Characterization of spaces of type W and pseudo ... - Springer Link

J. Pseudo-Differ. Oper. Appl. (2014) 5:215–230 DOI 10.1007/s11868-014-0092-6

Characterization of spaces of type W and pseudo-differential operators of infinite order involving fractional Fourier transform S. K. Upadhyay · Anuj Kumar · Jitendra Kumar Dubey

Received: 17 September 2013 / Revised: 13 March 2014 / Accepted: 15 March 2014 / Published online: 21 April 2014 © Springer Basel 2014

Abstract The characterization of W-type spaces is investigated and various properties of pseudo-differential operators are studied by using the fractional Fourier transform. Keywords Fractional Fourier transform · Pseudo-differential operator · Convex functions · Gel’fand and Shilov spaces Mathematics Subject Classification (2010)

46F12 · 46E35 · 35S05

1 Introduction The theory of pseudo-differential operators was introduced by Kohn–Nirenberg [8], Hörmander [6,7], Wong [13] and others. They applied pseudo-differential operators associated with different types of symbols on the Schwartz space S(Rn ) and studied many properties by exploiting the Fourier transformation. Cappiello [2], Zanghirati [14], Boutet de Monvel [1] and Upadhyay et al. [11,12] studied the concept of pseudodifferential operators of infinite order on Gevrey, Gelfand and Shilov types of spaces by using the Fourier transformation. The characterization of pseudo-differential operators associated with fractional Fourier transformation on the Schwartz space S(R) was

S. K. Upadhyay (B) DST-CIMS, Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] A. Kumar · J. K. Dubey DST-CIMS, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] J. K. Dubey e-mail: [email protected]

216

S. K. Upadhyay et al.

given by Pathak et al. [9], Prasad and Kumar [10]. Our main aim of this paper is to discuss the properties of W-type spaces and to study the properties of pseudo-differential operators involving fractional Fourier transformation on the space W M (Rn ), where Rn = {(x1 , . . . , xn ) : x j ’s are real numbers} is usual Euclidean space. If x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). Then the inner product of x and y is defined by n x, y = x.y = x j .y j (1.1) j=1

and the norm of x is defined by ⎛ |x| = ⎝

n

⎞1

2 1 2 x 2j ⎠ = x12 + · · · + xn2 .

(1.2)

j=1

Fractional Fourier transform is the generalization of Fourier transform. It is frequently applied in signal processing, optics, quantum physics, mathematical sciences and many other scientific fields. The n-dimensional fractional Fourier transform with parameter α of f (x) on x ∈ Rn is denoted by (Fα f )(ξ ) [3] and defined as ˆ (1.3) f α (x) = (Fα f )(ξ ) = K α (x, ξ ) f (x)d x, ξ ∈ Rn Rn

where K α (x, ξ ) =

⎧ i(|x|2 +|ξ |2 ) cot α ⎪ −ix,ξ csc α ⎨Cα e 2 if

α = nπ

⎪ ⎩

α=

1 n (2π ) 2

e−ix,ξ

if

π 2,

∀ n ∈ Z,

and −n

inα

Cα = (2πi sin α) 2 e 2 1 = n . [π(1 − e−2iα )] 2 The corresponding inversion formula is given by f (x) = K α (x, ξ ) fˆα (ξ ) dξ, x ∈ Rn Rn

where the kernel K α (x, ξ ) = Cα e

−i(|x|2 +|ξ |2 ) cot α +ix,ξ csc α 2

,

(1.4)

Characterization of spaces of type W and pseudo-differential operators

217

and n

inα

Cα = (2πi sin α)− 2 e 2 − n 2 = π(1 − e−2iα ) . Now, the Eq. (1.3) can be written as, Fα [ f (x)](ξ ) = Cα e

i|ξ |2 cotα 2

e

−ix,ξ csc α

Rn n

= (2π ) 2 Cα e Replacing f (x) = e Fα [e

i|x|2 cot α 2

−i|x|2 cotα 2

i|ξ |2 cot α 2

f (x)e

f (x)e

i|x|2 cot α 2

i|x|2 cotα 2

dx

ˆ (ξ csc α).

(1.5)

φ(x) in (1.5), we obtain n

φ(x)](ξ ) = (2π ) 2 Cα e

i|ξ |2 cot α 2

[φ(x)ˆ](ξ csc α).

(1.6)

Now substituting ξ = sin α · w, where w ∈ Rn in (1.6), we obtain Fα [e

−i|x|2 cot α 2

n

φ(x)](sin α · w) = (2π ) 2 Cα e

i|sin α·w|2 cot α 2

[φ(x)ˆ](w).

(1.7)

−i|x|2 cot α 2 φ(x) , then (1.7) can be written as Let ψ = Fα e n

ψ(sin α · w) = (2π ) 2 Cα e

i|sin α·w|2 cot α 2

[φ(x)ˆ](w).

(1.8)

Now we recall the definition of W-type spaces from [4,5], which are given below. Let μ j and w j , j = 1, . . . , n, be continuous and increasing functions on [0, ∞) with μ j (0) = w j (0) = 0 and μ j (∞) = w j (∞) = ∞. We define x j M j (x j ) = μ j (ξ j )dξ j , (x j ≥ 0) (1.9) 0

y j j (y j ) =

w j (η j )dη j ,

(y j ≥ 0)

(1.10)

0

where j = 1, . . . , n. The functions M j (x j ) and j (y j ) are continuous, increasing and convex with M j (0) = j (0) = 0 and M j (∞) = j (∞) = ∞, we have (1.11) M j (−x j ) = M j (x j ), M j (x j ) + M j (x j ) ≤ M j (x j + x j ), j (−y j ) = j (y j ),

j (y j ) + j (y j ) ≤ j (y j + y j ).

(1.12)

218


We set μ(ξ ) = (μ1 (ξ1 )), . . . , (μn (ξn )), w(η) = (w1 (η1 )), . . . , (wn (ηn )). The space W M (Rn ) consists of all C ∞ -complex valued functions φ(x) on Rn which for some a ∈ Rn+ satisfy the inequality k Dx φ(x) ≤ Ck ex p [−M(ax)] ,

(1.13)

where, ex p [−M(ax)] = ex p [−M1 (a1 x1 ) − · · · − Mn (an xn )] , Dxk = Dxk11 · · · Dxknn and a1 , . . . , an , Ck are positive constants depending on the function φ(x) and the space W M,a (Rn ) consists of all infinitely differentiable functions φ(x) on x ∈ Rn , which for any δ ∈ Rn+ satisfy the inequality k Dx φ(x) ≤ Ck,δ ex p [−M(a − δ)x]

(1.14)

where ex p [−M(a − δ)x] = ex p −M1 (a1 − δ1 )x1 − · · · − M j (a j − δ j )x j − · · · − Mn (an − δn )xn and a, δ ∈ Rn+ , depend on the function φ(x). The space W (Cn ) consists of all entire analytic functions φ(z), where z = x + i y n , satisfy the inequality and x, y ∈ Rn , which for some b ∈ R+ k z φ(z) ≤ Ck ex p [(by)] , where

(1.15)

z k = z 1k1 . . . z nkn ,

ex p [(by)] = ex p 1 (b1 y1 ) + · · · + j (b j y j ) + · · · + n (bn yn ) , and Ck , b1 , . . . , bn are positive constants depending on the function φ(x), and the space W ,b consists of all entire analytic functions φ(z) such that for k ∈ Zn+ , ρ ∈ Rn+ , there exists a constants Ck,ρ > 0 such that k z φ(z) ≤ Ck,ρ ex p [(b + ρ)y] ,

(1.16)

where ex p [(b + ρ)y] = ex p 1 (b1 + ρ1 )y1 + · · · + j (b j + ρ j )y j + · · · + n (bn + ρn )yn ,


219

(Cn ) consists of all entire analytic functions φ(z) such that there The space W M n exist a, b ∈ R+ and C > 0 such that

|φ(z)| ≤ C ex p [−M[(ax)] + [(by)]] ,

(1.17)

where ex p [−M(ax)] and ex p [(by)] have usual meaning like (1.13) and (1.15), ,b and the space W M,a (Cn ) consists of all entire analytic functions φ(z) such that for n ρ, δ ∈ R+ and Cρ,δ > 0, |φ(z)| ≤ Cρ,δ ex p [−M[(a − δ)x] + [(b + ρ)y]] ,

(1.18)

where ex p [−M[(a − δ)x]] and ex p [[(b + ρ)y]] have usual meaning like (1.14) and (1.16), and the constants Cρ,δ , a, b and ρ, δ, depend only on the function φ(z). Let M j (x j ) and j (y j ) be the functions defined by (1.9) and (1.10), where the functions μ j (ξ j ) and w j (η j ) which occur in these equations are mutually inverse, that is μ j (w j (η j )) = η j and w j (μ j (ξ j )) = ξ j , then the corresponding functions M j (x j ) and j (y j ) are said to be the dual in sense of Young. In this case, the Young inequality (1.19) x j y j ≤ M j (x j ) + j (y j ), holds for any x j ≥ 0, y j ≥ 0. Definition 1.1 Let m ∈ R. Then we define the symbol class U m to be the space of all entire analytic functions θ (x, ξ ) ∈ C ∞ (Rn × Cn ) in ξ such that for any two multiindices μ and ϑ , there is a positive constant Cμ,ϑ depending upon μ and ϑ such that μ ϑ (1.20) (Dx Dξ θ )(x, ξ ) ≤ Cμ,ϑ (1 + |ξ |)m−|ϑ| ex p[(a0 t)] ∀ x ∈ Rn , ξ ∈ Cn and ξ = u + it. If we put t = 0 in (1.20), then the symbol class U m reduces to the symbol class introduced by Kohn and Nirenberg [8], see also Wong [13]. Definition 1.2 Let θ (x, ξ ) be a symbol belonging to U m , then the pseudo-differential operator Aθ,α associated with θ (x, ξ ) is defined as (Aθ,α φ)(x) =

K α (x, ξ ) θ (x, ξ )φˆ α (ξ ) dξ , φ ∈ W M (Rn )

(1.21)

Rn

where φˆ α (ξ ) is the fractional Fourier transform of φ, defined by (1.3). 2 Characterization of W-type spaces In this section we study the characterization of W-type spaces by using the fractional Fourier transformation.

220


Theorem 2.1 Let M(x) and (y) be a pair of functions which are dual in sense of Young . Then 1 Fα W M,a ⊂ W , a . (2.1) i|x|2 cot α

Proof Let e 2 φ(x) ∈ W M,a (Rn ) and σ = w + iτ . Then by the technique of Gel’fand and Shilov [5, pp. 20–21] and (1.8), we have 2 k k i|sin α·σ2| cot α (sin α · σ ) ψ(sin α · σ ) = Cα (sin α · σ ) e e−iσ,x φ(x)d x Rn

= Cα (sin α)|k| e

i|sin α·σ |2 cot α 2

(i)|k|

Dxk e−iσ,x φ(x)d x.

Rn

Now using the integration by parts, we get (sin α · σ )k ψ(sin α · σ ) = Cα (sin α)|k| e

i|sin α·σ |2 cot α φ(x) 2

(i)|k| (−1)|k|

e−iσ,x Dxk φ(x)d x

Rn

Thus, we get −n inα (sin α · σ )k ψ(sin α · σ ) ≤ (2π sin α) 2 e 2 |sin α||k| e−τ,x Dxk φ(x) d x ≤ D |sin α|

|k|− n2

Rn

e|τ,x| Dxk φ(x) d x

Rn n

≤ Ck,δ D |sinα||k|− 2 ≤ Ck,δ,sin α

ex p(|x| |τ |)ex p [−M(a − δ)x] d x Rn

ex p [|x| |τ | − M(a − δ)x] d x.

(2.2)

Rn

Using the young inequality (1.19), then the exponent of (2.2) can be evaluated as follows τ |x| |τ | − M [(a − δ)x] ≤ −M [(a − δ)x] + M [(a − 2δ)x] + . (a − 2δ) Hence (2.2) can be written as (sin α · σ )k ψ(sin α · σ ) τ (a−2δ) ≤ Ck,δ,sin α e ex p [−M(a − δ)x + M(a − 2δ)x] d x. Rn


Therefore,

(sin α · σ )k ψ(sin α · σ ) ≤ C ex p

221

τ (a − 2δ)

.

1 The quantity a−2δ can be represented in the form a1 + ρ, where ρ = (ρ1 , . . . , ρn ), ρ j arbitrarily small. Therefore the last expression can be written as, 1 k (sin α · σ ) ψ(sin α · σ ) ≤ Cex p [( + ρ)τ ] . a

Theorem 2.2 Let M(x) be the function which is dual in the sense of Young of (y). Then Fα W ,b ⊂ W M, 1 . b

−i|z|2 cotα

Proof Let e 2 φ(z) ∈ W ,b (Cn )and σ = w + iτ . Then by the technique of [5, pp. 21–22] and (1.8), we have −i|z|2 cotα k k Dw Fα e 2 φ(z) (sin α · w) ψ(sin α · w) = Dw ˆ i|sin α·w|2 cotα n k 2 φ(z) (w) = Dw (2π ) 2 Cα e ⎫ ⎧ ⎬ ⎨ i|sin α·w|2 cotα k 2 = C α Dw e−iw,z φ(z)d x e ⎭ ⎩ Rn ⎞ ⎛ k i|sin α·w|2 cot α r k−r ⎝ 2 Dw e Dw = Cα e−iw,z φ(z)d x ⎠ r r ≤k Rn k i|sin α·w|2 cot α i |sin α|2 cot α 2 = Cα Pr ,w e r 2 r ≤k × (i z)k−r e−iw,z φ(z)d x Rn

= C(α) ×

k i|sin α·w|2 cot α 2 Cη wη e r η≤r r ≤k

(i z)k−r e−iw,z φ(z)d x,

Rn

where C(α) = Cα | sin α|2 cot α and Pr

−i|sin α|2 cot α ,w 2

is a polynomial of degree r.

222


Therefore, k ψ(sin α Dw

k i|sin α·w|2 cot α 2 · w) = C(α) Cη e (i)|k−r |−|η| r η≤r r ≤k Dzη e−iw,z z k−r φ(z)d x. × Rn

k i|sin α·w|2 cot α 2 Cη e (i)|k−r |−|η| r η≤r r ≤k ×(−1)|η| e−iw,z Dzη z k−r φ(z) d x

= C(α)

Rn

k i|sin α·w|2 cotα 2 Cη e (i)|k−r |−|η| = C(α) r η≤r r ≤k η Dzη−s z k−r Dzs φ(z) d x ×(−1)|η| e−iw,z s s≤η Rn k i|sin α·w|2 cotα 2 Cη e (i)|k−r |−|η| = C(α) r η≤r r ≤k η (k − r )! × eiw,z z k−r −η+s Dzs φ(z)d x. s (k − r − η + s)! s≤η Rn

Thus, we have k k Cη Dw ψ(sin α · w) ≤ |C(α)| r η≤r r ≤k η (k − r )! −w,y k−r −η+s s × Dz φ(z) d x. e z s (k − r − η + s!) s≤η Rn

Now using the inequality |z| p ≤ |z||x|2+|z| and (1.14), we get +1 k [(b+ρ)]y −w,y Dw ψ(sin α · w) ≤ D(Ck−r −η+s+2,ρ+Ck−r −η+s,ρ )e e p+2

p

≤ Ck,r,η,s,ρ ex p[− w, y + (b + ρ)y]

Rn

dx (|x|2 + 1)

So that, k Dw ψ(sin α · w) ≤ Ck,r,η,s,ρ ex p[−w1 y1 + 1 [(b1 + ρ1 )y1 ] · · · − w j y j + j [(b j + ρ j )y j ] · · · − wn yn + n [(bn + ρn )yn ]]. (2.3)


223

Arguing as in [5, p. 22], we choose the sign of y j in such a way that the following equality holds wj + (b j + ρ j )y j . w j y j = w j y j = M j (b j + ρ j ) The j-th term of the exponent in the expression (2.3), becomes wj . −w j y j + [(b j + ρ j )y j ] = −M j (b j + ρ j ) Hence, we have k Dw ψ(sin α · w)

wj w1 ≤ Ck,ρ,r,η,s ex p −M1 − ··· − Mj (b1 + ρ1 ) (b j + ρ j ) wn − · · · − Mn (bn + ρn ) w . = Ck,ρ,r,η,s ex p −M (b + ρ)

1 In the above expression, we get b+ρ = small, then we obtain the inequality

1 b

− δ, where δ = (δ1 , . . . , δn ), δ j is arbitrarily

1 k −δ w . Dw ψ(sin α · w) ≤ Ck,ρ,r,η,s ex p −M b This implies that ψ(sin α · w) ∈ W M, 1 . b

Theorem 2.3 Let M0 (x) and 0 (y) be dual in sense of Young to the functions M(x) and (y), respectively. Then 0 , 1 ,b ⊂ W 1a . Fα W M,a M0 , b

−i|x|2 cotα

,b (Rn ) and σ = w + iτ .Then by the technique of Proof Let e 2 φ(x) ∈ W M,a Gel’fand and Shilov [5, pp. 23–24] and (1.8), we have i|sin α·σ |2 cot α 2 e−iσ,z φ(z)d x ψ(sin α · σ ) = Cα e

Rn

= Cα e

i|sin α·σ |2 cot α 2

Rn

e−iw+iτ,x+i y φ(x + i y)d x.

224


Thus, we get −n inα |ψ(sin α · σ )| ≤ (2πi sin α) 2 e 2 eiw+iτ,x+i y |φ(z)| d x Rn

−n inα ≤ (2πi sin α) 2 e 2 e−w,y−τ,x |φ(z)| d x Rn

Using (1.18), above expression yields |ψ(sin α · σ )| ≤ Cρ,δ,α e−w,y ex p [[(b+ρ)y]]

ex p [−M[(a − δ)x]+|τ | |x| ] d x.

Rn

Now using the arguments of [2, p. 24], we have 1 1 |ψ(sin α · σ )| ≤ Cρ,δ,α ex p −M0 ( − δ1 )w + 0 ( + ρ1 )τ . b a This shows that ψ(sin α · σ ) ∈ W

0 , a1 M0 , b1

.

3 Pseudo-differential operators of infinite order In this section the mapping properties of pseudo-differential operators defined by (1.21) on W M (Rn ) are obtained. Theorem 3.1 Let θ (x, ξ ) ∈ U m , where m ∈ R. Then Aθ,α maps W M (Rn ) into itself. Proof Let φ ∈ W M (Rn ). Then for non-negative integers μ and ν, we prove that sup ex p [M(ax)] Dxμ Aθ,α φ (x) < ∞.

x∈Rn

(3.1)

From (1.21) for φ ∈ W M (Rn ), we have Dxμ

Aθ,α φ (x) =

= Cα

Rn

=

Cα

Dxμ

K α (x, ξ ) θ (x, ξ )φˆ α (ξ ) dξ ,

Rn

−i(|x|2 +|ξ |2 ) cot α μ +ix,ξ csc α 2 Dx e θ (x, ξ ) φˆ α (ξ ) dξ

μ−r −i(|x|2 +|ξ |2 ) cot α μ r +ix,ξ csc α 2 Dx θ (x, ξ )φˆ α (ξ ) dξ. Dx e r r ≤μ

Rn


225

So that, x ν Dxμ Aθ,α φ (x) −i(|x|2 +|ξ |2 ) cot α μ r r r −r ix,ξ csc α 2 = x ν Cα e D e D x x r r r ≤μ

Rn

r ≤r

(x, ξ )φˆ α (ξ ) dξ μ r −i(|x|2 +|ξ |2 ) cot α i cot α ix,ξ csc α ν 2 e e = x Cα Pr x, r r 2 r ≤μ ×

Dxμ−r θ

Rn

r −r

×(iξ csc α)

r ≤r

Dxμ−r θ (x, ξ )φˆ α (ξ ) dξ,

α where Pr x, i cot is a polynomial of degree r. 2 x ν Dxμ Aθ,α φ (x) μ r −i(|x|2 +|ξ |2 ) cot α 2 = x ν Cα a cot α e x s (i csc α)|r −r | s r r r ≤μ r ≤r s≤r Rn ×eix,ξ csc α Dxμ−r θ (x, ξ ) ξ r −r φˆ α (ξ ) dξ. Since Dξν+s eix,ξ csc α = (i x csc α)ν+s eix,ξ csc α , we have x ν Dxμ Aθ,α φ (x) −i(|x|2 +|ξ |2 ) cot α μ r ν+s ix,ξ csc α 2 e e = Cα a cot α D s ξ r r r ≤μ r ≤r s≤r Rn ×(i cos α)|r −r |−|ν+s| Dxμ−r θ (x, ξ ) ξ r −r φˆ α (ξ ) dξ. Using integration by parts, we have x ν Dxμ Aθ,α φ (x) μ r = Cα as cot α(−1)|ν+s| (i csc α)|r −r |−|ν+s| r r r ≤μ r ≤r

s≤r

−i(|x|2 +|ξ |2 ) cot α ix,ξ csc α ν+s μ−r r −r ˆ 2 e φα (ξ ) dξ Dx θ (x, ξ ) ξ e Dξ

× Rn

= Cα

μ r

r ≤μ

r

r ≤r

r

s≤r

as cot α(−1)|ν+s| (icscα)|r −r |−|ν+s|

226


×

e

β≤ν+s

Rn ν+s−β

×Dξ = Cα

ν + s β −i|ξ |2 cot α 2 Dξ e Dxμ−r θ (x, ξ ) β

−i|x|2 cot α +ix,ξ csc α 2

! ξ r −r φˆ α (ξ ) dξ

μ r r ≤μ

×

e

r

r ≤r

r

as cot α

s≤r

−i|x|2 cot α −ix,ξ cscα 2

ν + s (−1)|ν+s| (i csc α)|r −r |−|ν+s| β

β≤ν+s

β β −i|ξ |2 ) cot α β−β μ−r 2 D e D D θ (x, ξ ) x ξ ξ β

β ≤β

Rn ν+s−β

×Dξ

! ξ r −r φˆ α (ξ ) dξ

Thus, we have x ν Dxμ Aθ,α φ (x) μ r ν + s = Cα (−1)|ν+s| (icscα)|r −r |−|ν+s| as cotα r r β r ≤μ r ≤r

β≤ν+s

s≤r

β −i(|x|2 +|ξ |2 ) cot α +ix,ξ csc α 2 × a cot α e s 1 β β ≤β

s1 ≤β

β−β

×ξ s1 Dξ

Rn

ν+s−β

Dxμ−r θ (x, ξ )Dξ

! ξ r −r φˆ α (ξ ) dξ.

So that using the arguments of [7, p. 121], we get ν μ x D Aθ,α φ (x) x

ν + s μ r |a |csc α||r −r |−|ν+s| ≤ Cα cotα| s r r β r ≤μ r ≤r

s≤r

β≤ν+s

β −x,t csc α s as1 cotα × e |ξ 1 | β β ≤β

s1 ≤β

Rn

! ν+s−β r −r ˆ β−β μ−r φα (ξ ) dξ ξ Dx θ (x, ξ ) Dξ × Dξ ν + s μ r |as cotα| |csc α||r −r |−|ν+s| ≤ Cα r r β r ≤μ r ≤r

s≤r

β≤ν+s


×

227

β as cot α e−x,t csc α D β−β D μ−r θ (x, ξ ) (1 + |ξ |s1 ) 1 x ξ β

β ≤β

s1 ≤β

Rn

! ν+s−β r −r ˆ φα (ξ ) (1 + |ξ |k )(1 + |ξ |k )−1 dξ. ξ × Dξ Now for k > 0, using (1 + |ξ |k )−1 ≤ 2k−1 (1 + |ξ |)−k where ξ = u + it, we obtain ν μ x D Aθ,α φ (x) x

ν + s μ r |csc α||r −r |−|ν+s| |as cot α| ≤ Cα r r β r ≤μ r ≤r

s≤r

β≤ν+s

β k−1 as1 cot α Cβ−β ,μ−r 2 × (1 + |ξ |)m−k−|β−β | β β ≤β

s1 ≤β

Rn

! ν+s−β r −r ˆ ×e[(a0 t)] (1 + |ξ |s1 )(1 + |ξ |k )e−x,t csc α Dξ ξ φα (ξ ) dξ. Therefore, we have ν μ x D Aθ,α φ (x) x

ν + s μ r |csc α||r −r |−|ν+s| |a ≤ Cα cot α| s r r β r ≤μ r ≤r

s≤r

β≤ν+s

β k−1 as1 cot α Cβ−β ,μ−r 2 × (1 + |ξ |)m−k−|β−β | β β ≤β

s1 ≤β

Rn

ν+s−β r −r ˆ φα (ξ ) ξ ×ex p[[(a0 t)] − x, t csc α] Dξ ν+s−β ν+s−β + ξ k+s1 Dξ ξ r −r φˆ α (ξ ) + ξ k Dξ ξ r −r φˆ α (ξ ) ν+s−β + ξ s1 Dξ ξ r −r φˆ α (ξ ) dξ.

(3.2)

Now using the definition of W (Cn ) and making reference to Gel’fand and Shilov [5, pp. 13–14], we obtain ν μ x D Aθ,α φ (x) x ≤ D C0,ν+s−β + Ck+s1 ,ν+s−β + Ck,ν+s−β + Cs1 ,ν+s−β × (1 + |ξ |)m−k−|β−β | ex p[[(a0 + b)t] − x, t csc α] dξ, Rn

228


where D be a positive constant depending on α, ν, μ, r, r , s, s1 and β, β . Now using the Young enquality (1.19) inside of above integral, we get ν μ x D Aθ,α φ (x) x −1 ≤ Dα ex p −M[(b + a0 ) x csc α] (1 + |ξ |)m−k−|β−β | dξ. Rn

The last integral is convergent for sufficiently large k. Thus above expression can be written as ν μ x D Aθ,α φ (x) ≤ Dm,k ex p −M[(b + a0 )−1 x csc α] . (3.3) x This implies that ex p[M[(b + a0 )−1 x csc α]]Dxμ Aθ,α φ (x) ≤ Dm,k,ν (1 + |x|ν )−1 . Therefore, sup ex p[M[(b + a0 )−1 x csc α]]Dxμ Aθ,α φ (x) ≤ Dm,k,ν sup (1 + |x|ν )−1 < ∞.

x∈Rn

x∈Rn

This implies that Aθ,α maps W M (Rn ) into itself.

Theorem 3.2 Let θ (x, ξ ) ∈ U m be a symbol, where m ∈ R. Then Aθ,α maps W M (Rn ) continuously into itself. Proof Using (3.2), we have ν μ x D Aθ,α φ j (x) x ν + s μ r |csc α||r −r |−|ν+s| |as cot α| ≤ Cα r r β r ≤μ β≤ν+s r ≤r s≤r β as cot α Cβ−β ,μ−r 2k−1 (1 + |ξ |)m−k−|β−β | × 1 β β ≤β s1 ≤β Rn ν+s−β r −r ˆ ξ ×e[(a0 t)] e−x,t csc α Dξ φα, j (ξ ) ν+s−β ν+s−β + ξ k+s1 Dξ ξ r −r φˆ α, j (ξ ) + ξ k Dξ ξ r −r φˆ α j (ξ ) s1 ν+s−β r −r ˆ + ξ Dξ φα, j (ξ ) dξ ξ ν + s μ r |a |csc α||r −r |−|ν+s| ≤ Cα cot α| s r r β r ≤μ r ≤r

s≤r

β≤ν+s


229

β as cot α Cβ−β ,μ−r 2k−1 1 β β ≤β s1 ≤β ν+s−β r −r ˆ ν+s−β φα, j (ξ ) + ξ k+s1 Dξ ξ ξ r −r φˆ α, j (ξ ) × sup ex p[−(bt)] Dξ ×

ξ ∈Cn

ν+s−β ν+s−β + ξ k Dξ ξ r −r φˆ α j (ξ ) + ξ s1 Dξ ξ r −r φˆ α, j (ξ ) × (1 + |ξ |)m−k−|β−β | e[−x,t csc α+[(b+a0 )t]] dξ. Rn

Now using the Young inequality (1.19), we have ν μ x D Aθ,α φ j (x) x ν + s μ r |csc α||r −r |−|ν+s| |as cot α| ≤ Cα r r β r ≤μ β≤ν+s r ≤r s≤r β x csc α k−1 as1 cot α Cβ−β ,μ−r 2 ex p −M × β (b + a0 ) β ≤β s1 ≤β ν+s−β r −r ˆ ν+s−β φα, j (ξ ) + ξ k+s1 Dξ ξ ξ r −r φˆ α, j (ξ ) × sup ex p[−(bt)] Dξ ξ ∈Cn

k ν+s−β r −r ˆ s1 ν+s−β r −r ˆ + ξ D ξ φα j (ξ ) + ξ Dξ φα, j (ξ ) ξ ξ × (1 + |ξ |)m−k−|β−β | dξ. Rn

Therefore, μ ex p M x csc α Dx Aθ,α φ j (x) (b + a0 ) μ r ν + s |csc α||r −r |−|ν+s| |a ≤ Cα,ν cot α| s r r β r ≤μ β≤ν+s r ≤r s≤r β as cot α Cβ−β ,μ−r 2k−1 sup ex p[−(bt)] × 1 β ξ ∈Cn β ≤β s1 ≤β ν+s−β r −r ˆ ν+s−β × Dξ φα, j (ξ ) + ξ k+s1 Dξ ξ ξ r −r φˆ α, j (ξ ) k ν+s−β r −r ˆ s1 ν+s−β r −r ˆ + ξ Dξ φα, j (ξ ) + ξ Dξ φα, j (ξ ) ξ ξ × (1 + |ξ |)m−k−|β−β | dξ. Rn

230


Now using the technique of Gel’fand and Shilov [5, p. 6], as φα, j → 0 when j → ∞ in W (Cn ). Therefore, ν+s−β r −r ˆ ν+s−β φα, j (ξ ) + ξ k+s1 Dξ sup ex p[−(bt)] Dξ ξ ξ r −r φˆ α, j (ξ )

ξ ∈Cn

ν+s−β ν+s−β + ξ k Dξ ξ r −r φˆ α, j (ξ ) + ξ s1 Dξ ξ r −r φˆ α, j (ξ ) → 0,

as j → ∞. This implies that the right hand side of last integral tends to 0 as j → ∞, which proves that Aθ,α φ j → 0 as j → ∞ in W M (Rn ).

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 since December (2012). The authors are thankful to referee for his valuable comments and suggestions regarding this paper.

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