TAIBLESON OPERATORS, p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION. 8 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO 0 0 2 Abstract. We give a multimensional version of the p-adic heat equation, and show that its fundamental solution is the transition density of aMarkov n process. a J 6 1 1. Introduction ] h Inrecentyearsp−adicanalysishasreceivedalotofattentiondue to itsapplica- p tions in mathematical physics, see e.g. [1], [2], [4], [5], [11], [12], [13], [16], [19] and - references therein. One motivation comes from statistical physics, in particular h t in connection with models describing relaxation in glasses, macromolecules, and a proteins. Ithasbeenproposedthatthenonexponentialnatureofthoserelaxations m is a consequence of a hierarchical structure of the state space which can in turn [ be put in connection with p−adic structures ([4], [5], [16]). In [4] was demostrated 2 that the p-adic analysis is a natural basis for the construction of a wide variety of v models of ultrametric diffusion constrained by hierarchical energy landscapes. To 8 each of these models is associated a stochastic equation (the master equation). In 1 severalcases this equation is a p-adic parabolic equation of type: 0 1 . ∂u(x,t) +a(Au)(x,t)=f(x,t), x∈Qn, t∈(0,T], 2 ∂t p 1  (1.1) 7 u(x,0)=ϕ(x), 0 wherea is a positiveconstant,Ais pseudo-differentialoperator,andQ is the field v: of p-adic numbers. The simplest case occurs when n = 1 and A is thepVladimirov i X operator: ar (Dαϕ)(x)=Fξ−→1x |ξ|αp Fx→ξϕ(x) , α>0, where F is the Fourier transform. T(cid:16)he fundamenta(cid:17)l solution of (1.1) is density transition of a time- and space-homogeneous Markov process, that is consider the p−adic counterpart of the Brownian motion (see [13], [19]). Itisrelevanttomentionthatinthecasen=1,thefundamentalsolutionof(1.1) when A = Dα (also called the p−adic heat kernel) has been studied extensively, see e.g. [6], [8], [9], [10], [13], [19]. Anaturalproblemistostudytheinitialvalueproblem(1.1)inthen-dimensional case. Recently, the second author considered Cauchy’s problem (1.1) when (Aϕ)(x)=F−1 |f(ξ)|αF ϕ(x) , α>0, ξ→x p x→ξ (cid:16) (cid:17) 2000 Mathematics Subject Classification. Primary35R60,60J25; Secondary47S10,35S99. Key words and phrases. Parabolic equations, Markov processes, p-adic numbers, ultrametric diffusion. 1 2 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO heref(ξ)isanelliptichomogeneouspolynomialinnvariables,andthedatumϕisa locallyconstantandintegrablefunction. Underthesehypothesesitwasestablished the existence of a unique solution to Cauchy’s problem (1.1). In addition, the fundamental solution is a transition density of a Markov process with space state Qn (see [20]). p In this paper we study Cauchy’s problem (1.1) when A is the Taiblesonpseudo- differential operator which is defined as follows: β Dβϕ (x)=F−1 max |ξ | F ϕ(x) , β >0. (1.2) T ξ→x i p x→ξ (cid:16) (cid:17) (cid:18)1≤i≤n (cid:19) ! Recently Albeverio, Khrennikov, and Shelkovich studied Dβ in the context of the T Lizorkin spaces [3]. We prove existence and uniqueness of the Cauchy problem (1.1-1.2) in spaces of increasing functions introduced by Kochubei in [14], see Theorem 1. We also associate a Markov processes to equation the fundamental solution (see Theorem 2). These results constitute an extension of the corresponding results in [13], [19]. We want to mention here a relevant comment due to the referee. There exists a procedure, developed in [13] for elliptic equations, of reducing multi-dimensional problemsoverQ toone-dimensionalproblemsoverappropriatefieldextensions. In p particular,theTaiblesonoperatorisconnectedwiththeunramifiedextensionofQ p ofdegreen(seeLemma2.1in[13]). Thefundamentalsolutionscorrespondingtothe multi-dimensional Cauchy problem and the problemover the unramified extension should be obtained from each other, up to a linear change of variables, as in the formula(2.38)of[13]fortheellipticcase. Thenmanypropertiesofthefundamental solution would follow directly from those known in the one-dimensional case. In this paper we use an elementary and independent method that has its obvious advantages. Letusexplaintheconnectionbetweentheresultsofthispaperandthoseof[20]. There are infinitely many homogeneous polynomial functions satisfying d |f(ξ)| = max |ξ | , for any ξ ∈Qn, p i p p 1≤i≤n (cid:18) (cid:19) here d denotes degree of f (c.f. Lemmas 14-15). Hence the pseudo-differential operators considered here are a subclass of the ones considered in [20]. However, the function spaces for the solutions and initial data are completely different. In this paper the initial datum and the solution to Cauchy problem (1.1-1.2) are not necessarily bounded, neither integrable, but in [20] are. Finally, our results can be extended to operators of the form n (Aϕ)(x)=a (x,t)(Dαϕ)(x)+ a (x,t)(Dαkϕ)(x)+b(x,t)ϕ(x), (1.3) 0 T k T k=1 X α > 1, 0 < α < ... < α < α, where the a (x,t) ,and b(x,t) are bounded 1 n k continuousfunctions, usingthe techniques presentedin [13]-[15]. These resultswill appear later elsewhere. Theauthorswishtothanktherefereefortherelevantcommentmentionedabove. p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 3 2. Preliminary Results As generalreference for p-adic analysis we refer the reader to [17] and [19]. The field of p-adic numbers Q is defined as the completion of the field of rational p numbers Q with respectto the non-Archimedeanp-adic norm|·| which is defined p as follows: |0| = 0; if x ∈ Q×, x = pγa with a, b integers coprime to p, then p b |x| = p−γ. The integer γ = γ(x) is called the p-adic order of x, and it will be p denoted as ord(x). We use the same symbol, |·| , for the p-adic norm on Q . We p p extend the p-adic norm to Qn as follows: p kxk := max |x | , for x=(x ,...,x )∈Qn. p i p 1 n p 1≤i≤n Note that kxk =p−min1≤i≤n{ord(xi)}. p Any p-adic number x6=0 has a unique expansion of the form ∞ x=pγ x pj, j j=0 X where γ = ord(x) ∈ Z, and x ∈ {0,1,...,p−1}. By using the above expansion, j we define the fractional part of x∈Q , denoted as {x} , as the following rational p p number: 0, if x=0, or γ ≥0 |γ|−1 {x} := p  pγ x pj, if γ <0.  j j=0 X Denote by Bn(a) = x∈Qn |kx−ak ≤pγ , the ball of radius pγ with center γ p p at a = (a1,...,an) ∈nQnp, and Bγn(0)= Bγn, γo∈ Z. Note that Bγn(a) = Bγ(a1)× ...×B (a ), where B (a )= x ∈Q ||x −a | ≤pγ is the one-dimensional γ n γ j j p j j p ball of radius pγ with center atnaj ∈ Qp. The Ball B0noequals the product of n copies of B (0)=Z , the ring of p-adic integers. 0 p LetdnxdenotetheHaarmeasureonQn normalizedbythecondition dnx= p Bn 0 1. R A complex-valued function ϕ defined on Qn is called locally constant if for any p x∈Qn there exists an integer l(x)∈Z such that ϕ(x+x′)=ϕ(x), for x′ ∈Bn . p l(x) A function ϕ : Qn → C is called Schwartz-Bruhat function, or test function, if p it is locally constant with compact support. The C-vector space of the Schwartz- Bruhat functions is denoted by S(Qn). If ϕ ∈ S(Qn), there exist an integer l ≥ 0 p p such that ϕ(x+x′) =ϕ(x), for x′ ∈Bn , and x ∈Qn (see e.g. [19, VI.1, Lemma −l p 1]). The largestof such numbers l =l(ϕ) is called the exponent of local constancy of ϕ. Let S′(Qn) denote the set of all functionals (distributions) on S(Qn). All the p p functionals on S(Qn) are continuous (see e.g. [19, VI.3]). p Given ξ = (ξ ,...,ξ ), x = (x ,...,x ) ∈ Qn, we set ξ·x := n ξ x . The 1 n 1 n p i=1 i i Fourier transform of ϕ∈S(Qn) is defined as p P (Fϕ)(ξ)= Ψ(−ξ·x)ϕ(ξ)dnx, ξ ∈Qn, p ZQnp 4 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO where Ψ(−ξ · x) = n Ψ(−ξ x ) = exp 2πi n {−ξ x } . The function i=1 i i i=1 i i p Ψ(αx ) = exp 2πi Qn {αx } is called th(cid:16)e staPndard additive(cid:17)character of Q . j i=1 j p p The Fourier Tr(cid:16)ansform is a linea(cid:17)r isomorphism from S(Qn) onto itself. P p 2.1. The Taibleson Operator. We set 1−pα−n Γ(n)(α):= , α6=0. p 1−p−α This function is called the p-adic Gamma function. The function ||x||α−n k (x)= p , α∈R\{0,n}, x∈Qn, α Γ(n)(α) p p iscalledthe multi-dimensional Riesz Kernel;itdeterminesadistributiononS(Qn) p as follows. If α6=0, n, and ϕ∈S(Qn), then p 1−p−n 1−p−α hk (x),ϕ(x)i= ϕ(0)+ ||x||α−nϕ(x)dnx α 1−pα−n 1−pα−n p Z||x||p>1 1−p−α + ||x||α−n(ϕ(x)−ϕ(0))dnx. (2.1) 1−pα−n p Z||x||p≤1 Thenk ∈S′(Qn),forR\{0,n}. Inthecaseα=0,bypassingtothelimitin(2.1), α p we obtain hk (x),ϕ(x)i:= lim hk (x),ϕ(x)i=ϕ(0), 0 α α→0 i.e., k (x)=δ(x),the Diracdeltafunction, andthereforek ∈S′(Qn),forR\{n}. 0 α p It follows from (2.1) that for α>0, 1−pα hk (x),ϕ(x)i= ||x||−α−n(ϕ(x)−ϕ(0))dnx. (2.2) −α 1−p−α−n p ZQnp Lemma1 ([17,Chap. III,Theorem4.5]). Aselementsof S′(Qn), (Fk )(x)equals p α ||x||−α, α6=n. p Definition 1. The Taibleson pseudo-differential operator Dα, α>0, is defined as T (Dαϕ)(x)=F−1 ||ξ||αF ϕ , for ϕ∈S(Qn). T ξ→x p x→ξ p As a consequence of the previous(cid:0)lemma and(cid:1)(2.2), we have (Dαϕ)(x)=(k ∗ϕ)(x)= T −α 1−pα ||y||−α−n(ϕ(x−y)−ϕ(x))dny. (2.3) 1−p−α−n p ZQnp Theright-handsideof(2.3)makessenseforawiderclassoffunctions,forexample, for locally constant functions ϕ(x) satisfying ||x||−α−n|ϕ(x)|dnx<∞. p Z||x||p≥1 p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 5 3. The p-adic Heat Equation and the Taibleson Operator In this paper we consider the following Cauchy problem: ∂u(x,t) +a(Dαu)(x,t)=f(x,t), x∈Qn, t∈(0,T], ∂t T p (3.1)  u(x,0)=ϕ(x), wherea>0,α>0and Dα is the Taiblesonoperator. Inthis sectionwe showthat T (3.1) is a multi-dimensional analog of the p-adic heat equation introduced in [19]. 3.1. TheFundamentalSolution. Thefundamentalsolution fortheCauchyprob- lem (3.1) is defined as Z(x,t):= Ψ(x·ξ)e−at||ξ||αp dnξ. (3.2) ZQnp Lemma 2. The fundamental solution has the following properties: 1) Z(x,t)=(1−p−n)||x||−n ∞ q−kne−at(q−k||x||−p1)α −||x||−ne−at(p||x||−p1)α; p k=0 p 2) Z(x,t)= ∞ (−1)m 1−pαm (at)m||x||−αm−n for x6=0; m=1 m! 1−pP−αm−n p 3) Z(x,t)≥0, for all x∈Qn, t∈(0,T]. P p Proof. 1) By expanding Z(x,t) as ∞ Z(x,t)= Ψ(x·ξ)e−at||ξ||αp dnξ, k=−∞Z||ξ||p=pk X and applying pkn(1−p−n), if ||x|| ≤p−k p  Ψ(x·ξ)dnξ = −pknp−n, if kxk =p−k+1 Z||ξ||p=pk  p 0, if ||x|| >p−k+1, p (c.f. Lemma 4.1 in [17, Chap. III]), we obtain ∞ Z(x,t)=(1−p−n)||x||−n p−kne−at(p−k||x||−p1)α −||x||−ne−at(p||x||−p1)α. (3.3) p p k=0 X Note that by the previous expansion Z(x,t) is a real-valued function. 2) By using the Taylor expansion of ex in (3.3), and exchanging the order of summation, and sum the geometric progression,we find that ∞ (−1)m 1−pαm Z(x,t)= (at)m||x||−αm−n, for x6=0. m! 1−p−αm−n p m=1 X 3) LetΩ (x) denote the characteristicfunction of the ballBn (0). ThenFΩ = l −l l p−nlΩ . The last part follows from this observation by means of the following −l 6 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO calculation: ∞ Z(x,t)= e−atplα Ψ(x·ξ)dnξ l=−∞ Z||ξ||p=pl X ∞ = e−atplα(pn(l)Ω (x)−pn(l−1)Ω (x)) −l −l+1 l=−∞ X ∞ = pnl(e−atplα −e−atp(l+1)α)Ω (x)≥0 −l l=−∞ X (cid:3) Lemma 3. Z(x,t)≤Ct(t1/α+||x|| )−α−n, t>0, x∈Qn. (3.4) p p Proof. Let l an integer such that pl−1 ≤t1/α ≤pl. Then Z(x,t)≤ e−at||ξ||αp dnξ ≤ e−apα(l−1)||ξ||αp dnξ = e−a||p−(l−1)ξ||αp dnξ ZQnp ZQnp ZQnp =p−(l−1)n e−a||η||αp dη =C0(α)p−np−ln ≤C1t−n/α. (3.5) ZQnp On the other hand, if ||x|| ≥t1/α, by applying Lemma 2 (2), we have p ∞ Cm Z(x,t)≤||x||−n 2 (t||x||−α)m ≤C t||x||−α−n. (3.6) p m! p 3 p m=1 X The result follows from (3.5-3.6) as follows. If ||x|| ≥t1/α, by (3.6), p Z(x,t)≤C t||x||−α−n ≤2α+nC t(t1/α+||x|| )−α−n. 3 p 3 p If ||x|| <t1/α, by (3.5), p Z(x,t)≤C t−n/α ≤2α+nC t(t1/α+||x|| )−α−n. 1 1 p (cid:3) Inequality (3.4) shows in particular that the function Z(x,t) belongs, with re- spect to x, to L (Qn)∩L (Qn). 1 p 2 p Corollary 1. Z(x,t)dnx=1. (3.7) Z Qn p 3.2. The Spaces M and Pseudo-differentiability of the Fundamental So- λ lution. Definition 2. Denote by M , λ>0, the set of the complex-valued locally constant λ functions ϕ(x) on Qn such that p |ϕ(x)|≤C(ϕ)(1+||x||λ). p Ifthefunctionϕdepends alsoonaparametert,weshallsaythatϕ∈M uniformly λ with respect to t, if its constant C and its exponent of local constancy l(ϕ) do not depend on t. p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 7 Lemma 4. If ϕ∈M , λ<α, with α as in (3.1), then λ lim Z(x−ξ,t)ϕ(ξ)dnξ =ϕ(x). (3.8) t→0+ZQnp Proof. By Corollary (1) and Lemmas 2 (part 3) and 3 we have Z(x−ξ,t)ϕ(ξ)dnξ−ϕ(x) = Z(x−ξ,t)(ϕ(ξ)−ϕ(x)) dnξ (cid:12)(cid:12)ZQnp (cid:12)(cid:12) (cid:12)(cid:12)ZQnp (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ≤ Z(x−ξ,t)|ϕ(ξ)−ϕ(cid:12)(x)|(cid:12)dnξ (cid:12) ZQnp ≤C t(t1/α+||x−ξ|| )−α−n|ϕ(ξ)−ϕ(x)|dnξ :=I(x,t) p ZQnp Let η be the exponent of locally constancy of ϕ. Since ϕ ∈ M , λ < α, we can λ re-write I(x,t) as follows: I(x,t)=C t(t1/α+||ξ−x|| )−α−n|ϕ(ξ)−ϕ(x)|dnξ p Z||ξ−x||p>pη ≤I (x,t)+I (x,t), 1 2 with 1+||ξ||λ I (x,t):=C t p dnξ, 1 1 (t1/α+||x−ξ|| )α+n Z p ||ξ−x||p>pη I (x,t):=Ct|ϕ(x)| (t1/α+||ξ−x|| )−α−ndnξ. 2 p Z ||ξ−x||p>pη Now, since α>0, and t>0, I (x,t)≤C t|ϕ(x)|, 2 2 and since λ<α, ||x−τ||λ I (x,t)≤C t C + p dnξ ≤ 1 1 3 ||τ|| α+n Z||τ||p>pη p ! ||x−τ||λ ||x−τ||λ C t C + p dnξ+ p dnξ = 1 3 Zpη<||τ||ηp≤kxkp ||τ||pα+n Z||τ||p>kxkp ||τ||pα+n ! 1 C t C (x)+ dnξ =C (x)t. 1 4 ||τ|| α−λ+n 5 Z||τ||p>kxkp p ! Therefore lim Z(x−ξ,t)ϕ(ξ)dnξ−ϕ(x) ≤ lim C (x)t=0. 6 t→0+(cid:12)(cid:12)ZQnp (cid:12)(cid:12) t→0+ (cid:12) (cid:12) (cid:3) (cid:12) (cid:12) (cid:12) (cid:12) For further reference we summarize the properties of the fundamental solution in the following proposition. 8 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO Proposition 1. The fundamental solution has the following properties: (1) Z(x,t)≥0, for all x∈Qn, t∈(0,T]. p (2) Z(x,t) dnx=1, for any t>0; Qn p (3) Rif ϕ∈S Qnp , then lim(x,t)→(x0,0) Qnp Z(x−η,t)ϕ(η)dnη =ϕ(x0); (4) Z(x,t+t′)= Z(x−y,t)Z(y,t′)dny, for t, t′ >0. (cid:0) (cid:1) Qn R p Proof. (1),(2),andR(3)arealreadyestablished(c.f. Lemma2-part(3),Corollary1, and Lemma 4). The last assertion is proved as follows: since e−at||ξ||αp ∈L1 Qn , p Z(x−y,t )Z(y,t )dny =F−1(F(Z(y,t )∗Z(y,t ))) (cid:0) (cid:1) 1 2 1 2 ZQnp =F−1 e−at1||ξ||αpe−at2||ξ||αp =Z(x,(cid:16)t +t ). (cid:17) 1 2 (cid:3) Proposition 2. If b>0, 0≤λ<α, and x∈Qn, then p I(b,x)= (b+||x−ξ|| )−α−n||ξ||λdnξ ≤Cb−α 1+||x||λ , p p p ZQnp (cid:0) (cid:1) where the constant C does not depend on b, x. Proof. Let m be an integer such that pm−1 ≤b≤pm. Then (b+||x−ξ|| )−α−n ≤ pm−1+||x−ξ|| −α−n, p p and (cid:0) (cid:1) I(b,x)≤I(pm−1,x)= pm−1| +||x−ξ|| −α−n||ξ||λdnξ p p p ZQnp (cid:0) (cid:1) =p(m−1)(−α−n) 1+||pm−1x−pm−1ξ|| −α−n||ξ||λdnξ p p ZQnp (cid:0) (cid:1) =p(m−1)(λ−α) 1+||pm−1x−η|| −α−n||η||λdnη p p ZQnp (cid:0) (cid:1) =p(m−1)(λ−α)I(1,pm−1x). (3.9) Let pm−1x=y, ||y|| =pl. We have p I(1,y)=I (y)+I (y)+I (y), 1 2 3 where l−1 I (y)= (1+||y−η|| )−α−n||η||λdnη, 1 p p k=−∞Z||η||p=pk X I (y)= (1+||y−η|| )−α−n||η||λdnη, 2 p p Z||η||p=pl ∞ I (y)= (1+||y−η|| )−α−n||η||λdnη. 3 p p k=l+1Z||η||p=pk X The results follows from the following estimations: Claim A. I (y)≤C (1+||y|| )−α−n||y||λ+n; 1 0 p p p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 9 Claim B. I (y)≤C ||y||λ; 2 1 p Claim C. I (y)≤C . 2 2 Indeed, from the claims we have I(1,y)≤C (1+||y||λ), and by (3.9), 3 p I(b,x)≤C p(m−1)(λ−α)(1+p(1−m)λ||x||λ) 3 p ≤C p−mα(1+||x||λ)≤Cb−α 1+||x||λ . 3 p p (cid:0) (cid:1) We now prove the announced claims. Proof of Claim A. l−1 I (y)= (1+||y−η|| )−α−n||η||λdnη, 1 p p k=−∞Z||η||p=pk X l−1 =(1−p−n)(1+||y|| )−α−n p(λ+n)k p k=−∞ X ≤C (1+||y|| )−α−n||y||λ+n, 0 p p where (1−p−n)p−λ−n C = . 0 1−p−λ−n Proof of Claim B. Let y ∈Q such that |y| =pl =||y|| , then p p p I (y)= (1+||ey−η|| )−α−n||η||λednη 2 p p Z||η||p=pl =||y||λ 1+|y| ||y−1y−y−1η|| −α−n dnη p p p Z||η||p=pl (cid:0) (cid:1) =||y||λ−α ||y|e|−1+e ||u−ηe|| −α−n dnη, with u=y−1y. p p p Z||η||p=1 (cid:0) (cid:1) We set e A ={η ∈Qn |||η|| =1 and ||u−η|| =p−m}, for m∈N, m p p p and for I non-empty subset of {1,2,...,n}, A ={η ∈A ||u −η | =p−m for i∈I and |u −η | <p−m for i∈/ I}, m,I m i i p i i p where u=(u ,...,u ), η =(η ,...,η )∈Qn, with ||η|| =||u|| =1. 1 n 1 n p p p With this notation we have A ⊆ A , m I m,I vol(A )≤(p−m(S1−p−1))|I|(p−m−1)n−|I|, m,I here |I| denotes the cardinality of I, then n n vol(A )≤ (p−m(1−p−1))|I|(p−m−1)n−|I| =p−mn, m |I| |XI|=0(cid:18) (cid:19) 10 J.J.RODR´IGUEZ-VEGAANDW.A.ZU´N˜IGA-GALINDO and ∞ I (y)=||y||λ−α ||y||−1+||u−η|| −α−n dnη 2 p p p m=0ZAm X (cid:0) (cid:1) ∞ ≤||y||λ−α ||y||−1+p−m −α−np−mn p p m=0 X (cid:0) (cid:1) ||y||λ−α ∞ = p ||y||−1+||η|| −α−n dnη 1−p−n p p m=0Z||η||p=p−m X (cid:0) (cid:1) ||y||λ−α = p ||y||−1+||η|| −α−n dnη 1−p−n p p Z||η||p≤1 (cid:0) (cid:1) ≤C′||y||λ−α ||y||−1+||η|| −α−n dnη 1 p p p ZQnp (cid:0) (cid:1) =C′||y||λ−α ||y||−1+||y||−1||yη|| −α−n dnη 1 p p p p ZQnp (cid:0) (cid:1) =C′||y||λ+n (1+||yη|| )−α−n ednη 1 p p ZQnp =C′||y||λ (1+||τ||e)−α−n dτ ≤C ||y||γ. 1 p p 1 p ZQnp Proof of Claim C. ∞ I (y)= (1+||η|| )−α−n||η||λdnη, 3 p p k=l+1Z||η||p=pk X ≤ (1+||η|| )−α−n||η||λdnη =C. p p ZQnp (cid:3) Lemma 5. If α>0, then 1 ||x||α = ||y||−α−n(Ψ(−x·y)−1) dny (3.10) p (n) p Γp (−α)ZQnp for all x∈Qn. p Proof. The proof is a slightly variation of the proof of Proposition 2.3 in [13]. (cid:3) Lemma 6. Let 0<γ ≤α, then (DγZ)(x,t)= Ψ(x·η)||η||γe−at||η||αp dnη. T p ZQnp Proof. By Lemma 2 (2), Z(x−y,t) = Z(x,t), for ||y|| < ||x||. Then we can use (2.3) to calculate(DγZ)(x,t): T 1 (DγZ)(x,t)= ||y||−γ−n(Z(x−y,t)−Z(x,t))dny T Γ(pn)(−γ)ZQnp p 1 = ||y||−γ−n(Z(x−y,t)−Z(x,t))dny. (n) p Γp (−γ)Z||y||p≥||x||p