Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 DOI 10.1186/s13660-014-0544-9 RESEARCH Open Access Characterization of W p-type of spaces involving fractional Fourier transform Santosh Kumar Upadhyay1,2* and Anuj Kumar1 * Correspondence: [email protected] 1 DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221005, India 2 Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India Abstract The characterizations of W p -type of spaces and mapping relations between W- and W p -type of spaces are discussed by using the fractional Fourier transform. The uniqueness of the Cauchy problems is also investigated by using the same transform. MSC: 46F12; 46E15 Keywords: fractional Fourier transform; convex functions; Gel’fand and Shilov spaces of type W; Lp -space 1 Introduction The spaces of W -type were studied by Gurevich [], Gel’fand and Shilov [] and Friedman []. They investigated the behavior of the Fourier transformation on W -type spaces. The spaces of W -types are applied to the theory of partial diﬀerential equations. Pathak and p p ,p ,b,p Upadhyay [] investigated the spaces WM , WM,a , W ,b,p , W ,p , WM , WM,a in terms of Lp norms. Here M, are certain continuous increasing convex functions and a, b are positive constants and p ≥ . It was shown that the Fourier transformation F is to be a continuous p  ,b,p ,  ,r a r linear mapping as follows: F : WM,a → W , a ,r , F : W ,b,p → WM,  , F : WM,a → WM,  . b a Using the theory of the Hankel transform, Betancor and Rodriguez-Mesa [] gave a new p p characterization of the space of Weμ -type and established the results WeM,a = WeM,a , p,,b Wep,,b = We,b , WeM,a = We,b M,a . Upadhyay [] established the results of the following p p,,b ,b by exploiting the theory of Fourier types: WM,a = WM,a , W p,,b = W ,b , WM,a = WM,a transformations. Motivated by the work of Pathak and Upadhyay [] and Upadhyay [] we shall extend a similar type of results in n dimensions by using the theory of the fractional Fourier transformations. Let Rn be the usual Euclidean space given by Rn = (x , . . . , xn ): xj ’s are real numbers . Assume x = (x , . . . , xn ) and y = (y , . . . , yn ). Then the inner product of x and y is deﬁned by x, y = x · y = n xj · yj (.) j= and the norm of x is deﬁned by |x| = n  xj  = x + · · · + xn  . (.) j= © 2015 Upadhyay and Kumar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 2 of 15 The Lp norm of a function f in Lp (Rn ),  ≤ p ≤ ∞, is denoted by f p and deﬁned as f p = Rn f (x)p dx p . (.) The n-dimensional fractional Fourier transform (FrFT) with parameter α of f (x) on x ∈ Rn is denoted by (Fα f )(ξ ) [, ] and deﬁned as fˆα (ξ ) = (Fα f )(ξ ) = Rn ξ ∈ Rn , Kα (x, ξ )f (x) dx, (.) where Kα (x, ξ ) = ⎧   ⎨Cα e i(|x| +|ξ| ) cot α –ix,ξ csc α if α = nπ, ⎩ if α = π ,  n (π )  e–ix,ξ ∀n ∈ Z and –n Cα = (πi sin α)  e inα  =  n [π( – e–iα )]  . (.) The corresponding inversion formula is given by f (x) =  (π) n  Rn Kα (x, ξ )fˆα (ξ ) dξ , x ∈ Rn , (.) where the kernel Kα (x, ξ ) = Cα e– i(|x| +|ξ | ) cot α +ix,ξ csc α  , and Cα is deﬁned by (.). Now from the technique of [, p.], (.) can be written as i|ξ | cot α Fα f (x) (ξ ) = Cα e  n = (π)  Cα e Replacing f (x) = e– i|x| cot α  Rn i|x| cotα dx e–ix,ξ csc α f (x)e  i|ξ | cot α  f (x)e i|x| cot α  ˆ (ξ csc α). (.) φ(x) in (.), we obtain i|x| cotα i|ξ | cot α n φ(x) ˆ(ξ csc α). Fα e–  φ(x) (ξ ) = (π)  Cα e  (.) Now substituting ξ = w sin α, where w ∈ Rn in (.), we obtain i|x| cot α i|w sin α| cot α n  φ(x) ˆ(w). Fα e–  φ(x) (w sin α) = (π)  Cα e Let ψ = Fα [e– i|x| cot α  (.) φ(x)], then (.) can be written as n ψ(w sin α) = (π)  Cα e i|w sin α| cot α  φ(x) ˆ(w). (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 3 of 15 Now we recall the deﬁnitions of W - and W p -type of spaces from [–], which are given below. Let μj and wj , j = , . . . , n, be continuous and increasing functions on [, ∞) with μj () = wj () =  and μj (∞) = wj (∞) = ∞. We deﬁne xj Mj (xj ) = μj (ξj ) dξj (xj ≥ ), (.) wj (ηj ) dηj (yj ≥ ), (.)  yj j (yj ) =  where j = , . . . , n. The functions Mj (xj ) and j (yj ) are continuous, increasing, and convex with Mj () = j () =  and Mj (∞) = j (∞) = ∞, we have Mj (–xj ) = Mj (xj ), j (–yj ) = j (yj ), Mj (xj ) + Mj xj ≤ Mj xj + xj , j (yj ) + j yj ≤ j yj + yj . (.) (.) We deﬁne μ(ξ ) = μ (ξ ) , . . . , μn (ξn ) , w(η) = w (η ) , . . . , wn (ηn ) . The space WM,a (Rn ) consists of all C ∞ -complex valued functions φ(x) on x ∈ Rn , which for any δ ∈ Rn+ satisfy the inequality k D φ(x) ≤ Ck,δ exp –M (a – δ)x , x (.) p and the space WM,a (Rn ) consists of all inﬁnitely diﬀerentiable functions φ(x) on x ∈ Rn , which for any δ ∈ Rn+ satisfy the inequality Rn exp –M (a – δ)x Dk φ(x)p dx x p ≤ Ck,δ,p , p≥ (.) for each k ∈ Zn+ where Dkx = Dkx · · · Dkxnn , exp –M (a – δ)x = exp –M (a – δ )x – · · · – Mn (an – δn )xn and a , . . . , an , Ck,δ,p , Ck,δ are positive constants depending on the function φ(x). The space W ,b (Cn ) consists of all entire analytic functions φ(z), where z = x + iy and x, y ∈ Rn , which for any ρ ∈ Rn+ satisfy the inequality k z φ(z) ≤ Ck,ρ exp (b + ρ)y , where zk = zk · · · znkn , k ∈ Zn+ , (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 4 of 15 and b , . . . , bn , Ck,ρ are positive constants depending on the function φ(x) and the space W ,b,p consists of all entire analytic functions φ(z) such that for k ∈ Zn+ , ρ ∈ Rn+ , there exists a constant Ck,ρ,p >  such that Rn exp (b + ρ)y zk φ(z)p dx p ≤ Ck,ρ,p , (.) where exp (b + ρ) y = exp  (b + ρ )y + · · · + n (bn + ρn )yn . ,b (Cn ) consists of all entire analytic functions φ(z) such that there exist conThe space WM,a n stants ρ, δ ∈ R+ and Cδ,ρ >  such that φ(z) ≤ Cδ,ρ exp –M (a – δ)x + (b + ρ)y , (.) ,b,p and the space WM,a (Cn ) consists of all entire analytic functions φ(z) such that for ρ, δ ∈ Rn+ and Cρ,δ,p > , Rn exp M (a – δ)x – (b + ρ)y φ(z)p dx p ≤ Cρ,δ,p , (.) where exp[M[(a – δ)x]] and exp[–[(b + ρ)y]] have the usual meaning like (.) and (.), and the constants Cρ,δ,p , a, b, and ρ, δ depend only on the function φ(z). Let Mj (xj ) and j (yj ) be the functions deﬁned by (.) and (.), respectively, the functions μj (ξj ) and wj (ηj ) which occur in these equations are mutually inverse, that is, μj (wj (ηj )) = ηj and wj (μj (ξj )) = ξj , then the corresponding functions Mj (xj ) and j (yj ) are said to be the dual in the sense of Young. In this case, the Young inequality, xj yj ≤ Mj (xj ) + j (yj ), (.) holds for any xj ≥, yj ≥ . 2 Characterization of W p -type of spaces In this section we study the characterization of W p -type of spaces by using the fractional Fourier transformation. Theorem . Let M(x) and (y) be the pair of functions which are dual in the sense of Young. Then p  Fα WM,a ⊂ W , a ,r i|x| cot α for any  ≤ p, r < ∞. p (.) Proof Let e–  φ(x) ∈ WM,a (Rn ) and σ = w + iτ . Then for any p and r, using the technique of [, pp.-] and (.), we have (σ sin α)k ψ(σ sin α) = r Rn r (σ sin α)k ψ(σ sin α)r dw . Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Now using the inequality |σ ||k| ≤ |σ ||k+| +|σ ||k| , |w| + Page 5 of 15 we have (σ sin α)k ψ(σ sin α) r r |k+| r + |σ ||k| r |σ | dw ≤ | sin α|r|k| ψ(σ sin α) |w| +  Rn r |k+| r r |σ | r|k| ψ(σ sin α) dw ≤ | sin α|  Rn |w| +  r r |σ ||k| r dw . ψ(σ sin α) + | sin α|r|k|  Rn |w| +  Using (.), we get (σ sin α)|k| ψ(σ sin α) r ≤ Cα,k+ Rn + Cα,k Rn dσ (|w| + )r dw (|w| + )r exp x, τ Dk+ φ(x) dx Rn Rn exp x, τ Dk φ(x) dx r r r r r r dw exp x, τ Dk+ φ(x) dx  r Rn (|w| + ) Rn r r k dw exp x, τ D φ(x) dx + Cα,k  r Rn (|w| + ) Rn p p k+ dw exp M (a – δ)x D φ(x) dx ≤ Cα,k+  r Rn (|w| + ) Rn  r r p exp |x||τ | – M (a – δ)x p dx × ≤ Cα,k+ Rn + Cα,k × Rn Rn dw (|w| + )r Rn exp M (a – δ)x Dk φ(x)p dx exp |x||τ | – M (a – δ)x p dx p  r r p . Now using the Young inequality (.) and the arguments of [, p.], we get τ [ γτ ] (σ sin α)k ψ(σ sin α) ≤ D + Dk,ρ,α,r e[ γ ] . k+,ρ,α,r e r In the above expression, we set γ = ( a +ρ), since γ = a–δ and ρ is arbitrarily small together with δ. Therefore, we have [ γτ ] (σ sin α)k ψ(σ sin α) ≤ C r . k,ρ,| sin α| e r Theorem . Let M(x) and (y) be the pair of functions which are dual in the sense of Young. Then r Fα W ,b,p ⊂ WM,  b for any  ≤ p, r < ∞. Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 6 of 15 i|z| cotα Proof Let e–  φ(z) ∈ W ,b,p (Cn ) and σ = w + iτ . Then from the arguments of [, Theorem .], we have k D ψ(w sin α) ≤ Ck,r,η,s,ρ exp –y, w + (b + ρ)y w ≤ Ck,r,η,s,ρ exp –y w +  (b + ρ )y – · · · – yn wn + n (bn + ρn )yn . From the arguments of [, p.] we have  k D ψ(w sin α) ≤ Ck,ρ,r,η,s exp –M  – δ w – M ρ w . w b b Hence, exp M  – δ w Dk φ(w sin α) ≤ C r w k,ρ,r,η,s . b r This implies that r ψ(w sin α) ∈ WM, . b Theorem . Let M (x) and  (y) be the functions which are dual in the sense of Young to the functions M(x) and (y), respectively. Then ,b,p ,  ,r Fα WM,a, ⊂ WM , a  b i|x| cotα for any  ≤ p, r < ∞. ,b,p Proof Let e–  φ(x) ∈ WM,a (Rn ) and σ = w + iτ . Then by the technique of [, pp.] and (.), we have ψ(σ sin α) ≤ (πi sin α) –n e inα  –n inα ≤ (πi sin α)  e  ≤ DCρ,δ e–w,y ≤ Cρ,δ,α e–w,y Rn Rn Rn Rn iσ ,z e φ(z) dx –w,y–τ ,x e φ(z) dx –τ ,x e exp –M (a – δ)x + (b + ρ)y dx exp –M (a – δ)x + (b + ρ)y exp |τ ||x| dx = Cρ,δ,α exp –w, y + (b + ρ)y Rn exp –M (a – δ)x + |τ ||x| dx. Now using the arguments of [, p.], we have  ψ(σ sin α) ≤ Cρ,δ,α exp –M  – δ w +   + ρ τ – M ρ w . b a b Hence, exp M  – δ w –   + ρ τ ψ(σ sin α) ≤ C r . α,δ,ρ b a r Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 7 of 15 3 Relation between W- and W p -types of spaces In this section the mapping relations between W - and W p -types of spaces are discussed. Theorem . Let M(x), (y) be the pair of functions which are dual in the sense of Young. Then p  ≤ p < ∞. WM,a = WM,a , Proof Now, for showing the above theorem we shall prove the following lemma. p Lemma . Let  ≤ p < ∞. Then WM,a ⊂ WM,a . |x| cot α Proof Let e–  rem ., we get p φ(x) ∈ WM,a (Rn ) and σ = w + iτ . Then from the arguments of Theo- p  Fα WM,a ⊂ W , a . (.) From the inverse property of the fractional Fourier transform, we have  p WM,a ⊂ Fα– W , a . (.) Now, let φˆ α (σ ) ∈ W , a . Then by the technique of [, pp.-] and (.), we have  φ(x) = Cα Rn e– i(|x| +|σ | ) cot α +ix,σ csc α  φˆ α (σ ) dw. Therefore, φ (k) (x) = Cα = Cα Rn i|x| cot α i|σ | cot α Dkx e–  +ix,σ csc α e–  φˆ α (σ ) dw k Drx e– i|x| cot α   ix,σ csc α – i|σ | cot α ˆ Dk–r e φα (σ ) dw x e r k i|x| cot α Cη (cot α)xη e–  (iσ csc α)k–r = Cα r Rn η≤r Rn r≤k r≤k i|σ | cot α × eix,σ csc α e–  φˆ α (σ ) dw k i|x| cot α Cη (cot α) e–  (iσ csc α)k–r = Cα r η≤r Rn r≤k i|σ | cot α × Dησ eix,σ csc α e–  φˆ α (σ ) dw k i|x| cot α Cη (cot α)(i csc α)–|η|+|k–r| (–)|η| e–  eix,σ csc α = Cα r η≤r Rn r≤k i|σ | cot α × Dησ σ k–r e–  φˆ α (σ ) dw k i|x| cot α = Cα Cη (cot α)(i csc α)–|η|+|k–r| (–)|η| e–  eix,σ csc α r η≤r Rn r≤k Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 8 of 15 η i|σ | cot α k–r ˆ Dη–β φα (σ ) dw × Dησ e–  σ σ β β≤η k i|x| cot α –|η|+|k–r| |η| Cη (cot α)(i csc α) (–) e–  eix,σ csc α = Cα r η≤r Rn r≤k η i|σ | cot α × Cλ (cot α)σ λ e–  β λ≤η β≤η η – β η–β–m k–r Dσ × φˆ α (σ ) dw. Dm σσ m m≤η–β Hence, k (k) φ (x) ≤ |Cα | Cη (cot α)| csc α|–|η|+|k–r| r η≤r r≤k η (k – r)! Cλ (cot α) × β (k – r – m)! λ≤β β≤η η – β eix,σ csc α σ k–r–m+λ Dη–β–m φˆ α (σ ) dw. × σ m Rn m≤η–β Therefore, k (k) φ (x) ≤ |Cα | Cη (cot α)| csc α|–|η|+|k–r| r η≤r r≤k η (k – r)! Cλ (cot α) × β (k – r – m)! λ≤β β≤η |k–r–m+λ|+ η – β + |σ ||k–r–m+λ| –x,τ csc α |σ | e × (|w| + ) m Rn m≤η–β φˆ α (σ ) dw × Dη–β–m σ  ≤ Cα,k exp –x, τ csc α + +ρ τ . a (.) Now using the arguments of [, p.], we get (k) φ (x) ≤ C exp –M (a – δ)x csc α α,k (.) for arbitrarily small δ together with ρ. Hence the above expression gives  Fα– W , a ⊂ WM,a . (.) Thus (.) and (.) imply that p WM,a ⊂ WM,a . (.) p Lemma . Let  ≤ p < ∞. Then WM,a ⊂ WM,a . Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Proof Let e– follows that i|x| cot α  Page 9 of 15 φ(x) ∈ WM,a (Rn ) and σ = w + iτ ∈ Cn . Then from [, Theorem .], it  Fα (WM,a ) ⊂ W , a . (.) Now by the inverse property of the fractional Fourier transform we have  WM,a ⊂ Fα– W , a . (.) Again let φˆ α (σ ) ∈ W , a . Then from (.) we have  k φ (x) ≤ C exp –x, τ csc α +  + ρ τ . α,k a (.) Using (.) and [, p.] we get k φ (x) ≤ C exp –M (a – δ)x csc α – M a ρ  x csc α . α,k (.) Therefore, exp M (a – δ)x csc α φ k (x) ≤ C e–M[a ρ  x csc α] . α,k p p (.) This implies that  p Fα– W , a ⊂ WM,a . (.) From (.) and (.) we ﬁnd p WM,a ⊂ WM,a . (.) Now from (.) and (.) we get the result p WM,a = WM,a . (.) Theorem . Let M(x) and (y) be the same functions as in Theorem .. Then W ,b,p = W ,b , Proof Let e– i|z| cot α   ≤ p < ∞. (.) φ(z) ∈ W ,b,p . Then from Theorem . it follows that Fα W ,b,p ⊂ WM,  . a (.) By the inverse property of the fractional Fourier transform, we have W ,b,p ⊂ Fα– (WM,  ). a (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 10 of 15 Now let φˆ α (x) ∈ WM,  . Then from the technique of [, pp.-], we have b (iσ csc α)k φ(σ ) i(|x| +|σ | ) cot α  Dkx eix,σ csc α φˆ α (x) dx e– = Cα Rn = Cα (–) |k| Rn i|σ | cot α i|x| cot α eix,σ csc α Dkx e–  φˆ α (x) e–  dx k – i|σ | cot α i|x| cot α ˆ  Dk–r dx Drx e–  x φα (x) e r Rn r≤k k i|x| cot α i|σ | cot α |k| ix,σ csc α ˆ xβ Dk–r e Cβ (cot α)e–  e–  = Cα (–) x φα (x) dx. r β≤r Rn = Cα (–)|k| eix,σ csc α r≤k Therefore, k –x,τ csc α β k–r Cβ (cot α) e x D φˆ α (x) dx (σ csc α)k φ(s) ≤ |Cα | x r β≤r Rn r≤k k  Cβ (cot α) e[|x||τ csc α|–M[( b –δ)x]] dx. ≤ |Cα | r β≤r Rn r≤k Now using the arguments of [, p.], we get (σ csc α)k φ(σ ) ≤ Cα,k exp (b + ρ)τ csc α , where ρ is arbitrarily small together with δ. Thus we have Fα– (WM,  ) ⊂ W ,b . b (.) Therefore (.) and (.) yield W ,b,p ⊂ W ,b . Again we take e– (.) i|z| cot α  φ(z) ∈ W ,b (Cn ). Then from [, Theorem .], we have Fα W ,b ⊂ WM,  . b (.) By the inverse property of the fractional Fourier transform, we have W ,b ⊂ Fα– (WM,  ). b (.) ˆ ∈ WM,  (Rn ). Then from the arguments of [, pp.-], we Furthermore, we take φ(x) b have (iσ csc α)k φ(σ ) p = Rn p (iσ csc α)k φ(σ )p dw Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 11 of 15 p p |iσ csc α||k|+ + |iσ csc α||k| p φ(σ ) dw = (|w| + ) Rn p p p p |iσ csc α||k|+ p |iσ csc α||k| p dw + dw φ(σ ) φ(σ ) ≤ (|w| + ) (|w| + ) Rn Rn p p –x,τ csc α dw ˆ Cα,k ≤ e φα (x) dx  p Rn (|w| + ) Rn p p –x,τ csc α dw e ˆ φ + (x) dx C α α,k  p Rn (|w| + ) Rn dw ≤ Cα,k Ck+,δ + Cα,k Ck,δ  p Rn (|w| + ) p p  × exp –x, τ csc α – M b – δ x dx n R p dw exp –x, τ csc α – M  – δ x dx ≤ Cα,k,δ  p b Rn Rn (|w| + ) exp –x, τ csc α – M  – δ x dx. ≤ Cα,k,δ Cp (.) n b R Now using the Young inequality (.) and from the arguments of [, p.], we get (iσ csc α)k φ(σ ) ≤ Cα,k,δ,p exp (b + ρ)τ . p This implies that Fα– (WM,  ) ⊂ W ,b,p . (.) b Now (.) and (.) give W ,b ⊂ W ,b,p . (.) Finally, (.) and (.) give W ,b = W ,b,p . (.) Theorem . Let  (y) and M (x) be the functions which are dual in the sense of Young to the functions M(x) and (y), respectively. Then ,b,p ,b WM,a = WM,a , Proof Let e– i|x| cot α   ≤ p < ∞. (.) ,b,p φ(x) ∈ WM,a . Then from Theorem ., it follows that ,b,p , Fα WM,a ⊂ WM , a .  b (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 12 of 15 By the inverse property of the fractional Fourier transform we get , ,b,p WM,a ⊂ Fα– WM , a . (.)  b  , Now let φˆ α (z) ∈ WM , a . Then from the arguments of [, p.], we get  b φ(σ + iτ ) = Cα Rn e– i(|z| +|σ | ) cot α +iσ ,z csc α  φˆ α (z) dx. Therefore, φ(σ + iτ ) exp –w, y csc α – τ , x csc α φˆ α (z) dx ≤ |Cα | Rn ≤ Cα Cδ,ρ Rn exp –w, y csc α – τ , x csc α   × exp –M – δ x +  + ρ y dx b a  + ρ y – w, y csc α ≤ Cδ,ρ,α exp  a  – δ x + τ , x csc α dx. exp –M × b Rn Now using (.), we have φ(σ + iτ ) ≤ C exp –M w csc α + τ csc α δ,ρ,α  +ρ ( b + δ) a   ≤ Cδ,ρ,α – δ w csc α + + ρ τ csc α , exp –M a b where ρ and δ are arbitrarily small together with ρ and δ, respectively. This shows that ,b, ,b . Fα– WM ,a ⊂ WM,a (.) Thus from (.) and (.), we get ,b,p ,b . WM,a ⊂ WM,a (.) Similarly it is easy to show that ,b,p ,b ⊂ WM,a . WM,a (.) Finally, (.) and (.) imply that ,b,p ,b . WM,a = WM,a (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 13 of 15 4 Uniqueness class of a Cauchy problem In this section we apply the theory of the fractional Fourier transform which is discussed in (.) and (.) to establish a uniqueness theorem for the Cauchy problem: ∂u(x, t) = P(ix )u(x, t), ∂t ∀(x, t) ∈ Rn × [, T], u(x, ) = u (x), (.) (.) where kx = kx · · · kxnn k kn ∂ ∂ = – ix cot α ··· – ixn cot α ∂x ∂xn (.) (.) is a diﬀerential operator and u(x, t) is an N ×  column vector. Here P is an N × N polynomial matrix with constant coeﬃcients of order k. A similar problem has been investigated by Gel’fand and Shilov [], and Friedman [] by exploiting the theory of Fourier transforms. Also, Pathak [] studied the uniqueness of the Cauchy problem by using the theory of the Hankel transform. Theorem . The Cauchy problem (.) and (.) possesses a unique solution u(x, t) in the   , a–θ  )  , b–θ space (WM for the interval  ≤ t ≤ T, T < (cp )– (d/)p , θ < a, and for any initial function u (x) belonging to the same space, where p is the reduced order of the system (.) ∂ and (.) with ix replaced by i ∂x and c being a constant depending on P. Proof From the fundamental result [, p.], the Cauchy problem (.) and (.) will have a solution in the space  for  ≤ t ≤ T if there exists a solution of the adjoint problem, ∂ φ(x, t) = P˜ i∗x φ(x, t), ∂t (.) φ(x, t ) = φ (x) ∈ , (.) in the space  for  ≤ t ≤ t , where t is any point in the interval  ≤ t ≤ T, P˜ is the adjoint of P and ∗x is the conjugate of x . Applying the fractional Fourier transform to (.) and (.), we get d ˜ csc α)Ψ (σ , t), Ψα (σ , t) = P(σ dt (.) Ψα (σ , t ) = Ψα, (σ ), (.) where Ψα (σ , t) = (Fα φ)(x, t). A formal solution of (.) and (.) is given by ˜ csc α) Ψα, (σ ). Ψα (σ , t) = exp (t – t )P(σ (.) Let us write ˜ csc α) , Q(σ csc α, t , t) = exp (t – t )P(σ (.) Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 14 of 15 consisting of entire analytic functions of σ where σ = w + iτ . Since p is the reduced order of the system (.) and (.), using the inequality |σ csc α|p ≤ p |w csc α|p + |τ csc α|p and the arguments of [, p.] in (.) we obtain Q(σ csc α, t , t) ≤ C exp (p )– θ p |w csc α|p + |τ csc α|p under the assumptions t ≤ t ≤ t + T and p + cT < (p )– θ p . If we set M(w csc α) = |w csc α|p /p , (τ csc α) = |τ csc α|p /p , then Q(σ csc α, t , t) ≤ C exp M(θ · w csc α) + (θ · τ csc α) . Now, let us assume that , φ (x) ∈ = WM , a .  b Then ,b ψα, (σ ) = (Fα φ )(x) ∈ WM,a . We now apply the theorem [, p.] for given a. One can always choose the time interval  ≤ t ≤ T so small that the inequality θ < a holds; for such values of T the matrix ,b Q(σ csc α, t , t) will be a multiplier in the space WM,a which maps this space into the space ,b+θ WM,a–θ taking T suﬃciently small. Thus the Cauchy problem (.) and (.) has a unique ,b+θ . Also we can show that solution in WM,a–θ ,b+θ  ,  Fα– WM,a–θ =  = WM , a–θ  ,  b+θ   , a–θ and the Cauchy problem (.) and (.) has a unique solution in WM   , b+θ . Now using the arguments of [, Theorem , p.], we get the complete proof. Competing interests The authors declare that they have no competing interests. Authors’ contributions The main idea of this paper was proposed by SKU. AK prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the ﬁnal manuscript. Acknowledgements 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. Received: 25 November 2014 Accepted: 23 December 2014 Upadhyay and Kumar Journal of Inequalities and Applications (2015) 2015:31 Page 15 of 15 References 1. Gurevich, BL: New types of test function spaces and spaces of generalized functions and the Cauchy problem for operator equations. Dissertation, Kharkov (1956) 2. Gel’fand, IM, Shilov, GE: Generalized Functions, vol. 3. Academic Press, New York (1967) 3. Friedman, A: Generalized Functions and Partial Diﬀerential Equations. Prentice Hall, New York (1963) 4. Pathak, RS, Upadhyay, SK: W p -Space and Fourier transformation. Proc. Am. Math. Soc. 121(3), 733-738 (1994) 5. Betancor, JJ, Rodriguez-Mesa, L: Characterization of W-type spaces. Proc. Am. Math. Soc. 126(5), 1371-1379 (1988) 6. Upadhyay, SK: W-Spaces and pseudo-diﬀerential operators. Appl. Anal. 82, 381-397 (2003) 7. De Bie, H, De Schepper, N: Fractional Fourier transforms of hyper complex signals. Signal Image Video Process. 6, 381-388 (2012) 8. Upadhyay, SK, Kumar, A, Dubey, JK: Characterization of spaces of type W and pseudo-diﬀerential operators of inﬁnite order involving fractional Fourier transform. J. Pseud.-Diﬀer. Oper. Appl. 5(2), 215-230 (2014) 9. Prasad, A, Mahato, A: The fractional wavelet transform on spaces of type W. Integral Transforms Spec. Funct. 24(3), 239-250 (2012) 10. Pathak, RS: On Hankel transformable spaces and a Cauchy problem. Can. J. Math. 34(1), 84-106 (1985)