We show the existence of fundamental solutions for p-adic pseudo-differential
operators with polynomial symbols.Comment: To appear in Rend. Sem. Mat. Univ. Padov

Zuniga-Galindo, W. A.

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REND. SEM. MAT. UNIV. PADOVA, Vol. 109 (2003)Fundamental Solutions of Pseudo-DifferentialOperators over p-Adic Fields.W. A. ZUNIGA-GALINDO (*)ABSTRACT - We show the existence of fundamental solutions for p-adic pseudo-dif-ferential operators with polynomial symbols.1. Introduction.Let K be a p-adic field, i.e. a finite extension of Qp the field of p-adicnumbers. Let RK be the valuation ring of K , PK the maximal ideal of RK ,and K 4RK / PK the residue field of K. The cardinality of K is denoted byq. For zK , v(z)ZN ]1Q( denotes the valuation of z , NzNK4q 2v(z)and ac(z) 4zp2v(z) where p is a fixed uniformizing parameter for RK . LetC denote an additive character of K trivial on RK but not on PK21.A function F : K nKC is called a Schwartz-Bruhat function if it is lo-cally constant with compact support. We denote by S(K n ) the C-vectorspace of Schwartz-Bruhat functions over K n.The dual space S8 (K n ) isthe space of distributions over K n. Let f4 f (x)K[x], x4 (x1 , R , xn ),be a non-zero polynomial, and b a complex number satisfying Re (b) D0.If x4 (x1 , R , xn ), y4 (y1 , R , yn )K n , we set [x , y] »4 !i41nxi yi .A p-adic pseudo-differential operator f (¯ , b), with symbol NfNKb , is an(*) Indirizzo dell’A.: Department of Mathematics and Computer Science,Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161,USA. E-mail: wzunigaHmail.barry.eduW. A. Zuniga-Galindo242operator of the formf (¯ , b) : S(K n )FKKS(K n )F21 (NfNKb F (F) ),(1.1)whereF : S(K n )FKKS(K n )K nC(2[x , y] ) F(x) dx(1.2)is the Fourier transform. The operator f (¯ , b)has self-adjoint extensionwith dense domain in L 2 (K n ). We associate to f (¯ , b) the following p-adic pseudo-differential equation:f (¯ , b) u4g , g S(K n ).(1.3)A fundamental solution for (1.3) is a distribution E such that u44E˜g is a solution.The main result of this paper is the following.THEOREM 1.1. Every p-adic pseudo-differential equationf (¯ , b) u4g , with f (x)K[x1 , R , xn ] 0K , g S(K n ), and bC ,Re (b) D0, has a fundamental solution E S8 (K n ).The p-adic pseudo-differential operators occur naturally in p-adicquantum field theory [11], [6]. Vladimirov showed the existence of a fun-damental solutions for symbols of the form NjNKa , aD0 [10], [11]. In [7],[6] Kochubei showed explicitly the existence of fundamental solutions foroperators with symbols of the form Nf (j 1 , R , j n )NKa , aD0, wheref (j 1 , R , j n ) is a quadratic form satisfying f (j 1 , R , j n ) c0 ifNj 1 NK1R1Nj nNKc0. In [8] Khrennikov considered spaces of func-tions and distributions defined outside the singularities of a symbol, inthis situation he showed the existence of a fundamental solution for a p-adic pseudo-differential equation with symbol NfNKc0. The main resultof this paper shows the existence of fundamental solutions for operatorswith polynomial symbols. Our proof is based on a solution of the divisionproblem for p-adic distributions. This problem is solved by adapting theideas developed by Atiyah for the archimedean case [1], and Igusa’s the-orem on the meromorphic continuation of local zeta functions [3], [4].The connection between local zeta functions (also called Igusa’s local ze-Fundamental solutions of pseudo-differential operators etc. 243ta functions) and fundamental solutions of p-adic pseudo-differential op-erators has been explicitly showed in particular cases by Jang and Sato[5], [9]. In [9] Sato studies the asymptotics of the Green function G of thefollowing pseudo-differential equation(f (¯ , 1 )1m 2 ) u4g , mD0.(1.4)The main result in [9, theorem 2.3] describes the asymptotics of G(x)when the polynomial f is a relative invariant of some prehomogeneousvector spaces (see e.g. [3, Chapter 6]). The key step is to establish a con-nection between the Green function G(x) and the local zeta function at-tached to f.All the above mentioned results suggest a deep connection betweenIgusa’s work on local zeta functions (see e.g. [3]) and p-adic pseudo-dif-ferential equations.2. Local zeta functions and division of distributions.The local zeta function associated to f is the distributionaNfNKs , Fb4 K nF(x)Nf (x)NKs dx ,(2.1)where F S(K n ), sC , Re (s) D0, and dxis the Haar of K n normalizedso that vol(RKn ) 41. The local zeta functions were introduced by Weil[12] and their basic properties for general f were first studied by Igusa[3], [4]. A central result in the theory of local zeta functions is thefollowing.THEOREM 2.1 (Igusa, [3, Theorem 8.2.1]). The distribution NfNKs ad-mits a meromorphic continuation to the complex plane such thataNfNKs , Fb is a rational function of q 2s for each F S(K n ). In additionthe real parts of the poles of NfNKs are negative rational numbers.The archimedean counterpart of the previous theorem was obtainedjointly by Bernstein and Gelfand [2], independently by Atiyah [1]. Thefollowing lemma is a consequence of the previous theorem.LEMMA 2.1. Let f (x)K[x1 , R , xn ] be a non-constant polynomial,and b a complex number satisfying Re (b) D0. Then there exists a dis-tribution T S8 (K n ) satisfying NfNKb T41.W. A. Zuniga-Galindo244PROOF. By theorem 2.1 NfNKs has a meromorphic continuation to Csuch that aNfNKs , Fb is a rational function of q 2s for each F S(K n ).LetNfNKs 4 !mZcm (s1b)m(2.2)be the Laurent expansion at 2b with cm S8 (K n ) for all m. Since thereal parts of the poles of NfNKs are negative rational numbers by theorem2.1, it holds that NfNKs1b4NfNKb NfNKs is holomorphic at s42b. ThereforeNfNKb cm40 for all mE0 andNfNKs1b4c0 NfNKb 1 !m41Qcm NfNKb (s1b)m .(2.3)By using the Lebesgue lemma and (2.3) it holds thatlimsK2baNfNKs1b , Fb4 K nF(x) dx4 a1, Fb(2.4)4c0 NfNKb .Therefore we can take T4c0 . rIf T S8 (K n ) we denote by FT S8 (K n ) the Fourier transform of thedistribution T , i.e. aFT , Fb4 aS , F (F)b, F S(K n ).3. Proof of the main result.By lemma 2.1 there exists a T S8 (K n ) such that NfNKb T41. We setE4F21 T S8 (K n ) and assert that E is a fundamental solution for (1.3).This last statement is equivalent to assert that F (F) 4 (FE) F ( g) satis-fies NfNKb F (F) 4F ( g). Since NfNKb F (F) 4NfNKb (FE) F ( g) 4NfNKb T F ( g) 44F ( g), we have that E is a fundamental solution for (1.3).4. Operators with twisted symbols.Let x : RK3KC be a non-trivial multiplicative character, i.e. ahomomorphism with finite image, where RK3 is the groupof units of RK .We put formally x(0) 40. If f (x)K[x1 , R , xn ] 0K , we say thatx(ac( f ) )NfNKb , with bC , Re (b) D0, is a twisted symbol, and call theFundamental solutions of pseudo-differential operators etc. 245pseudo-differential operatorFK f (¯ , b , x) F4F21 (x(ac( f ) )NfNKb F (F) ), F S(K n ),(4.1)a twisted operator. Since the distribution x(ac( f ) )NfNKb satisfies all theproperties stated in theorem 2.1 (cf. [3, Theorem 8.2.1]), theorem 1.1generalizes literally to the case of twisted operators. In [6, chapter 2]Kochubei showed explicitly the existence of fundamental solutions fortwisted operators in some particular cases.R E F E R E N C E S[1] M. F. ATIYAH, Resolution of singularities and division of distributions,Comm. Pure Appl. Math., 23 (1970), pp. 145-150.[2] I. N. BERNSTEIN - S. I. GELFAND, Meromorphic property of the functions P l ,Functional Anal. Appl., 3 (1969), pp. 68-69.[3] IGUSA JUN-ICHI, An introduction to the theory of local zeta functions,AMS/IP studies in advanced mathematics, v. 14, 2000.[4] J. IGUSA, Complex powers and asymptotic expansions, I Crelles J. Math.,268/269 (1974), pp. 110-130; II, ibid., 278/279 (1975), pp. 357-368.[5] Y. JANG, An asymptotic expansion of the p-adic Green function, TohokuMath. J., 50 (1998), pp. 229-242.[6] A. N. KOCHUBEI, Pseudodifferential equations and stochastics over non-archimedean fields, Marcel Dekker, 2001.[7] A. N. KOCHUBEI, Fundamental solutions of pseudo-differential equationsassociated with p-adic quadratic forms, Izvestiya Math., 62 (1998), pp. 1169-1188.[8] A. KHRENNIKOV, Fundamental solutions over the field of p-adic numbers,St. Peterburgh Math. J., 4 (1993), pp. 613-628.[9] FUMIHIRO SATO, p-adic Green functions and zeta functions, CommentariiMathematici Universitatis Sancti Pauli, 51 (2002), pp. 79-97.[10] V. S. VLADIMIROV, On the spectrum of some pseudo-differential operatorsover p-adic number field, Algebra and Analysis, 2 (1990), pp. 107-124.[11] V. S. VLADIMIROV - I. V. VOLOVICH - E. I. ZELENOV, p-adic Analysis andmathematical physics, World Scientific, Singapore, 1994.[12] A. WEIL, Sur la formule de Siegel dans le théorie des groupes classiques, Ac-ta Math., 113 (1965), pp. 1-87.Manoscritto pervenuto in redazione il 2 dicembre 2002.

Fundamental solutions of pseudo-differential operators over p-adic fields

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