In this article we solve the Cauchy problem associated to a radial symbol constant on a ball of radius pr and we show that the fundamental solution, Zr(x, t), vanishes outside the ball of radius p−r.En este artículo encontramos la solución al problema de Cauchy asociado a un símbolo radial que es constante en bolas de radio pr, y mostramos que la solución fundamental Zr(x, t) se anula fuera de una bola de radio p−r

Casas-Sánchez, Oscar

Rodríguez-Vega, Jhon Jaime

Spanish

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Bolet́ın de Matemáticas 22(1) 97–106 (2015) 97Parabolic type equations on p−adic ballsEcuaciones de tipo parabólico sobre bolas p−ádicasO. F. Casas-Sánchez1,a, J.J. Rodŕıguez-Vega2,bAbstract. In this article we solve the Cauchy problem associated to a radialsymbol constant on a ball of radius pr and we show that the fundamentalsolution, Zr(x, t), vanishes outside the ball of radius p−r.Keywords: p−adic numbers, pseudodifferential equations, Cauchy problem.Resumen. En este art́ıculo encontramos la solución al problema de Cauchyasociado a un śımbolo radial que es constante en bolas de radio pr, y mostramosque la solución fundamental Zr(x, t) se anula fuera de una bola de radio p−r.Palabras claves: Números p−ádicos, ecuaciones pseudodiferenciales, proble-ma de Cauchy.Mathematics Subject Classification: 35S99, 47S10.Recibido: marzo de 2015 Aceptado: junio de 20151. IntroductionA p−adic pseudodifferential operator f(∂, α), with symbol f(ξ), is an operatorof the formf(∂, α)u := F−1ξ→x(|f(ξ)|αpFx→ξu), α > 0, (1)where ϕ ∈ S, S denotes the C-vector space of Bruhat-Schwartz functions overQnp , F denotes the Fourier transform on L2(Qnp ), and f(ξ) is a p-adic valuedfunction on Qnp .If the symbol |f(ξ)|αp is continuous, then the operator f(D) is continuousand has a self-adjoint extension with dense domain in L2(Qnp ). This operatoris considered to be a p-adic analogue of a linear partial elliptic differentialoperator with constant coefficients.In [4] Kochubei studies the Cauchy problem{∂u(x,t)∂t +Dαu(x, t) = g(x, t), x ∈ Qp, n ≥ 1, t ∈ (0, T ],u(x, 0) = ϕ(x), x ∈ Qp,(2)1Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia2Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombiaaoscasas@uniandes.edu.cobjjrodriguezv@unal.edu.co98 O. F. Casas-Sánchez & J.J. Rodŕıguez-VegawhereDαu := F−1ξ→x(|ξ|αpFx→ξu), α > 0, (3)is the Valdimirov operator. He finds a fundamental solution in certain spaceMλ, and gives some properties of it.In [5] Rodŕıguez-Vega and Zúñiga-Galindo study the same problem, but indimension n. They consider the Taibleson operator, Dαu := F−1ξ→x(||ξ||αpFx→ξu),for α > 0, and x ∈ Qnp , and by using the same method and the same spaceMλ, they find a fundamental solution for the Cauchy problem and some of itsproperties.In [7] Zúñiga-Galindo considers a more general class of pseudodifferentialoperadors, defined asf(∂, α)u := F−1ξ→x(|f(ξ)|αpFx→ξu), α > 0, (4)where f is an elliptic polynomial and x ∈ Qnp . He build a fundamental solutionfor the homogeneous Cauchy problem{∂u(x,t)∂t + f(∂, α)u(x, t) = 0, x ∈ Qnp , n ≥ 1, t ∈ (0, T ],u(x, 0) = ϕ(x), x ∈ Qnp ,(5)with α > 0 and ϕ a bounded and continuous function. This fundamentalsolution is called the heat kernel and it is given byZ(x, t) =∫Qnpχ(−x · ξ)e−t|f(ξ)|αp dnξ, (6)where χ is a standard additive character on Qnp .In [3] Casas-Sánchez and Zúñiga-Galindo study the Cauchy problem withvariable coefficients associated to an elliptic quadratic form in dimension twoand four, more recently Casas-Sánchez, Galeano-Peñaloza and Rodŕıguez-Vega[2], studied the Cauchy problem associated with an elliptic quadratic form ofdimension three.The main goal of this article is to solve the Cauchy problem associated to aradial symbol constant on a ball of radius pr, see Definition 3.1, and to see thatthe fundamental solution, Zr(x, t), vanishes outside a ball, Proposition 3.4.The article is organized as follows. In Section 2, we give the definitions andsome preliminars results. In Section 3, we define the heat kernel Z(x, t) andwe show that the heat kernel is a fundamental solution of the Cauchy problem,see Theorem 3.14.2. Preliminary ResultsAlong this article p will denote a prime number different from 2. Let Qp bethe field of the p-adic numbers, and let Zp be the ring of p-adic integers. Forx ∈ Qp, let v(x) denote the valuation of x normalized by the condition v(p) = 1,Bolet́ın de Matemáticas 22(1) 97–106 (2015)Parabolic type equations on p−adic balls 99and let |x|p = p−v(x), |0|p = 0 be the normalized absolute value. We extendthe p-adic absolute value to Qnp as follows:||x||p := max{|x1|p, . . . , |xn|p}, for x = (x1, . . . , xn) ∈ Qnp .The Bruhat-Schwartz space S(Qnp ) consists of all complex-valued and locally-constant functions with compact support, such functions are called test func-tions, where the exponent of local constancy of ϕ(x) ∈ S(Qnp ) is the smallestinteger, l ≥ 0, with the property:ϕ(x+ x′) = ϕ(x) if ||x′||p ≤ p−l, for any x ∈ Qnp .A linear functional f defined over S(Qnp ), f : S(Qnp ) −→ C, φ 7→ 〈f, φ〉 iscalled a generalized function or a distribution. Denote by S′(Qnp ) the set of alldistributions over Qnp . We note that all distributions over Qnp are continuous.The Fourier transform of ϕ ∈ S(Qnp ) is defined as(Fϕ)(ξ) =∫Qnpχ(ξ · x)ϕ(x)dx for ξ ∈ Qnp ,where ξ · x :=∑nj=1 ξjxj , dx is the Haar measure on Qnp normalized by thecondition vol(Bn0 ) = 1 and χ denote an additive character of Qp, trivial on Zp,but not on p−1Zp. The Fourier transform is a linear isomorphism from S(Qnp )onto itself satisfying (F(Fϕ))(ξ) = ϕ(−ξ).For a more detailed explanation, the reader may consult [1] and [6].3. A General Type of EquationsIn this section we consider the following integral.Definition 3.1. We define the heat kernel asZr(x, t) :=∫Qnpχ(−x · ξ) e−t(f(||ξ||p)−λ)dξ, (7)where f(||ξ||p) satisfies the following conditions(i) f(||ξ||p) = λ if ||ξ||p ≤ pr.(ii) f(||ξ||p) is a non-decreasing function.(iii) There are positive constants A1, A2, γ1 and γ2, such thatA1||ξ||γ1p ≤ f(||ξ||p) ≤ A2||ξ||γ2p , for ||ξ||p  1.We first note that the function e−tf(||ξ||p) belongs to L1(Qnp ).Lemma 3.2. For ρ ≥ 1, and fixed t > 0, we have e−tf(||ξ||p) ∈ Lρ(Qnp ).Bolet́ın de Matemáticas 22(1) 97–106 (2015)100 O. F. Casas-Sánchez & J.J. Rodŕıguez-VegaProof. It is a consequence of the fact that∫Qnpe−ρt||ξ||αp dξ < +∞for any ρ, t, α > 0. Indeed|| exp(−tf(||ξ||p))||ρLρ =∫||ξ||p>pre−tρf(||ξ||p) dξ +∫||ξ||p≤pre−λρt dξ≤∫Qnpe−ρtA1||ξ||γ1p dξ + e−λρt < +∞.Lemma 3.3. For all x ∈ Qnp and t > 0,Zr(x, t) ≥ 0.Proof.Zr(x, t) = eλt∞∑k=−∞e−tf(pk)∫||ξ||p=pkχ(−x · ξ) dξ= eλt∞∑k=−∞e−tf(pk)(pnkΩ−k(x)− pn(k−1)Ω−k+1(x))= eλt∞∑k=−∞pnk(e−tf(pk) − e−tf(pk+1))Ω−k(x),where Ωk is the indicator function of the ball {x ∈ Qnp | ||x||p ≤ pk}.Proposition 3.4. If x 6= 0 and t > 0, then we have thatZr(x, t) =pnr − ||x||−np e−t(f(p||x||−1p )−λ)+(1− p−n)||x||−npv(x)−r−1∑l=0p−nle−t(f(p−l||x||−1p )−λ) if ||x||p < p−r,pnr − ||x||−np e−t(f(p||x||−1p )−λ) if ||x||p = p−r,0 if ||x||p > p−r.Proof. As f(||ξ||p) = λ for ||ξ||p ≤ pr, thenZr(x, t) =∫||ξ||p≤prχ(−x · ξ) dξ + eλt∫||ξ||p>prχ(−x · ξ) e−tf(||ξ||p) dξ, (8)the first integral is equal to the Fourier transform of Ωr(ξ), i.e., pnrΩ−r(x).For the second integral note that if ||x||p > p−r, then for ||ξ||p = pυ, υ > r, wehave ∫||ξ||p=pυχ(−x · ξ) dξ = 0,Bolet́ın de Matemáticas 22(1) 97–106 (2015)Parabolic type equations on p−adic balls 101therefore Zr(x, t) = 0 for ||x||p > p−r.Now∫||ξ||p>prχ(−x · ξ) e−tf(||ξ||p) dξ =∞∑ν=r+1e−tf(pν)∫||ξ||p=pνχ(−x · ξ) dξ,the last sum has only a finitely number of terms not equal to zero. In fact if||x||p = p−v(x) (v(x) > r), then ||ξ||p can be equal to pr+1, pr+2, . . . , pv(x)+1.In the last case if ||ξ||p = ||x||−1p p we have thate−tf(||x||−1p p)∫||ξ||p=||x||−1p pχ(−x · ξ) dξ = −e−tf(||x||−1p p)||x||−np ,and that ||ξ||p varies between pr+1, . . . , pv(x) is equivalent to ||ξ||p varies be-tween||x||−1p pr+1−v(x), . . . , ||x||−1p p−1, ||x||−1p ,and therefore∫pr<||ξ||p<||x||−1pχ(−x · ξ) e−tf(||ξ||p) dξ=v(x)−r−1∑l=0∫||ξ||p=p−l||x||−1pχ(−x · ξ) e−tf(||ξ||p) dξ=v(x)−r−1∑l=0e−tf(p−l||x||−1p )p−ln||x||−np (1− p−n)= (1− p−n)||x||−npv(x)−r−1∑l=0e−tf(p−l||x||−1p )p−ln. (9)Corollary 3.5.Zr(x, t) ≤ Ct||x||−np(f(p||x||−1p )− λ). (10)Proof. (i) If ||x||p < p−r, we have that pnr < ||x||−n, andZr(x, t) ≤ ||x||−n − ||x||−ne−t(f(p||x||−1)−λ)1− (1− p−1) v(x)−r−1∑l=0p−nl≤ ||x||−n − ||x||−ne−t(f(p||x||−1)−λ)(1− (1− p−1)1− p−n(v(x)−r)1− p−n)≤ ||x||−n(1− pn−1 − 1pn − 1 e−t(f(p||x||−1)−λ))≤ Ct||x||−np(f(p||x||−1p )− λ).Bolet́ın de Matemáticas 22(1) 97–106 (2015)102 O. F. Casas-Sánchez & J.J. Rodŕıguez-Vega(ii) If ||x||p = p−r, thenZr(x, t) = ||x||−n − ||x||−ne−t(f(p||x||−1)−λ)≤ ||x||−n(1− e−t(f(p||x||−1)−λ))≤ Ct||x||−np(f(p||x||−1p )− λ).Lemma 3.6. For α > 0, t > 0, the following assertions hold(i) Zr(x, t) ∈ C(Qnp ,R) ∩ L1(Qnp ) ∩ L2(Qnp ), for t > 0.(ii)∫QnpZr(x, t) dx = 1.(iii) limt→0+Zr(x, t) ∗ ϕ(x) = ϕ(x), for ϕ ∈ S(Qnp ).(iv) Zr(x, t) ∗ Z(x, t′) = Z(x, t+ t′), for t, t′ > 0.Proof. (i) It is a direct consequence of Lemma 3.2.(ii) It follows from (i) by the inversion formula for the Fourier transform.(iii) By using (ii), we have that∫QnpZr(x− ξ, t)ϕ(ξ) dξ − ϕ(x) =∫QnpZr(x− ξ, t)(ϕ(ξ)− ϕ(x)) dξ.Since ϕ(x) is locally constant, there exists m ∈ Z such that ϕ(ξ)−ϕ(x) =0 if ||ξ − x||p ≤ pm. ThenI : =∣∣∣∣∣∣∣∫QnpZr(x− ξ, t)(ϕ(ξ)− ϕ(x)) dξ∣∣∣∣∣∣∣=∣∣∣∣∣∣∣∫||ξ−x||p>pmZr(x− ξ, t)(ϕ(ξ)− ϕ(x)) dξ∣∣∣∣∣∣∣≤ Ct∫||ξ−x||p>pm||ξ − x||−np f(p||ξ − x||−1p ) |ϕ(ξ)− ϕ(x)| dξ≤ A2Ctp∫||ξ−x||p>pm||ξ − x||−n−γ2p |ϕ(ξ)− ϕ(x)| dξ= C2t∫||z||p>pm||z||−n−γ2p |ϕ(x− z)− ϕ(x)| dzBolet́ın de Matemáticas 22(1) 97–106 (2015)Parabolic type equations on p−adic balls 103≤ C2t∫||z||p>pm||z||−n−γ2p |ϕ(x− z)− ϕ(x)| dz≤ C2t(C3 + C4|ϕ(x)|)→ 0, as t→ 0+.(iv) By (i) the formula is equivalent toFx→ξ(Zr(x, t))· Fx→ξ(Zr(x, t′))= Fx→ξ(Zr(x, t+ t′))butFx→ξ(Zr(x, τ))= e−τ(f(||ξ||p)−λ).Definition 3.7. We define the following pseudodifferential operatorsJϕ = F−1ξ→x(f(||ξ||p)Fy→ξ ϕ(y)), ϕ ∈ S(Qnp ) (11)andHϕ = Jϕ− λϕ, ϕ ∈ S(Qnp ). (12)Lemma 3.8. We have the following(i) The map J : S(Qnp )→ S(Qnp ), sending ϕ to Jϕ is a homeomorphism.(ii) The map H : S(Qnp )→ S(Qnp ), sending ϕ to Hϕ is continuous.Proof. For the first claim recall that ϕ 7→ Fϕ is a homeomorphism of S(Qnp ),the results follows from this observation since ϕ 7→ f(||ξ||p)Fϕ is a homeomor-phism.The second statement follows from (i).Definition 3.9. We set for ϕ ∈ S(Qnp )u(x, t) ={Zr(x, t) ∗ ϕ(x), if t > 0,ϕ(x), if t = 0.(13)Lemma 3.10. For t ≥ 0, u(x, t) ∈ S(Qnp ).Proof. Since Zr(x, t) ∗ ϕ(x) ∈ L1(Qnp ) ∩ L2(Qnp ) for t > 0Fx→ξ(u(x, t)) = e−t(f(||ξ||p)−λ)ϕ̂(ξ) ∈ S(Qnp ),since the Fourier transform is a homomorphism on S, we conclude that u(x, t) ∈S for t > 0.Remark 3.11. For t ≥ 0, α > 0H(u(x, t)) = F−1ξ→x[(f(||ξ||p)− λ)e−t(f(||ξ||p)−λ) ϕ̂(ξ)].Bolet́ın de Matemáticas 22(1) 97–106 (2015)104 O. F. Casas-Sánchez & J.J. Rodŕıguez-VegaLemma 3.12. Let ϕ ∈ S(Qnp ) and u(x, t), t ≥ 0 defined by (13). Then u(x, t)is continuously differentiable in time for t ≥ 0, in the pointwise sense, and thederivative is given by∂u∂t(x, t) = −H(u(x, t)).Proof. The results follow from the following claimClaim 3.13. (i) For t0 ≥ 0limt→t0e−t(f(||ξ||p)−λ) − e−t0(f(||ξ||p)−λ)t− t0− (f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ) = 0.(ii) For t, t0 ∈ [0, T ]∣∣∣e−t(f(||ξ||p)−λ) − e−t0(f(||ξ||p)−λ)t− t0+ (f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ)∣∣∣ ≤ C(t)||ξ||γ2p .Indeed, note thatlimt→t0u(x, t)− u(x, t0)t− t0+ F−1((f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ)ϕ̂(ξ))= limt→t0F−1[ϕ̂(ξ)(e−t(f(||ξ||p)−λ) − e−t0(f(||ξ||p)−λ)t− t0+ (f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ))]The first part follows from the Claim (i)−(ii) by using the Dominated Conver-gence Theorem. The second part follows from Claim (i) by the Parseval-SteklovTheorem.Proof of the Claim. The Claim is a consequence of the following calculation.By applying the Mean Value Theorem twice we obtaine−t(f(||ξ||p)−λ) − e−t0(f(||ξ||p)−λ)t− t0+ (f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ)= −(f(||ξ||p)− λ)e−t′(f(||ξ||p)−λ) + (f(||ξ||p)− λ)e−t0(f(||ξ||p)−λ)= (f(||ξ||p)− λ)2(t′ − t0)e−t′′(f(||ξ||p)−λ),where t′ = t′(||ξ||p) is a point between t0 and t and t′′ = t′′(||ξ||p) is a pointbetween t0 and t′ (and thus between t0 and t).Theorem 3.14. Consider the following Cauchy problemu ∈ C([0,∞],S(Qnp ))∩ C1([0,∞], L2(Qnp )),∂u∂t(x, t) +(H u)(x, t) = 0, x ∈ Qnp , t ∈ (0, T ], α > 0,u(x, 0) = ϕ(x), ϕ ∈ S(Qnp ).(14)Then the function u(x, t) defined in (13) is a solution to Problem (14).Proof. This is a consequence of Lemmas 3.6 and 3.12.Bolet́ın de Matemáticas 22(1) 97–106 (2015)Parabolic type equations on p−adic balls 1054. Heat equation on a ballConsider the following pseudodifferential operatorf(∂, α, r)ϕ := F−1ξ→x(max(||ξ||p, pr)αFy→ξ ϕ(y)), ϕ ∈ S(Qnp ), (15)and the following equation∂u∂t(x, t) +(f(∂, α, r)u)(x, t) = λu(x, t), (16)then the fundamental solution to this equation is given byZr(x, t) =∫Qnpχ(−x · ξ) e−t(max(||ξ||p,pr)α−λ)dξ, λ = prα (17)andZr(x, t) = eλtZ(x, t) + pnr − eλt(1− p−n)pnr∞∑n=0(−1)nn!tnprαn1− p−αn−1,where Z(x, t) is given by (6) in the case |f(ξ)|p = ||ξ||p.We note that our fundamental solution (17) agrees with the solution ob-tained by Kochubei in the case λ = prα, see [4], Theorem 4.9.References[1] S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, Theory of p-adicdistributions: linear and nonlinear models, Cambridge University Press,2010.[2] O. F. Casas-Sánchez, J. Galeano-Peñaloza, and J. J. Rodŕıguez-Vega,Parabolic-type pseudodifferential equations with elliptic symbols in dimen-sion 3 over p−adics, p−Adic Numbers, Ultrametric Analysis and Applica-tions 7(1) (2015), 1–16.[3] O. F. Casas-Sánchez and W. A. Zúñiga-Galindo, p-adic elliptic quadraticforms, parabolic-type pseudodifferential equations with variable coefficients,and Markov processes, p-Adic Numbers Ultrametric Analysis and Applica-tions 6(1) (2014), 1–20.[4] Anatoly N. Kochubei, Pseudo-differential equations and stochastics overnon-archimedean fields, Marcel Dekker, Inc., New York, 2001.[5] J. J. Rodŕıguez-Vega and W. A. Zúñiga-Galindo, Elliptic pseudodifferentialequations and Sobolev spaces over p-adic fields, Pacific J. Math. 246 (2010),407–420.Bolet́ın de Matemáticas 22(1) 97–106 (2015)106 O. F. Casas-Sánchez & J.J. Rodŕıguez-Vega[6] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic analysis andmathematical physics, World Scientific, 1994.[7] W. A. Zúñiga-Galindo, Parabolic equations and Markov processes over p-adic fields, Potential Anal. 28 2 (2008), 185–200.Bolet́ın de Matemáticas 22(1) 97–106 (2015)

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