QUANTIZATIONS and SYMBOLIC CALCULUS OVER the P-ADIC NUMBERS
Ann. Inst. Fourier, Grenoble 43, 4 (1993), 997-1053
QUANTIZATIONS AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS
by Shai HARAN
INTRODUCTION
We shall be concerned with functions / : VQ —^ C, defined in some vector space Vo OYer the p-adic numbers Qp and taking values in the complex numbers C. One of the most basic problems encountered when trying to imitate the classical theory — where the domain VQ is a vector space over R or C — is the lack of derivatives. Indeed, the derivation 9/Qx is nothing but T~^xF^ where T is the Fourier transform, and «a;» is multiplication by the function x which is an additive homomorphism from VQ to C; and there are no such homomorphisms from Vo to C when VQ is a vector space over Qp. This problem repeatedly makes its appearance in various disguises; for example, given a unitary representation of a p-adic analytic group on a Hilbert space, one cannot associate with it the derived representation of a p-adic Lie algebra. From a different perspective, the derivatives {Q/Qx^ correspond to the extra poles of the oo-component of the zeta function, while the p-components have a unique pole. There are thus no differential operators over Qp. But as we will show in this paper, there is a meaningful theory of pseudodifferential operators over the p-adics, which parallels the classical theory over the real numbers R. In fact, the theory over Qp is better behaved than the one over R, in as much as in all estimates the numbers 2 = [C : R] for the reals can be replaced by oo = [Qp : Qp] for the p-adics, and one encounters here the phenomenon expounded in [15], [16].
Key words : p-adic integers - Heisenberg groups - Weyl quantization - Besov spaces - Elliptic operators - Spectral asymptotics. A.M.S. Classification : 11E95 - 11F85 - 11M41 - 11R42 - 11S80 - 12J25 - 20G25 - 35S99 - 41A65 - 43A25 - 81Q99 - 81S10 - 81S30. 998 SHAIHARAN
The pseudodifferential operator p(f) associated with the « symbol» /, a function defined on the «phase space)) V = VQ (D VQ^ is given via the action of the Heisenberg group (cf. 0.1.10). The composition of operators correspond to «twisted multiplication)) of their symbols : p(/i)p(/2) = ^(/i#/2)- The main goal of the symbolic calculus is to control the twisted multiplication over appropriately defined symbol classes, thus enabling the comparison of the resulting function algebra and operator algebra. Classically, the symbol classes are defined using the Mihiin condition which incorporates the global growth condition with the local smoothness condition on the symbols. To define symbol classes in the p-adic setting we encounter once again the lack of derivatives. Fortunately, this time the problem can be overcome if one uses in place of derivatives a Besov condition. Chapter 0 is devoted to the Heisenberg group and its fundamental representation. For the analogous theory over R see [6], [9], [21], [23], [32]; some of the material over Qp can be found in [II], [36]. In paragraph 0, we fix our notations and recall the Schrodinger model, the Weyl quantization procedure and twisted multiplication. In paragraph 0.2, we review the lattice model and the Bargmann isomorphism [I], and we describe the Wick symbols [4]. In paragraph 0.3, with an eye towards connection with [14], [15], we offer a third realization of the fundamental representation in the «positive)) subspace of Z/2 of the p-adic circle, or the space of «p-adic particles)), giving explicit formulas for the Hermite functions, and explaining why the Toeplitz operators over Qp have index zero. In paragraph 0.4, we give a «mannerized)) version of the lattice model, and we present the Wigner transform, the uncertainty principle and the wave- packet-expansions [7] in the p-adic setting. Chapter 1 is devoted to the exposition of the various functions spaces over Qp. For the analogous theory over R, we refer to [3], [25], [33], [34]; a good reference for some of the material over Qp can be found in [31]. In paragraph 1.1, we recall the Bessel potentials and Sobolev spaces. In paragraph 1.2, we explain the problematics of these — the lack of the Leibniz rule for differentiation, and offer a way out via Besov spaces. After reviewing the elementary properties of the ordinary Besov spaces, we present the concept of a metric-covering, and the associated « mannerized)) Besov spaces. In paragraph 1.3, we define the « smooth)) functions over Qp, and the (analytic) Schwartz spaces. We quote the Schwartz kernel theorems and define our symbol classes (in analogy with [18]). In paragraph 1.4, we collect the basic properties of temperate metric-coverings needed in the QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 999 sequel, and offer as examples the Toeplitz symbols (discussed over the reals in [12], [17], [21], [30], [35]) and the pseudodifferential symbols. The Toeplitz symbols E^V) consist of functions / on V which satisfy
\f(x)\< Co -|W, and for all f3 > 0,
\f(x)-f(x-^y)\
Schwartz space. The operator p(f) is thus the sum of p(/), an explicitly diagonalizable operator, and of p(fo), a trace-class operator (in fact, a « smoothening » operator in the sense that it maps 5" into 5). As a corollary we see that if / ^ 0 is positive, then the operator p(/) is positive modulo trace-class operators and is bounded from below. If / in E^V) is elliptic, in the sense that \f(x)\ > C ' ^.px^ for x outside a compact set, then p(f) is a Fredholm operator of index zero; if / is real valued, then p(f) is an essentially self-adjoint operator, if, moreover, a > 0, then p{f) has a discrete point spectrum tending to infinity and there is an orthonormal basis for L^Vo) consisting of p(/)-eigenvectors all of which are in the Schwartz space. Assuming further that both / and p(f) are positive, we can defined their zeta functions
CpCoO^tr^/)-8) and ^(s) = [/(x)-8 dx Jv both of which are holomorphic in the right half plane Re{s) > - dim V, have meromorphic continuation and moreover, their difference is entire. Denoting by N(X) the number of eigenvalues < A, we have the following precise estimate :
N(\) = vo\{x I f(x) ^ A} + 0(A6), for any e > 0.
Our motivation in developing the theory of pseudodifferential operators in an arithmetical situation was derived from [14], [15], where it was shown that such a theory could have bearings on the problem of the Riemann hypothesis. Much of what we do here carries over to the global situation of the adeles, where the zeta function not only controls the symbolic calculus, but to some extent is controlled by it. We note that the content of this paper generalizes easily to the case of an arbitrary local field. Other future applications of our theory could include the local Howe's duality conjectures. The theory can also be made to work over varieties where it might be useful as a tool in defining the zeta functions of such varieties by analytic means (a basic ingredient for trying to do Tale's thesis for such varieties). Since we wrote this paper, we have found some attempts to construct quantum mechanics over the p-adics in the literature of physics, e.g. [42], but these are always wrong and lead to strange phenomenon (e.g. the existence of two vacuum states), because the physicists were insensitive to the basic difference between the p-adics and the reals - the two dimensional absolute value, which for the reals is the L^ QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1001 one : \x,y\ = (|a:|2 + H2)172, should be replaced in the p-adic setting by its Loo analog : \x,y\ = max(|a;|, \y\). In the one dimensional theory, VQ = Qp, V = Qp © Qp, we can take as a basic group of symmetries the group G = Sp(l,Qp) = SL(2,Qp), and identify V \ {(0,0)} with the symmetric space G/N, where N is the subgroup of matrices of the form ^ I * ] ^, the action of G on V, and the resulting action on symbols, correspond to the metaplectic representation on L2(Qp). In a similar fashion we can follow the Unterbergers, cf. [39], [40], [41], and take as our phase space G/K, where K = SL(2,Zp) is a maximal compact subgroup, obtaining quantizations which are relevant to the other discrete series representations of G; or we can take as phase space G/T = P^Qp) x P^Qp) \ diagonal, where T = {(a ° M is a [\0a-1^) maximal torus, obtaining quantizations relevant to the principal and the complementary series. Again, these constructions could be carried in an adelic situation, and will ultimately lead to the quantization of G(A)/G(Q), i.e. quantization of automorphic forms. This paper was written in may-July 1990, while the author was a guest of the department of mathematics at Cambridge, England. Special thanks are due to Professor John Coates who gave constant encouragement and support.
0. THE HEISENBERG GROUP
0.1. The Schrodinger model and twisted multiplication.
We denote by '0 the additive character of Qp, given by
exp(27r%a;) (O.I.I) ^ : Qp —— Qp/Zp ^ Z^-^/Z <——^——-. C*. We let dx denote Haar measure on Q^, self-dual with respect to d ^(x ' y) = ^{Y, Xi' yi), so the Fourier transform 1=1
^d^y) = j (p(x)^(xy) dx satisfy J^d^d^W = ^(-^). The symbol 0 will denote the characteristic function of Z^, so that T^ = (/). Weshall usually assume p ^ 2, and leave it to the reader to make the easy modifications needed for p = 2. 1002 SHAIHARAN
We let V, () denote a symplectic vector space over Qp, and El = V xQp the associated Heisenberg groups with multiplication law given by (0.1.2) (^i)-(^2)= (^i+^i+^2 + j^i,^)), viev, ^eQp.
El is two-step nilpotent group, with center Qp, and the Stone-von Neumann theorem states that it has a unique irreducible representation with central character ip (up-to-equivalence). The Schrodinger model for this representation, p : HI -> (/(^(Vo)), is associated with any complete polarization V == VQ (BVi, Vi Lagrangian subspaces. Fixing coordinates we can focus our attention on the case
Vo = Vi = Q^, <(:KI, 2/1), (a;2,2/2)) = a;i2/2 - ^22/i,
so that p is given by
(0.1.3) p(x, y, t) ^p(z) = ^(t + x(z - \y)} . The integrated representation, p : Li(EI) —> B^L^Vo)), factors through the algebra (0.1.4) Li(E[,^) = {/ e Li(H) [ f(h • ^) = ^(^). f(h) for ^€Qp =center(H)l. Identifying V with the subspace of El, via the isotopic cross-section v ^ (z^O), we can identify Li(]HI,'0) with Li(V) via restriction. Thus for /€Li(V)and^€L2(Vb), (0.1.5) p(/M^) = [ dvf{v)p(v^)^p(z) = / dyKf(z^y)^(y) JY Jvo with associated kernel given by (0.1.6) Kf^y)= fdxf(x^z-y)^x(z+y)). We have p(/i)p(/2) = p(/i t] ^2) with twisted convolution t) given by (0.1.7) fi\\f2(v)= f dv'f^f^v-v^^^v^v-v^. Since (0.1.6) is essentially a partial Fourier transform, we see that p induces a unitary isomorphism from the algebra L^(V), \\ onto the algebra QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1003 of Hilbert-Schmidt operators ^(L^Vo)); hence it takes Li(V), \\ into the algebra of compact operators )C(L^(Vo)). We have f IIA \\f2\\L^ HA |k -ll/alk, (0-1-8) < II/I h/2 |k. < || A |k-H/2 |k, III/I ^21| L»^ || A |k •||/2 |k. We denote the symplectic Fourier transform^ or isotropic symbol by (0.1.9) ^/(z;) =: f(v) = /^'/(^((z/^)). T is a unitary operator on L^ (V), satisfying J:2 = id. The Weyl quantization procedure associates with a function / on V the operator p(f) = p(f) on L^(Vo), which is given by (0.1.10) p(f)^(x)= ffdydU^^{x+y))^(^x-y))^y). We have p(/i)p(/2) = p(/i#/2), where twisted multiplication # is given by (0.1.11) /i #/2^) = ^(^/l ^/2)M = yy d^i d^/i (^+^i)/2(^+^2)^(2(^1,^2)) =^(W))/1^/2(^). Here /i (g) /2(^i^2) == /i(^i) • h{v^}, and ^(Q(D)) is the unitary operator on L2(V C V) = 1/2 (V) 0 V2(^) given by (0.1.12) ^{Q{D))=^^^^{Q{v^v^)J^^^ with Q(-ui, ^2) = ^ (^i, V2) the quadratic form on V C V associated with the symplectic form ( , } on V. The adjoint operator /?(/)* is given by : (0.1.13) p(/)* == p(f), f the complex conjugate of/. (0.1.14) Iff(^x) = f(x) is independent of ^ then p(f)^p(x) = f(x)'^(x). (0.1.15) If f(^x) = /(O is independent of x, then p{f)y(x) = ^f^Wx). 1004 SHAIHARAN Setting, for example, ^y^y) = p-^^^ix - x,)) ^{y - 2/0)), ^o(^) = ^^ - ^o)) ^(p-^y - yo)) ^{(x - xo)(y - ^o)), a straightforward calculation gives : /r. -i -<^\ ( al'bl • Cb^,^ (0.1.16) -'-^1,2/1 -*-a;2,2/2 = ^i ' ^y\ if ^1 + ^ h + 02 > 0, ^;^ #(^ = ^ ^i;^ • S if ai + 62 ^ 0 > &i + 02, ^015^2 . /ha2,^l xi,y2 -*-a;2,2/i ifai+62<0^&i +02, ^ai,fc2 . $02^1 < a;i,2/2 --^2,2/1 ifai +&2,&i +^2^0. For a lattice 0 C Y, we denote the c^a^ lattice by (0.1.17) ^ = {^ e v | {?;, 0) c Zp} and say that 0 is certain if C^ C 0. If 0 is a certain lattice, and /i, /2 are 0-locally-constant, i.e., they factor through V/0, then an easy calculation shows that /i # /2 == fi' /2, i.e., twisted multiplication reduces to ordinary multiplication. More generally, if V = ]J 2^ is written as a disjoint union j of translates of certain lattices, ^ = Xj + Oj, Oj D OJ, and /i,/2 are constant on each Si^, then again (0.1.18) /i#/2=/r/2. 0.2. The lattice model. If 0 C V is a certain lattice, then we can always find a self-dual lattice 0° = (0°)^ such that C^ C 0° C 0. The representation p can be realized in the 0-lattice model (0.2.1) p ^ p^ ind^o^Q/Q = ind^(po) where -0 is viewed as a character of the abelian subgroup 0° x Qp/Zp C HI/Zp, via ^(v,t) = -0(t), and po is the representation ind^Q W of 0 x Qp, which is finite dimensional of dimension (0.2.2) [0 : 0°} = [0 : C^]1/2. QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1005 In particular for V = Q^, OQ = Z92^, we obtain a realization of p in L^V)^ where the subscript -0 indicates that we are looking on the subspace of 1/2 (V) consisting of i^-holomorphic functions, i.e., functions / € L^(V) satifying the following « Cauchy-Riemann equation)) (0.2.3) f(v+vo)=f(v)^(^(v,vo)) for all veV, VQ € Oo. The representation p^ : HI —^ U{L^(V)^) is given by (0.2.4) P^t)f(vf)=^{t^^{v^f))f(vf-v). To obtain the Bargmann isomorphism B : L^(V)^ —^—> L^(VQ)^ inter- twining p^ and p, we notice that L^(V)^ is also gotten by taking the coefficients of p with respect to (f) C L'z(Vo). Thus (0.2.5) Bf{x) = p{f)^x) = (f^pW(x))^ , (0.2.6) B-l^v)=^p(v)^^^ For any v G V there is a unique function ^>y C L-^(V)^ which is supported in v + Oo, and normalized by ^v{v) = 1; the functions B^>y = p(v)(f)^ v G V/OQ^ form an orthonormal basis for 1/2(^0); i11 particular we have (0.2.7) (!>o=B~l(|)=(f)^(t)= characteristic function of OQ. For / C L^(V)^ we have f(v) = (f,^v), i.e., I/2(^)i/; is a reproducing- kernel-Hilbert space. The orthogonal projection P^ : L^(V) —>- L^(V)^ is given by (0.2.8) P^f(v)=f^^v)= ( dvof(v-vo)^(^{v^o}). JOo For a function / € Loo(V), letting Mf denote multiplication by /, we obtain the operator P^Mf on L^(V)^ having anti-Wick symbol /; but an easy calculation gives (0.2.9) P^Mf = M^ f(v) = { dvo f(v - ^o), JOo i.e. the algebra of such operators reduces to the commutative algebra Loo(y/0o) acting on L^ (V)^ by multiplication. 1006 SHAI HARAN For any bounded operator T C B^L^V)^)^ we denote its Wick symbol by (0.2.10) Er(^)=(r^,^). E is a norm-decreasing positive projection B(L'z(V)^) —» Loo(V/Oo). The integrated representation associated with p^ is given by (0.2.11) ^(/) /o = / \\ /o, /o e L2(V)^ and the Weyl quantization p^(f) = p^(p) == B~lp(f)B is given by (0.2.12) ^(/) /o = f # /o, /o e L2(V)^. The relations between the Weyl, Wick, and anti-Wick symbols are easy to obtain (Ep^(f)(v)=f(v)^ (0.2.13) \Mf=p^(f) forfeL^V/Oo). 0.3. The Toeplitz model. In this section we take d = 1, so that we are dealing with operators on L2(Qp) and ^(QpCQp)^. Choose e C Z;\(Z;)2, and identify V ^ Qp[v^L the unramified quadratic extension of Qp, via (x^y) <-^ x 4- \/^y\ thus OQ = Zpiv^] is the ring of integers of V, and the associated norm is given by (0.3.1) \x,y\= \x+^ey\ = \x2 - ^2]1/2 = max{|a;|, \y\}. (0.3.2) Let £/i == ker{N : V* -^ Q^} = {rr+y^ e 0^ | x2 -ey2 = 1} = ^p+i x exp^J^p). By Hilbert lemma 90, we have a commutative diagram with exact rows ]^u=u-u * —— Ui —— 0*0 ——— Z; —— * "21 II Ix2 * —— Ui —— Oo* —— Z; —— * U/U •<———I U QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1007 Hence we may also identify (0.3.4) U, = 0^/Z; == V^/Q; = P^Qp). U\ may also be identified with the isometry group of the Hertmitian form on V, given by vi' vz = qe(vi,V2) + Ve {vi,v^} with symmetric part qe{x^ + \fey\,x^ + ^1/2) (0.3.5) = x^ - eyiy2, [ and symplectic part (^1,^2) = tr((2\/?)~1 v^). Note that the metric induced on U\ via U\ <-^ V is given, under the above identification, by any of the following metrices f^le^wl,6ev/ipw2)=r if^^ Upwi-pW2| if $1=6, ^ ^ ^ where £/i ^ ^p+i x exp(y/ipZp); <^i, ^2) = l^i/^i - ^/^h wheret/i ^ V/Q^ ^i:^i^2:i/2)=T^2—^^. where ^i^P^Qp). l^i, 2/i| • |^2,2/2| The group Sp(V) = SL2(Qp) acts on HI as automorphisms inducing the identity on the center, hence by the uniqueness of the representation p, we obtain a projective representation R of Sp(V) such that (0.3.7) RAPWA1 = PW^ ^ ^ Sp(V), v e V. R in fact lifts to a true representation of the double cover of Sp(V), the metaplectic group Sp(V). We have an embedding (0.3.8) U^—SL2(Zp), x-^-Vey——(x ^Y \y x ) by which we view U\ as a maximal abelian subgroup of Sp(V); in the terminology of [22] : ((7i, U^) is a dual reductive pair. The metaplectic group splits over SL2(Zp), hence we obtain a true representation R of SL2(Zp), and a posteriori of U\. This can be summarized in the following diagram : Sp(V) -^ U{L^Qp)) (0.3.9) ^ c— SL2(Zp) -— Sp(V) 1008 SHAI HARAN Explicitly, R: U^ -^ U(L^((Qp)) is given by the Mehler kernel (O.»0) ^,(.,^^,(^^) for x + Vey € (7i \ {±1} and R±i^(z) = <^(±z). Here (a;, ^/) denotes the quadratic symbol, ^(y) denotes Well's root of unity, and they are given by (p -^ 2) (0.3.11) (P-71^1,?72^2) == (-l)4(P-l)jlJ2+fcij2+Jlfc2 (0.3.12) 7(p^)=41 , it J is even, " / l(-z)i(P-D^ if j is odd. Denote the dual group of £/i by Z = (C/i^ ^ (^p+i)" x Qp/Zp, and write a typical character \ e Z as (0.3.13) X^e^-) = ^(0 . ^(j^Pw), where ^ e (^p+l)v and ^ e Qp/p-^Zp. We put ^ = U ^(N), ^(0) = {X0} the identity, Z^ = {^ ^ ^0} N>0 the non-trivial tame characters, and for N > 2 : Z^ = {^; |g | = p1^}. A straightforward calculation gives that -^(Qp) decomposes under the £/i action given by R as (0.3.14) L2(Qp) = ^ C • ^, a,^ = ^(u) . ^, xe^+ where Z+ = ]J z(2AQ. In particular, ^° = 0 is the unique [/i-invariant N>0 vector. For ^ e Z^.N > 1, the Hermite function ^ is the unique X-covariant vector (up-to a constant of absolute value one), and is given explicitly by (0.3.15) ^=(l+p-l)l/2pN/2 f yux{u)p(u'z^) Comparing the orthonormal basis {0^ | \ e Z~^} with the orthonormal basis {p{x,y)(f) \ x,y e Qp/Zp} of the lattice model, we note that (0.3.16) #Z^ == p^d - p-2) =#{(^)ey/0o; 1^1=^}, N>^ and each ^ is a linear combination of pN(l +p~1) p(x,y)(f)'s, and vice versa. More precisely, (0.3.17) ^(v) = B-1^^) = (<^,p(^)0) is supported in Vx = {v € V \ Nz» = ^ (modp'^Zp)} == U^ ' z^ + Oo, where it is given by (0.3.18) ^(^•^+^)=((l+P-l)pN)~l/2.xM^(j^•^^o)), with u € U\ and vo ^ Oo. Using (0.3.14) we obtain the Tceplitz model pr : 1HI -^ [/(^(^i)-^), where L2(^7l)+ C L^Ui) is the subspace corresponding to ^(^+) ^ ^(^) under Fourier transform L^ (£/i) ^ ^(^). The orthogonal projection P'^ : 1/2(^1) ^> 1/2 (^i)"^ is easily seen to be given by the singular integral (0.3.19) P~^f(uo) == p.v. / yuf(u)6{u^uo)-\ Jui where 6{u, uo) == (—p)"^ if 6(u^ uo) = p~N. If / € C{U\Y is a continuous non-vanishing function, the associated Toeplitz operator P^Mf is Fredholm of index zero. Indeed, any such function is homotopic to 1, the constant function 1, since U\ is totally disconnected and compact. This can be partially explained by considering the problem from a dual point of view : if \ e Z^, translation by \ followed by the projection P+ : ^{Z} —» <2(^+) induces a Fredholm operator P?^ on £^(Z^)^ which has index zero since, (0.3.20) dimkerP^ = dim coker P?^ = p^-1, N ^ 1. Remark. — The material in paragraph 0.3. works equally well for the case of a totally ramified quadratic extension of Qp. 1010 SHAIHARAN 0.4. Wave packets. We can consider more general coefficients of the representation p than those of (f>. Thus for (^i,y?2 € L^Vo) let c^^(v) = (^2,^)^2). After Fourier inversion we obtain (0.4.1) c^^(x,y)= jdz^(xz)^(z+ \y) ^{z - \y}. A direct calculation gives (0.4.2) ^i^^^^L^V) = (^1^3)L2(Vo) • (^2,^4)L2(Vo) showing that p is square integrable modulo the center. The Schwartz inequality gives (OA3) II^^IlL^ <\ML^\ML^ Hence the map y?i 0 y?2 ^-^ ^1,^2 mduces c : L^Vo) 0 L2W)) — L2(^) n Co(y). Using (0.4.2) we get (^',p(c^^)^) = (c^^,c^,^) = (^, (^^2)^1), i.e., (0.4.4) P(c^^) is the rank 1 operator p{c^^)y = ( Hence Cy,,^ tic^,^ = (^2,^3) • c^,^. Taking the symplectic Fourier transform of Cy,,^, we obtain after Fourier inversion (0.4.5) c^(^x)= [dz^z)^(x^ \^^{x-\z\ ^1^2 satisfies (0.4.2), (0.4.3), and again induces c : L^Vo) ^ Wo) —— L^V) n Co(V). Moreover, by direct verification we have (OA6) ^d^l ,^^2 (^ x) = ^, ,^ (X, -0 , ^U.4./J ^i,^ ==c<^2,<^l• QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1011 We define the Wigner transform by W^p = c^^ e L^(V) n Co(V), and by (0.4.7) it is real valued. It also satisfies the following properties : (0.4.8) f^W^x)=\^(x)\\ (0.4.9) fdxW^x) = |^(012, (0.4.10) W^i = W^ ^=^ Note that W(p{^x) -^ 0 implies that there exists z such that x ± z C supp(^, and hence x C A(supp^) ^{jQ/i + y^) \ Vi e supp(^}, thus we have (0.4.11) proj^(suppW^) C A(supp^), and using (0.4.6), (0.4.12) proj^(suppW^) C A(supp.F^). We have the following inequalities : (0.4.13) II^H^ ^ II^H^ = jf dxd^W^x), < \\W\L^ -vol(suppW^) from which we obtain the uncertainly principle (0.4.14) vol(suppl^) > IMIL2 > 1. 11^11^ We have further (0.4.15) Wy = c^^ = ^o = characteristic function of OQ, (0.4.16) W(p(vo)^)(v) = W^{v - vo), (0.4.17) W{n^){v} = W^A^v), A G Sp(V), by (0.3.7). Since Sp(V) acts transitively on the collection of self-dual lattices, we deduce using (0.4.15), (0.4.16), (0.4.17) that for any translate of a self-dual lattice 21 = VQ + 0, 0^ = 0, there exists a unit vector (^ e L^(Vo) such 1012 SHAIHARAN that W(p^ = characteristic function of 21; by (0.4.10) (^ is unique up to a constant of absolute value one. For any unit vector (po e L^(Vo), the map C^o(^) = c^^o is by (0.4.2) an isometry c^ : L^Vo) ^ L^V). The adjoint map C^:W)——L2(Vo) is given by C^f(x) = p{f)^{x), f G L^V) H Li(V). Since C^C^ = [d^fYo)^ we obtain the wave packet expansion of (p as superposition of the p{v)(po's (0.4.18) (p{x) = dv {(p, p(v)^o) • p(v) ^o{x). For / 6 Loo(V) n Li(V), denoting by Mf multiplication by /, we get the operator C^MfC^ on L^Vo). We have (C^MfC^^^L^Vo) = (MfC^i,C^2)L2{V) = dv f(v)c^^(v)c^^{v) = ^ d^/(^)((^i,/9(v)^o) • (^2,P(^)^0) = [dvf(v){c^^W(p{v)^o)) by (0.4.2) = ^d^ /d^i /(^) ^,,^(^i) W^o(^i - v) by (0.4.16) = ^ d^i/*W^o(^i)^,<^(vi) = dv^^f ^W(po)(-v)(^,p(v)^2) = dvJ:'(f^W^o)(v){p{v)^^-2)= (p(/*W^o)^i^2). (0.4.19) Hence : C^MfC^ = p{f * W^o). (0.4.20) If y = ]j2lj is written as a disjoint union of translates of self- 3 dual lattices, Sly = ^ + O^, 0] == 0^ and if for each j, ^ € L^Vo) is a unit vector such that W(pj = ^j is the characteristic function of 2lj, then QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1013 the (pj form an orthonormal basis for L^(Vo). Indeed, we have by (0.4.2), for .71 ^ j2 K^i^L^yj2 = (^1^2)1^) -O; moreover, letting ^ = p(-Xj)ipj, so that by (0.4.16), (0.4.21) ^ = p(l)^= ^p(^)^ = E^° * ^ j j -E^^^^ 3 =^C^(v).^,p(v)^w} 3 = Y^ j dv^(v) • {^,p(v)^j)L^Vo) • PW^pj =Y^^^3)^W^3' 3 If / is a function on V, assuming a constant value, say A^, on each of the 21^'s, so that / = ^ \j ' <^-, we have as in (0.4.21) : p(f)^pj = Aj • (/^, 3 i.e., p(f) is diagonalizable with respect to the orthonormal basis ^ with associated eigenvalues the A/s. 1. FUNCTION SPACES OVER Qp 1.1. Soboleff spaces. The local zeta function of Qp is denoted by (1.1.1) C(5) = (l-p-6)"1 = ( 0(^)1^1^*3;, Res>0. ^Q; It is a meromorphic function of s, (27rz/logp)Z periodic, with a unique simple pole at s = 0. We denote by I0' the operator of multiplication by the function (1.1.2) |l,a; -a = max{l, l^}"^ 1014 SHAI HARAN and we denote by J^ the operator F^I^F^, which for Re a > 0 is given by convolution with the L\ function (1.1.3) Ja{x)=^{\l^\-a){x) = c——Al.rr-1 -p"-1)^), where Re a > 0 and a ^ 1 mod (27n/logp)Z. The J^'s form an entire family of distributions, and we have J0' * J^ = J^^. For real a > 0, the J^'s form a semi-group of probability measures associated with the negative definite function (1.1.4) log|U|= [{l-W))da(x)^ where the Levi measure o~(x) is given by the divisor function (1.1.5) a(x) = logp -Y^P71 • ^(p^x)' n^O Note that, since J^ is supported in Zp, and 1^ is Zp-locally-constant, we have (1.1.6) Z^J0 == J0^- All this extends to the higher dimensional case VQ = Q9^; using the metric |:ci,...,a^ = max{|.ri ,...Ja:^|} we have commuting operators J^, Ja = J^^lIaJ^d 5 J^ forming an entire family of distributions, J^ * J73 == JQ;+^; for a > 0, J^ being a probability measure supported in Z^; and (1.1.7) J^x) = c(^(|^|a-d-pa-d) .^), for Re a > 0, a ^ d mod (27n/logp)Z, (1.1.8) ^^)=^_(i-iog^^|).^). Note that (1.1.9) J^^L^Vo) for Rea>d'(l-q-1) and in particular : J^ G Li, Rea > 0; J" G Z/2, Rea > jd; J^ e Loo, Re a > d. QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1015 The Soboleff spaces (or Lebesgue spaces of Bessel potentials [31]) are defined by (1.1.10) ^(Vo) = (1.1.11) L^(Vo) = ^(Wo)), 11/11^1. = P-^-'/HL,. The spaces L^, L^ are Banach spaces isomorphic to Lg, and they satisfy the following list of basic properties [31] : (1.1.12) Embedding L^(Vo) ^ L^(Vo) continuously for 0 < q^ < (h1 < 1, d' (q^~1 - q^1) $ ai - 02 ; 0 ^ q^ <: gf1 ^1, d' (gf1 - q^1) < ai - 02. (1.1.13) Duality (L^)' = L^01 for 1 < g < oo, ^-1 + q'~1 = 1. (1.1.14) Relation with the Riesz potential ^ ^ ^^-^ ^ c(d-a) 1^1^-^ for Rea > 0, C(Q;) L^(Yo) ^ domain of R^ acting on Lq(Vo), Re a > 0, i.e.ll/llL^II/ll^+ll-R-'/IlL,. In particular, let us note that we have (1.1.15) L^{Vo) -—— Co(Vo), tor a > |d, and that we have quasinuclear (i.e. Hilbert-Schmidt) embeddings (1.1.16) 1/^(21) c—> L^(2l), ai - 02 > ^, 21 bounded domain, (1.1.17) ^^(yo)-^21^), oi-02> jd, /3i-/?2> ^. 1016 SHAI HARAN Recalling the Bargmann isomorphism B : L^(Y)^ -^ L^Vo), where V = Vo © VQ, (0.2.5), (0.2.6), we note that using (0.1.14), (0.1.15), and (0.2.13), we have (1.1.18) B~lIQB=M^-c., (1.1.19) B-lJQB=M|l^-a. It becomes natural to define the Schrodinger operators by (1.1.20) B-^K^B = M|I^|-. and the associated spaces (1.1.21) L^(Vo) = ^"(Wo)), \\fh^ = II^-'/HL, since lUl-" . 11,^ ^ [1,^1-° < 11,^1-^ and ll,^-0, for a > 0, we have (1.1.22) L^ ^ L^ ^ L^0 H L0^. The operators J^, J0, T^0' belong to the algebra BL^(V/Oo)B-1, for Re a > 0, and hence they admit the lattice basis {p{v)(f) \ v C V/Oo} as eigenfunctions. In particular, the operators J^J0', and K^, are trace class in the following domains, with trace (1.1.23) tr^J^^^.*.^, Rea,Re/?>d, (1.1.24) tr^^^^^, Rea>d. The operators J^20, for d = 1, also admit the Hermite functions {c^} as eigenfunctions : (1.1.25) K^^ = (condx)"" • ^ where cond^ = p1^ for ^ C Z^\ One can further generalize the above spaces by considering more general « metrics)) i.e. functions g : VQ —^ {0} Uj^ satisfying (g(vi +^2) ^ max{^(^i),^(z;2)}, Vz C Vo, (1.1-26) <{ ^(A^)=|A|.^),AeQ^, veV^ g(v) = 0 <^=^ V = 0. QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1017 Recall that such metrics correspond bijectively to lattices 0 C VQ, the correspondence given by (1.1.27) 0 = {v € V | g(v) ^ 1}, g(v) = inf{|A|; A-1 'v e 0}. If Oi C Vi corresponds to ^, i = 1, 2, then Oi @0^ C Vi 0^2 corresponds to (1.1.28) 91^92(^1^2) =max{^i (^1)^2(^2)}. If 0 C V corresponds to g, the dual lattice 0" C V^, defined in the dual vector space by (1.1.29) 0" = {^v e V I {v\0) C Zp} corresponds to the dual-metric ^v, given by (1.1.30) ^v)^rK^ ;^y}. I ^(2;) J This holds in particular when V is a symplectic vector space identified with its own dual by means of the symplectic form. Given V, g we can define the operators f 1^ == multiplication by max^^a;)}"0, J^^-1^, (1.1.31) ^-^-wr"^ | yCt _ DTOi D—l I, Kg - Bl^^gB , and the associated spaces (1.1.32) L^), L^^), LM(y^)^ and ^(21^) for a domain 31 such that 91 + 0 C 51 (note that since J^v is C^-locally- constant, suppj^ C 0). Since any metrics are equivalent by an affine transformation (i.e. given g^,g^ on V, there exists A C GL{V) such that pi(v) = ^(^)), it is clear that the results (1.1.12) through (1.1.17) remain valid for the spaces (1.1.32). 1018 SHAIHARAN 1.2. Besov spaces. The problem with operators J", and the spaces L^, is that unlike the real case, where we have ./-2N ==(!-+- 92/9x2)N, we lack in the p-adic setting an analogue of the Leibnitz rule 9 N 9 9 Uf ^^ ^•^-^E^JU^'U f\ V ( \( Vf f ^'^ h which follows upon taking the Fourier transform from the Newton rule : (i.2.i) 0/1+2/2)^= E (^yi-y^3- 0 Thus, for example, in the p-adic setting /i,/2 e L^ does not imply /i • /2 ^ L0^. To see the problem more clearly, we notice that we can write for Re a > 0 and yi € Qp, /.27T/10g«27T/logp 1 |^+^=p.v./ dt.-^ Jo 27r ,J^ 1 CO) , C(-l)C(-(l+a)n ^^ 2C(l+a)+ C(a) ; (1.2.2) xd^r^'12/21-^+12/11-^ •12/2^} ((l ) (l+ ) + C(-^)^" E^-l)•(-^)- ' x r710^XT^ • ^I . x^i) iz/ii^2^ • x^) i^i072-^ Jo 27r where ^ varies over all non-trivial characters of Z* but the sum ^ is x actually finite for ^1+^/2 7^ 0. Thus, (1.2.2) like (1.2.1), expresses \y\ -^-y^ as «sum» of ^1(2/1) • X2(^/2), with ^i • ^2 = | I0. To overcome this problem we introduce the Besov spaces B^ (or Lipschitz spaces) of functions that are well approximated by locally- constant-functions. We shall use the following notations : V is a Qp-vector space with a metric g corresponding to the lattice 0 C V, and C^ C V^ is QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1019 the dual lattice. We let ' (f)g ^ = Fourier transform of the characteristic function of p~k0", k ^ 0; ^g,k = 0<7,fc - ^g^k-l for k > 1, (Rg^o = (f)gft ; dgX = Haar measure in V normalized by d^(0)=l; (1.2.3) < L,=L,(V,d^); d^ - C(c0 • ^Q/)^ • d<^, ^ = dimY, so that d^y is the «multiplicative Haar measure)) with ^y{pk0\pk+10} = 1; L; =L,(0,d^); t-^y. is the usual ^r-space on A; € {0,1,2,...}. The Besov space B^(V, g) consists of functions / (or distributions, ifa<0) such that (1.2.4) ll/llB-^^ll^.fc*/^)!!^ < oo. We shall only consider these spaces for 1 <. q < oo, 1 <. r <_ oo. The B" = B°(V,g) are Banach spaces with the following basic properties : Embeddings (1.2.5) B^, %.„ n^ra; (1.2.6) B^ B^, ai>Q2; f n^ ^g.g ^ B^, Kq^2, (1.2.7) DO: B^ 2 (1.2.9) Interpolation [B^, B^J^ = B^^, for the complex method; here 0 < 0 < 1, q~1 = (1 - 0) • Qo1 + 0 ' q^\ r-1 = (l-^.To^+^rf1, a = (l-0)'ao+0'a^ao ^ ai; (1.2.10) (L^°,L^1)^ = B^, for the real method; (1.2.11) Duality (B^)7 = B^^, with g-1 + q'~1 = 1 = r-1 + r'-i 1020 SHAI HARAN (1.2.12) Lifting J^ : B^ -^ B^ ; (1.2.13) Characterization via approximation : for a > 0, \\f\\B^^\f\\L,+\\Pka•\\f(x)-^^fW\\^ (1.2.14) Characterization via differences : for a > 0, I f\\B^ ^ \\f\\L, + ff(2/)-° • ||/(;r) - f(x+y)\\L, L* (1.2.15) Pointwise multiplier : for a > 0, \\fl'f2\\B^ (1.2.16) Bqy a and B^ are quasi Banach algebras of continuous functions. (1.2.17) Remark. — Note that if supp/ C 0 then suppy?^ * / c 0. Hence we can define B^(0,g) = {f € B^(V,g) \ supp/ C 0}, and similary we can define B^(^,g) for any domain such that 51 + 0 = 21. All the above properties remain valid for the spaces B^^g). When 21 = x + 0, we shall write B^(2l) for B^(2i, ^). (1.2.18) Remark. — The only discrepancy from the analogous spaces over the real numbers is (1.2.14); for the reals, one has to use instead of the difference Ay f(x) = f{x) - f(x + y), the TV-fold-difference ^/c^)- E2^ (^(-W^+M J 0 B^{V, x V^g^g,) d^ B^(V^g,) 0 B^(V^g,)^ (1.2.19) B^{^ x ^gi^g2) d^ B^(2li,^i) 0 B^(<^2), QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1021 with 21^ + Oi = 21^, where 0 denotes the completed (in the crossed- product topology) tensor product. These are again Banach spaces, satisfying analogous properties, their norm being given by (1.2.20) ii/iia,— = llp'1'1^2'2!!^^ ^^wJ*/||^ In particular, we can restate the pointwise multiplier assertions of (1.2.16) as trace theorems : (1.2.21) Res : B^^^fV x V, g (g) g) —> B^^^V, g), a > 0, (1.2.22) Res : B^I^V x V^g^g) —. B^^g) are continuous, where Res denotes the restriction to the diagonal. The spaces in (1.2.21), (1.2.22) are spaces of continuous functions, and we can view the ^-functions as elements of their dual. We have (1.2.23) ll^ci^llfR^2^/2^ = ll^ci^ll^-^2'-^2 = 1- '•2,1 / 2,oo We can also define the weighted Besov space B^(V^g) given by the norm (1.2.24) H/ll^i, = ^ll^^^*/)^)!!^!!^ - \\I^f\\B^ noting that Ig^^g^ * /) = ^g,k * (-^A since supp^,fc c 0. B^/ will satisfy the analogous properties, e.g. . interpolation [B^^B^1], = B^^ as in (1.2.9) with f3 = (1 - 0)f3o + 0 • /3i; . duality (B^^B,^^ . lifting ^:B^°-^^°^; • pointwise multiplier ||/i •/2||Dc.i/3i+/32 < C'||/i||D^l^i • II^HRC^I^? -"g.r ^oo^r A metric-covering of V is an assignment for each x 6 V of a metric g^ such that (1.2.25) gx(y}<^ =^ gx=gx^y. 1022 SHAIHARAN Letting Ox denote the lattice correponding to g^, and setting 34 = x + Ox, i.e. (1.2.26) ^={yeV\gx(x-y)<^l} we can choose a sequence xj e V such that V = U 24 , a disjoint union. j Conversely, given a covering V = U^, by disjoint sets 21^ of the form j ^ = ^j + ^j? ^j a lattice, we obtain a metric covering by setting (1.2.27) Qx = the metric corresponding to Oj, for x € 21^. A g-continuous weight is a function m : V —)- R4', such that for some constant C, (1.2.28) ^(^/) < 1 =^ C-1 • m(^) < m{x + ^/) < G • m(x). Given such g and m, we define the Banach space JE?^y.(m,^), with norm (1.2.29) ll/llB,^,,) = sup ——, - \\MB^(^) x^V m'\J-') where fx denote the restriction of / to Sla;. Choosing a sequence Xj e V such that V = ]_j2lj, 2lj = .Tj + 0^., _ 3 and putting for x € 5ij : m(rc) = m(^) = m^-, it follows from (1.2.28) that B^(m,g) = B^^(m,g). Hence we may as well assume from the start that m is g-locally-constant in the sense that (1.2.30) gx(y) < 1 =^ m{x) = m{x + y) and then the norm of B^(m, g) is given by (L2-31) ll/llB^(m,<7) =SUp^-||^||^^(^.^.) where fj denotes the restriction of / to 2^. The spaces B^(m, g) will inherit many of the properties of B^tV), in particular : QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1023 Embeddings (1.2.32) B^^(m,g)^B^(m,g), n < ^ ; (1.2.33) B^(m,g)^B^(m,g), ai > 03 ; (1.2.34) B^dW (1.2.35) B^(mi,ff) -^ 5^(m2,ff), mi < C-ma. Liftings (1.2.36) ^ : B^(m,g) ^ B^^m, g), where J^f(x}=J^f(x); (1.2.37) J^:B^(mo,g)^^B^(m.mo,ff), where Imf{x) = m(a;) • /(a;). Pointwise multiplier : for a > 0, (1.2.38) ||/i . h\\B^^g) ^C-\\fl\\B^(m,,g) • \\h\\B^(^g) and characterizations via approximation and difference similar to (1.2.13) and (1.2.14) of which we note : for o; > 0, (1.2.39) ||/||^ ,oo(^)^^l^)l "+ sup ———.1/(^A^. xo,xiev rn{xo) gxo[xo - x^ 0<9xo{xo-xi) 1.3. Symbol classes and the analytic Schwartz space. For a lattice 0 C V we denote by Coo(O) the Frechet space defined by the norms ||/HL^? a > 0. We can also use the norms H/HL^ ^ >. 0, for any 1 < q < oo, and similary we can use the norms ||/HBC< , a ^ 0, for any 1 < q,r < oo. In particular, we see that Coo(O) is an algebra under pointwise multiplication, consisting of continuous functions, and that it is a nuclear space. 1024 SHAIHARAN (1.3.1). — For any open set 21 C V, we denote by Coo (21) the Frechet space defined by the norms H/Hz,0 (rc+o)? o;>0,a'4-OC2la translate of a lattice contained in 21. Again we can use the norms associated with L^x-^-O), or with Ba(x-^0)^ and Coo (21) is a nuclear algebra of continuous functions. In particular, we have Coo(2li x 212) = Coo(2li) ^> Coo (212). Just as Coo(VQ = lim Coo^), the inverse limit taken over all lattices 0 C V with respect to the restriction mapping Coo(Oi) —^ €00(02)? Oi 2 02? we can define the space of compactly supported smooth functions Coo^c(Y) == lim Coo (0), where now we take the direct limit with respect to the mappings Coo (02) ^ Coo(Oi), Oi 3 Oii given by extending by zero. Coo,c(Y) is similary a nuclear algebra of compactly supported continuous functions. (1.3.2). — We define the (analytic) Schwartz space S(V) to be the Frechet space defined by the norms H/H^1^, Q^/3 >. 0. Equivalently, we can use the norms associated with L^, a,/? >: 0, for any g; or the norms of L^ , a > 0, associated with the Schrodinger operator (1.1.21); or the norms of B^/{V^g)^ a, (3 > 0, for any 1 < g,r < oo and any metric g on V. S(V) is a nuclear algebra of continuous functions. Fixing some non-degenerate quadratic form, giving an identification V ^ V^ of V with its dual, we can view the Fourier transform T as a mapping ^:L^(V) -^^(y), hence F defines an automorphism of S(V), which consequently forms an algebra also under convolution. We have again We note that J Hom(Coo(2li),C^(2i2)) ^ C^ x Sli), (1.3.3) f Rom(S(V,)^Sf(V^^S\V2(BV,). An operator A on functions on VQ will be called smoothening if A : ^(Vo) —— S{Vo) QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1025 continuously, or equivalently if A is given by a kernel A{x, y) e S(Vo C Vo). A still equivalent condition for A to be smoothering is that A = p(f) with / e S{VoOVo) = S(V), since A(x,y) and /(^.r) are obtained from each other by a partial Fourier transform which takes S(Vo C Vo) isomorphically onto S(VQ^ Vo). Given a metric-covering g, and a ^-locally-constant weight function m, we define the symbol class S{m, g) to be the Frechet space defined by the norms ||/HB^ ^(m,^)? <^ > 0. These norms are given in an equivalent form in (1.2.39), so that / e S(m, g) if and only if for some constants C, Co, we have (1.3.4) \f(x)\ < C ' m{x) for all x e V, (1.3.5) \f(xo) - f(x^\ ^ G, • m{xo) . g^xo - x^, for all XQ,X\ C V, gxo(xo - x-i) < 1, and for any given a > 0. Using the pointwise-multiplier, or by using (1.3.4), (1.3.5), we have : (1.3.6) /, e 5(m,, g) => /i . ^ e 5(mi . m^ g), Note that Coo,c c S{m,g) C Goo, but the first inclusion is not dense, hence a continuous linear form on S(m^g) is not determined by its restriction to Coo,c; such a form will be called weakly continuous if its restriction to a bounded subset of 5(m, g) is continuous in the Coo topology. 1.4. Temperate metric-coverings. We return to a symplectic vector space V, ( ), and we shall deal with a metric-covering ^, corresponding to V = [J2lj, 2lj = Xj + Oj, where g^ j is always assumed to be certain : g^ < g^, or equivalently Oj D OJ • g^ will be said to be temperate if there exist no? N >, 0, such that (1.4.1) g^(x) ^no .^) .^(l,^ -^ ^0^1^ e V, where ^(l,^) = max{l,^v(a;)}. Using the definition (1.1.30) of the dual metric, one can write (1.4.1) in the equivalent form, (1.4.2) g,, (x) < p710 • g^ (x) . g^ (1, x, - x^. 1026 SHAIHARAN Taking x =XQ-X\ in (1.4.1) we have in particular, (l-4^) 9U^X^^^Pno'9^^-x^\ From (1.4.1) we may also deduce that 9^(xi - xo) < p^ . g^^_,(x, - xo) . 9^^-^X2 - ^o)^ ^^-.0(^1 - ^o) < P^ ' 9^(x, - xo) . ^(l,^i - ^ ^^•^(l^i-o-o)^1 and similary with x\,x^ interchanged, from which we obtain (1.4.4) ^ (m -^o) < P^^0 < (1, rci -.ro)^1 •^ (1, ^ -.z-o)^^^. Multiplying (1.4.4) with the similar expression obtained by inter- changing rci, a;2 we finally get, (1A5) 9^ (a-i - ^o) • 9^2 - xo) <^(2^)no .^(^^ _^W .^(^^ _^)(^1)2. LEMMA (1.4.6). — For a > d(27V + 1), ^^.(l,^. - ^-a < p^o . C(a - d(27V + 1)). 3 Proof. — Set ^(fc) = {j | ^. (1, x, - x) < p^. For j 6 ^(^) we have 9^(y) 9x(x - x,) ^ g^(x - x,) ^ p^ • g^x, - x)^ < p^^i^ and hence ^ C a: +p-^o+fc(^v+i)] . ^ Denoting by vol the Haar measure in V normalized by vol(Cy = 1, we get from the above two inclusions : #J(k)= ^ 1= ^ vol(^•+pno+fcN(^).pd(no+fcN) iej{k) iej(k) ^VOl( ]_J ^Ypd^kN) 3^JW < \0\{x +p-[no+A;(N+l)]0^) .p^o+fcN) ^ d(no+fc(A^+l)) . d{no+kN) ^ 2dno . d(2N+l)fc QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1027 Hence ^9^x,-x)-a=^#(J{k)\J{k-l)).p-k•a 3 J>0 ^ p2dno . y^(d(2N+l)-a).fc fe>0 ^^^(a-^Tv+l)). D Using (1.4.5) we deduce the following COROLLARY (1.4.7). — For (3 > d' {2N + 1) • {N + I)2, E ^ (1^ - ^ • ^, (l^i - ^-/3 < oo. J'l J2 The weight function m will be called temperate if there exist C and /3, such that (1.4.8) m(^i) < C • m(xo) . ^ (1, ^i - x^. If m is temperate, so is m" for all a > 0, and using (1.4.3) also for a < 0. We also have (1.4.9) h^^s^p93^- is temperate. y 9'xW Indeed, using (1.4.1) and (1.4.2), we have h^^p^^h^) .^(l^i-^o)^. Given two temperate weights mi, 7722, then mi • m^ is also temperate; moreover we have mi(:ri). 7712(^2) Co • mi(.ro) • m2(a;o) • ^(l^i - xo)00 . g^x^ - xof° for some new constants Go, A)- Hence, 0 v m :c (i 14 10)lu c^ nl JTo/ - Tn^ • ^ n± rx. XT }-^ < r l( o) •^2(^0) V- --"- ^ /rEl^ 5 - 2 ^O^ 9x2\ ^ ^~ O) S ^0 •———;——^——;———•mi(a;i)m2(.T2) 1028 SHAI HARAN When g and m are temperate, we have l/m{x) < C/m{0) ' g^l^x)0, and using (1.4.2) we easily obtain B^^^^go) ^ B^^(m,g), and hence S(V) c-^ S(m^g). Conversely, given g^ the functions rriQ^x) = go(l,x)~1, m\(x) = (^(l,^)"1, are temperate, and an easy estimate gives %,oo(m^ • m^^.g) ^ B^(V,go). Thus we conclude that (1.4.11) 0, m temperate. Beside the uniform metric-covering gx = go < 9o^ which we studied in paragraph 0.2, basic metrics are the Toeplitz metric and the pseudodifferential metric which we next describe. The Toeplitz symbols are defined by (1.4.12) E0 = 5(7^), g^y) = |1,^|-1 • H, m^rc) = 11,?^. It is easily verified that g^ is a metric-covering (i.e. it satifies (1.2.25)), that it is certain (i.e. g^(y) < g^(y) = |l,p.r| • \y\) with h(x) = |l,pa;|~2 = m"2^), and that it is temperate; indeed, we have a stronger inequality than (1.4.1) r g^W <, g^{x)' \i,p(xo - a;i)|, (1.4.13) [ \l,p(xo -x^)\ < ^(l,^i -xo). This follows from the basic inequality (1.4.14) |l,2/o|<|l^iHl^o-2/i|. Explicitly, / € E0' if and only if (1.14.15) |/(^)|< Co •11,^1" for all x € V, and for all (3 > 0 : \f(x) - f(x + y)\ ^ Cf,. 11,^1^ • \yf for \y\ ^ |1,^| QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1029 Letting %(V) = Ft B^(V) denote the Frechet space defined by (3>0 the norms of B^10^, /? > 0, we note that since l^llB^^m^) = sup 11,^1-". |/(;c)|+ sup II^^^M-^I/^-^+Z/)! a; ' ' a;,2/ ' \y\<\l,px\ ^ sup [l^l-0. |/(a:)| +sup |l,p^|-aH-^|/(r^)-/(^+^/)| x x^y M<1 >/3|-Q "B^-'00,00 we have (1.4.16) E" -^ CTV). In particular, S0 consists of bounded smooth functions, and ^a for a < 0 consists of smooth functions vanishing at infinity. Letting E~00 = Q E" a denote the Prechet space defined by the norms of all the E^'s, we have by (1.4.11) and (1.4.16) : (1.4.17) E-°° = S(V). The pseudodifferential symbols are defined by 5° =5(7^), (1.4.18) { 9{^x)(r],y) =max{)l,p$|-1 • IT/I.II/I}, m^^ll,?^, where (^, a;) and ('rj, y) refer to the symplectic coordinates associated with the complete polarization V = VQ (B V\. Again it is easy to verify that g^x is a metric-covering, certain with h(^,x) = |l,p^|~1 = m"1^,^), and that it is temperate with in fact a stronger inequality ^(so^o^'y) ^ ff(Sl,^l)(77>y) • |i,p(^o -^i)|, (1.4.19) { |1,P($0 - $i)| < ff(^,.i)(l^o - $i,.ro - xi). Explicitly, / € S0' if and only if ' |/($,a-)| ^ Co • 11,^1° for all ($,.r) e V, and for all /3 > 0 : |/($, a;) -f^+-n,x+y)\ For the norm associated with (1.4.20) for a fixed (3, we have sup ll.^l-". \f(^x)\ ^cTLll?pra max^ll?prll^ l^l}^' 1^^) - f^ + ^ + v)\ \y\<.^\^\\^\ > sup 11,^1-|/(^)| + sup \^P^~a'^y\-^\f^x)-f(^^x+y)\ x,y^,r) 1 |2/^1<1 -ll^llB^- with I^f(r],y) = ll^^l-0 • f(rj,y). Hence, letting ^(-"'^fv) - n i-^B^ ^00 V / — I I ^ ^OO,^? /3>0 the Prechet space defined by the norms H^/IL^ , (3 > 0, we have - —'00,00 (1.4.21) 5°-—C^°)(V). In particular, 6'° C C°^ consists of bounded smooth functions, and 5°' for a < 0 consists of smooth functions vanishing at oo in the ^-direction. 2. THE SYMBOLIC CALCULUS 2.1. Estimates for twisted multiplication. We fix a temperate, certain, metric-covering g^ in our 2d-dimensional symplectic vector space V, so V = [J ^;, ^; = Xj + Oj, Oj^ D 0^. In V C V 3 3 we have the metric covering g (g) g corresponding to vev=Y[ ^^=^x^ = (^,^)+o^ eo,,. Jl J'2 We begin by giving a short range estimate for the restriction to a, of (2.1.1) fi#f2(x)==^(Q(D))f^f^x,x) = .F® ^(Q(vi, V2))^<8 ^/i ® /2(a;,a;), QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1031 where /i,/2 € B^(2lj), x e 2lj, and Q is as in (0.1.12). We have for .EI , a;2 € 2lj 'IWOT)/! 0/2(^2) | < ||WW)/i ® /2||^(vew,-) by (1.2.23), f2 1 2) < ^ ^fl (2> •^^•i'1^®^®^) (since WW)is "Qitary on Z/2 and commutes with convolution) = 11/1 f ||^,,fcl ® Vg,M * V'(0(£>))/1 ® /2||^(^,,) = ||^(Q(£>))^,,fc, * /I ®^,,fe3 * /2||^(ai,,,) (2.1.3) ^ llya,,fei * /illa^^a,) • II Vg,^ * /2|lB^^(a,) =Pfcl'dlly5„fcl */ilk(a,) •P^'^Vg.M */2||L2(a,). Multiplying (2.1.3) by pC^i+^a^ and taking ^r-norm, we conclude that (2.1.4) ||^(Q(Z?))/i ® /2||^,^a,,,) ^ IIAIlB^(a,-) • ll/2|lB^(a,)- Using (1.2.21), we obtain from (2.1.4) (2.1.5) HA #/2||B^(a,) < ca • ll/i|lB^"(a,) • \\M\B^W Note that Haar measure is normalized by (1.2.3) as da;(2lj) = 1, so that £oo(2lj) ^ -£'2(2l.») with norm 1, and hence \\f\\B^^,) <. 11/llag, ^(a,)> so that (2.1.5) gives {\\h#h\\B^) ^\\h#h\\B^,) (2.1.6) < Ca • ll/l|lB^°(a,) • ll/2|lB^°.(a^) ^ Co • 11/1 llB^(a,) • ll/2|lB^(a,-)- To estimate the «error », that is the difference between /i #/2 and /i • /2 we notice that for y 6 Qp we have f2 ify^Zp, (2.1.7) m(y) - 1 < < """ l-l0 ifyeZp, 1032 SHAI HARAN and hence in particular for any 7 >, 1 (2.1.8) |^(y) - 1| < \y\\ Also, using ffj" = gj, we have (2.1.9) sup \(xl,X2}\=p(kl+k2)sup—[{x^^ ff;(^)=^ ^X^i)-<^2) i^^)^2 . . :=p(^+^)sup^W=p(fcx+fc2).^. x 9j (x) where hj is defined as in (1.4.9) with the metric gj. (2.1.10) Applying these remarks we can now estimate |^(<9(£»))/i 8> f2(xi,X2) - fi <8> /2(a;i,a-2) < \\^(Q(D)) - 6)f, ® /2||^(ve^^) ^ (L2-23) = E ^^ll^^^^^-Wi^i'^))-l)^/i ®^/2||^^^ fci,fc2^0 (by the definitions (1.2.4), (0.1.12) and Plancherel) = U ' II /I II B^ (21,.) • II/21| B^ (a,) (by Plancherel again, and the definition (1.2.4)). Using (2.1.10), we can now deduce as in (2.1.3), (2.1.4), (2.1.5) and (2.1.6), that we have f 11/1 #/2 - /I • f2\\B^^j) <\\fl#f2-fl'f2\\B^W (2.1.11) < Ca • h] • || A lla^-a ^ • ||/2|lB^y+«(^.) [ < Co • h] ' ||/i|[^+;y+a^ • \\f2\\B^+ot^y QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1033 We next give the long range estimate for the restriction to ^ of /i#/2? where supp/^ C 21^, and at least one of the ji's is different from j. We recall from (0.1.11) that we can write V?(0(D))/i (g) f^ also as a convolution (2.1.12) ^(Q(D))f, 0/2(^2) = // ^x'^xllf^xl)f^x")^(2{xf-x^xll-x^. JJ ^31^32 We set (2.1.13) 9nJ^X)= mm ^(1^+z/s) 2/2 & ^j'a so that we have (2.1.14) 9n,J2 is ^Ji + 0^-locally constant, and (2.1.15) ^9^^x)^g^x). Hence if we put (2tL16) ^-^r^^F then J^ acting on S^V), preserves by (2.1.14) C^(5l^), the distributions supported in 21^ ; moreover, it commutes with convolution, and by (2.1.15), (1.2.12), and Plancherel, we have for f3 > 0 (^•^ ll^/illa^,) < 11^-fAll^,) = ll/illB^^,)- Similary we can define ^J^, and ^^, and they satisfy the same properties. Notice that we have in the distributional sense, (2.1.18) F^^x'-x^'-x^ ^V^i,^'-^)) -^-^ and hence, letting J^^ act through the variable x1\ we have (2.1.19) J^^x'-x^^-x^} -r^(^.xll-x^^{xt-x^xll-x^Y 1034 SHAIHARAN Putting, (2-1.20) ^2^)= min ^(1,^-^) x ^•"72 = ^Ji J2 (1'x" ~~ ^2), for any x" e %„, we obtain from (2.1.12) and (2.1.19), (2.1.21) ^(0(2?))/i®/2^i,a-2) = 9n (^2, S^)-^ . V>(Q(I?)) [^^A ® /a] (a-i, ^2). Similarly, using 5^ ^ and J^, and the analogous formula to (2.1.19), we have (2.1.22) ^(Q(D))/i<8/2(a;i,a;2) =^(^2,^)-^(^,a,J-^ •^^(^K.A®^,^^,^). Now we can estimate, as in the short range-estimate (2.1.2), flWCD))/i®/2(a;i,;E2)| = gj^W0 • g^W13 •IWW^lA^l^i,^)! < 9j^W0 • g^x,,^)-'3 (2.1.23) • \\^QW)J^f, ^ J^MB^V^^ ^ 9n^W3 • 9j^^)-i3 •K^^^nf^B^y,^) ^ ff;,(^,a,j-/3. ^(^,a,j-^ • IIAIlB^(a,,) • ll/2|la^(^) by (2.1.17). Using the fact that by our normalization we have £oo(2li,ffj) ^-> ^2(^,5.,) with norm 1, and the embedding (1.2.6), we obtain from (2.1.23) that for any /3' > (3 + d (2.1.24) \W(D))f,®f^,X2)\ ^ C • g^x^)-^g^(x^)-'3 IIA I B^(^)MB^(^)- QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1035 This, of course gives an estimate for ||/i # h\\L^{^), but to estimate the Besov norms of /i -#- f^ we need a further argument since we have different metrics ^ and ^. We remark that since we have, by (1.4.2), for any xi e ^ (2-1-25) Q^^gM'p^'gUx^y an easy estimate gives (2;.i.26) sup QAyY^Q-^^f^B^v^ 9j{y)<^- < fi • fnnooiny (^r- 9( \No. \\ r [| -^ ° P Q^^z^^i) '1^1'B^+^(V,^•.)• Hence from (2.1.24) and (2.1.26) we deduce, since ^(Q(D)) commutes with translations, that for x e 2lj, ft' > f3 + d, we have f SUp ^)-a|/l#/2^)-/l#/2^+^| | 9 o (?/) su p ( )-QII? ) /21 \.^•(^)^) i < C' • ov (T 21 • ^"^r^ fr 21 't-/3(^v fr 91 • 'l7^ ^^ ^Jl^? ^27 ^V^'^1^ yj2^x^^2) ' II^HB^ rsi- ) ' ll^ll^+a^ ^ -'-'oc^oo ^Ji ^ -Doo,oo V-aJ2 I -/3 v 7VQ! --\-r C'^ y^a' ^(r , 21^2- "l""^/ y^^ fV^r 2^j1 i ^'t <^^•i ^fr5 9^ji1 • ;^ '^^IB^+^^^'II^HB^^^) 77 Na v a -<^ ^(7 • yji^^n^ (r 21 J"l"^^ yj^^^ji fr 91 • )^"^ i^^l\^ fx^•>^^l) 91 • } n9j2\ (vx^^ 9afj2 )^ ' lljlllD^+c^of. \ ' ^HD^'+a.oj ^ -"oo.ool^i; -"oo,^^^^^ where the constant C" depends only on a, {/3 — f31), no and d. 1036 SHAI HARAN 2.2. The main theorem of symbolic calculus. We now give a global estimate for /i # /2? where fi € S(mi, g) and the weights 7711,7712 are temperate, i.e. they satisfy (1.4.8). We let /^) denote the restriction of fi to 9lj, and we note that the partial sums ^ /^) j (2.2.1) |/i,o,) #/2,(^)| Taking (3 sufficiently large so that (1.4.7) applies, we obtain (2.2.2) El/i.^^W^I < C' • mi . m^x) . \\fi\\^^) • 11^11<^,,). From the estimate (2.2.2), being valid for all fi in a bounded subset of S(mi,g) we deduce that the form /i (g) /2 ^ /i#/2 defined on Coo^^Q^Coo,^^) = Coo,c(^®^) extends uniquely to a weakly continuous linear form on S{m\^g) (g) S(m'z,g), given by /1#/2(^=^/1,00#/2,0.)(^). 31 J2 Similarly, note that the functions (2.2.3) mi{y)=mi(yVgy(x^y)-Na are again temperate, with the corresponding constants in (1.4.8) depending only on 77^, g and a (and independent of re); letting (3a denote the associated constant in (1.4.10) (with mi replaced by m^), and using (1.4.5), we deduce from (2.1.27) that for any f3' > (3 + d + /?a + {N + I)2 • Na, we have (2.2.4) ||/i,0.)#/2,a.)(^l|^^^) ^CIII'g]^x^)-^g]^x^)-ftm^(x) ' II^VujJiB^o^) • ll^1^^)!!^^^,)- QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1037 Taking again a /3 satisfying (1.4.7), we conclude that with some new constant C (2.2.5) ^ ||/i,(ji) # /2,(j2) HB^O^) J'l J2 < C . m^(x) • ll/ill^^,,) • WB^(^Y Thus the map /i <8) /2 '-^ /i # /2 is weakly continuous when viewed as a map S{m^,g)^S(m^g) —> 5(mi -m^g). Moreover, since the function x ^ h(x) of (1.4.9) is temperate, and since for x ^ 21^ (2.2.6) 1 < j^n g^x - y) <, ^ • ^(^), we see that, using (1.4.5), we can bound for x ^ 51^ H 21^, (2.2.7) ^(^^)-/31•7^(^,^)-/31•7 ^ h(xr for some /^i > 0. Hence, by taking (3 large, we can replace m\ ' m^(x) in the right hand side of (2.2.4) by W ' mi • m^x), for x ^ 21^ n 21^, and a posteriori we can similarly replace m\ ' m^ by W ' m\ • ms in (2.2.5) if we sum only over j\ ^ j^. When ji = j^ = j, we can use the estimate (2.1.11). Thus we conclude that the map f\ 0/2 '—> A # /2 — /i • /2 is weakly continuous when viewed as a map 5'(mi, g) (g) S{m^,g) —> S'(^7 • mi • 7722, p), for any 7. We summarize the above discussion in the following MAIN THEOREM (2.2.8). — Given a temperate^ certain, metric- covering g in the symplectic space V, and two temperate weights mi, 7722, then # : S'(mi,^)(g) S{m^g) —> S(m^m^g) is weakly continuous. If h(x) = s\ipgx(y)/g^(y)^ then for any 7 the map y fi ^ /2 '—^ fi # /2 — fi' /2 is weakly continuous S(m^,g) (g) S(m'2,g) —> S(h^ ' mi 'm^,g). 1038 SHAI HARAN 2.3. Bounds for operators. In this section we study the continuity of the operators p(f) on the spaces S(Vo) c—^ L^(Vo) c—^ ^'(Vo), for / e S(m,g), g^m temperate. We first remark that since by (0.1.13) p takes complex conjugates to adjoints, the continuity in THEOREM (2.3.1). — -For a temperate certain-metric-covering g^ and a temperate weight m, the operators p(f) are continuous in S(m^) ^ Kom{S(Vo)^(Vo)) = S(Vo) ^^(Vo), (2.3.2) S(m^) ^ Hom^^Yo)^'^)) = ^(Vo) ^ 5(Yo). For the continuity on L^(Yo) we take / G S(l, g) and write / = ^ fi i for the corresponding decomposition of / with respect to the metric covering g. We shall use the lemma of Cotlar-Knapp-Stein-Calderon- Vaillancourt, so that we need to estimate the operator norms (2.3.3) ||pCA)W)|lop = Wallop = llpW.#/.))||op < IW^^)HL.(V) < ^ • F^-lk,^) for a sufficiently large. From (2.2.4) we obtain for f3' ^> f3,a (2.3.4) ||/^/,-[|^^^ ^G.^^^r^^^-^ll/ll^^. By using repeatedly (1.4.1), and the inequality of (1.1.26), one easily obtains that for some constants Go, M, and all x G V (2.3.5) ^(^.^^min ^(W - V"} < Co • g^x^ . g^x^. y ^i y"^j QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1039 Putting (2.3.3), (2.3.4), (2.3.5) together we obtain, with some new constant G, for 7 > /? > 0, (2.3.6) WiYW\\^ < C • gn^M^ • 11/11^(1,,). By an analogous proof to the proof of Lemma (1.4.6) we have, for f3 » 0 (2.3.7) ^^(^.^^-^oo, ^^(^,2l,)-^ i 3 From (2.3.6) and (2.3.7), we obtain by applying the lemma of Collar et al., (2.3.8) ||p(/)||op ^ C ' \\f\\B^(i^ for 7 sufficiently large. THEOREM (2.3.9). — For a temperate certain metric-covering g, and a temperate bounded weight m, the operators p{f) are bounded on L^VQ) for every f € (2.3.10) \\P(m^ If m vanishes at infinity^ the operators p(f) with f € *S'(m, g) are all compact. Proof. — The inequality in (2.3.10) follows immediately from (2.3.8), since ifm (2.3.11) Remark. — We note that if fj is a bounded sequence in 5(1,?) converging to / in Coo(V) then for any (p € ^2(^0)5 pUj)^ —^ p{f)^P in L^(Yo). Indeed, we may assume / = 0, and in view of the uniform bound (2.3.8) we can take (p € its characteristic function, we know from (0.4.15), (0.4.17) that there is a unit vector yo e L^Vo) such that Wyo = $ = $. If / is such that supp/ C 0, then by (0.4.19), we have p(f) = p(f * $) = C^MyC^,, and hence ||p(/)||op ^ ||/HL^,. This also holds for_/ such that supp/ C x + 0, since the function U(y) = f(y + x) = J-(^ . f)(y) is supported in 0, and so ll^llop-ll^-^)^)^-^)!^ = l|p(^ •/)|1^< 11^ •/ll^= ll/ll^. Thus for a self-dual metric covering g^ = g^, corresponding to V = ]_[2lj, 2lj = Xj + Oj, Oj = O], letting $j denote the characteristic function ofSlj, we have for any function / lollop-|KE^*/)|L, (2-^) ll^)|lop ^ 11/11^(,) < ^^ . ||/||^.(^, From this we deduce the following THEOREM (2.3.13). — For a self-dual metric g, the operators p{f) are bounded on L^Vo) for every f e L^(V,g), a > dimV. If such an f vanishes at infinity, then p(f) is compact. For a ^-locally constant function m, we define the operator J^ = p(m-1); if m > 1, it is bounded on L^Vo) by (2.3.8). We define the m-th Soboleff space by (2.3.14) L^(V,) = J^(W)), |M|^^ = \\p(m)4^. QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1041 Since for p-locally constant functions m\^m^ we have m\ # m^ = mi • m2, we deduce that J^J^2 = J^'1'1712, and in particular, we see that J^ = J^2' (J^Y is positive, and J^J^~1 = id, so that J^ is invertible. For example, taking g and m as in (1.4.12) (or as in (1.4.18)) we get essentially the operator AT" of (1.1.20) (respectively, the usual J01), and the space L^^Vo) of (1.1.21) (respectively, L^(Yo)). For a temperate weight mo if / € 5(mo,^), we have f#m'Q1 € S(l,g), hence by (2.3.8) P{f # rno1) = p(/)J^° is bounded. Similarly replacing / by m^1 # / # mi C 5'(mo, g), we obtain the COROLLARY (2.3.15). — For temperate weights mo,mi, every f in 5'(mo,^) defines a bounded operator p(f) : L^107711^) -^ I^O^). 3. ELLIPTIC OPERATORS 3.1. The Garding inequalities. Fix a temperate certain metric covering g in V corresponding to v = U^ ^j = Xj + Oj, Oj 3 (9J, and write ^ = p-^ for the quantity 3 defined in (1.4.9) for the metric g^. For each j choose a self-dual lattice 0° such that Oj 3 0^ 3 OJ, and moreover such that we have (3.1.1) p^Oj D(^°, with bj = [j^-] = (the greatest integer < ja^-). To see that such a choice for 0°j is possible, write in some symplectic basis Oj = p'^Zp e • • • ep'^Zp e Zp e - • • e Zp, o < ei < • • • ^ e^, 0] = Zp C ... C Zp e p^Zp e ... C P^Zp, so ei = o^, and take o^0 = p-^Zp e •.. ep-^Zp ep61 e • • • ep^Zp, with c, = [|e,]. We denote by ^ the characteristic function of 0° which is by (0.4.15), (0.4.17) the Wigner transform of a unit vector (f)j e ^2(^0)5 ^j = W(pj. For a function / in V, we denote, as usual, by fj the restriction of / to 21^, and we let f{x) = ^ * fj(x) for x C ^, / is the average of / over the x + O^s. We have the following : 1^^ SHAIHARAN LEMMA (3.1.2). — For / € S(m,g), we have f-fe S(mh^,g), for any 7. Proof. — It is enough to prove that for any a, 7 > 0, we have for all j (3.1-3) /l7711^•-^•llB_(.^2.||/,||^^, With the notations of (1.2.3), using (3.1.1), we have (3.1.4) cf>g^ ^ * / = <^. ^ * /, and hence ^ ^ * / = ^. ^ * /, for A; = 0,1,..., 6y. From (3.1.4) we obtain ^M-MB^, = p7^ supp^ sup |^,,, * (/, - /,)(^)[ fc^O a;€2tj = p1-^ SUp p0^ SUp |^,fc * (/, - /,)(^)| fc>bj a;G2lj ^ p7-^ sup p0^ [sup |^,,fc * /.(a;) I + sup |^,,fc * $, * /,(^)|] k>b, xefHj xe^j < 2 • p^ sup p"^ sup ly^fc * /,(a;)| fc>frj .1:621, ^ 2 • SUP p(a+2^ g^ j^^ ^ ^.(^ A;>6j a;62lj < 2. W^^,, * /,(.)| = 2. ||/,||,^^,. n The operator p(/) corresponding to / is expressed by (0.4.19) as (3.1.5) p(f) = ^p(/, * ^,) = ^p(^. * TV^) = ^ ^.M^.G^,. ^ ^ 3 In particular, if / > 0 is positive, p(f) is positive in the sense that (3.1.6) (p(f)^ ^)L^Va) ^ 0 for all ^ e Thus we have obtained the following COROLLARY (3.1.7). — If f e S{m,g) is positive, f > 0, then f = f + /o, where the operator p(f), is positive, and fo e S(mh^,g) for all 7. QUANTIZATION AND SYMBOLIC dALCULUS OVER THE p-ADIC NUMBERS 1043 COROLLARY (3.1.8). — Iff Proof. — Writing / = /+ /o as in (3.1.7), we have (/)(/), ^)^^ > (p(/oW)^) - (Krn-^h-^^f^^m1/2^)^)^ > -\\p{m-^h-^p(f^ \^ . \p(m^h^\\^ >-^-11<^.^ where C denotes the norm of p(fo) : L^2^^) -> L^"172^, which is bounded by (2.3.15). Q As a special case of (3.1.8) we note that we have COROLLARY (3.1.9). — If f e S(h~^,g) is positive, then, for all ^eS(Vo), W^^L^^-G'Mwy We say that / e S(m, g) is strongly-elliptic if for some A > 0, we have (3.1.10) Re f(x) > A' m{x) for x outside a compact subset of V. SHARP GARDING INEQUALITY (3.1.11). — For a strongly-elliptic f in S(m, g), we have for any 7 : M/K/W)^ > A. IMI^v^ -c. M^^^y Proof. —Set/i = Re/ - A • m + <^ where ^> C C^(^) is large enough to make /i ^ 0. Applying (3.1.8) to /i, and noting that by (0.1.13) ^O^/)^ ^L^Vo) = (p(Ref)y, ^)L2(Vo)^ we g^ the desired result. D We can write 21, = [J ^-^ with 21^ = Xj, + 0^ z = 1,..., [0^ ^°], so that the decomposition V = U^iji corresponds to a self-dual metric _ _ JZ covering g^ with g^ ^ g^ ^ ^. For any / G S(m,g), denoting by fjz = f^ji) the average of / over ^-^ the operator p(f) is diagonalizable with respect to the orthonormal basis (^ e ^2(^0), W^jz = characteristic function of ^, cf. (0.4.20) : p(/)(^ = /„ • ^. Thus we have «almost diagonalized)) p{f). 1044 SHAIHARAN 3.2. Fredholm operators. By the Stone-von Neumann theorem, p : L^(V) —^ ^2(^2(^0)) is an isometry onto the ideal of Hilbert-Schmidt operators. Since an operator is trace-class if and only if it is a product of two Hilbert-Schmidt operators, we conclude that (3.2.1) p{f) is trace-class, if and only if / e L^V) # L^ (V), and note that L^V) # L^(V) C L^(V) H Co(V). Given an invertible trace-class operator /C, if IC^p^f) is bounded then p(f) is trace-class, so we can use the criteria for boundedness of paragraph 2.3 to get criteria for p(f) to be trace-class. In particular, using (1.1.23), (1.1.24) we get : LEMMA (3.2.2). — The operator p(f) will be of trace-class provided either p(f #^(1,^,^)°') is a bounded operator for some metric g in V, and some a > 2d == dimY, or if p{f # go^l,^)^ • 9o{^^x)oi2) ls a bounded operator for some metric go in VQ^ and some ai, 0:2 > d. For example, using the first criteria of the lemma, we see that the Toeplitz symbols S01 correspond to trace-class operators if a < —2d. A symbol / € S{m, g) will be called elliptic if (3.2.3) \f{x)\ > ^' m{x) for x outside a compact set K of V. For such an / we define /,,,. f-z, . ifW ^x€V\K, (3.2.4) JK (a-) == \ 10 if x € K, where we can take K to be a finite union of the sets 21, corresponding to the metric-covering g. Since we have for x £ V \ K, and for y such that 9x(y) ^ 1, I/^I^A-1.^)-1, n (y}-a\f-l(r} f-^r I ^| - ^(^""l^ + J/) - A^)l 9x{y' \fKW-fK^+V)\- \f^f(x+y)\ (3.2.6) h(x) < C ' go(l, x)~6 for some 6 > 0. We first remark that from (3.2.6) we obtain for any temperate weight m, m{x) < Ci . m(0) • ^(1, xf1 by (1.4.8) (3.2.7) { l-f#fK^ l-^^/^^V), and they correspond by (3.2.2) to trace-class operators. Hence we have, ' Index(p(/)) = tr(l - p(f^ # /)) - tr(l - p{f # f^)) (3-2-8) < =^(p{f#fKl)-P(fKl#f)) = fdx(f#fj,\x)-f^#f(x)). Writing / = / + /o, f~K1 = f]K1 + A as in Lemma (3.1.2), with fo^fi ^ S(Y) by (3.2.7), so that p(fi) are trace-class, we obtain from (3.2.8) Index(p(/)) = f dx(f#f^{x)^\J -tt-JK \^) ~- fKf^1 #/(^)). j v But since / and f^1 are ^-locally-constant, we have f # f^1 /p1 #f = f ' /K1^ an^ we have proved the following THEOREM (3.2.9). — For global p, every elliptic f e S(m, g) defines a Fredholm operator of index zero ^^(yo)-^^). 1046 SHAIHARAN 3.3. Zeta functions and spectral asymptotics. In this section we fix a global metric-covering g, temperate and certain, so that we have from (3.2.7) : ?] S(mh^,g) = p(m)^ = p(m # f^) p(f) ^ + p{m # Ro) ^, showing that p(m)(p e L^{Vo), and so (^ e L^^Vo). From the continuity of p(f) in ^'(Vb), we deduce that the operator A is closed. We claim that A is self-adjoint, and to prove it we shall show that A equals the closure of its restriction to k)^ -^ ^ in ^(Vb); moreover, since ^k ^ S(V) the operators p(^/c) are smoothening, and so p(^/c)^ e ^(Vb). For^G47n)(yo)wehave Ap(^ = p(f #^k# f^W) ^ + P(f # ^ # -Ro) ^. The / # ^k # fK\ and / # ^ # J?o, are bounded in 5'(1, ^), and converge to f#f~K1^ and to /^o, in Coo(y), respectively, hence by (2.3.11) we have Ap^k}^ — p(f # f^)?^ + p(f # Ro)^ = A^ in La (Vo)- Thus A is indeed self-adjoint. We note that if / ^ 0 is positive, so that by Garding inequality (3.1.11) A is bounded from below, and hence for some real number Ao, {A - \o) is invertible. Its inverse (A - \o)~1 is a self-adjoint injective operator with image L^^Vo). Assume now that m~1 e Co(V) vanishes at infinity. From Theorem (2.3.9) we see that (A - \o)~1 is a compact operator. From the spectral theorem for compact, self-adjoint, injective operators, we deduce that there is an orthonormal basis {(^} for L^Vo) consisting of eigenfunctions for (A - Ao)~1, with real non-zero eigenvalues ^ converging to 0. Hence the 0/s are eigenfunctions QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1047 of A with the real eigenvalues Xj = Ao + /^•1, converging to 4-00. Moreover, writing (/-A,)^#(/-A,)=1-^,, with R\^ e S(V), we have <^ = p(R^)cf)j is in «S(Vb). We summarize the above discussion by the following THEOREM (3.3.1). — For a global certain temperate metric-covering g, and a temperate weight m, every real-valued elliptic f € S(m, g) defines an essentially self-adjoint operator p(f). Denoting by A the unique self-adjoint extension of p(f), Domain(A) = L^^Vo), and iff>.0 is positive, A is bounded from below. If moreover, m~1 e Co(V) vanishes at infinity, then A has a discrete point spectrum, and there is an orthonormal basis for L'z(Vo) consisting of eigenfunctions (f)j € S{Vo},A(t)j = \j ' (/)j, with eigenvalues \j converging to +00. To study the asymptotic behaviour of the eigenvalues \j, we may assume that f(x) > c • m(x) > e > 0, and moreover that minAj > e, since we can always achieve this by adding to/a large constant. For A G C\ [e, oo] we can write (3.3.2) f(/-A)-^(/-A)=l-^ la-A^cf-A)-^!-^ with R\,R\ £ S(V). From (3.3.2) we obtain (3.3.3) (.4-A)-i-p((/-A)-i) = p{R^ #(f- A)-1) + pW{A - A)-^(^). For A ^ {\j} the operator (A - A)"1 is bounded, so that the right hand side of (3.3.3) is a smoothening operator and can be written as p(r[\\) for some r^j € S(V); thus we conclude that for A ^ {Aj}, (3.3.4) (A-Xr^^x}) for some /^} € S(m-\g) with f{^ - (f - A)-1 = r^} € S(V). Note that for Re(A) ^ £-1 < e we have the inequalities ' sup | (/-A)-1 (a;) | <, |c-mj-A|-1 <. |e-A|-1, X^j sup g,{x, - a;2)-Q|(/ - A)-^^!) - (/ - A)-^) (3.3.5) Xl,X2E^.j =SUD ffj^l-^)-0!/^!)-/^)! ^ea, |/(a:i) - A| • \f(x^ - A| <, Ca • \C • mj — \\~2 <, Ca\£ - A|~2. ^S SHAIHARAN Hence in particular, (3.3.6) ||(/-A)-i[|^^^=o(|A|-1) for|A|-.oo, Re(A) < e,. From (3.3.5) and the error estimates of the main theorem (2.1.11), (2.2.4), (2.2.7), a straightforward calculation gives for Rx = (f - A)-1 •/-(/- A)-1 #/, Re(A) < e, < e, the inequalities /Q Q ^ I ll^ll^ooC^) ^ c^^ and similarly ^o.o.t) ^ Ul^llB^^^-y^) ^C^. From (3.3.6) and (3.3.7), combined with (2.3.8), and with (2.3.15), we have as |A[ —> oo, Re(A) ^ ^i, fllw-Ar^iL^odAi-1), (3.3.8) J ll/3(^)|lop^)=0(l)' lll^^llop^-O^). where ||A||op(A^ denotes the norm of R : L^''\Vo) -> L^^Vo). From (3.3.8) we have for the right hand side of (3.3.3) the estimate (3-3-9) ll^{A})|lop^) = M - A)-1 - P((/ - A)-1)!^ == 0(\X\-1) as |A| —^ oo, Re(A) < £1. For Re(s) < 0 we have (3.3.10) As=— r^00 ^(^ - A)-l dA 2^ ^,-zoc which converges in the operator norm since \\A - (e^ + ^)||op = Od^"1). From (3.3.5) we see that for Re(^) < 0 we have 1 pe-i-\-ioo (3•3•ll) r:=27^/ ^(/-^-^Ae^m^)^). -" l' J e^—ioo QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1049 From (3.3.9) we see that for Re(s) < 0 1 /•e-1+ioo (3.3.12) r^ = —— / X8^ dX € S(V) 27r21 Jci-ioo since the norms || ||op(/i^)? A > 0, define the space S(V) by the remarks after (1.3.3). Combining (3.3.10), (3.3.11), (3.3.12) with (3.3.4) we conclude that for Re(s) < 0 f A8 = p(f^8) with f^8 = /s +r<^ € ^(m1^,^, (3.3.13) { [r^ C Since for 5 an integer (3.3.13) holds with f:^s=f#-•'^f,s times, we see that (3.3.13) actually holds for all s € C. Thus we have proved the following THEOREM (3.3.14). — For a global certain temperate metric covering (7, a temperate weight m with m~1 € Co(V), and a positive elliptic symbol f € 5(m, 5'), denoting by A the unique self-adjoint extension ofp(/), and assuming that A is positive^ A generates a holomorphic group of operators A8 (bounded only for Re{s) <: 0) and given by A8 = p(/^5), /^^4-r^} e^m1^5)^),^ CS{V). Note in particular that s ^—> r^ is an entire holomorphic function of s with values in S(V); for Re(5) < 0 this follows from (3.3.9) (3.3.12), and we have a «step by step » holomorphic continuation by means of the recursion formula rW^f^rW+a^f8-/^8). Assume now that we have (3.3.15) m'1^) < C ' ^(l^)""0 for some OQ > 0. It follows from Lemma (3.2.2), and from Theorem (2.3.9), that the symbols in S^m"5,^) correspond to trace-class operators for s > 2d/OQ. Thus we can define the zeta function^ (3.3.16) C.400 = tr^-5) = ^A78 = f Ar/^-^), J Jv 1050 SHAI HARAN which is a holomorphic function of s in the right half plane Re(s) > 2d/ao. Similarly we can define (3-3.17) Q(s)^ [ dxf(x)-8 Jv which is holomorphic in the same right half plane Re(s) > 2d/o;o. Setting NAW= #{A, Nf{\)=vo\{x\f(x) we have the Mellin transform expressions for our zetas (3.3.18) C^^A-dA^A), (3.3.19) ^(s)= fx^dNfW. Their difference is given via (3.3.14) by (3.3.20) ^(s)-(:f(s)= f X-8 d(NA(\) - Nf{\)) = / dxr^(x), and is an entire function of 5. Applying the inverse Mellin transform we obtain the following THEOREM (3.3.21). — For a global certain temperate metric- covering g, a temperate weight m such that mW^C'go^x)^, ao>0, and a positive elliptic f e S(m,g), denoting by \j the eigenvalues of the self-adjoint extension A of p(/), we have for every e > 0 NAW = # {A, < A} = 0(A2d/QO+£), as A ^ +00. Moreover, we have for every e > 0 N^W = vo\{x I f{x) < A} + 0(A^ as A -> +00. This theorem should be compared with the remark at the end of paragraph 3.1. Using the notation of the same paragraph, / is ^-locally QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1051 constant and the operator A = p(f) is explicitly diagonalizable; denoting by \j its eigenvalues, we have {\j} = f{V), so that for any positive real A we have an equality # {Xj ; |A^| < A} = yo\{x ; \f{x)\ < X} if finite, and more precisely we have an equality of measures (3.3.22) ^^.=/.(d.r). 3 Here there are no ellipticity or positivity assumptions on /. If however the assumptions of Theorem (3.3.21) are satisfied, then / is also positive elliptic, and so for any e > 0, combining (3.3.21) with (3.1.2) we have as A —> +00, f NA(X) = vo\{x | f(x) ^ A} + 0^) (3.3.23) ^ = vol{rr | f(x) < A} + O^) [ =A^(A)+0(A£). For particular examples of Theorem (3.3.21) one can take the Toeplitz symbols defined in (1.4.12). BIBLIOGRAPHY [1] V. BARGMANN, On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Comm. Pure Appl. Math., 14 (1961), 187-214. [2] R. BEALS, A General Calculus of Pseudodifferential Operators, Duke Math. J., 42 (1975), 1-42. [3] J. BERGH, J. LOFSTROM, Interpolation Spaces, Berlin-Heidelberg-New York, Springer, 1976. [4] F.A. BEREZIN, Wick and anti-Wick Operator Symbols, Math. USSR Sb., 15 (1971), 577-606. [5] A. CALDERON, R. VAILLANCOURT, On the Boudedness of Pseudodifferential Operators, J. Math. Soc. Japan, 23 (1971), 374-378. [6] P. CARTIER, Quantum Mechanical Commutation Relations and Theta Functions, Proc. Symp. Pure Math., 9, AMS, Providence, 1966, 361-383. [7] A. CORDOBA, C. FEFFERMAN, Wave Packets and Fourier Integral Operators, Comm. Partial Diff. Eq., 3 (1978), 979-1005. [8] C. FEFFERMAN, D.H. PHONG, The Uncertainty Principle and Sharp Carding Inequalities, Comm. Pure. Appl. Math., 34 (1981), 285-331. [9] G.B. FOLLAND, Harmonic Analysis in Phase Space, New Jersey, Princeton University Press, 1989. [10] L. CARDING, On the Asymptotic of the Eigenvalues and Eigenfunctions of Elliptic Differential Operators, Math. Scand., 1 (1953), 237-255. 1052 SHAIHARAN [11] S. GELBART, Weil's Representation and the Spectrum of the Metaplectic Group, Lectures Notes in Math. Springer 530, Berlin—Heidelberg—New York, 1976. [12] A. GROSSMAN, G. LOUPIAS, E.M. STEIN, An Algebra of Pseudodifferential Operators and Quantum Mechanics in Phase Space, Ann. Inst. Fourier (Grenoble), 18-2 (1968), 343-368. [13] V. GUILLEMIN, S. STERNBERG, The Metaplectic Representation, Weyl Operators, and Spectral Theory, J. Funct. Anal., 42 (1981), 128-225. [14] S. HARAN, Riesz Potentials and Explicit Sums in Arithmetic, Invent. Math., 101 (1990), 697-703. [15] S. HARAN, Index Theory, Potential Theory, and the Riemann Hypothesis Proc. Durham Symp. on L-functions and Arithmetic, Cambridge Univ. Press, 1991. [16] S. HARAN, Analytic Potential Theory over the p-adics, Ann. Inst. Fourier (Grenoble), 43-4 (1993). [17] B. HELFFER, Theorie spectrale pour des operateurs globalement elliptiques, Asterisque, 112 (1984). [18] L. HORMANDER, The Weyl Calculus of Pseudodifferential Operators, Comm. Pure Appl. Math., 32 (1979), 359-443. [19] L. HORMANDER, The Analysis of Linear Partial Differential Operators, III, Springer, Berlin-Heidelberg-New-York Tokyo, 1985. [20] L. HORMANDER, On the Asymptotic Distribution of Eigenvalues of Pseudodif- ferential Operators in R", Arkiv for Math., 17 (2) (1979), 296-313. [21] R. HOWE, Quantum Mechanics and Partial Differential Equations, J. Funct. Anal., 38 (1980), 188-254. [22] R. HOWE, Theta Series and Invariant Theory, Proc. Symp. Pure Math. 33, AMS Providence 1979, part. 1, 275-285. [23] R. HOWE, On the Role of the Heisenberg Group in Harmonic Analysis, Bull. AMS, 3 (1980), 821-843. [24] A.W. KNAPP, E.M. STEIN, Intertwining Operators for Semisimple Groups, Ann. of Math., 93 (1971), 489-578. [25] J. PEETRE, New Thoughts on Besov Spaces, Duke Univ. Math. Series, 1976. [26] J. PEETRE, The Weyl Transform and Laguerre Polynomials, Le Mathematiche (Catania), 27 (1972), 301-323. [27] D. ROBERT, Proprietes spectrales d'operateurs pseudodifferentiels, Comm. Partial Diff. Eq., 3 (1978), 755-826. [28] R.T. SEELEY, The Complex Powers of an Elliptic Operator, Proc. Symp. Pure Math. 10, AMS, Providence, 1967, 308-315. [29] J.-P. SERRE, Local Fields, Springer, Berlin-Heidelberg-New York, 1979. [30] M.A. SUBIN, Pseudodifferential Operators and Spectral Theory, Nauka, Moscow, 1978. [31] M.H. TAIBLESON, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975. [32] M.E. TAYLOR, Noncommutative Harmonic Analysis, AMS Providence, 1986. [33] F. TREVES, Topological Vector Spaces, Distribution, and Kernels, Academic Press, New York, 1967. [34] H. TRIEBEL, Theory of Functions Spaces, Monogr. in Math. 78, Basel-Boston- Stuttgart, Birkhauser, 1983. QUANTIZATION AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS 1053 [35] A. VOROS, An Algebra of Pseudodifferential Operators and the Asymptotics of Quantum Mechanics, J. Funct. Anal., 29 (1978), 104-132. [36] A. WEIL, Sur certains groupes d'operateurs unitaires, Acta. Math., Ill (1964), 143-211; also in Wen's (Euvres Scientifiques, vol. Ill, 1-69, Springer, Berlin-Heidelberg-New York, 1980. [37] H. WEYL, The Theory of Groups and Quantum Mechanics, New York, Dover, 1950. [38] R. HOWE, The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132. [39] A. UNTERBERGER, J. UNTERBERGER, La serie discrete de SL(2,R) et les operateurs pseudo-differentiels sur une demi-droite, Ann. Scient. Ecol. Norm. Sup., 17 (1984), 83-116. [40] A. UNTERBERGER, J. UNTERBERGER, Quantification et analyse pseudod- ifferentielle, Ann. Scient. Ecol. Norm. Sup., 21 (1988), 133-158. [41] A. UNTERBERGER, J. UNTERBERGER, Serie principale et quantification, C.R. Acad. Sci. Paris, 312, Serie 1 (1991), 729-734. [42] V.S. VLADIMIROV, I.V. VOLOVICH, p-adic Quantum Mechanics, Comm. Math. Phys., 123 (1989), 659-676. Manuscrit recu Ie 24 aout 1992, revise Ie 3 mai 1993. Shai HARAN, Dept of Mathematics Technion Institue of Technology Technion City 32000 Haifa (Israel).