Table Of ContentTAIBLESON 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 numbers. 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
