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A HEAT EQUATION ON ADIC COMPLETIONS OF Q AND ULTRAMETRIC ANALYSIS VICTOR A. AGUILAR–ARTEAGA∗, MANUEL CRUZ–LO´PEZ∗∗ AND SAMUEL ESTALA–ARIAS∗∗∗ Abstract. The productring Q = Q , where S is a finite setofprime numbers andQ is the S p∈S p p field of p–adic numbers, is a second countable, locally compact and totally disconnected topological Q ring. This work introduces a natural ultrametric on Q that allows to define a pseudodifferential S 7 operator Dα and to study an abstract heat equation on the Hilbert space L2(Q ). The fundamental 1 S 0 solution of this equation is a normal transition function of a Markov process on Q . As a result, the S 2 techniquesdevelopedhereprovidesageneralframeworkoftheseproblemsonotherrelatedultrametric n groups. a J 0 3 Introduction ] A The theory of pseudodifferential operators over the field Q of p–adic numbers has been deeply F p . explored by several authors. Many classical equations have been formulated on the n-dimensional h t version of this non–Archimedean space and their solutions have been relevant to some models of a m modern physics (see e.g., [VVZ], [Koc] and [Zun1] and the references therein). In particular, sev- [ eral analogous heat equations on Qn are now well understood: the fundamental solutions of these p 1 equations give rise to transition functions of Markov processes on Qn, which are non–Archimedean v p 2 counterparts to the classical Brownian motion. 3 7 Pursuing ageneralization, numerous Markov stochasticprocesses have beendefinedandelaborated 8 on several different ultrametric and algebraic spaces, such as the finite adèle ring A , and on general 0 f 1. locally compact topological groups, such as the complete adèle ring A, as documented in [CE1], 0 [CE2], [BGPW], [VVZ], [Koc], [KKS], [KR], [TZ] and [Zun], among others. 7 1 This article deals with a Markov process related to the fundamental solution of a heat equation : v on the direct product ring Q = Q , where S is a fixed finite set of finite prime numbers. i S p∈S p X The techniques developed here are different from the well known ones: they are geometrical and Q r a very simple. In addition, this method apply in the case of a selfdual, second countable, locally compact, Abelian, topological group G with a selfdual filtration {H } by compact and open n n∈Z subgroups, such that the indices [H : H ] are uniformly bounded. The authors show how to n n−1 construct many examples of these groups. (See Section 7 for details and also [BGPW] for other general setting and techniques.) The ring Q is asecond countable, locally compact, totallydisconnected, commutative, topological S ring and contains, as a maximal compact and open subring, the direct product ring Z = Z , S p∈S p Q Date: February 6, 2017. 2010 Mathematics Subject Classification. Primary: 35Kxx, 60Jxx Secondary: 35K05,35K08, 60J25. Key words and phrases. adic completions, ultrametrics, pseudodifferential equations, heat kernels, Markov processes. 1 2 V.A.AGUILAR–ARTEAGA,M. CRUZ–LO´PEZ AND S.ESTALA–ARIAS where Z is the ring of p–adic integers. Furthermore, as a topological group, it has a Haar measure p dµ normalised to be a probability measure on Z , it is selfdual in the sense of Pontryagin and there S exists an additive character χ(·) such that x 7→ χ(x·) gives an explicit isomorphism. Using this identification, the subgroup Z coincides with its own annihilator. S Let ψ(n) be the second Chebyshev function (see Section 2.2). This function determines a “sym- metric” filtration of Q by open and closed subgroups: S eψ(m)Z ⊂ eψ(n)Z ⊂ Z ⊂ eψ(−n)Z ⊂ eψ(−m)Z (m ≥ n ≥ 1). S S S S S There is a unique additive invariant ultrametric d on Q such that it has the filtration above as the S set of balls centred at zero and the Haar measure of any ball is equal to its radius. The ultrametric d leads to define a pseudodifferential operator Dα on the Hilbert space L2(Q ), S whichissimilartotheTaiblesonoperatoronQn butlooksalsoastheVladimirovoperatoronQ . The p p operator −Dα is a positive selfadjoint unbounded operator and allows to state the abstract Cauchy problem on L2(Q ) for the classical heat equation. This problem is well posed and the normalization S propertyoftheultrametricdallowstogiveclassicalboundsoftheheatkernel Z(x,t), usingproperties of the Archimedean Gamma function, and to find and explicit solution of this problem. The main result on the solution of the Cauchy problem reads as follows (see Section 4 for details and also Theorem 7.1). Theorem 4.9: If f is a complex valued square integrable function on Q , which belongs to the S domain of −Dα, the Cauchy problem ∂u(x,t) +Dαu(x,t) = 0, x ∈ Q , t ≥ 0, ∂t S  u(x,0) = f(x),  has a classical solution u(x,t) determined by the convolution of f with the heat kernel Z(x,t). In  addition, Z(x,t) is the transition density of a time and space homogeneous Markov process which is bounded, right–continuous and has no discontinuities other than jumps. The exposition is organized as follows. In Section 1 the results of the theories of p–adic numbers and Fourier analysis on Q are collected. The description of the ring Q and several important p S properties related to the non–Archimedean metric d are considered in Section 2. Section 3 presents the theory of Fourier analysis and introduces the pseudodifferential operator Dα defined on Q and S in Section 4 the homogeneous heat equation is treated. Section 5 provides the description of the Markov process derived from the fundamental solution to the heat equation. The general Cauchy problem on Q is presented in Section 6, and finally, in Section 7 different aspects on generalization S of the theory presented here are analysed. 1. The field of p–adic numbers This section outlines the basic elements of the theory of p–adic numbers Q as well as the rele- p vant function spaces defined on Q and some aspects of the Fourier analysis on this space. For a p comprehensive introduction to these subjects we quote [VVZ]. Let N = {1,2,...} be the set of natural numbers and let P be the set of prime numbers. Fix a a prime number p ∈ P. If x is any nonzero rational number, it can be written uniquely as x = pk , b A HEAT EQUATION ON SOME ADIC COMPLETIONS OF Q AND ULTRAMETRIC ANALYSIS 3 with p not dividing the product ab and k ∈ Z. The function p−k if x 6= 0, |x| := p  0 otherwise,  isanon–ArchimedeanabsolutevalueonQ. Thefieldofp–adicnumbersQ isdefinedasthecompletion  p of Q with respect to the distance induced by |·| . p Any nonzero p–adic number x has a unique representation of the form ∞ pγ a pi, i i=0 X where γ = γ(x) ∈ Z, a ∈ {0,1,...,p − 1} and a 6= 0. The value γ, with γ(0) = +∞, is called i 0 the p–adic order of x. Any series of the above form converges in the topology induced by the p–adic metric. The fractional part of a p–adic number x is defined by pγ −γ−1a pi if γ < 0, {x} = i=0 i p  0 if γ ≥ 0. P  The field Q is a locally compact topological field. The unit ball p  Z = {x ∈ Q : |x| ≤ 1} p p p is the ring of integers and the maximal compact and open subring of Q . Denote by dx the Haar p measure of the topological Abelian group (Q ,+) normalized to be a probability measure on Z . p p The algebraic and topological properties of the ring Z and the field Q can be expressed, respec- p p tively, by an inductive and a projective limit (1) Z = lim Z /plZ , Q = lim p−lZ . p ←− p p p −→ p l∈N∪{0} l∈N∪{0} In addition to the limits above, to the ring Z and the field Q , there corresponds, respectively, p p an infinite rooted tree T(Z ) with constant ramification index p, and an extended tree T(Q ) with p p constant ramification index p. The endpoints of these trees corresponds, respectively, to Z and Q . p p A function φ : Q −→ C is locally constant if for any x ∈ Q , there exists an integer ℓ(x) ∈ Z such p p that φ(x+y) = φ(x), for all y ∈ B , ℓ(x) where B is the closed ball with centre at zero and radius pℓ(x). ℓ(x) The set of all locally constant functions of compact support on Q forms a C–vector space denoted p by D(Q ). The C–vector space D(Q ) is the Bruhat–Schwartz space of Q and an element φ ∈ D(Q ) p p p p is called a Bruhat–Schwartz function (or simply a test function) on Q . p If φ belongs to D(Q ) and its not zero everywhere, there exists a largest ℓ = ℓ(φ) ∈ Z such that, p for any x ∈ Q , the following equality holds p φ(x+y) = φ(x), for all y ∈ B . ℓ This number ℓ is called the parameter of constancy of φ. 4 V.A.AGUILAR–ARTEAGA,M. CRUZ–LO´PEZ AND S.ESTALA–ARIAS Denote by Dℓ(Q ) the finite dimensional vector space consisting of functions with parameter of k p constancy is greater or equal than ℓ and whose support is contained in B . k A sequence (f ) in D(Q ) is a Cauchy sequence if there exist k,ℓ ∈ Z and M > 0 such that m m≥1 p f ∈ Dℓ(Q ) if m ≥ M and (f ) is a Cauchy sequence in Dℓ(Q ). That is, m k p m m≥M k p Dℓ(Q ) = limDℓ(Q ) and D(Q ) = limDℓ(Q ). p −→ k p p −→ p k ℓ With this topology the space D(Q ) is a complete locally convex topological algebra over C. It is p also a nuclear space because Dℓ(Q ) is the inductive limit of countable family of finite dimensional p algebras and D(Q ) is the inductive limit of countable family of nuclear spaces Dℓ(Q ). p p For each compact set K ⊂ Q , let D(K) ⊂ D(Q ) be the subspace of test functions whose support p p is contained in K. The space D(K) is dense in C(K), the space of complex–valued continuous functions on K. An additive character of the field Q is defined as a continuous function χ : Q −→ C such that p p χ(x+y) = χ(x)χ(y) and |χ(x)| = 1, for all x,y ∈ Q . The function χ (x) = exp(2πi{x} ) defines p p p a canonical additive character of Q which is trivial on Z and not trivial outside Z . In fact, all p p p characters of Q are given by χ (x) = χ (ξx) with ξ ∈ Q . p p,ξ p p The Fourier transform of a test function φ ∈ D(Q ) is given by the formula p F φ(ξ) = φ(ξ) = φ(x)χ (ξx)dx, (ξ ∈ Q ). p p p ZQp The Fourier transform is a continbuous linear isomorphism of the space D(Q ) onto itself and the p inversion formula holds: φ(x) = φ(ξ)χ (−xξ)dξ (φ ∈ D(Q )). p p ZQp b The Parseval – Steklov equality reads as: φ(x)ψ(x)dx = φ(ξ)ψ(ξ)dξ, (φ,ψ ∈ D(Q )). p ZQp ZQp b b Remark 1.1. From expressions (1) or the description of Q as collection of endpoints of the tree p T(Q ), it follows that the Hilbert space L2(Q ) has a numerable Hilbert base, which is an analogous p p of a wavelet base, and therefore it is a separable Hilbert space (see e.g. [AKS]). Remark 1.2. The extended Fourier transform F : L2(Q ) −→ L2(Q ) is an isometry of Hilbert p p spaces and the Parseval – Steklov identity holds on L2(Q ). p 2. The S–adic ring of Q This section introduces the ring Q and an ultrametric on Q invariant under translations by S S elements of Q and under multiplication by units of Z . We will describe Q as topological ring S S S itself, however other proofs of our statements can be given by looking at the product structure of Q S (see [CE1] for these kind of construction, where it is done for the finite adèle ring of Q). A HEAT EQUATION ON SOME ADIC COMPLETIONS OF Q AND ULTRAMETRIC ANALYSIS 5 2.1. The adic ring Q . Fix a finite subset S ⊂ P and define Q as the direct product S S Q = Q . S p p∈S Y The Tychonoff topology and the componentwise operations provide Q with a structure of a topo- S logical ring. The topological ring Q is commutative, second countable, locally compact and totally S disconnected. ThemaximalcompactandopensubringofQ isthedirectproductringZ = Z . S S p∈S p The additive group(Q ,+) is a locallycompact Abelian groupand therefore it has a Haar measure S Q dµ which can be normalized to be a probability measure on Z . The measure dµ can be expressed S in terms of the measures dx on the groups (Q ,+) as the direct product measure p p dµ = dx . p p∈S Y With regard to characters, there is a canonical additive character χ on Q , which is trivial on Z S S and not trivial outside Z , given by S χ(x) = χ (x ), x = (x ) ∈ Q , p p p p∈S S p∈S Y (cid:0) (cid:1) where χ (x ) is the canonical character of Q . Recall that these characters are given by χ (x ) = p p p p p e2πi{xp}p, where {x } is the p–adic fractional part of x . p p p For ξ ∈ Q , the application S χ (x) = χ(ξ ·x) = χ (ξ x ), ξ = (ξ ) ∈ Q , ξ p p p p p∈S S p∈S Y (cid:0) (cid:1) defines a character on Q . Moreover, since Q is a direct product of some p-adic fields, any arbitrary S S character on Q has the form χ , for some ξ ∈ Q . Therefore Q is a selfdual group with isomorfism S ξ S S given by ξ 7→ χ . ξ Recall that the annihilator of a compact and open subgroup H of Q is the set of characters that S are trivial in H. From the expression of the characters on Q , the relation S Ann (B ) = B , QS n −n where Ann (B ) is the annihilator of B in Q , holds. In particular, Z coincides with its own QS n n S S annihilator. Remark 2.1. For the scope of this work the relevant properties of Q are that it a topological ring S which is second countable, locally compact, and totally disconnected. In addition, as a topological group, Q is selfdual. S 2.2. An ultrametric on Q . Let us introduce an ultrametric d on Q compatible with its topology, S S making (Q ,d) a complete ultrametric space. The set of rational numbers Q is diagonally embedded S in Q with dense image. In this sense, the ring Q can be thought of as an S–completion of Q (see S S [Bro] for a description of this topology on the ring of integer numbers Z). 6 V.A.AGUILAR–ARTEAGA,M. CRUZ–LO´PEZ AND S.ESTALA–ARIAS We start by defining two arithmetical functions which are related to the set S and similar to the second Chebyshev and von Mangoldt functions (see e.g., [Apo]). For any natural number n, write logp if n = pk for some p ∈ S and integer k ≥ 1, Λ(n) = Λ (n) = S  0 otherwise.  Likewise, let ψ(n) denote the function implicitly given by  eψ(n) = eψS(n) = lcm pl ≤ n : p ∈ S, l ∈ N∪{0} (n ∈ N). These arithmetical functions are rela(cid:8)ted by the equations (cid:9) n n ψ(n) = Λ(k) and eψ(n) = eΛ(k) (n ∈ N). k=1 k=1 X Y For any integer number n, define n ψ(|n|) if n 6= 0, ψ(n) = |n|  0 if n = 0,  and  Λ(n) if n > 0, Λ(n) =  Λ(|n−1|) = Λ(|n|+1) if n ≤ 0.  Therelationsbetweenthesefunctionsonthenaturalnumbersextendtotheintegersinthefollowing  way: for any integers n > m n n ψ(n)−ψ(m) = Λ(k) and eψ(n)/eψ(m) = eΛ(k). k=m+1 k=m+1 X Y The collection {eψ(n)Z } of compact and open subgroups is a neighbourhood base of zero for S n∈Z the Tychonoff topology on Q and determines a filtration (see figure 1) S {0} ⊂ ··· ⊂ eψ(n)Z ⊂ ··· ⊂ Z ⊂ ··· ⊂ eψ(m)Z ⊂ ··· ⊂ Q n > 0 > m , S S S S satisfying the properties: (cid:0) (cid:1) eψ(n)Z = {0} and eψ(n)Z = Q . S S S n∈Z n∈Z \ [ Remark 2.2. From the properties of the above filtration, the topology of the ring Q is expressed by S the inductive and projective limits Q = limeψ(−n)Z , Z = limZ /eψ(n)Z . S −→ S S ←− S S n∈N n∈N In addition, we have the identity eψ(n)Z = pordp(eψ(n))Z , S p p∈S Y where ord (·) is the p–adic order function on Q . p p A HEAT EQUATION ON SOME ADIC COMPLETIONS OF Q AND ULTRAMETRIC ANALYSIS 7 Figure 1. The decomposition of Q by the filtration {eψ(n)Z} for S = {2,3}. S For any element x ∈ Q define the order of x as: b S max n : x ∈ eψ(n)Z if x 6= 0, S ord(x) :=  ∞ (cid:8) (cid:9) if x = 0.  Notice that this order satisfies the following properties:  • ord(x) ∈ Z∪{∞} and ord(x) = ∞ if and only if x = 0, • ord(x+y) ≥ min{ord(x),ord(y)} and • it takes the values SZ = −pl : l ∈ N and p ∈ S ∪ pl −1 : l ∈ N and p ∈ S ∪{∞}. The nonnegative f(cid:8)unction d : Q ×Q −→(cid:9)R+(cid:8)∪{0} given by (cid:9) S S d(x,y) = e−ψ ord(x−y) (cid:0) (cid:1) is an ultrametric on Q . This ultrametric d takes values in the set {eψ(n)} ∪ {0}, any ball B S n∈Z n centred at zero with radius eψ(n) is precisely the subgroup B = B(0,eψ(n)) = e−ψ(n)Z (n ∈ Z), n S and any sphere centred at zero and radius eψ(n) is S = S(0,eψ(n)) = B \B . n n n−1 The norm induced by this ultrametric is given by kxk = e−ψ(ord(x)) and kxk = eψ(n) if and only if x ∈ S . n It is worth to notice that the radius of any ball on Q is equal to its Haar measure: S dx = dx = dx = eψ(n) (y ∈ Q , n ∈ Z). S ZBn+y ZBn Ze−ψ(n)ZS 8 V.A.AGUILAR–ARTEAGA,M. CRUZ–LO´PEZ AND S.ESTALA–ARIAS Using this fact, the area of any sphere is given by dx = dx = eψ(n) −eψ(n−1) (y ∈ Q , n ∈ Z). S ZSn+y ZSn Remark 2.3. If eψ(n) < eψ(n+1), B is strictly contained in B ; otherwise, if eψ(n) = eψ(n+1), n+1 n B = B . This behaviour is controlled by the function Λ(n) and there exists a unique increasing n+1 n bijective function ρ : Z −→ SZ, such that eψ(ρ(n)) is a strictly increasing function with eψ(ρ(0)) = 1. For this reason, in the sequel, we suppose that eψ(n) < eψ(n+1) for any integer number n. 3. Function spaces and pseudodifferential operators on Q S The relevant spaces of test functions as well as the basic facts of Fourier analysis on Q are S described in this section. It also introduces a pseudodifferential operator Dα on Q . The description S made here follows closely the account made in [CE2] for the finite adèle ring of Q. A second point of view is obtained by looking at the product structure of Q . S The reader can consult these topics in the excellent books [VVZ], [AKS], [Igu]. The theory of general topological vector spaces can be found in [Sch]. 3.1. Bruhat–Schwartz test functions on Q . The Bruhat–Schwartz space D(Q ) is the space S S of locally constant functions on Q with compact support. Being Q a totally disconnected space, S S D(Q ) has a natural topology which can be described by two inductive limits, given any filtration S by compact and open sets in Q . S Let us describe the topology of D(Q ) using the ultrametric d. If φ ∈ D(Q ), there exists a S S smallest ℓ = ℓ ∈ Z such that, for every x ∈ Q , φ S φ(x+y) = φ(x), for all y ∈ B = e−ψ(ℓ)Z . ℓ S This number ℓ is called the parameter of constancy of φ. The set of all locally constant functions on Q with common parameter of constancy ℓ ∈ Z and support in B forms a finite dimensional S k complex vector space of dimension eψ(ℓ)/eψ(k). Denote this space by Dℓ(Q ). The topology of D(Q ) k S S is expressed by the inductive limits Dℓ(Q ) = limDℓ(Q ) and D(Q ) = limDℓ(Q ), S −→ k S S −→ S k ℓ which essentially states that every finite dimensional vector space Dℓ(Q ) is open in D(Q ). There- k S S fore, D(Q ) is a complete locally convex topological algebra over C and a nuclear space. S Finally, for each compact subset K ⊂ Q , let D(K) ⊂ D(Q ) be the subspace of test functions S S with support on a fixed compact subset K. The space D(K) is dense in C(K), the space of complex valued continuous functions on K. The space D(Q ) is dense in L2(Q ). S S 3.1.1. Bruhat–Schwartz test functions as a tensor product. Since the Tychonoff topology on Q is S the box topology, any function φ ∈ D(Q ) can be written as a finite linear combination of elementary S functions of the form φ(x) = φ (x ), x = (x ) ∈ Q , p p p p∈S S p∈S Y (cid:0) (cid:1) where each factor φ (x ) belongs to the space of test functions D(Q ). p p p A HEAT EQUATION ON SOME ADIC COMPLETIONS OF Q AND ULTRAMETRIC ANALYSIS 9 Moreover, from the finite group identification B /B = Bp /Bp , l k ℓp kp p∈S Y where ℓ = ord (eψ(l)) and k = ord (eψ(k)), the following identity between finite dimensional spaces p p p p holds: Dℓ(Q ) = Dℓp(Q ). k S kp p p∈S O Commuting the induced limits with the tensor products, it is possible to show that Dℓ(Q ) cor- S responds to the algebraic and topological tensor product of nuclear spaces {Dℓ(Q )} , that is to p p∈S say Dℓ(Q ) = Dℓp(Q ). S p p∈S O Furthermore, D(Q ) corresponds to the algebraic and topological tensor product of nuclear spaces S {D(Q )} , viz., p p∈S D(Q ) ∼= D(Q ). S p p∈S O 3.2. The Fourier transform on Q . The Fourier transform of φ ∈ D(Q ) is defined by S S φ(ξ) = F[φ](ξ) = φ(x)χ(ξx)dx, (ξ ∈ Q ). S ZQS In particular, the Fourierbtransform of any elementary function φ = φ is given by p∈S p Q F[φ](ξ) = φ (x )χ (ξ x )dx , φ(x) = φ (x ), ξ = (ξ ) ∈ Q . p p p p p p p p p p∈S S p∈SZQp p∈S ! Y Y The following two integrals are of main importance for the properties of the Fourier transform on the Bruhat-Shwartz space, D(Q ). S Lemma 3.1. eψ(n) if kξk ≤ e−ψ(n), χ(−ξx)dx = ZBn 0 if kξk > e−ψ(n). Proof. Thisfollowsfromtheequality, Ann (B ) = B andthefactthatinanycompacttopological QS n −n (cid:3) group, the integral of a nontrivial character over the group is zero. Lemma 3.2. For any n ∈ Z, eψ(n) −eψ(n−1) if kξk ≤ e−ψ(n), χ(−ξx)dx = −eψ(n−1) if kξk = e−ψ(n−1), ZSn 0 if kξk ≥ e−ψ(n−2).  Proof. The integral can be decomposedas  χ(−ξx)dx = χ(−ξx)dx− χ(−ξx)dx, ZSn ZBn ZBn−1 (cid:3) and the last proposition provides the result. 10 V.A.AGUILAR–ARTEAGA,M. CRUZ–LO´PEZ AND S.ESTALA–ARIAS Remark 3.3. From the well known computations of analogous integrals on Q , another proof of the p last lemmas can be derived directly. From last propositions, definitions of the spaces Dℓ(Q ) and Fourier transform, it follows k S F : Dℓ(Q ) −→ D−k(Q ). k S −ℓ S Moreover, the Fourier transform F is a continuous linear isomorphism from D(Q ) into itself. It is S worth to notice that this property of the Fourier transform follows directly by construction, because Ann (B ) = B . QS n −n Remark 3.4. Recall that, in the case of Q , the Fourier transform F sends Dℓ(Q ) into D−k(Q ) p p k p −ℓ p and the identification of Dℓ(Q ) with the tensor product, ⊗ Dℓp(Q ), of finite dimension spaces. k S p∈S kp p Therefore, F : Dℓ(Q ) −→ D−k(Q ) and the Fourier transform gives and isomorphism D(Q ) ∼= k S −ℓ S S D(Q ). S From the description of Q and Z , respectively, as an inductive and a projective limit or from the S S description of Q as the endspace of a regular infinite tree it follows that the Hilbert space L2(Q ) S S has a numerable Hilbert base which is a counterpart of a wavelet bases. Therefore, L2(Q ) is a S separable Hilbert space. In addition, the Fourier transform F : L2(Q ) −→ L2(Q ) S S is an isometry. Remark 3.5. Notice that L2(Q ) ∼= L2(Q ), where the tensor denotes the Hilbert tensor S p∈S p product, because each L2(Q ) is a separable Hilbert space, then p N F : L2(Q ) −→ L2(Q ) S S is an isometry. 3.3. Lizorkin space of test functions on Q . Another space of test functions which is useful S in the study of the heat equation is the following: Let Ψ(Q ) be the space of test functions which S vanish at zero Ψ(Q ) = {f ∈ D(Q ) : f(0) = 0}. S S This means that for any element f ∈ Ψ(Q ) there exists a ball B with centre at zero and radius S n eψ(n) such that f ≡ 0. The image of Ψ(Q ) under the Fourier transform is the space |Bn S Φ(Q ) = {g : g = F[f], f ∈ Ψ(Q )} ⊂ D(Q ) S S S called the Lizorkin space of test functions of the second kind. The space Φ(Q ) is nontrivial and, as a S subspace of D(Q ), it has the subspace topology which makes it a complete topological vector space. S