QUANTIZATIONS AND SYMBOLIC CALCULUS OVER THE p-ADIC NUMBERS

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 / Q x 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

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)\<C(3' [1,^1^.1^

for |z/| <, |l,p^|. We note that our (analytic) Schwartz space, 5, which can be defines as the intersection over all a of E0, is different from the usual (algebraic) Schwartz space of locally constant compactly supported functions that one encounters in the literature, though it shares much of its nice properties such as nuclearity, closure under multiplication, convolution and Fourier transform.

Chapter 2 is devoted to the proof of the main theorem of the calculus (2.2.8) and the basic boundedness theorems, along the lines of [18], [19].

In paragraph 2.1, we prepare the grounds by giving the short range estimate (2.1.6), the error estimate (2.1.11) and the Long range estimate (2.1.24) and (2.1.27). Then in paragraph 2.2, we put these together and prove the main theorem of the calculus (2.2.8). In paragraph 2.3, we establish the continuity of our operators on S' and S", (2.3.1), and their boundedness on L^ (2.3.9), and give a proof of the Calderon-Vaillancourt estimate [5] (along the lines of [21]).

In chapter 3 we establish the basic properties of elliptic operators.

In paragraph 3.1, we prove the sharp Garding inequality (3.1.11) and the improved Fefferman-Phong inequality (3.1.9) (cf. [8]), as consequences of the wave-packets theory (cf. [7]). In paragraph 3.2,, we show that every elliptic symbol defines a Fredholm operator of index zero (3.2.9). In paragraph 3.3, we consider positive elliptic symbols, show that they give rise to self-adjoint operators with discrete spectra (3.3.1); study their complex powers (3.3.21) (in analogy with [28]); and determine the precise asymptotic behavior of their eigenvalues (3.3.14) (see [10], [13], [17], [20], [27], [30] for some of the literature on the analogous theory over R).

Thus the operators on Z/2(Vo) corresponding to symbols in E^V) are bounded for a <^ 0, Hilbert-Schmidt for a < — d i m V o ? 8Ln(^ trace-class for a < —2dimyo- For / in E^V), we can write / = / + fo, where f(x) is the average of / over the unit ball centered at x^ and /o ^s in the

1000 _SHAIHARAN

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^

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, X i ' 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)} . <p(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 ^ € Q p =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 / € L i ( V ) a n d ^ € L 2 ( V b ) ,

(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

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 S H A I H A R A N

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 - < ^ \ ( <I>a l'b l • 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, i f a i + 6 2 < 0 ^ & i +02,

^ 0 1 5 ^ 2 . /ha2,^l xi,y2 -*-a;2,2/i

^ai,fc2 . $02^1

< a;i,2/2 --^2,2/1

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

i f a i + & 2 , & i + ^ 2 ^ 0 .

(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.

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(v^f)=^{t^^{v^^f))f(v^f-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.

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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

(0.2.13) (Ep^(f)(v)=f(v)^

\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^}

= { r r + 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; —— *

" ² 1 II I ^x2

* —— Ui —— Oo* —— Z; —— * U/U •<———I U

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

(0.3.5)

v i ' vz = qe(vi,V2) + Ve {vi,v^}

with symmetric part qe{x^ + \fey\,x^ + ^1/2)

= 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^w l,6ev / i p w 2)=r i f^ ^ 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 - ^ - V e y — — (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 . 1 1 ) (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^)<f>

J u\

where q^ = N^, and d*u is Haar measure on £/i of total mass 1.

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)

= # { ( ^ ) e y / 0 o ; 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)p^N)~^{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)= j d z ^ ( x z ) ^ ( 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 = (<y?, (^2) • ^i.

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•

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^ ^=^ <Pi=\'^2 for some A, |A| = 1.

Note that W(p{^x) -^ 0 implies that there exists z such that x ± z C supp(^, and hence x C A(supp^) ^ { j Q / 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^ ^ I I ^ H ^ = jf dxd^W^x),

< \\W\L^ -vol(suppW^) from which we obtain the uncertainly principle (0.4.14) vol(suppl^) > I M I L 2 > 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

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), <E^ = W^° is the characteristic function of 0^ ^ * ^j = <1>^, and for any locally constant compactly supported function (p in ^(Vo)? we have by (0.4.19)

(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;, R e s > 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^, J^a = J^{^}^^lI^aJ^{^}d 5 J^ forming an entire family of distributions, J^ * J⁷³ == J^Q;+^; for a > 0, J^ being a probability measure supported in Z^; and

(1.1.7) J^x) = ^c(^(|^|^a-^d-p^a-^d) .^), for Re a > 0, a ^ d mod (27n/logp)Z,

(1.1.8) ^ ^ ) = ^ _ ( i - i o g ^ ^ | ) . ^ ) . Note that

(1.1.9) J^^L^Vo) for R e a > d ' ( l - q -1)

and in particular : J^ G Li, Rea > 0; J" G Z/2, Rea > jd; J^ e Loo, Re a > d.

The Soboleff spaces (or Lebesgue spaces of Bessel potentials [31]) are defined by

(1.1.10) ^(Vo) = <nW)), 11/11^ = ll^/lk,

and in a similar way, we can define 1^(21), where 21 C Vo is any domain such that 21 -j- Z®^ = 21. We also define the weighted Soboleff spaces by

(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 . l l / l l L ^ I I / l l ^ + l l - R - ' / I l L , .

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^), o i - 0 2 > jd, / 3 i - / ? 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-^lJ^QB=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 | . ^ ) , A e Q ^ , veV^

g(v) = 0 <^=^ V = 0.

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) = i n f { | 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 ) ^ r K ^ ;^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,

(1.1.31) ^{J^^-}

1^,

^-^-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

f

^^ f\ V (

\ (

V f f

^'^

U ^•^-^E^JU^'U ^h

which follows upon taking the Fourier transform from the Newton rule :

(i.2.i) 0/1+2/2)^= E (^yi-y^

-

0<j<N v J /

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,

(1.2.2)

« 2 7 T / l o g p /.27T/10gp 1

| ^ + ^ = p . v . / dt.-^

Jo 27r

,J^ 1 CO) , C(-l)C(-(l+a)n

^^ 2 C ( l + a )+ C(a) ;

xd^r^'12/21-^+12/11-^ •12/2^}

+ ^((l

^" ⁾ E^-l)•(-^)- ^(l+ ' ⁾

x r

⁷¹⁰ ^ • ^. x^i) iz/ii^

² ^ • x^) i^i ⁰⁷² -^

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

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, i f a < 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^, (1.2.6) B^

%.„ n ^ r a ; B^, a i > Q 2 ;

f n^

^g.g ^ B^, Kq^2,

(1.2.7)

DO:

^,2 ^ B^ 2 < g < c x ) ; (1.2.8) B^^-^ —— B^ ^^o;

(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^<C\\fl\\B^-\\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^)- E (^(-W^+M 2^

0<j<n n<^<-.n \ J /

N an integer greater than a. The proofs of (1.2.5) through (1.2.12) are immediate translations of the corresponding proofs in the real case (e.g. [3]).

The proof of the equivalence of the norms (1.2.4), (1.2.13), (1.2.14) follows as in [31], theorem (2.2), p. 180. The proof of (1.2.15), (1.2.16) follows as in the real case (cf. [25]), or as a consequence of (1.2.14).

We shall also need the Besov spaces of a product space given by (1.2.19)

B^{V, x V^g^g,) d^ B^(V^g,) 0 B^(V^g,)^

B^{^ x ^gi^g2) d^ B^(2li,^i) 0 B^(<^2),

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'

'

^

'

!!^^ ^^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 I g ^ ^ 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 • I I ^ H R C ^ I ^ ?

-"g.r ^oo^r <l,r

for a > 0;

etc.

A metric-covering of V is an assignment for each x 6 V of a metric g^

such that

(1.2.25) gx(y}<^ =^ g x = g x ^ y .

1022 S H A I H A R A N

Letting Ox denote the lattice correponding to g^, and setting 34 = x + Ox, i.e.

(1.2.26) ^ = { y e V \ g x ( 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) l l / l l B ^ ( 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 :

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<t-^(m,g)^B^m,g), q <, go ; (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)<l

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 / H L ^ ^ ^>. 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 - O C 2 l a 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

<S(Vi © V^) = S(Vz) 0<?(V2). Denoting by S^\g(V) the algebraic Schwartz space of locally-constant compactly-supported functions, we have proper dense inclusions

<?alg C C^ C S C S ' C C^ C S^

We note that <Saig^ C'oo,c? S, Goo, as well as their duals, are all stable under the action of the Heisenberg group given by p.

We have the Schwartz kernel theorems (cf. [33] for the similar proof over M) :

J Hom(Coo(2li),C^(2i2)) ^ C^ x Sli), f Rom(S(V,)^Sf(V^^S\V2(BV,).

(1.3.3)

An operator A on functions on VQ will be called smoothening if A : ^(Vo) —— S{Vo)

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)<pn^kN9x{y)

by (1.4.2), and hence Xj +p^o+feN . ^^ ^ ^. ^_ ^ ^ ^ Similarly, using

^ < ^ and (1.4.1), for j ^ J(k),

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)

^ V O l ( ]_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

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 - ^ T v + 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) <Cl'C^'m^xo)m2{xo)g^{l,x^-xo)f31'g^(l,X2-xo)f32

which by (1.4.5) is smaller than

Co • mi(.ro) • m2(a;o) • ^(l^i - xo)⁰⁰ . g^x^ - xof°

for some new constants Go, A)- Hence,

(i 4 10) ^ n To - Tn^

• ^

n r. T }-^ < r

l(

o) •^2(^0)

V-1--"-l u^ c/ r E l ^l5J-/2 ^O^ 9 x 2 \±^x^ ~XO ) 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) <?(V)=n5(m,<7),

m

where the intersection is taken over all temperate weights m, i.e. the topology of S(V) is defined by all the norms B°^^{rn,g)^ a > 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,^|

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 I I ^ ^ ^ M - ^ I / ^ - ^ + Z / ) !

a; ' ' a;,2/ '

\y\<\l,px\

^ sup [l^l-0. |/(a:)| + s u p |l,p^|-aH-^|/(r^)-/(^+^/)|

x x^y

M<1

"B^-> / 3 | - Q '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

{

(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)(

7>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)\

<Cf,. 11,^1°. maxdl.p^l-1.^!,^^

. f o r | 7 ? | < | l , p $ | a n d | y | ^ l .

103^ SHAI HARAN

For the norm associated with (1.4.20) for a fixed (3, we have sup ll.^l-". \f(^x)\

^cTL

^

l^ 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;),

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)^{i s} "Qitary

on Z/2 and commutes with convolution)

= 11/1 <S f2\\B'i''d'{-&,,) since ^PPfi c ^j

= II /i II B^ (a,) • ll/alls^ (a^- Usmg (2.1.2) we also obtain

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

(2.1.6)

{\\h#h\\B^)

^\\h#h\\B^,)

< 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

(2.1.7) f 2 i f y ^ Z p ,

m(y) - 1 < <

" " " l - l 0 i f y e Z p ,

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||^^^

f c i , f c 2 ^ 0

(by the definitions (1.2.4), (0.1.12) and Plancherel)

<h] ^ ^^•^^ll^^^^^^/i^^ll^^^

A;i,fc2^0

(by using (2.1.8) and (2.1.9), with any 7 > 1),

= 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

(2.1.11)

f 11/1 #/2 - /I • f2\\B^^j)

<\\fl#f2-fl'f2\\B^W

< Ca • h] • || A lla^-a ^ • ||/2|lB^y+«(^.) [ < Co • h] ' ||/i|[^+;y+a^ • \\f2\\B^+ot^y

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