾_𝛼(𝑥,𝜉)𝑢(𝑥)𝑑𝑥, (1) where 𝐾_𝛼()=𝐶𝑥,𝜉^𝛼𝑒^{𝑖(𝑥2+𝜉2)cot𝛼/2−𝑖𝑥𝜉cosec𝛼}1if𝛼≠𝑛𝜋√𝑒2𝜋^{−𝑖𝑥𝜉}𝜋if𝛼=2,𝐶∀𝑛∈𝕀,_𝛼=(2𝜋𝑖sin𝛼)^−1/2𝑒^𝑖𝛼/2=1−𝑖cot𝛼.2𝜋 (2) The corresponding inversion formula is given by 𝑢(𝑥)=12𝜋_ℝ𝐾_𝛼(𝑥,𝜉)𝑢_𝛼(𝜉)𝑑𝜉, (3) where the kernel 𝐾_𝛼=𝐶(𝑥,𝜉)_′𝛼𝑒^{−𝑖(𝑥2+𝜉2)cot𝛼/2+𝑖𝑥𝜉cosec𝛼}1if𝛼≠𝑛𝜋√𝑒2𝜋^𝑖𝑥𝜉𝜋if𝛼=2,𝐶∀𝑛∈𝕀,_′𝛼=(2𝜋𝑖sin𝛼)^1/2𝑒sin𝛼^{−𝑖𝛼/2}=2𝜋(1+𝑖cot𝛼). (4) Zayed [3] and Bhosale and Chaudhary [4] studied fractional Fourier transform of distributions with compact support. Pathak and others [5] defined the pseudo-differential operator involving fractional Fourier transform on Schwartz space 𝑆(ℝ) and studied many properties.

Our main aim in this paper is to generalize the results of Zaidman [6] and to find an asymptotic series of general symbol of pseudo-differential operator involving fractional Fourier transform.

Now we are giving some definitions and properties which are useful for our further investigations.

Linearity of fractional Fourier transform is given as 𝐹_𝛼𝑐₁𝑢₁(𝑥)+𝑐₂𝑢₂(𝑥)=𝑐₁𝐹_𝛼𝑢₁(𝑥)+𝑐₂𝐹_𝛼𝑢₂(𝑥), (5) where 𝑐₁ and 𝑐₂ are constants and 𝑢₁(𝑥) and 𝑢₂(𝑥) are two input functions.

Let 𝐿^𝑝(ℝ) denote the class of measurable functions 𝑓 defined on ℝ such that 𝑓_{𝐿𝑝(ℝ)}=_ℝ𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝<∞, (6) where 1≤𝑝<∞.

From [5], generalized Sobolev space 𝒢_𝛼^𝑠,𝑝(ℝ) involving fractional Fourier transform is defined by ‖𝑢‖_𝑠,𝑝=_ℝ1+𝜉^2𝑠𝑝/2𝑢_𝛼(𝜉)^𝑝𝑑𝜉^1/𝑝, (7)𝑠∈ℝ and 𝑢∈𝑆^′(ℝ).

The convolution of two functions 𝑓∈𝐿¹(ℝ) and 𝑔∈𝐿¹(ℝ) is defined [5, 7] as 𝑓∗𝑔(𝑥)=_ℝ𝑓𝑦𝑔𝑥−𝑦𝑑𝑦, (8) provided that the integral exists.

Let 𝒢 be a class of all measurable complex-valued functions 𝑎(𝑥,𝜉) which are defined on ℝ×ℝ−{0}. Then, we assume the following properties.(i)lim_𝑥→∞𝑎(𝑥,𝜉)=𝑎(∞,𝜉) exists for all 𝜉∈ℝ−{0} and is bounded to mesaurable function.(ii)We define 𝑎^′(𝑥,𝜉)=𝑎(𝑥,𝜉)−𝑎(∞,𝜉), then 𝑎^′()=𝐶𝑥,𝜉^′𝛼2𝜋_ℝ𝑒^{−𝑖(𝑥2+𝜉2)cot𝛼/2+𝑖𝑥𝜉cosec𝛼}𝐹_𝛼𝑎^′∀(){0},𝜂,𝜉𝑑𝜂,𝑥,𝜉∈ℝ×ℝ− (9)where (𝐹_𝛼𝑎^′)(𝜂,𝜉) is complex-valued function defined on ℝ×ℝ−{0}, which is measurable in 𝜂 and 𝜉 for all (𝜂,𝜉)∈ℝ×ℝ−{0} and satisfies the estimate: 𝐹_𝛼𝑎^′𝜂,𝜉≤𝐾𝜂,∀𝜂∈ℝ, (10) where (1+|𝜂|²)^𝑙𝐾(𝜂)∈𝐿¹(ℝ)  ∀𝑙=0,1,2,3,….

Let (𝑟_𝑗)^∞₀ be a strictly decreasing sequence; that is, 𝑟₀>𝑟₁>𝑟₂>⋯>𝑟_𝑗→−∞ as 𝑗→∞ and 𝜓∈𝐶^∞(ℝ) such that 0≤𝜓(𝜉)≤∞ for all 𝜉∈ℝ−{0}, 𝜓(𝜉)=10if0<𝜉≤2,1if𝜉≥1. (11) Let {𝑎_𝑗(𝑥,𝜉)}^∞₀ be an infinite sequence of function defined on ℝ×ℝ−{0}.

Then, we define a function 𝑎(𝑥,𝜉)=^∞_𝑗=0𝜓𝜉𝑡_𝑗𝜉^𝑟𝑗𝑎_𝑗(𝑥,𝜉), (12) where (𝑡_𝑗)^∞₀ is a sequence of positive real numbers such that 𝑡_𝑗→∞ as 𝑗→∞.

From (12), it is clear that 𝑎(𝑥,𝜉)=0, for |𝜉|≤1/2, 𝑥∈ℝ, and 𝑡∈[1,∞).

The global estimate of the above defined function 𝑎(𝑥,𝜉) and of remainders of order 𝑁 is given as 𝑏_𝑁(𝑥,𝜉)=𝑎(𝑥,𝜉)−^𝑁−1_𝑗=0𝜓𝜉𝑡_𝑗𝜉^𝑟𝑗𝑎_𝑗()=𝑥,𝜉_∞𝑗=𝑁𝜓𝜉𝑡_𝑗𝜉^𝑟𝑗𝑎_𝑗(𝑥,𝜉),∀𝜉∈ℝ−{0}. (13)

Theorem 1. Let {𝑡_𝑗}^∞₀ be a sequence of positive real numbers such that the following inequalities: 𝑎(𝑥,𝜉)≤𝐶𝜉^𝑟0,𝑏_𝑁(𝑥,𝜉)≤𝐶𝜉^𝑟𝑁,∀(𝑥,𝜉)∈ℝ×ℝ−{0}, (14) are satisfied for 𝑁=1,2,3…. In particular the estimates are as follows: 𝑎(∞,𝜉)≤𝐶𝜉^𝑟0,𝑏_𝑁(∞,𝜉)≤𝐶𝜉^𝑟𝑁,∀𝜉∈ℝ−{0}. (15)

Proof. The proof of the above theorem is obvious from [6, pages 233-234].

Theorem 2. Let {𝑡_𝑗}^∞₀ be a sequence of positive real numbers such that the following estimates: 𝐹_𝛼𝑎^′(𝑥,𝜉)≤𝐾(𝜆)𝜉^𝑟0,𝐹_𝛼𝑏^′_𝑁(𝑥,𝜉)≤𝐾(𝜆)𝜉^𝑟𝑁,∀(𝜆,𝜉)∈ℝ×ℝ−{0}, (16) are satisfied for 𝑁=1,2,3…. In particular the estimates are as follows: 𝐹_𝛼𝑎^′(∞,𝜉)≤𝐶𝜉^𝑟0,𝐹_𝛼𝑏^′_𝑁(∞,𝜉)≤𝐶𝜉^𝑟𝑁,∀𝜉∈ℝ−{0}, (17) where (1+|𝜆|²)^𝑝𝐾(𝜆)∈𝐿¹(ℝ), (1+|𝜆|²)^𝑝𝐾_𝑁(𝜆)∈𝐿¹(ℝ), ∀𝑝=0,1,2,3…, and 𝑁=1,2,3,….

Proof. The proof of above theorem is also obvious by using the same arguments from [6, pages 133–135].

2. Asymptotic Expansion of Pseudo-Differential Operator Associated with General Symbol

Definition 3. Let 𝑎(𝑥,𝜉) be a general symbol belonging to 𝒢. Then pseudo-differential operator 𝑎(𝑥,𝐷)=𝐴_𝑎,𝛼 associated with symbol 𝑎(𝑥,𝜉) is defined by (𝑎()𝑢𝑥)=1𝑥,𝜉)(2𝜋_ℝ𝐾_𝛼(𝑥,𝜉)𝑎(𝑥,𝜉)𝑢_𝛼(𝜉)𝑑𝜉, (18) where 𝑢_𝛼(𝜉) is defined in (3) ∀(𝑥,𝜉)∈ℝ×ℝ−{0}.

Definition 4. An infinitely differentiable complex-valued function 𝑢 is member of 𝑆(ℝ) if and only if for every choice of 𝜇 and 𝜈 of non-negative integers, it satisfies 𝛾_𝜇,𝜈(𝑢)=sup_𝑥∈ℝ𝑥^𝜇𝐷^𝜈𝑢(𝑥)<∞. (19)

Lemma 5. A function 𝑢∈𝐶^∞(ℝ) satisfies (19) if and only if 𝜏_𝜇,𝜈(𝑢)=sup_𝑥∈ℝ1+|𝑥|^2𝑚/2𝐷^𝛽𝑢(𝑥)<∞,∀𝑚,𝛽∈𝑁₀. (20)

Lemma 6 (Peetre). For any real number t and for all 𝜉,𝜂∈ℝ, the estimate (1+𝜉²)(1+𝜂²)^𝑡≤2^|𝑡|1+𝜉−𝜂^|𝑡| (21) is satisfied.

Theorem 7. Let 𝑎^′(𝑥,𝜉)∈𝒢; then one has the following relation: 𝐹_𝛼𝑒^{𝑖𝑥2𝑐𝑜𝑡𝛼/2}𝑎^′()𝑢(𝑥𝜉)=𝐶𝑥,𝐷)(_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)𝑐𝑜𝑡𝛼}𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂, (22) where 𝑢∈𝑆(ℝ), 𝑥∈ℝ.

Proof. By the definition of fractional Fourier transform (1), we have 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢(𝑥)(𝜉)=𝐶_𝛼ℝ𝑒^{𝑖(𝑥2+𝜉2)cot𝛼/2−𝑖𝑥𝜉cosec𝛼}×𝑎^′(𝑥,𝜉)𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥)𝑑𝑥=𝐶_𝛼ℝ𝑒^{𝑖(𝑥2+𝜉2)cot𝛼/2−𝑖𝑥𝜉cosec𝛼}𝑎^′(𝑥,𝜉)𝑒^{𝑖𝑥2cot𝛼/2}×𝐶^′_𝛼2𝜋_ℝ𝑒^{−𝑖(𝑥2+𝜂2)cot𝛼/2+𝑖𝑥𝜂cosec𝛼}𝑢_𝛼𝜂=𝐶𝑑𝜂𝑑𝑥_𝛼𝐶^′_𝛼2𝜋_ℝ𝑒^{𝑖(𝑥2+𝜉2−𝑥2−𝜂2+𝑥2)cot𝛼/2−𝑖𝑥(𝜉−𝜂)cosec𝛼}×𝑎^′(𝑥,𝜉)𝑢_𝛼𝜂=𝐶𝑑𝜂𝑑𝑥_𝛼𝐶^′_𝛼2𝜋_ℝ𝑒^{𝑖(𝑥2+𝜉2−𝜂2)cot𝛼/2−𝑖𝑥(𝜉−𝜂)cosec𝛼}×𝑎^′(𝑥,𝜉)𝑢_𝛼𝜂=𝐶𝑑𝜂𝑑𝑥_𝛼𝐶^′_𝛼2𝜋_ℝ𝑒^{𝑖(𝑥2+𝜉2+𝜂2−2𝜉𝜂)cot𝛼/2−𝑖𝑥(𝜉−𝜂)cosec𝛼}𝑎^′(𝑥,𝜉)×𝑒^{𝑖(𝜉𝜂−𝜂2)cot𝛼}𝑢_𝛼𝜂=𝐶𝑑𝜂𝑑𝑥_𝛼𝐶^′_𝛼2𝜋_ℝ𝑒^{𝑖(𝑥2+𝜉2+𝜂2−2𝜉𝜂)cot𝛼/2−𝑖𝑥(𝜉−𝜂)cosec𝛼}𝑎^′(𝑥,𝜉)×𝑒^{𝑖(𝜉𝜂−𝜂2)cot𝛼}𝑢_𝛼𝜂=𝐶𝑑𝜂𝑑𝑥_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}×𝐶_𝛼ℝ𝑒^{𝑖[𝑥2+(𝜉−𝜂)2]cot𝛼/2−𝑖𝑥(𝜉−𝜂)cosec𝛼}×𝑎^′(𝑥,𝜉)𝑑𝑥𝑢_𝛼𝜂=𝐶𝑑𝜂_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (23)

Theorem 8. If 𝑎(𝑥,𝜉)∈𝒢 is a symbol and 𝑎(𝑥,𝐷)=𝑎(∞,𝐷)+𝑎^′(𝑥,𝐷) is the associated operator, then, one has the following relation: 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎()𝑢(𝑥𝜉)()𝐹𝑥,𝐷)(=𝑎∞,𝜉_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥𝜉)+𝐶)(_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂, (24) where 𝑢∈𝑆(ℝ), 𝑥∈ℝ.

Proof. We have 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎(𝑥,𝐷)𝑢(𝑥)(𝜉)=𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎(∞,𝐷)+𝑎^′(𝑥,𝐷)𝑢(𝑥)(𝜉). (25) By linearity of fractional Fourier transform (5) we get 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎(𝑥,𝐷)𝑢(𝑥)(𝜉)=𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎(∞,𝐷)𝑢(𝑥)(𝜉)+𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢(𝑥)(𝜉). (26) Now using (1) and (22), we get the required result.

Theorem 9. Let 𝑎(𝑥,𝜉)∈𝒢 be a symbol and 𝑎(𝑥,𝐷) the associated operator; then one has the following relation: ‖(𝑎(𝑥,𝐷)𝑢)‖_𝑠,𝑝≤𝐶_𝑠,𝑝‖𝑢‖_{𝑠+𝑟0,𝑝}, (27) where 𝑢∈𝑆(ℝ), 𝑠∈ℝ.

Proof. Firstly from (7) we have𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(∞,𝐷)𝑢)_𝑠,𝑝=_ℝ1+𝜉^2𝑠𝑝/2𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎(∞,𝐷)𝑢(𝑥)(𝜉)^𝑝𝑑𝜉^1/𝑝=_ℝ1+𝜉^2𝑠𝑝/2𝑎(∞,𝜉)^𝑝𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥)(𝜉)^𝑝𝑑𝜉^1/𝑝. (28)Now the argument of (15) yields 𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(∞,𝐷)𝑢)_𝑠,𝑝≤𝐶_ℝ1+𝜉^2𝑠𝑝/21+𝜉^2𝑟0𝑝/2×𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥)(𝜉)^𝑝𝑑𝜉^1/𝑝=𝐶_ℝ1+𝜉^{2(𝑠+𝑟0)𝑝/2}×𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥)(𝜉)^𝑝𝑑𝜉^1/𝑝=𝐶𝑒^{𝑖𝑥2cot𝛼/2}𝑢_{𝑠+𝑟0,𝑝}. (29) Thus we have 𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(∞,𝐷)𝑢)_𝑠,𝑝≤𝐶𝑒^{𝑖𝑥2cot𝛼/2}𝑢_{𝑠+𝑟0,𝑝},𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(∞,𝐷)𝑢)_𝑠,𝑝≤𝐶‖𝑢‖_{𝑠+𝑟0,𝑝}, (30) for every 𝑢∈𝑆(ℝ), 𝑠∈ℝ.
Now we consider the function 𝑈_𝑠,𝛼(𝜉)=(1+|𝜉|²)^𝑠/2𝐺_𝛼(𝜉), where 𝑈_𝑠,𝛼(𝜉)=0 for |𝜉|≤1/2 and for |𝜉|≥1/2 and 𝐺_𝛼(𝜉) is defined as 𝐺_𝑠(𝜉)=𝐶^′_𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (31) Therefore, 𝑈_𝑠,𝛼(𝜉)=𝐶^′_𝛼𝜉2𝜋1+^2𝑠/2_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂=𝐶𝑑𝜂_′𝛼2𝜋_ℝ1+𝜉^2𝑠/21+𝜂^2𝑠/2𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}×1+𝜂^2𝑠/2𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (32) Using (21) we get 𝑈_𝑠,𝛼(𝜉≤2)^|𝑠|/2𝐶^′_𝛼2𝜋_ℝ1+𝜉−𝜂^2|𝑠|/21+𝜂^2𝑠/2×𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (33) From Theorem 2 we get 𝑈_𝑠,𝛼(𝜉≤2_|𝑠|/2)𝐶^′_𝛼2𝜋_ℝ1+𝜉−𝜂^2|𝑠|/21+𝜂^2𝑠/2×𝐾𝜉−𝜂1+𝜉^2𝑟0/2𝑢_𝛼𝜂=2𝑑𝜂,_|𝑠|/2𝐶^′_𝛼2𝜋_ℝ1+𝜉−𝜂^2|𝑠|/21+𝜂^{2(𝑠+𝑟0)/2}𝜉×𝐾𝜉−𝜂1+^2𝑟0/21+𝜂^2𝑟0/2𝑢_𝛼𝜂=2𝑑𝜂,_{(|𝑠|+|𝑟0|)/2}𝐶^′_𝛼2𝜋_ℝ1+𝜉−𝜂^2|𝑠|/21+𝜉−𝜂^2|𝑟0|/2×1+𝜂^{2(𝑠+𝑟0)/2}𝐾𝜉−𝜂𝑢_𝛼𝜂=2𝑑𝜂,_{(|𝑠|+|𝑟0|)/2−1}𝐶^′_𝛼𝜋×_ℝ1+𝜉−𝜂^{2(|𝑠|+|𝑟0|)/2}1+𝜂^{2(𝑠+𝑟0)/2}×𝐾𝜉−𝜂𝑢_𝛼𝜂=𝑑𝜂,_ℝΦ𝜉−𝜂ℎ_𝛼𝜂=𝑑𝜂Φ∗ℎ_𝛼(𝜉). (34) Here Φ(𝜉−𝜂)∈𝐿¹(ℝ) by (10) for all 𝑠∈ℝ and since 𝑢_𝛼(𝜂)∈𝑆(ℝ), therefore ℎ_𝛼(𝜂)∈𝐿^𝑝(ℝ).
Thus, we have (Φ∗ℎ_𝛼)(𝜉)∈𝐿^𝑝(ℝ) and the inequality Φ∗ℎ_{𝛼𝐿𝑝(ℝ)}≤‖Φ‖_𝐿¹_(ℝ)ℎ_{𝛼𝐿𝑝(ℝ)}. (35) This implies that 𝑈_𝑠,𝛼(𝜉)_𝑝≤𝐶_𝛼‖𝑢‖_{𝑠+𝑟0,𝑝},∀𝑢∈𝑆(ℝ),𝑠∈ℝ. (36) So that 𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢_𝑠,𝑝=1+𝜉^2𝑠/2𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢(𝑥)_𝑝=𝑈_𝑠,𝛼(𝜉)_𝑝. (37) Using (36) we get 𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢_𝑠,𝑝≤𝐶_𝛼‖𝑢‖_{𝑠+𝑟0,𝑝}. (38) Therefore, 𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(𝑥,𝐷)𝑢)_𝑠,𝑝=𝑒^{𝑖𝑥2cot𝛼/2}𝑎(∞,𝐷)𝑢+𝑎^′(𝑥,𝐷)𝑢_𝑠,𝑝≤𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(∞,𝐷)𝑢)_𝑠,𝑝+𝑒^{𝑖𝑥2cot𝛼/2}𝑎^′(𝑥,𝐷)𝑢_𝑠,𝑝. (39) Now using (30) and (38) we get 𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(𝑥,𝐷)𝑢)_𝑠,𝑝≤𝐶+𝐶_𝛼‖𝑢‖_{𝑠+𝑟0,𝑝}. (40) This implies that 𝑒^{𝑖𝑥2cot𝛼/2}(𝑎(𝑥,𝐷)𝑢)_𝑠,𝑝≤𝐶_{𝑠,𝑝,𝛼}‖𝑢‖_{𝑠+𝑟0,𝑝}. (41) This implies the required result (27).

Theorem 10. One has the following estimates: 𝜓_𝑟1𝑡𝐷𝑢_𝑠,𝑝≤𝐶‖𝑢‖_{𝑠+𝑟,𝑝},∀𝑢∈𝑆(ℝ),𝑠,𝑟∈ℝ. (42)

Proof. Let 𝜓∈𝐶^∞(ℝ), then by using (11) and (12) we find that 𝜓_𝑟(𝜉/𝑡)=𝜓(𝜉/𝑡)|𝜉|^𝑟 for |𝜉|≥𝑡/2. Therefore, we have 𝜓_𝑟𝜉𝑡≤𝐶1+𝜉^2𝑟/2,∀𝜉∈ℝ. (43) Now we define the operator 𝜓_𝑟((1/𝑡)𝐷) given on 𝑆(ℝ) by 𝜓_𝑟1𝑡𝐷𝑢=𝐹_𝛼⁻¹𝜓_𝑟𝜉𝑡𝑢_𝛼(𝜉). (44) Hence, 𝜓_𝑟1𝑡𝐷𝑢_𝑠,𝑝=_ℝ1+𝜉^2𝑠𝑝/2𝜓_𝑟𝜉𝑡^𝑝𝑢_𝛼(𝜉)^𝑝𝑑𝜉^1/𝑝≤𝐶_ℝ1+𝜉^2𝑠𝑝/2𝑢_𝛼(𝜉)^𝑝1+𝜉^2𝑟𝑝/2𝑑𝜉^1/𝑝=𝐶_ℝ1+𝜉^{2(𝑠+𝑟)𝑝/2}𝑢_𝛼(𝜉)^𝑝𝑑𝜉^1/𝑝. (45) Thus, 𝜓_𝑟1𝑡𝐷𝑢_𝑠,𝑝≤𝐶‖𝑢‖_{𝑠+𝑟,𝑝}. (46) Since 𝒢^∞_𝛼=∩_𝑠∈ℝ𝒢_𝛼^𝑠,𝑝(ℝ), 𝜓_𝑟(((1/𝑡)𝐷)𝑢) is a linear operator 𝒢^∞_𝛼 into itself.

Definition 11. A linear operator 𝐿 with 𝑟∈ℝ, and ∀𝑠∈ℝ, there exists a constant 𝐶_𝑠>0 such that ‖𝐿𝑢‖≤𝐶_𝑠‖𝑢‖_{𝑠+𝑟,𝑝},∀𝑢∈𝒢^∞_𝛼. (47) The infimum of all orders 𝑟 of 𝐿 is called true order of 𝐿.

Definition 12. Let 𝜓_𝑟(((1/𝑡)𝐷)𝑎(𝑥,𝐷)) be a linear operator from ∀𝑢∈𝒢^∞_𝛼 into itself and satisfy the following inequality: 𝜓_𝑟1𝑡𝐷𝑎(𝑥,𝐷)_𝑠,𝑝≤𝐶‖𝑢‖_{𝑠+𝑟,𝑝},∀𝑠∈ℝ,𝑢∈𝒢^∞_𝛼(ℝ). (48) Then 𝜓_𝑟((1/𝑡)𝐷)𝑎(𝑥,𝐷) is said to be a canonical operator of degree 𝑟, where 𝑟∈ℝ.

Definition 13. Let (𝑟_𝑗)^∞₀ be a strictly decreasing sequence of real numbers and {𝜓_𝑟𝑗((1/𝑡)𝐷)𝑎_𝑗(𝑥,𝐷)} a sequence of canonical operators of degree 𝑟_𝑗. Then, corresponding a sequence of positive real numbers (𝑡_𝑗)^∞₀ and {𝑎_𝑗(𝑥,𝜉)}^∞₀⊂𝒢, a linear operator 𝑀∶𝒢^∞_𝛼→𝒢^∞_𝛼 is asymptotically expanded into the series ∑^∞_𝑗=0𝜓_𝑟𝑗((1/𝑡)𝐷)𝑎_𝑗(𝑥,𝐷) if it satisfies the following inequality: t.o𝑀−^𝑁_𝑗=0𝜓_𝑟𝑗1𝑡_𝑗𝐷𝑎_𝑗(𝑥,𝐷)<𝑟_𝑁. (49)

Theorem 14. Let {𝑎_𝑗(𝑥,𝜉)}^∞₀ be a sequence of symbols belonging to 𝒢 and (𝑟_𝑗)^∞₀ a strictly decreasing sequence of real numbers that tends to −∞. Then, there exists a sequence of canonical operators of degree 𝑟_𝑗,𝐾_𝑗,𝛼 and a linear operator 𝐴_𝑎,𝛼 in 𝒢^∞_𝛼 such that(i)t.o (𝐴_𝑎,𝛼)≤𝑟₀;(ii)𝐴_𝑎,𝛼∼∑^∞_𝑗=𝑜𝐾_𝑗,𝛼, that is, t.o  [𝐴_𝑎,𝛼−∑^𝑁_𝑗=𝑜𝐾_𝑗,𝛼]<𝑟_𝑁.

Proof. Note that (i) is obviously true by using Theorem 9 and for (ii) by using the arguments of [6, pages 241-242]. We can define the canonical operator by the following way: 𝐾_𝑗,𝛼𝑢(𝑥)=𝑒^{−𝑖𝑥2cot𝛼/2}𝐹_𝛼⁻¹×𝜓_𝑟𝑗𝜉𝑡_𝑗𝑎_𝑗(∞,𝜉)𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥)(𝜉)+𝐶^′_𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}×𝑎^′_𝑗,𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂, (50) where 𝑎(𝑥,𝜉)=∑^∞_𝑗=𝑜𝜓_𝑟𝑗(𝜉/𝑡)[𝑎_𝑗(∞,𝜉)+𝑎^′_𝑗(𝑥,𝜉)] belongs to class of symbol 𝒢. Also we have 𝐴_𝑎,𝛼𝑢(𝑥)=𝑒^{−𝑖𝑥2cot𝛼/2}𝐹_𝛼⁻¹×𝑎(∞,𝜉)𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥𝜉)+𝐶)(_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}×𝑎^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (51) Therefore, 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝐴_𝑎,𝛼𝑢−^∞_𝑗=0𝐾_𝑗,𝛼𝑢(𝜉)=𝑎(∞,𝜉)−^𝑁_𝑗=0𝜓_𝑟𝑗𝜉𝑡_𝑗𝑎_𝑗(∞,𝜉)×𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥𝜉)+𝐶)(_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}×𝑎^′_𝛼𝜉−𝜂,𝜉−^𝑁_𝑗=0𝜓_𝑟𝑗𝜉𝑡_𝑗𝑎^′_𝑗,𝛼×𝑢𝜉−𝜂,𝜉_𝛼𝜂𝑑𝜂. (52) Using the previous definitions (12) and (13), we have 𝑎(∞,𝜉)−^𝑁₀𝜓_𝑟𝑗𝜉𝑡_𝑗𝑎_𝑗(∞,𝜉)=𝑏_𝑁+1(∞,𝜉). (53) From (15) and (16), we get |𝑏_𝑁+1(∞,𝜉)|≤𝐶(1+|𝜉|²)^𝑟𝑁+1/2, where 𝜉∈ℝ−{0} and 𝑎^′_𝛼(𝜉−𝜂,𝜉)−∑^𝑁_𝑗=0𝜓_𝑟𝑗(𝜉/𝑡_𝑗)𝑎_𝛼,𝑗(𝜉−𝜂,𝜉)=𝑏̂^′_𝛼,𝑁+1(𝜉−𝜂,𝜉).
Also |𝑏̂^′_𝛼(𝜉−𝜂,𝜉)|≤𝐶𝐾_𝑁+1(𝜉−𝜂)(1+|𝜉|²)(𝑟_𝑁+1/2). Now using these arguments in (52) we get 𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝐴_𝑎,𝛼−^𝑁_𝑗=0𝐾_𝑗,𝛼𝑢(𝜉)=𝑏_𝑁+1(∞,𝜉)𝐹_𝛼𝑒^{𝑖𝑥2cot𝛼/2}𝑢(𝑥𝜉)+𝐶)(_′𝛼2𝜋_ℝ𝑒^{−𝑖(𝜂2−𝜉𝜂)cot𝛼}𝑏̂^′_𝛼𝜉−𝜂,𝜉𝑢_𝛼𝜂𝑑𝜂. (54) Using Theorem 9, we get 𝐴_𝑎,𝛼𝑢−^𝑁_𝑗=0𝐾_𝑗𝑢_𝑠,𝑝≤𝐶‖𝑢‖_{𝑠+𝑟𝑁+1,𝑝},∀𝑢∈𝑆(ℝ),𝑠∈ℝ. (55) Hence, we get t.o𝐴_𝑎,𝛼−^𝑁_𝑗=0𝐾_𝑗,𝛼≤𝑟_𝑁+1<𝑟_𝑁. (56)

Acknowledgments

The first author is thankful to DST-CIMS, Banaras Hindu University, Varanasi, India, for providing the research facilities, and the second author is also thankful to DST-CIMS, Banaras Hindu University, Varanasi, India, for awarding the Junior Research Fellowship from December 2012.