Fourier transforms in the “classical sense”, Schur spaces and a new formula for the Fourier transforms of slowly increasing, O(p,q) -invariant functions

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Indagationes Mathematicae 24 (2013) 142–160 www.elsevier.com/locate/indag

Fourier transforms in the “classical sense”, Schur spaces and a new formula for the Fourier transforms of slowly increasing, O( p, q)-invariant functions Norbert Ortner ∗ University of Innsbruck, Institute of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria Received 18 February 2011; received in revised form 8 November 2011; accepted 27 July 2012 Communicated by E. van den Ban

Abstract The calculation of Fourier transforms F T of integrable distributions T ∈ D′ 1 gives rise to the question L of its “pointwise” calculation, i.e., the question if the relation lim j→∞ ⟨ϕ, T j ⟩ = ⟨ϕ, T ⟩ for each ϕ ∈ D L ∞ , ′ is sufficient to prove the convergence of the sequence (T j ) j∈N , T j ∈ D 1 , to the limit T. Since D′ 1 is L L a Schur space, pointwise convergence suffices (for sequences). The fact that D′ 1 is a Schur space can be L ˆ ′ . A generalization of this reasoning is given in Chapter 2. derived from the isomorphism D′ 1 ≃ ℓ1 ⊗s L    In Chapter 3, a representation of the Fourier transform F f [x, x] is given, [, ] denoting the quadratic form x12 + · · · + x 2p − x 2p+1 − · · · − xn2 and f a slowly increasing C ∞ –function. ′ ) The representation is a vector-valued integral ⟨1σ , F f (σ )K (σ,ξ )⟩ with F f (σ )K (σ,ξ ) ∈ D′ 1 (OC,ξ L ,σ and with an explicitly given kernel K (σ, ξ ).

c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: (vector-valued) distributions; Fourier transforms; Locally convex spaces; Fundamental solutions; Sobolev’s operator

∗ Tel.: +43 0 512 507; fax: +43 0 512 507.

E-mail address: [email protected]. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights 0019-3577/$ - see front matter ⃝ reserved. doi:10.1016/j.indag.2012.07.004

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0. Introduction and notation The present paper arose in an attempt to make precise the following statement in Gelfand’s and Shilov’s treatise on generalized functions [7, p. 190]: “If f (x) is a function that has a Fourier transform in the classical sense, then f˜(= F f ) is the regular functional corresponding to the Fourier transform of f (x).” First I shall illustrate the calculation of “Fourier transforms of functions in the classical sense” by six examples (see 1.1) for which I define six corresponding distributions (see 1.2) whose “distributionally correct” Fourier transforms in the space S ′ of temperate distributions are deduced in 1.3. As a matter of fact, formulae for Fourier transforms are often valid in “smaller” spaces of distributions than in S ′ (see 1.4); this means that, e.g., the limit relation    N −ixξ e −1 1 dx = −π|ξ | F Pf 2 = lim 2 N →∞ −N x x is valid not only in S ′ but also in O0M , the space of slowly increasing continuous functions (equipped with the usual inductive limit topology). The simple Propositions 1 and 2 (in 1.5 and 1.6) allow the calculation of Fourier transforms F T for integrable distributions T ∈ D′L 1 by approximating F T by a sequence F Tk such that F Tk can be calculated “more easily”. The convergence Tk −→ T in S ′ (for k → ∞) is equivalent with “pointwise convergence”: ⟨ϕ, Tk ⟩ −→ ⟨ϕ, T ⟩ in C for k → ∞ for each ϕ ∈ S individually. This is not necessarily the case in “smaller” spaces of distributions, e.g., in D′L p , 1 < p ≤ ∞. But it is valid for the space of integrable distributions D′L 1 since it is a Schur space. Hence, I shall investigate Schur spaces in Section 2. The main result (Proposition 3) implies that completed tensor products of nuclear spaces with Schur spaces are Schur spaces. It is an easy consequence of a “permanence”-assertion in Grothendieck’s thesis. Finally, I give a new representation of the Fourier transform F ( f ([x, x])) for functions f ∈ O M (R). The indefinite quadratic form [x, x] is invariant under the action of a pseudoorthogonal group O( p, q). The new representation (Proposition 5) consists in vector-valued integration of F f (σ )K (σ, ξ ) with respect to σ involving the oscillating kernel n

K (σ, ξ ) =

π 2 −1 2|σ |

n 2

ei

π 1 4 ( p−q) sign σ − 4σ

[ξ,ξ ]

′ ˆ C,ξ ∈ OC,σ ⊗O .

ˆ C′ ) An application of (a vector-valued version of) Proposition 5 (involving f ∈ O M (R)⊗O is given in Proposition 6 where the fundamental solution of Sobolev’s operator ∂t − ∂x3yz is calculated. Some of my considerations were presented at the workshop “Generalized Functions and Partial Differential Equations” (Innsbruck 2009, 19.20./02). Hereby I posed the question whether ˆ 1 is a Schur space if E is nuclear. The answer is given in Proposition 3. E ⊗ℓ Notation Constant use of Schwartz’ theory of distributions is made as explained in his treatises [30,28], cf. also Horv´ath’s textbook [14]. Thus the spaces D ⊂ S ⊂ O M denote compactly supported, rapidly decreasing and slowly growing C ∞ -functions respectively. OC′ and S ′ denote the spaces of rapidly decreasing and temperate distributions, respectively. The spaces D L p , 1 ≤ p < ∞, and B˙ denote the C ∞ functions whose derivatives are in L p (Rn ) and C0 (Rn ), respectively. ˙ ′ of integrable distributions whereas D′ p = The strong dual of B˙ is the space D′L 1 = (B) L 1 1 ′ (D L q ) , p + q = 1, 1 ≤ q < ∞.

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D L ∞ = B is the dual of D′L 1 . If B is equipped with the topology of uniform convergence on compact subsets of D′L 1 , it is denoted by Bc . Hence, (Bc )′ = D′L 1 and Bc is semireflexive. The spaces OC′ ⊂ D′L p ⊂ S ′ ⊂ D′ are endowed with their strong topologies, respectively. χ A denotes the characteristic function of the subset A ⊂ Rn and Y = χ(0,∞) is the Heaviside 1 function. For x, ξ ∈ Rn we write |x| = (x12 + · · · + xn2 ) 2 and xξ = x T ξ = x1 ξ1 + · · · + xn ξn . B N = B N (0) is the ball in Rn with centre at 0 and with radius N ; τa denotes the translation x −→ τa (x) = x − a.N(N0 ) is the set of natural numbers (including 0). J0 , N0 , K 0 denote the Bessel, the Neumann and the McDonald function of order 0, respectively; Si is the function sine integral. In contrast to Schwartz’ convention, the Fourier transform is defined by  Fϕ(ξ ) = e−ixξ ϕ(x) dx, ϕ ∈ S(Rn ). Rn

For tensor products of locally convex spaces and for the theory of vector-valued distributions, I use Schwartz’ and Grothendieck’s treatises [11,28,29] and Treves’ text book [34], for locally convex topological vector spaces [4,24,25]. Whereas O M (Rn ) is the space of complex-valued C ∞ functions whose derivatives grow at most like polynomials, the space O M (OC′ ) denotes the space of slowly growing C ∞ functions f , defined on Rn , with values in OC′ , i.e., ˆ C′ f ∈ O M (OC′ ) ≃ Lε (OC , O M ) ≃ Lε (O′M , OC′ ) ≃ O M ⊗O wherein Lε (E, F) is the space of linear and continuous mappings endowed with the topology of uniform convergence on equicontinuous sets. The partial Fourier transform is used as explained in [28, I, p. 123] and in [29, Lecture 8, p. 37]. 1. Functions which have Fourier transforms in the classical sense 1.1. Six classical integral formulae To illustrate our subsequent considerations let us consider the following formulae: ∞ x) dx = π2 , ξ > 0 [9, 333, (14a), p. 119]; (a) 0 sin(ξ x  ξ cos x−1 ∞ (b) 0 dx + ξ cosx x dx = −C − log ξ, ξ > 0 ([9], 333, 31, p. 123; C: Euler’s constant); x ∞ )−1 (c) 0 cos(xξ dx = − π2 ξ, ξ ≥ 0 [9, 333, (20a), p. 120]; x2   ∞ sin( xt −ξ x ) 0 if t > 0, ξ > 0, (d) 0 dx = π J0 2t|ξ |  if t > 0, ξ < 0; ([9, 334,11a,11b, p. 132]. Note that the limit x t ↘ 0 does not reduce formula (d) to (a).)     −ixξ −ixξ (e) |x|≤1 e |x|2−1 dx + |x|≥1 e|x|2 dx = 2π log |ξ2| − C , ξ ∈ R2 \ {0}; [36, pp. 117,118]   n inπ i|ξ |2 2 (f) Rn e−it|x| −ixξ dx = πt 2 e− 4 + 4t , t > 0, ξ ∈ Rn ; ([13, Theorem 7.6.1, p. 206]; [39, p. 160]; [6, Theorem 2, p. 257]). 1.2. Six associated distributions Up to multiplication by constants the formulae (a)–(d) can be interpreted as Fourier transforms i t/x 1 , x12 , e x in dimension 1, formula (e) represents the Fourier transform of the “functions” x1 , |x| of the “regularized” function

1 |x|2

in R2 , and formula (f) is the well-known limiting case of the 2

Fourier transform of the Gauß kernel e−z|x| , Re z > 0, in Rnx .

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Intending to interpret the formulae (a)–(f) as Fourier transforms in distribution theory we must associate suitable distributions to these “classical” functions: In case (a), the principal value distribution vp x1 , defined by    1 ϕ(x) − ϕ(0)χ(−1,1) (x) ϕ, vp = dx, ϕ ∈ D(R); x x R 1 in case (b), the distribution Pf |x| , defined by    1 ϕ(x) − ϕ(0)χ(−1,1) (x) ϕ, Pf = dx, |x| |x| R

in case (c), the distribution Pf x12 , defined by    ϕ(x) − ϕ(0) 1 dx, ϕ, Pf 2 = x x2 R  i t/x  in case (d), the distribution Pf e x , t ∈ R, defined by 

ϕ, Pf



ei t/x x



t

 = R1

 ei x  ϕ(x) − ϕ(0)χ(−1,1) (x) dx, x

in case (e), the distribution Pf |x|1 2 , for ϕ ∈ D(R2 ) defined by    ϕ(x) − ϕ(0)χ B1 (0) (x) 1 ϕ, Pf 2 = dx, |x|2 = x12 + x22 , dx = dx1 dx2 , |x| |x|2 R2 and in case (f), the “regular” distribution, defined by the locally integrable function   2 eit|x| ∈ L ∞ (Rn ) ⊂ L 1loc (Rn ) , t ∈ R. The symbols “vp” and “Pf” mean Cauchy’s “valeur principale” and Hadamard’s “partie finie” respectively which are defined as the finite part of a meromorphic distribution-valued function (cf. [21, Definition 1.5.12, p. 26]). In the concrete cases above these finite parts reduce to the definitions given in (a)–(e). 1.3. Six distributional Fourier transforms Since the distributions, defined in 1.2, (a)–(f), are temperate, their Fourier transforms in S ′ (Rn ) are defined by replacing test functions ϕ ∈ S(Rn ) by Fϕ: (a)′ 



1 ϕ, F vp x





 dx Fϕ(x) − Fϕ(0)χ(−1,1) (x) x R     dx −ixξ = e ϕ(ξ ) dξ − χ(−1,1) (x) ϕ(ξ ) dξ x R R R     N sin(ξ x) = −2i lim ϕ(ξ ) dx dξ N →∞ R x 0 = −iπ ⟨ϕ, sign ξ ⟩. =



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Thus, the distributional content of formula (a) is the equation   1 = −iπ sign ξ in S ′ (R). F vp x (For a derivation using the homogeneity and the parity, see [40, Ex. 73, p. 144, and Sol., p. 159].) ′ (b) Similarly, we obtain   1 = −2C − 2 log |ξ | in S ′ (R), F Pf |x| (c)′

see [16, p. 85] or [15, p. 154]. Due to Pf x12 ∈ D′L 1 , we obtain      1 1 cos(ξ x) − 1 −iξ x = F Pf 2 = e , Pf 2 dx = −π |ξ |, x x D′ x2 R B 1 L

valid in O0M (the space of slowly increasing continuous functions [31, p. 117]; in the notation  ∞ 0 of [21, p. 86, Definition 3.1.1] and of [27, 5◦ , p. 99]), O0M = ∞ k=0 (BC)−k = k=0 (B )−k 0 ′ due to F(D L 1 ) ⊂ O M . Obviously, the last result follows also by use of the homogeneity of Pf x12 or by differentiation. ′ (d) For t > 0 and ϕ ∈ S(R) we have    i t/x  e ϕ, F Pf x   i t/x  e = Fϕ, Pf x   i t/x   e = e−iξ x ϕ(ξ ) dξ − χ(−1,1) (x) ϕ(ξ ) dξ dx R x R R    i( xt −ξ x ) i t/x e − e χ(−1,1) (x) dx dξ = lim ϕ(ξ ) N →∞ x |x|≤N      t N sin t − ξ x − χ (0,1) (x) sin x x = 2i lim ϕ(ξ ) dx dξ N →∞ R x 0       1 t ∞ sin t − ξ x sin x x = 2i ϕ(ξ ) dx − dx dξ x x R 0 0       π  = 2i ϕ(ξ ) π Y (−ξ )J0 2 t|ξ | + Si(t) − dξ, i.e. 2 R   i t/x       e F Pf = i 2π Y (−ξ )J0 2 t|ξ | + 2Si(t) − π x for t > 0 in S ′ (R). In accordance with formula (a) we obtain for t ↘ 0     1 1 F Pf = F vp = i [2π Y (−ξ ) − π ] = −iπ sign ξ. x x

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(e)′ The formula (e) is the two dimensional special case of the Fourier transform   F Pf |x|−n in S ′ (Rn ), given in [30, (VII,7;15), p. 258], i.e.,     n   Γ ′ n2 π2 2 −n F Pf |x| =  n  2 log + n − C . |ξ | Γ 2 Γ 2 (For n = 1, the formula reduces to the result in (b)′ .) (f)′ For t ∈ R \ {0} and ϕ ∈ S(Rn ) we have      −it|x|2 −it|x|2 −iξ x ϕ, F(e ) = e e ϕ(ξ ) dξ dx Rn Rn    −it|x|2 −iξ x = lim ϕ(ξ ) e dx dξ N →∞ Rn [−N ,N ]n     N n  −it|x j |2 −iξ j x j = ϕ(ξ ) lim e dx j dξ. Rn

Due to 

∞

−it x 2 −iξ x

e

 dx =

−∞

j=1

π −i π e 4 |t|

N →∞ −N

sign t+i

ξ2 4t

,

t ∈ R \ {0},

we conclude that −it|x|2

F(e

)=



π |t|

n

2

e−i

nπ 4

sign t+i

|ξ |2 4t

in S ′ (Rn )

for t ∈ R \ {0}. Since, for t ∈ R \ {0}, 2

e−it|x| ∈ OC′ (Rn ) ⊂ D′L 1 (Rn ), the formula is valid in O M (Rn ). A remark concerning the choice of the six integral formulae (a)–(f) and (a)′ –(f)′ : The formulae (a)–(c) are merely chosen for pedagogical reasons, a formula similar to (d) will be used in Section 3; the distributional version (e)′ of (e) furnishes a fundamental solution to the two-dimensional Laplacean: The Fourier transform of the equation ∆2 E = δ furnishes − |ξ |2 F E = 1. Hence, F E = −Pf |ξ1|2 is a solution to the division problem (∗). Therefore,   1 1 1 C − log 2 E = − 2 F Pf 2 = log |x| + 2π 2π 4π |ξ | (cf. [30, (II,3;14), p. 46; (VII,10;21), p. 288]).

(∗)

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Similarly, formula (f)′ furnishes the fundamental solution to the Schr¨odinger operator ∂t − i∆n : By the partial Fourier transform, the uniquely determined, temperate fundamental solution E with support in t ≥ 0, satisfies ∂t (F E) + i |ξ |2 (F E) = δt , and therefore 2

F E = Y (t)e−it|ξ | , or E=

Y (t) −inπ + i|x|2 Y (t) −it|ξ |2 4 4t F(e ) = n e (2π )n (4π t) 2

(cf. [35, (6.10), p. 44]; [30, (VII,10;31), p. 290]). Our considerations in 1.1–1.3 confirm the assertion of Gelfand and Shilov [7, p. 190]: “If f (x) is a function that has a Fourier transform in the classical sense, then f˜(= F f ) is the regular functional corresponding to the Fourier transform of f (x).” Only the expressions “function” and “regular functional” should be made more precise as we indicated in Section 1.2. We shall attempt to sharpen the statement of Gelfand and Shilov in Propositions 1 and 2 (Sections 1.5 and 1.6). 1.4. The Fourier transforms (a)′ –(e)′ in “smaller” spaces of distributions Improper integrals are limits which yield the validity of the formulae (a)–(e) pointwise (if regarded as functions of the parameters). The derivations in (a)′ –(e)′ yield the validity in S ′ (Rn ). Let us explain the difference, guided by the formulae (a), (a)′ : N (a) means lim N →∞ 0 sinxξ x dx = π2 sign ξ , if ξ ∈ R \ {0} is fixed. The derivation in (a)′ shows that   N     1 sin ξ x = −2 i lim ϕ(ξ ) dx dξ ϕ, F vp N →∞ R x x 0 = −i π ⟨ϕ, sign ξ ⟩, for ϕ ∈ S(R), i.e., by Proposition 2 in [14, p. 231], in S ′ (R),    N 1 sin ξ x F vp = −2 i lim dx = −2 i lim (sign ξ Si (N |ξ |)) = −i π sign ξ N →∞ 0 N →∞ x x (since S ′ is a Montel space). The decomposition vp x1 = vp Y (1−|x|) + Y (|x|−1) ∈ E ′ (R) + L p (R) (where the multiplication x x Y (1 − |x|) · vp x1 is well-defined due to sing supp Y (1 − |x|) = {±1},

sing supp vp

1 = {0} x

and where 1 < p ≤ ∞) shows that the limit relation, π  π lim Si(N ξ ) = Si(ξ ) + sign ξ − Si(ξ ) = sign ξ, N →∞ 2 2

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N. Ortner / Indagationes Mathematicae 24 (2013) 142–160 q

holds in O M (R) + L q (R) ⊂ L loc (R), 2 ≤ q < ∞, i.e.  N sin ξ x π q lim dx = sign ξ in L loc (R). N →∞ 0 x 2 (This result was derived by the application of general theorems on the Fourier transformation. A direct inspection of the limit relation shows that the convergence holds even in L q (R), 1 < q < ∞.) If we consider Example(c)′ we observe that lim Pf

N →∞

Y (N − |x|) 1 = Pf 2 in D′L 1 (R), 2 x x

since D′L 1 is a Schur space (cf. Chapter 2) and since for ϕ ∈ B      ϕ(x) − ϕ(0) 1 Y (N − |x|) = lim dx = ϕ, Pf . lim ϕ, Pf N →∞ |x|≤N N →∞ x2 x2 x2 Therefore,  lim

N

N →∞ −N

  1 cos(xξ ) − 1 dx = F Pf = −π |ξ | x2 x2

in O0M (Rξ ) (⊃ FD′L 1 ); this relation implies, in particular, that:  N cos(xξ ) − 1 dx = −π|ξ | in L ∞ lim loc (R), N →∞ −N x2 i.e., locally uniformly. Concerning Example(f)′ we have: 2

2

lim e−it x χ(−N ,N ) = e−it x ,

N →∞

t ∈ R \ {0}, in D′L 1 (Rx ) 2

2

since D′L 1 is a Schur space and since lim N →∞ ⟨ϕ, e−it x χ(−N ,N ) ⟩ = ⟨ϕ, e−it x ⟩ for ϕ ∈ B, namely  2 lim ϕ(x)e−it x dx = 0 for ϕ ∈ B. N →∞ |x|≥N

Hence, 2

2

e−it|x| = lim e−it|x| χ(−N ,N ) (x1 ) . . . χ(−N ,N ) (xn ), N →∞

in D′L 1 (Rn ), and therefore,  −it|x|2 F(e ) = lim

N →∞ [−N ,N ]n

 =

π |t|

n

2

e−i

nπ 4

e−it|x|

2 −ixξ

sign t+ i |ξ4t|

It was remarked by Cayley that the integral  2 e−it|x| −ixξ dx, t ∈ R \ {0}, Rn

2

dx in O0M .

t ∈ R \ {0},

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is not an improper integral in the usual sense — its convergence depends on the kind of approximation of Rn (cf. [2, pp. 624–626]): If B1 (0) is the unit ball in Rn , then lim N →∞  −it|x|2 −ixξ dx, t ∈ R \ {0}, does not exist for ξ ∈ Rn if n > 1. N B1 (0) e 1.5. Fourier transforms of locally integrable functions which also are integrable distributions In order to treat Fourier transforms as in Example (f)′ we obtain immediately the following. Proposition 1. If T ∈ L 1loc (Rn ) ∩ D′L 1 (Rn ) and if T · χ N K −→ T in D′L 1 for N → ∞, then  e−iξ x T (x) dx in O0M (Rnξ ). F T (ξ ) = lim N →∞ N K

Herein, K is a (measurable) neighbourhood of 0 in Rn . Proof. For T ∈ D′L 1 , F T (ξ ) = B ⟨e−ixξ , Tx ⟩D′ = lim ⟨e−ixξ , T χ N K ⟩ L1

with T χ N K ∈

L 1 (Rn ).

N →∞



L 1loc (Rn )

The assumption T ∈ has two reasons: By multiplying T by χ N K we obtain the “usual” cut-off approximations whose Fourier transforms are O M -functions, given by an “ordinary” Lebesgue integral. In the derivations in (a)′ and (c)′ we even used the cut-off approximations though the distributions vp x1 , Pf x12 ̸∈ L 1loc . Nevertheless we had   χ  1 Y (N − |x|) N B1 vp = lim vp = lim vp in E ′ + L p , N →∞ N →∞ x x x and Pf

χ  1 N B1 = lim N →∞ x2 x2

in D′L 1 .

1.6. Fourier transforms of integrable distributions 2

The “usual” way of computing the Fourier transform F(eit|x| ) in Example(f)′ is by means of the Gauß–Weierstraß approximation 2

e−it|x| = lim e−ε|x|

2 −it|x|2

ε↘0

,

valid in D′L 1 ,

2

due to limε↘0 e−ε|x| = 1 in Bc , and due to the partial continuity of the multiplication mapping Bc × D′L 1 −→ D′L 1 ,

(ϕ, T ) −→ ϕ · T.

Hence, 2

2

F(e−it|x| ) = lim F(e−(ε+it)|x| ) ε↘0

in O0M (Rn )

wherein the last Fourier transform is obtained as analytic continuation of 2

F(e−z|x| )

for Im z = 0, z > 0.

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Obviously we have, more generally, 2

F T = lim (e−ε|x| T ) in O0M (Rn ), ε↘0

for T ∈ D′L 1 (Rn ), and 2

F T = lim (e−ε|x| T ) in O M (Rn ), ε↘0

for T ∈ OC′ (Rn ). Hence, the limit relation 2

2

F(e−it|x| ) = lim F(e−(ε+it)|x| ), ε↘0

t ∈ R \ {0},

even holds in O M (Rn ). Noteworthy enough, even less strong summability procedures allow representations of F T for T ∈ D′L 1 (Rn ). We obtain, e.g., a representation of the Fourier transform of integrable distributions by means of spherical Riesz means in the next proposition, i.e., a representation as a limit of Fourier transforms of distributions with compact support. Proposition 2. Let T ∈ D′L 1 (Rn ) be an integrable distribution. There exists m ∈ N0 such that   m |x|2 F T = lim F 1− 2 χ BN T in O0M . N →∞ N Proof. According to [30, Th´eor`eme XXV, p. 201], we have T = m ∈ N and f α ∈ L 1 (Rn ). Hence, T ∈ D′Lm−1 , 1 wherein the spaces D′Lm1 are defined in [27, p. 99]. Due to m  |x|2 χ B N ∈ B m−1 1− 2 N   2 m the product 1 − |x| χ B N T is well-defined. Since N2  m |x|2 1− 2 χ B N f α −→ f α N

in L 1 for N → ∞,

it follows   m |x|2 α ∂ 1− 2 χ B N f α −→ ∂ α f α N

in D′L 1

and  m |x|2 1− 2 χ B N ∂ α f α −→ ∂ α f α N

in D′L 1



|α|≤m−1 ∂

α

f α for some

152

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if |α| ≤ m − 1, due to   m |x|2 β ∂ χ BN → 0 1− 2 N if 0 < |β| ≤ m − 1. Thus m  |x|2 χ B N T −→ T 1− 2 N

in D′L 1

in D′L 1 .

The continuity of the Fourier transform F : D′L 1 −→ O0M 

finishes the proof.

2

Remarks. (a) In Example (f)′ we have taken m = 0 for T = e−it x , t ∈ R \ {0}, n = 1. As 2 the corresponding example in 1.4 indicates, we cannot take m = 0 if n ≥ 2 and T = e−it|x| , t ∈ R \ {0}. (b) One could argue that the Fourier transforms of integrable distributions can always be calculated as finite sums of monomials multiplied by Fourier transforms of integrable functions, i.e.,  FT = (iξ )α F f α . |α|≤m−1

But the simple example of 2

T = eix = f 1 + f 2′′ ∈ D′L 1 (R1 ), 1 3 2 2 2 f 2 = 2 (1 − eix ), f 1 = 4 (eix − 1 − ix 2 eix ), 2x 4x shows that the actual calculation by means of the decomposition F T = F f1 − x 2F f2 is a very complicated one. Without proof let me finally mention that for T ∈ E(Rn ) ∩ S ′ (Rn ),  F T = lim T (x)e−iξ x dx in S ′ (Rn ). N →∞ B N

This representation yields, e.g., F(eλx+i e ) = Γ (λ − iξ ) e(ξ +iλ)π/2 x

in S ′ (R) for 0 < λ <

1 . 2

Note that ⟨ψ, eλx+i e ⟩ for ψ ∈ S(R1 ) cannot be written as the Lebesgue integral  x ψ(x)eλx+i e dx, x

R

since |eλx+i e | = eλx . x

Wagner has found this example of a function in R1 not belonging to any L p (R)-space, but with a Fourier transform in C0 (R).

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2. Schur spaces If we want to calculate the Fourier transform of a distribution T , belonging to one of the spaces E′ ∪ D

⊂ OC′ ∪ ⊂ S

⊂ ⊂

S′ ∪ OM ,

by approximation, using a convergent sequence (T j )N in these spaces, it is necessary and sufficient to prove that ⟨ψ, T j ⟩ −→ ⟨ψ, T ⟩,

j → ∞,

for each ψ in the dual space: All these spaces are Montel spaces and thus, weakly convergent sequences converge strongly [4, IV. 18, Proposition 8]; [14, Proposition 2, p. 231, Corollary, p. 232]). This is not the case if T, T j ∈ D′L p , 1 < p, since weakly convergent sequences in D′L p are not necessarily strongly convergent: Modifying the example given in [30, p. 200], we observe that the sequence (τ j δ) j∈N converges weakly in D′L p , 1 < p, due to ⟨ψ, τ j δ⟩ = ψ( j) → 0

if j → ∞, ψ ∈ D L q ,

1 1 + = 1. p q

But the sequence (τ j δ)N does not converge strongly: There exist bounded sets B in D L q such that   sup ⟨ψ, τ j δ⟩ 9 0 for j → ∞. ψ∈B

      − n+1 2 N ∈N . Take, e.g., B = τ N 1 + |x|2 Note that the sequence (τ j δ) j∈N does not converge weakly in D′L 1 since there exists ψ ∈ D L ∞ = B such that ⟨ψ, τ j δ⟩ = ψ( j) 9 0

for j → ∞.

This is a consequence of the fact that, in D′L 1 , weakly convergent sequences are also strongly convergent — due to the so-called Schur property which we are going to define now. Definition. A Hausdorff locally convex topological vector space E is a Schur space (E has the Schur property) if weakly compact sets in E are relatively compact. With respect to the sequence space ℓ1 , the definition is given in [23, 3.5.3, p. 81], with respect to the space of integrable distributions D′L 1 in [5, p. 52]. Generalizing these examples we use the name “Schur space” analogously, e.g., to “Mackey space”. A Hausdorff locally convex topological vector space is a Mackey space if and only if equicontinuous and weakly relatively compact, absolutely convex sets in the dual space E ′ coincide [25, p. 132]. Examples. (a) The classical example of a Schur space is ℓ1 : [23, 3.5.3, p. 81], [10, pp. 161,175, 406]; [34, p. 451], [26, p. 82, III]. (b) Every semi-Montel space [14, Definition 1, p. 231] is a Schur space since weakly compact sets are bounded and bounded sets are relatively compact.

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(c) The space D′L 1 (Rn ) of integrable distributions is a Schur space [5, Corollary (3.5a), p. 71, and Remark 3.9, p. 74]. The Schur property of D′L 1 is contained in Th´eor`eme XXV, Remarque 2◦ , [30, p. 202]: “une forme plus fine . . . ”, in [12, p. 546], and in Teorema 2 [1]. The proofs (in [30] and in [12] only sketched) invoke the parametrix method and the Theorem of Eberlein–Shmulyan, whereas Remark 3.9 in [5, p. 74], makes use of the isomorphism ˆ ′ D′L 1 ≃ ℓ1 ⊗s ([38, 3.2 Theorem, p. 415]; [37, Theorem 1, p. 766]) and of Proposition 24.3 in [11, p. 116]. ˆ ′ follows from A further proof of the Schur property by means of the representation ℓ1 ⊗s Proposition 2 in [3, p. 293]. I gave a proof of the Schur property of D′L 1 without the parametrix method but by making use of completed tensor products and de Wilde’s closed graph theorem [17, 8.4, p. 77–79]. (d) In [17, 7.3, p. 67], it is also shown that D L 1 is a Schur space. (e) L 1 (Rn ) is not a Schur space since the set    ei N x χ(0,1)  N ∈ N is weakly compact in L 1 (R) but not relatively compact. Due to the representations ˆ ℓ1 ≃ ℓ1 ⊗C, ˆ D L 1 ≃ ℓ1 ⊗s, ˆ ′, D′L 1 ≃ ℓ1 ⊗s the above mentioned results can be recovered from the following. ˆ ε F is Proposition 3. Let E be a quasicomplete Schwartz space and F a Schur space. Then E ⊗ a Schur space. Recall that a locally convex space is a Schwartz space if and only if all continuous linear maps into any Banach space are compact ([10, pp. 332–333]; [14, Definition 1, p. 275]). The topologies ε and π are defined in [34, Definition 4.3.1, 4.3.2, p. 434]. The proof is an easy consequence of Proposition 24.3 in [11, p. 116]: ˆ ε F = Lε (E τ′ , F) is weakly compact if M is bounded and if the sections A set M ⊂ E⊗  ′ ′ M(x ) = u(x )  u ∈ M , x ′ ∈ E ′ , are weakly relatively compact in F. Since F is a Schur ˆ ε F.  space, M(x ′ ) is relatively compact, and hence, M is relatively compact in E ⊗ Grothendieck states, in Proposition 13 [11, p. 76] some permanence properties for completed ˆ of a nuclear space E and a Hausdorff, locally convex, topological vector tensor products E ⊗F ˆ has the same property, space F: if F is semi-reflexive or semi-Montel or nuclear then E ⊗F respectively. A complement to this proposition is the following. ˆ is also a Schur space. Corollary. Let E be a nuclear space and F a Schur space. Then, E ⊗F Proof. Analogously to the reasoning in the proof of Proposition 13 in [11, p. 76], the space E can ˆ remains the same space if E is replaced by its completion. be assumed to be complete since E ⊗F ˆ The assertion follows from Proposition 3 by taking into account that E ⊗F ≃ Lε (E τ′ , F) (Theorem 6 in [11, p. 34], and that E is a complete Schwartz space [11, p. 38, Corollary 1]). 

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155

Apart from the examples given above let me point out that the following spaces are Schur spaces: (a) The space of partially summable distributions ˆ ′1 D′ ⊗D L ([28, Chapter I, p. 130]; [12, p. 536]; [19, p. 319]). (b) The space of semitemperate and partially summable distributions ˆ ′1 S ′ ⊗D L ([28, Chapter I, p. 123 and p. 130]; [12, p. 549]; [19, p. 325]; [20]). (c) The convolution algebra ˆ ′ 1 = O M (D′ 1 ). O M ⊗D L L (d) The convolution algebras E ′ (D′L 1 )

and

OC′ (D′L 1 ),

wherein the convolution is defined according to Proposition 34 in [28, Chapter II, p. 151]. (e) The convolution algebra of holomorphic, integrable distributions ˆ ′ 1 = H(Ω , D′ 1 ), H(Ω )⊗D L L Ω ⊂ Cn open. An immediate consequence of Proposition 3 is the next assertion on weakly continuous mappings. Proposition 4. Let X be a metrizable topological space and F a Schur space, f : X −→ F a map. Then, f is weakly continuous if and only if it is continuous.    Proof. Let (x j ) j∈N be a sequence in X converging to x0 in X . The set K = x j  j ∈ N ∪ {x0 } is compact and hence f (K ) is weakly compact. By the Schur property of F, f (K ) is relatively compact which proves the continuity of f.  Remarks. (a) Proposition 4 is a generalization of the corresponding Proposition 36.11 for Montel spaces [34, p. 377, Exercise 36.4]. (b) Proposition 4 simplifies the proof of Proposition 6 in [19, p. 330], considerably. (c) If F is a complete Schur space, Proposition 4 can be expressed as ˆ ε F = C(X )⊗ ˆ ε Fσ , C(X )⊗ wherein C(X ) = E 0 (X ) is the space of continuous functions endowed with the topology of compact convergence and Fσ the space F with the weak topology σ (F, F ′ ). Note that ˆ ε F = C(X, F) C(X )⊗ by Theorem 44.1 in [34, p. 449]. 3. A new formula for the Fourier transform of a slowly increasing, O( p, q)-invariant function As in [33, Section 2, p. 500], let us denote by [, ] the bilinear form [, ] : Rn × Rn −→ R,

(x, y) −→ x1 y1 + · · · + x p y p − x p+1 y p+1 − · · · − xn yn ,

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p ∈ N0 , n ∈ N, q = n − p ∈ N, and by = O( p, q) the group of linear transformations preserving the quadratic form [x, x]. If f ∈ O M (R) ⊂ S ′ (R) we obviously have f ([x, x]) ∈ O M (Rn ). The next proposition gives a new formula for F ( f ([x, x])). Proposition 5. Let p ∈ N0 , n ∈ N, q = n − p, f ∈ O M (R) and n

K (σ, ξ ) =

π 2 −1 2|σ |

n 2

ei

π i 4 ( p−q) sign σ − 4σ

[ξ,ξ ]

′ ˆ C,ξ . ∈ OC,σ ⊗O

Then, ′ F ( f ([x, x])) = ⟨1σ , F f (σ )K (σ, ξ )⟩ ∈ OC,ξ .

For the proof let us apply the theory of vector-valued distributions [28,29]. (a) We have ˆ M,σ,ξ , δ(σ − s) ⊗ δ(ξ − x) ∈ O′M,s,x ⊗O by [28, Chapter I, p. 102]. Hence, by F(O′M ) = OC and by Proposition 8.1 in [29, p. 37], ′ ˆ ′M,s,x . e−iσ s−iξ x ∈ OC,σ,ξ ⊗O

The multiplication with ˆ C,σ f (s)eiσ [x,x] ∈ O M,s,x ⊗O ˆ M,x ) yields [28, Chapter II, Proposition 2, p. 18] (due to eiσ [x,x] ∈ OC,σ ⊗O ′ ′ ˆ C,σ,ξ f (s)e−iσ s−iξ x+iσ [x,x] ∈ O′M,s,x ⊗O ⊂ OC,σ,ξ,s,x ,

taking into account the hypocontinuity of the multiplication mappings •

•

O M × O′M − → O′M ,

OC × OC′ − → OC′ .

By Fubini’s theorem for integrable distributions (cf. [28, Chapter I, Corollary, p. 136]) and by observing that OC′ ⊂ D′L 1 , we obtain, on the one hand,   ′ OC,ξ ∋ 1σ,s,x , f (s)e−iσ s−iξ x+iσ [x,x] = 2π ⟨1σ , (Fs f )(σ )K (σ, ξ )⟩ 1 1 Fx (eiσ [x,x] ) = 2π ⟨1x , e−ixξ +iσ [x,x] ⟩. if K (σ, ξ ) := 2π The multiplication Fs f (σ )K (σ, ξ ) therein is defined by Theorem 7.1 in [29, p. 31] (Proposition 21 bis, p. 70, in [28, Chapter I]) as the image under the map •

′ ′ ′ ′ ′ OC,σ × OC,σ (OC,ξ )− → OC,σ (OC,ξ ) = OC,σ,ξ .

On the other hand, we have, by Fubini’s theorem,     1σ,s,x , f (s)e−iσ s+iσ [x,x]−iξ x = 2π 1s,x , f (s)δ (s − [x, x]) e−ixξ   = 2π 1x , e−iξ x f ([x, x]) = 2π F ( f ([x, x])) . (b) In order to deduce the formula for K , we use Example (f)′ in Chapter 1 and we obtain iσ (x12 +···+x 2p −x 2p+1 −···−xn2 )

eiσ [x,x] = e

ˆ M,x1 ⊗ · · · ⊗ O M,xn ), ∈ OC,σ ⊗(O

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and n

K (σ, ξ ) =

1 π 2 −1 i π ( p−q) sign σ − i 4 4σ Fx (eiσ [x,x] ) = n e 2π 2|σ | 2

[ξ,ξ ]

.

Note that the distribution ′ ˆ C,ξ = L(OC,ξ , OC,σ ) K (σ, ξ ) ∈ OC,σ ⊗O

is given by the iterated integral    ⟨ψ, K ⟩ = K (σ, ξ )ψ(σ, ξ ) dξ dσ, R

Rn

ψ ∈ S(Rσ × Rnξ ).



Remarks. (a) Under the more restrictive assumption f ∈ S(R), another formula for F ( f ([x, x])) is given in [33, Theorem 1, p. 509]. (b) The representation of F ( f ([x, x])) in Proposition 5 is a distribution-valued integral representation ⟨1σ , F f (σ )K (σ, ξ )⟩ , wherein the integrand, ′ ′ ′ ′ ˆ C,ξ F f (σ )K (σ, ξ ) ∈ OC,σ (OC,ξ ) = OC,σ ⊗O ′ -valued, integrable distribution. is an OC,ξ Hence, F ( f ([x, x])) is given as ⟨1, T ⟩ with T ∈ D′L 1 (E), E ∈ OC′ [28, Chapter I, p. 128]. In concrete situations we can calculate ⟨1, T ⟩, e.g., by Gauß–Weierstraß approximations, i.e., 2

⟨1, T ⟩ = lim ⟨1, e−ε|x| T ⟩ ε↘0

for T ∈ D′L 1 (E).

(c) The assumption f ∈ O M (R) is forced only by the method of proof: The multiplier space of OC′ is the space OC , and of O′M the space O M . In many applications the assumption f ∈ O M is sufficient, for example: (i) for the calculation of the temperate fundamental solution of the iterated Klein–Gordon operators (∂t2 − ∆n − ia)m , m ∈ N, a > 0, by the Fourier transform, take 1 f (s) = ; (s + ia)m (ii) for the calculation of the temperate fundamental solutions (with support in t ≥ 0) of the Schr¨odinger and the vibrating plate operators ∂t ±i∆n and ∂t2 +∆2n by the partial Fourier transform, take sin(ts 2 ) , t > 0. f (s) = e±its and f (s) = s2 However, for the calculation of the temperate fundamental solution (with support in t ≥ 0) of the metaharmonic or the heat operator, −∆n + µ, µ > 0, and ∂t − ∆n , respectively, by 1 the (partial) Fourier transform, we had to take f (s) = |s|+µ and f (s) = e−t|s| , respectively. Though these functions do not belong to O M , Proposition 5 is applicable also in these cases if we impose suitable conditions on f and if we modify the proof.

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More generally, the Poisson–Bochner formula [30, (VII,7;22), p. 259] for rotationally invariant (i.e., O(n, 0)-invariant) functions and distributions can be deduced analogously to the proof of Proposition 5. (d) That the formula in Proposition 5 is not always suitable for the calculation of the Fourier transform of a given distribution is seen by the simple example of a polynomial:   (−[∂, ∂])m δ = (2π )−n F [x, x]m   = (2π )−n F (x12 + · · · + x 2p − x 2p+1 − · · · − xn2 )m . In order to present a further non-trivial example of the applicability of Proposition 5 let us calculate the fundamental solution E of Sobolev’s operator ∂t − ∂x3yz which is uniquely determined by the conditions    t         x 4 ′ 4   E ∈ S (R ), supp E ⊂   ∈ R t ≥ 0 . y       z Sobolev [32] presented an expression for E by finding a solution of the ordinary differential equation of the third order for the function f in the ansatz 1  x yz  E= f . t t Later on, I calculated E by Fourier transformation ([18]: the result for E differs from that of Sobolev by two signs). A further different derivation is given in [22]. Proposition 6. The uniquely determined, temperate fundamental solution E of Sobolev’s operator ∂t − ∂x3yz with support in t ≥ 0 is given by        x yz π Y (t) x yz − Y (x yz)N0 2 . E = 2 Y (−x yz)K 0 2 − t 2 t π t Proof. By the partial Fourier transform with respect to x and y we obtain (∂t + ξ η∂z )Fx y E = δ(t, z), and thus Fx y E = Y (t)δ(z − ξ ηt). Herein, the distribution Y (t)δ(z − ξ ηt) is defined by  ∞ ⟨ϕ, Y (t)δ(z − ξ ηt)⟩ = ϕ(t, ξ ηt) dt, ϕ ∈ S(R2t z ). 0

Due to δ(z − ξ ηt) =

1 2π

Fζ−1 (e−itξ ηζ )

and to e−itξ ηζ ∈ O M , we have

′ ′ ′ ′ ˆ C,z ˆ M,ξ,η = O M,ξ,η (D+,t ˆ C,z Y (t)δ(z − ξ ηt) ∈ D+,t ⊗O ⊗O ⊗O ).

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Proposition 5 (n = 2, p = q = 1) and the linear transformation 1 ξ1 = (ξ + η), 2

1 η1 = − (ξ − η), 2

ξη =

ξ12

− η12

 =

ξ1 η1

   ξ1 , η1

yield ix y   Y (t) e− σ Y (t) F − ξ ηt)) = 1 , − st))) (σ ) (δ(z (F (δ(z ξ,η σ s |σ | 4π 2 4π 2    ∞   ix y iσ z Y (t) 1 Y (t) x y dσ z = = σ+ 1σ , e− t − σ cos |σ | t σ σ 4π 2 t 2π 2 t 0        Y (t) −x yz x yz = 2Y (−x yz)K 0 2 − π Y (x yz)N0 2 t t 2π 2 t

E =

(cf. formulae 334, (11a) and (11b) in [9, p. 132] and [8], 3.868, 2.,4., p. 470). ′ ′ ⊗O ˆ C,x,y,z The deduction above implies that E ∈ D+,t .  Acknowledgements I owe many thanks to my friend P. Wagner for fruitful discussions and, in particular, for the final formulation of Propositions 1 and 2 appearing in a joint book on distributions and fundamental solutions. References [1] S. Abdullah, S. Pilipovi´c, Bounded subsets in spaces of distributions of L p -growth, Hokkaido Math. J. 23 (1994) 51–54. [2] J. Bass, Cours de Math´ematiques. Vol. 1, Masson, Paris, 1968. [3] J. Bonet, M. Maestre, A note on the Schwartz space B(Rn ) endowed with the strict topology, Arch. Math. (Basel) 55 (1990) 293–295. [4] N. Bourbaki, Espaces Vectoriels Topologiques, Masson, Paris, 1981. ˚ )′ , B(Ω ˚ )′′ ⟩, Pacific J. Math. 108 (1983) 51–82. [5] P. Dierolf, S. Dierolf, Topological properties of the dual pair ⟨B(Ω [6] G.B. Folland, Harmonic Analysis in Phase Space, in: Annals Math. Studies, N. 122, Princeton Univ. Press, Princeton, New Jersey, 1989. [7] I.M. Gelfand, G.E. Shilov, Generalized Functions. Vol. 1, Academic Press, New York, 1964. [8] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, fifth ed., Academic Press, New York, 1972. [9] W. Gr¨obner, N. Hofreiter, Integraltafel. 2. Teil: Bestimmte Integrale, fifth ed., Springer, Wien, 1973. [10] A. Grothendieck, Espaces Vectoriels Topologiques., second ed., Sociedade de Matematica de S. Paulo, S. Paulo, 1958. [11] A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucl´eaires, third print, in: Memoirs of the American Mathematical Society, AMS, Providence, RJ 1955, 1973. [12] Y. Hirata, R. Shiraishi, On partial summabililty and convolutions in the theory of vector-valued distributions, J. Sci. Hiroshima Univ., Ser. A 24 (1960) 535–546. [13] L. H¨ormander, The Analysis of Linear Partial Differential Operators I, Grundlehren-Springer, Berlin, 1983. [14] J. Horv´ath, Topological Vector Spaces and Distributions, Addison-Wesley Publ, Reading, Mass, 1966. [15] R.P. Kanwal, Generalized Functions. Theory and Applications, Birkh¨auser, 2004. [16] J. Lavoine, Transformation de Fourier des Pseudo-fonctions, Centre Nat. Recherche Scientifique, Paris, 1963. [17] I. Maak, Die kompakten Mengen in Folgen-, Funktionen- und Distributionenr¨aumen. Diploma Thesis, Univ. Innsbruck, 2009. [18] N. Ortner, Construction of Fundamental Solutions. Innsbruck, 1986. Manuscript. [19] N. Ortner, On convolvability conditions for distributions, Monatsh. Math. 160 (2010) 313–335. [20] N. Ortner, The Fourier exchange theorem for kernels, Appl. Anal. 90 (2011) 1635–1650.

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[21] N. Ortner, P. Wagner, Distribution-Valued Analytic Functions. Theory and Applications, Max-Planck-Institut., Leipzig, LN 37, 2008, http://www.mis.mpg.de/preprints/ln/lecturenote-3708.pdf. [22] N. Ortner, P. Wagner, On the Fourier transform of Lorentz invariant distributions, Funct. Approx. Comment. Math. 44.1 (2011) 133–151. [23] A. Pietsch, History of Banach Spaces and Linear Operators, Birkh¨auser, Boston, 2007. [24] A.P. Robertson, W.I. Robertson, Topological Vector Spaces, second ed., Cambridge Univ. Press, Cambridge, 1973. [25] H.H. Schaefer, Topological Vector Spaces, third print, Springer, New York, 1971. ¨ [26] I. Schur, Uber lineare Transformationen in der Theorie der unendlichen Reihen, J. Reine Angew. Math. 151 (1921) 79–111. [27] L. Schwartz, Espaces de fonctions diff´erentiables a` valeurs vectorielles, J. Anal. Math. 4 (1953/54) 88–148. [28] L. Schwartz, Th´eorie des distributions a` valeurs vectorielles. Chapter I, I, Ann. Inst. Fourier 7 (1957) 1–141. 8(1959), 1–209. [29] L. Schwartz, Lectures on Mixed Problems in Partial Differential Equations and Representations of Semi-groups, Tata Inst., Bombay, 1957. [30] L. Schwartz, Th´eorie des Distributions, Nouvelle e´ dition, Hermann, Paris, 1966. [31] L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford Univ. Press, 1973. ∂ 3 u − 1 ∂u = F(x, y, z, t), Dokl. [32] S.L. Sobolev, The fundamental solution of Cauchy’s problem for the equation ∂ x∂ y∂z 4 ∂t Adad. Nauk SSSR 129 (1959) 1246–1249. [33] R. Strichartz, Fourier transforms and non-compact rotation groups, Indiana Univ. Math. J. 24 (1974) 499–526. [34] F. Treves, Topological Vector Spaces, Distributions and Kernels, second print, Academic Press, New York, 1970. [35] F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975. [36] V.S. Valdimirov, Generalized Functions in Mathematical Physics, Mir Publ., Moskow, 1979. [37] M. Valdivia, On the space D L p , in: L. Nachbin (Ed.), Mathematical Analysis and Applications, Part. B, Academic Press, New York, 1981, pp. 759–767. [38] D. Vogt, Sequence space representations of spaces of test functions and distributions, in: G.I. Zapata (Ed.), Functional Analysis, Holomorphy and Approximation Theory, M. Dekker, New York, 1983, pp. 405–444. [39] V.A. Zorich, Mathematical Analysis. Vol. II, Springer, Berlin, 2004. [40] C. Zuily, Problems in Distributions and Partial Differential Equations, Elsevier, Paris, 1988.