ANNALES DE L INSTITUT OURIER - Centre MersenneThere are thus no differential operators over Qp. But...

ANNALES DE L’INSTITUT FOURIER

SHAI HARANQuantizations and symbolic calculus overthe p-adic numbersAnnales de l’institut Fourier, tome 43, no 4 (1993), p. 997-1053<http://www.numdam.org/item?id=AIF_1993__43_4_997_0>

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Ann. Inst. Fourier, Grenoble43, 4 (1993), 997-1053

QUANTIZATIONS AND SYMBOLIC CALCULUSOVER THE p-ADIC NUMBERS

by Shai HARAN

INTRODUCTION

We shall be concerned with functions / : VQ —^ C, defined in somevector space Vo OYer the p-adic numbers Qp and taking values in the complexnumbers C. One of the most basic problems encountered when trying toimitate the classical theory — where the domain VQ is a vector space over Ror C — is the lack of derivatives. Indeed, the derivation 9 / Q x is nothingbut T~^xF^ where T is the Fourier transform, and «a;» is multiplicationby the function x which is an additive homomorphism from VQ to C; andthere are no such homomorphisms from Vo to C when VQ is a vector spaceover Qp.

This problem repeatedly makes its appearance in various disguises;for example, given a unitary representation of a p-adic analytic group ona Hilbert space, one cannot associate with it the derived representation ofa 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 showin this paper, there is a meaningful theory of pseudodifferential operatorsover the p-adics, which parallels the classical theory over the real numbersR. In fact, the theory over Qp is better behaved than the one over R, inas much as in all estimates the numbers 2 = [C : R] for the reals can bereplaced by oo = [Qp : Qp] for the p-adics, and one encounters here thephenomenon 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 viathe action of the Heisenberg group (cf. 0.1.10). The composition ofoperators correspond to «twisted multiplication)) of their symbols :p(/i)p(/2) = (/i#/2)- The main goal of the symbolic calculus is to controlthe twisted multiplication over appropriately defined symbol classes, thusenabling the comparison of the resulting function algebra and operatoralgebra. Classically, the symbol classes are defined using the Mihiincondition which incorporates the global growth condition with the localsmoothness condition on the symbols. To define symbol classes in the p-adicsetting we encounter once again the lack of derivatives. Fortunately, thistime the problem can be overcome if one uses in place of derivatives a Besovcondition.

Chapter 0 is devoted to the Heisenberg group and its fundamentalrepresentation. 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, wefix our notations and recall the Schrodinger model, the Weyl quantizationprocedure and twisted multiplication. In paragraph 0.2, we review thelattice model and the Bargmann isomorphism [I], and we describe theWick symbols [4]. In paragraph 0.3, with an eye towards connection with[14], [15], we offer a third realization of the fundamental representationin 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. Inparagraph 0.4, we give a «mannerized)) version of the lattice model, andwe 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 spacesover 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. Inparagraph 1.2, we explain the problematics of these — the lack of theLeibniz rule for differentiation, and offer a way out via Besov spaces.After reviewing the elementary properties of the ordinary Besov spaces, wepresent 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 theoremsand define our symbol classes (in analogy with [18]). In paragraph 1.4, wecollect 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 realsin [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, whichcan be defines as the intersection over all a of E0, is different from theusual (algebraic) Schwartz space of locally constant compactly supportedfunctions that one encounters in the literature, though it shares much of itsnice properties such as nuclearity, closure under multiplication, convolutionand 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 rangeestimate (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 andprove the main theorem of the calculus (2.2.8). In paragraph 2.3, weestablish the continuity of our operators on S' and S", (2.3.1), and theirboundedness on L^ (2.3.9), and give a proof of the Calderon-Vaillancourtestimate [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 theimproved Fefferman-Phong inequality (3.1.9) (cf. [8]), as consequences of thewave-packets theory (cf. [7]). In paragraph 3.2,, we show that every ellipticsymbol 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-adjointoperators with discrete spectra (3.3.1); study their complex powers (3.3.21)(in analogy with [28]); and determine the precise asymptotic behavior oftheir eigenvalues (3.3.14) (see [10], [13], [17], [20], [27], [30] for some of theliterature 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 < —dimVo? 8Ln(^ trace-classfor 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 explicitlydiagonalizable 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 corollarywe see that if / ^ 0 is positive, then the operator p(/) is positive modulotrace-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, thenp(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) hasa discrete point spectrum tending to infinity and there is an orthonormalbasis for L^Vo) consisting of p(/)-eigenvectors all of which are in theSchwartz space. Assuming further that both / and p(f) are positive, wecan defined their zeta functions

CpCoO^tr^/)-8) and ^(s) = [ / ( x ) - 8 dxJv

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 followingprecise estimate :

N(\) = vo\{x I f(x) ^ A} + 0(A6), for any e > 0.

Our motivation in developing the theory of pseudodifferential operators inan arithmetical situation was derived from [14], [15], where it was shownthat such a theory could have bearings on the problem of the Riemannhypothesis. Much of what we do here carries over to the global situation ofthe 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 caseof an arbitrary local field. Other future applications of our theory couldinclude the local Howe's duality conjectures. The theory can also be madeto work over varieties where it might be useful as a tool in defining the zetafunctions of such varieties by analytic means (a basic ingredient for tryingto do Tale's thesis for such varieties). Since we wrote this paper, we havefound some attempts to construct quantum mechanics over the p-adics inthe literature of physics, e.g. [42], but these are always wrong and lead tostrange phenomenon (e.g. the existence of two vacuum states), because thephysicists were insensitive to the basic difference between the p-adics andthe 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 byits Loo analog : \x,y\ = max(|a;|, \y\).

In the one dimensional theory, VQ = Qp, V = Qp © Qp, we can takeas 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 representationon 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 amaximal compact subgroup, obtaining quantizations which are relevant tothe 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 thecomplementary series. Again, these constructions could be carried in anadelic 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 guestof the department of mathematics at Cambridge, England. Special thanksare due to Professor John Coates who gave constant encouragement andsupport.

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 tod

^(x ' y) = {Y, X i ' yi), so the Fourier transform1=1

^d^y) = j (p(x)^(xy) dx

satisfy J^d^d^W = (-^). The symbol 0 will denote the characteristicfunction of Z^, so that T^ = (/). Weshall usually assume p ^ 2, and leaveit 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 xQpthe associated Heisenberg groups with multiplication law given by

(0.1.2) (î)-(^2)= (î+î+^2 + jî,^)), viev, êQp.El is two-step nilpotent group, with center Qp, and the Stone-von Neumanntheorem states that it has a unique irreducible representation withcentral character ip (up-to-equivalence). The Schrodinger model for thisrepresentation, p : HI -> (/(^(Vo)), is associated with any completepolarization V == VQ (BVi, Vi Lagrangian subspaces. Fixing coordinateswe 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 throughthe 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-sectionv ^ (zÔ), 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 pinduces a unitary isomorphism from the algebra L^(V), \\ onto the algebra


of Hilbert-Schmidt operators ^(L^Vo)); hence it takes Li(V), \\ intothe 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 quantizationprocedure 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 # isgiven by

(0.1.11) /i #/2^) = (^/l /2)M

= yy dî d^/i (^+î) /2(^+^2)^(2(^1,^2))=^(W))/1^/2(^).

Here /i (g) /2(î^2) == /i(î) • h{v^}, and ^(Q(D)) is the unitary operatoron L2(V C V) = 1/2 (V) 0 V2(^) given by

(0.1.12) ^{Q{D))=^^^^{Q{v^v^)J^^^

with Q(-ui, ^2) = ^ (î, V2) the quadratic form on V C V associated with thesymplectic 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-^^îx - x,)) {y - 2/0)),ô(^) = ^ - o)) ^(p-^y - yo)) {(x - xo)(y - o)),

a straightforward calculation gives :

/r. -i -<^ \ ( <I>al'bl • Cb^,^(0.1.16) -'-^1,2/1 -*-a;2,2/2

= i ' y\ if ^1 + h + 02 > 0,

^;^ #(^ = ^ î;^ • S if ai + 62 ^ 0 > &i + 02,

i f a i + 6 2 < 0 ^ & i +02,^015^2 . /ha2,^lxi,y2 -*-a;2,2/i

âi,fc2 . $02^1< a;i,2/2 --^2,2/1

For a lattice 0 C Y, we denote the câ^ 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 are0-locally-constant, i.e., they factor through V/0, then an easy calculationshows that /i # /2 == fi' /2, i.e., twisted multiplication reduces to ordinarymultiplication. More generally, if V = ]J 2^ is written as a disjoint union

jof translates of certain lattices, = Xj + Oj, Oj D OJ, and /i,/2 areconstant on each Si^, then again

ifai + & 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-duallattice 0° = (0°)^ such that C^ C 0° C 0. The representation p can berealized in the 0-lattice model

(0.2.1) p ^ p^ indô^Q/Q = ind^(po)

where -0 is viewed as a character of the abelian subgroup 0° x Qp/Zp CHI/Zp, via ^(v,t) = -0(t), and po is the representation ind^Q Wof 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 pin L^V)^ where the subscript -0 indicates that we are looking on thesubspace 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 thecoefficients 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 issupported in v + Oo, and normalized by ^v{v) = 1; the functionsB^>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)^ isgiven 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 /, weobtain the operator P^Mf on L^(V)^ having anti-Wick symbol /; but aneasy 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 algebraLoo(y/0o) acting on L^ (V)^ by multiplication.

1006 SHAI HARAN

For any bounded operator T C B^L^V)^)^ we denote its Wicksymbol 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 areeasy 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 onL2(Qp) and ^(QpCQp)^. Choose e C Z;\(Z;)2, and identify V ^ Qp[v^Lthe unramified quadratic extension of Qp, via ( x ^ y ) <-^ x 4- \/^y\ thusOQ = Zpiv^] is the ring of integers of V, and the associated norm isgiven 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


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 Hertmitianform 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 theabove identification, by any of the following metrices

f^le^wl,6ev/ipw2)=r if^Ûpwi-pW2| if $1=6,

^ ^ ^ where £/i ^ ^p+i x exp(y/ipZp);<î, ^2) = lî/î - /^h wheret/i ^ V/Q^

î:î^2:i/2)=T^2—^^. where î^P^Qp).lî, 2/i| • |^2,2/2|

The group Sp(V) = SL2(Qp) acts on HI as automorphisms inducingthe 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), themetaplectic 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 theterminology of [22] : ((7i, U^) is a dual reductive pair. The metaplectic groupsplits 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 ofunity, 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, andwrite 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 £/iaction 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-invariantN>0

vector. For ^ e Z^.N > 1, the Hermite function is the uniqueX-covariant vector (up-to a constant of absolute value one), and is givenexplicitly 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 theorthonormal basis {p{x,y)(f) \ x,y e Qp/Zp} of the lattice model, wenote 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 viceversa. 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^•^ô)),

with u € U\ and vo ^ Oo.

Using (0.3.14) we obtain the Tceplitz model pr : 1HI -^ [/(^(î)-^),where L2(^7l)+ C LÛi) is the subspace corresponding to (^+) ^ ^(^)under Fourier transform L^ (£/i) ^ ^(^). The orthogonal projectionP' : 1/2(^1) ^> 1/2 (î)"^ is easily seen to be given by the singular integral

(0.3.19) P~^f(uo) == p.v. / yuf(u)6{uûo)-\Jui

where 6{u, uo) == (—p)"^ if 6(u^ uo) = p ~ N .

If / € C{U\Y is a continuous non-vanishing function, the associatedToeplitz operator P^Mf is Fredholm of index zero. Indeed, any such functionis homotopic to 1, the constant function 1, since U\ is totally disconnectedand compact. This can be partially explained by considering the problemfrom a dual point of view : if \ e Z^, translation by \ followed bythe projection P+ : ^{Z} —» <2(^+) induces a Fredholm operator P?ôn £^(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 thecase of a totally ramified quadratic extension of Qp.

1010 SHAIHARAN

0.4. Wave packets.

We can consider more general coefficients of the representation p thanthose of (f>. Thus for (î,y?2 € L^Vo) let c^^(v) = (^2,^)^2). AfterFourier 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) î^^^^L^V) = (^1^3)L2(Vo) • (^2,^4)L2(Vo)

showing that p is square integrable modulo the center. The Schwartzinequality gives

(OA3) II^ÎlL^ <\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) • î.

Hence Cy,,^ tic^,^ = (^2,^3) • c^,^.

Taking the symplectic Fourier transform of Cy,,^, we obtain afterFourier 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 ,

Û.4 . / J î,^ = = c<^2,<^l•


We define the Wigner transform by W^p = c^^ e L^(V) n Co(V), andby (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) Wî = W^ ^=^ <Pi=\'^2 for some A, |A| = 1.

Note that W(p{^x) -^ 0 implies that there exists z such thatx ± 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^ ^ I I^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^^ = ô = characteristic function of OQ,

(0.4.16) W(p(vo)^)(v) = W^{v - vo),

(0.4.17) W{n^){v} = WÂ^v), A G Sp(V), by (0.3.7).

Since Sp(V) acts transitively on the collection of self-dual lattices, wededuce using (0.4.15), (0.4.16), (0.4.17) that for any translate of a self-duallattice 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 aconstant of absolute value one.

For any unit vector (po e L^(Vo), the map Cô(^) = c^ô 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 superpositionof the p{v)(po's

(0.4.18) (p{x) = dv {(p, p(v)ô) • p(v) o{x).

For / 6 Loo(V) n Li(V), denoting by Mf multiplication by /, we get theoperator C^MfC^ on L^Vo). We have

(C^MfC^^^L^Vo)

= (MfCî,C^2)L2{V)

= dv f(v)c^^(v)c^^{v)

= ^ d^/(^)((î,/9(v)ô) • (^2,P(^)^0)

= [dvf(v){c^^W(p{v)ô)) by (0.4.2)

= ^d^ /dî /(^) ,,^(î) Wô(î - v) by (0.4.16)

= ^ dî/*Wô(î)^,<^(vi)

= dv^^f W(po)(-v)(^,p(v)^2)

= dvJ:'(f^Wô)(v){p{v)^^-2)= (p(/*Wô)î^2).

(0.4.19) Hence : C^MfC^ = p{f * Wô).

(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 aunit 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 thecharacteristic function of 0^ * j = <1>^, and for any locally constantcompactly 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 ofthe 21^'s, so that / = \j ' <^-, we have as in (0.4.21) : p(f)^pj = Aj • (/^,

3i.e., p(f) is diagonalizable with respect to the orthonormal basis withassociated 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 uniquesimple pole at s = 0. We denote by I0' the operator of multiplication by thefunction

(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 givenby 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 haveJ0' * J ^ = J^^. For real a > 0, the J^'s form a semi-group of probabilitymeasures 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 commutingoperators J^, Ja = J ^ ^ l I a J ^ d 5 J^ forming an entire family of distributions,J^ * J73 == JQ;+^; for a > 0, J^ being a probability measure supportedin 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 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]) aredefined 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 domainsuch 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 theysatisfy 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 lL^II/ l l^+ll-R- ' /I lL, .

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 -02> jd, /3i-/?2> ^.

1016 SHAI HARAN

Recalling the Bargmann isomorphism B : L^(Y)^ -^ L^Vo), whereV = 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, forRe a > 0, and hence they admit the lattice basis {p{v)(f) \ v C V/Oo} aseigenfunctions. In particular, the operators J^J0', and K^, are trace classin 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 moregeneral « 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.


Recall that such metrics correspond bijectively to lattices 0 C VQ, thecorrespondence 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 dualvector 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 identifiedwith 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—lI, 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 affinetransformation (i.e. given g^,g^ on V, there exists A C GL{V) suchthat 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 unlikethe real case, where we have ./-2N ==(!-+- 92/9x2)N, we lack in the p-adicsetting an analogue of the Leibnitz rule

f9^^ f\ V ( N \ ( 9 V f f9^'Û •^-Ê^JU^'U h

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

(i.2.i) 0/1+2/2)^= E (^yi-y^3-0<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 writefor Re a > 0 and yi € Qp,

(1.2.2)

«27T/logp/.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+')C(-^)XTÎ

x r710^ • . xî) iz/ii^2^ • x^) iî072-^Jo 27r

where ^ varies over all non-trivial characters of Z* but the sum ^ isx

actually finite for ^1+^/2 7^ 0. Thus, (1.2.2) like (1.2.1), expresses \y\ -^-yâs «sum» of ^1(2/1) • X2(^/2), with î • ^2 = | I0.

To overcome this problem we introduce the Besov spaces B^ (orLipschitz spaces) of functions that are well approximated by locally-constant-functions. We shall use the following notations : V is a Qp-vectorspace 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 characteristicfunction of p~k0", k ^ 0;

^g,k = 0<7,fc - g^k-l for k > 1, (Rgô = (f)gft ;

dgX = Haar measure in V normalized byd^(0)=l;

(1.2.3) < L,=L,(V,d^);

d^ - C(c0 • ^Q/)^ • d<^, ^ = dimY, sothat d^y is the «multiplicative Haarmeasure)) 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. TheB" = B°(V,g) are Banach spaces with the following basic properties :

Embeddings

(1.2.5) B^,

(1.2.6) B^

%.„ n^ra;

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^ ^ô;

(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'aâo ^ 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 continuousfunctions.

(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}, andsimilary we can define B^(^,g) for any domain such that 51 + 0 = 21.All the above properties remain valid for the spaces B^^g). When21 = x + 0, we shall write B^(2l) for B^(2i, ).

(1.2.18) Remark. — The only discrepancy from the analogous spaces overthe real numbers is (1.2.14); for the reals, one has to use instead of thedifference Ay f(x) = f{x) - f(x + y ) , the TV-fold-difference

^/c^)- E (^(-W^+M2^0<j<nn<^<-.n \ J /

N an integer greater than a. The proofs of (1.2.5) through (1.2.12) areimmediate 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) followsas in [31], theorem (2.2), p. 180. The proof of (1.2.15), (1.2.16) follows as inthe 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, satisfyinganalogous 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 assertionsof (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 : BÎ^V x V^g^g) —. B^^g)

are continuous, where Res denotes the restriction to the diagonal. Thespaces in (1.2.21), (1.2.22) are spaces of continuous functions, and we canview the ^-functions as elements of their dual. We have

(1.2.23) l l^ci^l l fR^2^/2^ = ll^ci^ll^-^2 '-^2 = 1-' • 2 , 1 / 2,oo

We can also define the weighted Besov space B^(V^g) given bythe norm

(1.2.24) H/llî, = ^ll^^^*/)^)!!^!!^ - \\I^f\\B^

noting that I g ^ ^ g ^ * /) = ^g,k * (-Â since supp^,fc c 0.B^/ will satisfy the analogous properties, e.g.

. interpolation [B^^B^1], = B^^ as in (1.2.9) withf3 = (1 - 0)f3o + 0 • /3i;

. duality (B^^B,^^

. lifting :B^°-^^°^;

• pointwise multiplier ||/i •/2 | |Dc.i /3i+/32 < C'| |/i | |D^lî • I I ^ H R C ^ I ^ ?-"g.r ôo^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 SHAIHARAN

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 formj

^ = ^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 someconstant 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 thestart 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 / l lB^(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 definedby 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, forany 1 < q,r < oo. In particular, we see that Coo(O) is an algebra underpointwise multiplication, consisting of continuous functions, and that it isa nuclear space.

1024 SHAIHARAN

(1.3.1). — For any open set 21 C V, we denote by Coo (21) the Frechetspace defined by the norms H/Hz,0 (rc+o)? o ; > 0 , a ' 4 - O C 2 l a translateof a lattice contained in 21. Again we can use the norms associated withL^x-^-O), or with Ba(x-^0)^ and Coo (21) is a nuclear algebra of continuousfunctions. In particular, we have Coo(2li x 212) = Coo(2li) ^> Coo (212). Justas Coo(VQ = lim Coo^), the inverse limit taken over all lattices 0 C Vwith respect to the restriction mapping Coo(Oi) —^ €00(02)? Oi 2 02?we can define the space of compactly supported smooth functionsCoo^c(Y) == lim Coo (0), where now we take the direct limit with respect tothe mappings Coo (02) Coo(Oi), Oi 3 Oii given by extending by zero.Coo,c(Y) is similary a nuclear algebra of compactly supported continuousfunctions.

(1.3.2). — We define the (analytic) Schwartz space S(V) to be theFrechet space defined by the norms H/H^ 1 ^ , Q^/3 >. 0. Equivalently, wecan use the norms associated with L^, a,/? >: 0, for any g; or thenorms 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 anymetric g on V. S(V) is a nuclear algebra of continuous functions. Fixingsome 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 Schwartzspace of locally-constant compactly-supported functions, we have properdense 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 stableunder the action of the Heisenberg group given by p.

We have the Schwartz kernel theorems (cf. [33] for the similar proofover 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 obtainedfrom 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 thenorms ||/HB^ ^(m,^)? <^ > 0. These norms are given in an equivalent formin (1.2.39), so that / e S(m, g) if and only if for some constants C, Co, wehave

(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 notdense, hence a continuous linear form on S(m^g) is not determined byits restriction to Coo,c; such a form will be called weakly continuous if itsrestriction 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 witha metric-covering , corresponding to V = [J2lj, 2lj = Xj + Oj, where g^

jis 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 dualmetric, 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,î - ^^•^(lî-o-o)^1

and similary with x\,x^ interchanged, from which we obtain

(1.4.4) (m -ô) < 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 < pô . 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 +pô+feN . ^ ^ . ^_ ^ Similarly, using^ <^ and (1.4.1), for j ^ J(k),

9x(x - x,) g^(x - x,) p^ • g^x, - x)^ < p^î^

and hence C a: +p-ô+fc(^v+i)] . Denoting by vol the Haar measurein 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^) .pô+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-^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^ - • , (lî - -/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(î) < 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^) .^(lî-ô)^.

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) • (lî - xo)00 . g^x^ - xof°

for some new constants Go, A)- Hence,

(i 4 10) ^ n To - Tn^0 • ^v n r. T }-^ < r ml(:co) •^2(^0)V-1--"- lu^ c /rEl^ l5J- /2 Ô^ 9 x 2 \ ± ^ x ^ ~ X 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), andhence 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 estimategives %,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. thetopology 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 westudied in paragraph 0.2, basic metrics are the Toeplitz metric and thepseudodifferential metric which we next describe. The Toeplitz symbols aredefined 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 itis 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,î -xo).

This follows from the basic inequality

(1.4.14) |l,2/o|<|lîHlô-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 II^^^M-Î/^-^+Z/)!a; ' ' a;,2/ '

\y\<\l,px\

^ sup [l^l-0. |/(a:)| +sup |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 < 0consists of smooth functions vanishing at infinity. Letting E~00 = Q E"

adenote the Prechet space defined by the norms of all the E^'s, we haveby (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 • I T / I . I I / I } ,

m^^ll,?^,

where (^, a;) and ('rj, y ) refer to the symplectic coordinates associated withthe complete polarization V = VQ (B V\. Again it is easy to verify that g^xis a metric-covering, certain with h(^,x) = |l,p^|~1 = m"1^,^), and thatit is temperate with in fact a stronger inequality

{ ^(soô^'y) ^ ff(Sl,^l)(77>y) • |i,p(ô -î)|,(1.4.19)

|1,P($0 - $i)| < ff(^,.i)(lô - $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.^!,^^.for |7?|<| l ,p$|and|y|^l .

103^ SHAI HARAN

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-dimensionalsymplectic 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 havefor .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 "Qitaryon Z/2 and commutes with convolution)

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

= I I /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^â,) • II Vg,^ * /2|lB^^(a,)

=Pfcl'dlly5„fcl */ilk(a,) •P^'^Vg.M */2||L2(a,).

Multiplying (2.1.3) by pCî+â^ and taking ^r-norm, we conclude that

(2.1.4) ||^(Q(Z?))/i ® /2||^,â,,,) ^ 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, sothat £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 • /2we 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;(^)=^ ^Xî)-<^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^^^^^-Wiî'^))-l)^/i ®^/2||^^^fci,fc2^0

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

<h] ^ ^^•^^ll^^^^^^/i^^ll^^Â;i,fc2^0

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

= U ' I I /I I I 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 differentfrom j. We recall from (0.1.11) that we can write V?(0(D))/i (g) f^ also as aconvolution

(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^^Fthen J^ acting on S^V), preserves by (2.1.14) C^(5l^), the distributionssupported 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 sameproperties.

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

Putting,

SHAIHARAN

-1.20) ^2^)= min ^(1,^-^)(2x ^•"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®/2î,a-2)

= 9n (^2, S^)-^ . V>(Q(I?)) [^Â ® /a] (a-i, 2).

Similarly, using 5^ and J^, and the analogous formula to (2.1.19), wehave

(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^lî,^)!< 9j^W0 • g^x,,^)-'3

• \\^QW)J^f, J^MB^V^^(2.1.23)

^ 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^(^)-


This, of course gives an estimate for ||/i # h\\L^{^), but to estimatethe Besov norms of /i -#- f^ we need a further argument since we havedifferent metrics and . We remark that since we have, by (1.4.2), forany xi e

(2-1-25) Q^^gM'p^'gUx^y

an easy estimate gives

;.i.26) sup QAyY^Q-^^f^B^v^(29j{y)<^-

< fi • fnnooiny ( ^ r - 9( \No. \\ r [|-^ ° P Q^^z^î) '1^1'B^+^(V,^•.)•

Hence from (2.1.24) and (2.1.26) we deduce, since ^(Q(D)) commuteswith translations, that for x e 2lj, ft' > f3 + d, we have

f SUp ^)-a|/l#/2^)-/l#/2^+^|| 9 o (?/)<!

< sup gAyy\^QW)h^{6Q-6,y}^f^x^x)\93 (?/)<!

+ sup gj{y)~a^(Q{D))(6o-6y)^f^f^x^y^+y)\9j (?/)<!

<C.^(^^)-^(^^)-^||A||^,^^^

\.su)p<l^(')-QII?~^)*/211<-(^^•(^)î+C.^(^2l,J-^(^,^)-P

(2.1.27) • s u p ^r'liô-^)*/!!!^ ^ , . ) - i i / 2 i i ^ .,.g - ( y ) < ^ l "-"oo,^^?^!; -DOO, 00(^2)

< 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^ ^-'-'ocôo Ji ^ -Doo,oo V-aJ2 I

-\- C' a ' (r 21- "l""^^ fr 21 't-/3<^v fr 91 • ^7VQ!-r ^ y^ , 2 / y^ V^ ji ^•i 5 ji;

'^ÎB^+^^^'II^HB^^^)

< (777 • n^ (r 21 "l"^^ fr 91 • ^"^^ f^ 91 • }Nanv (v 9f â-^ ^ yji^^J y j ^ ^ ^ j i ) i^^l\x•>^^l) 9 j 2 \ x ^ ^ a j 2 )

' lljlllD^+côf. \ ' ^HD^'+a.oj ^-"oo.oolî; -"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 theweights 7711,7712 are temperate, i.e. they satisfy (1.4.8). We let /^) denotethe restriction of fi to 9lj, and we note that the partial sums ^ /^)

j<M

are bounded in S(mi,g)^ and they converge to fi in Coo(V), since they areultimately equal to fi on any compact set. Letting /3o denote the constantin (1.4.10), we deduce from (2.1.24) that for any f3' > /3 + d + /?o,

(2.2.1) |/i,o,) #/2,(^)|<C.^(a:,^)-^(^,^)-^.mi(a;)m2(a;)

• IKVi^JI^^) • || 2 'fw^B^^y

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 subsetof S(mi,g) we deduce that the form /i (g) /2 ^ /i#/2 defined onCoo^^Q^Coo,^^) = Coo,c(^®^) extends uniquely to a weakly continuouslinear 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) dependingonly on 77^, g and a (and independent of re); letting (3a denote the associatedconstant in (1.4.10) (with mi replaced by m^), and using (1.4.5), we deducefrom (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^^)!!^^^,)-


Taking again a /3 satisfying (1.4.7), we conclude that with some newconstant 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 amap

S{m^,g)^S(m^g) —> 5(mi -m^g).

Moreover, since the function x ^ h(x) of (1.4.9) is temperate, and sincefor 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) inthe right hand side of (2.2.4) by W ' mi • m^x), for x ^ 21 n 21^, anda posteriori we can similarly replace m\ ' m^ by W ' m\ • ms in (2.2.5) ifwe 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 weaklycontinuous 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 mapy

fi /2 '—^ fi # /2 — f i ' /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 thespaces S(Vo) c—^ L^(Vo) c—^ ^'(Vo), for / e S(m,g), g^m temperate. Wefirst remark that since by (0.1.13) p takes complex conjugates to adjoints,the continuity in <S'(Vo) follows from the continuity in «S(Vo)- Secondly, wenote that the Bargman isomorphism B~1 : L^Vo) -^ L'z{Vo)^ induces anisomorphism B~~1 : <S(Vo) —^ S(Vo)^ = S(V) n L^(V)^ onto the subspaceof S(V) consisting of ^-holomorphic functions. By the description of S(V)given in (1.4.11), and the description of B~~lp(f)B given in (0.2.12), wededuce from our main theorem (2.2.8) the continuity in S(Vo). Thus wehave :

THEOREM (2.3.1). — -For a temperate certain-metric-covering g^ anda temperate weight m, the operators p(f) are continuous in <S(Vo)? ^d inS^Vo)^ for every f G 5'(m, g). Moreover^ the map f \—^ p(f) induces weaklycontinuous embeddings

(2.3.2)S(m^) Kom{S(Vo)^(Vo)) = S(Vo) ^(Vo),

S(m^) Hom^^Yo)^'^)) = (Vo) 5(Yo).

For the continuity on L^(Yo) we take / G S(l, g) and write / = fii

for the corresponding decomposition of / with respect to the metriccovering 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 easilyobtains that for some constants Go, M, and all x G V

(2.3.5) ^(^.^^min ^(W - V " } < Co • g^x^ . g^x^.y ^iy " ^ j


Putting (2.3.3), (2.3.4), (2.3.5) together we obtain, with some newconstant 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,)-^<oo.i 3

From (2.3.6) and (2.3.7), we obtain by applying the lemma of Collaret 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, anda temperate bounded weight m, the operators p{f) are bounded on L^VQ)for every f € <S'(m, g). Moreover^ for some (7,7 we have

(2.3.10) \\P(m^<C-\\f\\B^(m^

If m vanishes at infinity^ the operators p(f) with f € *S'(m, g) are allcompact.

Proof. — The inequality in (2.3.10) follows immediately from (2.3.8),since i f m < C 7 , then ||/HB^(I^) < C • ||/HB^(^^)- e compactnessassertion follows since the operators p(fi) are all compact, and by (2.3.8)

11^) - E )|| c. ||i>L_^ < c • \\f\\B_^ • which converges to 0 as j —> oo by assumption. D

(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)^Pin L^(Yo). Indeed, we may assume / = 0, and in view of the uniformbound (2.3.8) we can take (p € <S(Vo)? then by the weak-continuity assertionof (2.3.1), p(fj)^p is bounded in 5(Vo), and converges to zero in <S'(Vo),hence a posteriori in <S(Vb) and inZ^Vo)-

The inequality (2.3.8) can be sharpened in various ways, one ofwhich we describe next. For a self-dual lattice 0 = C^, denoting by <I>

1040 SHAIHARAN

its characteristic function, we know from (0.4.15), (0.4.17) that there isa unit vector yo e L^Vo) such that Wyo = $ = $. If / is such thatsupp/ C 0, then by (0.4.19), we have p(f) = p(f * $) = C^MyC^,, andhence ||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 toV = ]_[2lj, 2lj = Xj + Oj, Oj = O], letting $j denote the characteristic

function ofSlj, we have for any function /

lollop-|KE^*/)|L,<Ellw*/)llopÊii^^ôoîi/ii^^,).

j

Here the right hand defines a norm for a Banach space B^ ^(g)analogous to a Besov space. If for example ^ = g is independent°°of x,^.^ g = g^ is a self-dual metric corresponding to a lattice 0 C V andy = U (x + ) is the associated covering, then an easy calculation gives

x€V/0 °

ll^*/!!^^^!^)-^!!^-0/!!ând since ^ ^(1, )-a = ^(a - 2d)/C(a) is finite for a > 2d == dim V

xev/o 'we obtain

(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 fvanishes at infinity, then p(f) is compact.

For a ^-locally constant function m, we define the operatorJ^ = p(m-1); if m > 1, it is bounded on L^Vo) by (2.3.8). We define them-th Soboleff space by

(2.3.14) L^(V,) = J^(W)), |M| ^ = \\p(m)4^.


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 seethat J^ = J ^ 2 ' (J^Y is positive, and J^J^~1 = id, so that J^ isinvertible. 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 temperateweight 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 in5'(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 tov = 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 choicefor 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. Fora function / in V, we denote, as usual, by fj the restriction of / to 21 , andwe 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), forany 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 <?(Vo).

Thus we have obtained the following

COROLLARY (3.1.7). — If f e S{m,g) is positive, f > 0, thenf = 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 <E S(m,g) is positive, then for any 7

(^^^^^^-^•MIL^-)^forallyeS(Vo).

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^, whichis 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 allêS(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 inS(m, g), we have for any 7 :

M/K/W)^ > A. IMI v^ -c. M^^^yProof. — S e t / 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)Ô^/)^ ^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 = Uîji corresponds to a self-dual metric_ _ JZ

covering g ^ with g^ ^ g ^ ^ ^. For any / G S(m,g), denoting byfjz = f^ji) the average of / over ^-^ the operator p(f) is diagonalizablewith respect to the orthonormal basis (^ e ^2(^0), W^jz = characteristicfunction of , cf. (0.4.20) : p(/)(^ = /„ • . Thus we have «almostdiagonalized)) p{f).

1044 SHAIHARAN

3.2. Fredholm operators.

By the Stone-von Neumann theorem, p : L^(V) —^ ^2(^2(^0)) is anisometry onto the ideal of Hilbert-Schmidt operators. Since an operator istrace-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 boundedthen p(f) is trace-class, so we can use the criteria for boundedness ofparagraph 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 providedeither 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 boundedoperator for some metric go in VQ^ and some ai, 0:2 > d.

For example, using the first criteria of the lemma, we see that theToeplitz 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 tothe 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^)l9x{y' \fKW-fK^+V)\- \f^f(x+y)\

<A~2 •Ca •m(x)~1,


we see that f~^ e 5'(m~1,^). By the main theorem (2.2.8) we get,

(3.2.5) l-f#f^\ l-f^^f^S^h^g) for all 7.

If h vanishes at infinity we conclude from (2.3.9), (2.3.15), that p(f)defines a Fredholm operator : L^\Vo) -^ L^(Vo). Assume now that gis global, in the sense that we have

(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 temperateweight m,

{ m{x) < Ci . m(0) • (1, xf1 by (1.4.8)

(3.2.7) <C2'go(l,xf2 by (1.4.3)

<C3'h{x)-^ by (3.2.6)so that n S(mh^,g) = ^ S{h^,g) = S(V) by (1.4.11).

'y>0 7>0

To calculate the index of p(f), f e S(m,g) elliptic, we may assumethat m = 1 by the logarithmic law for the index. So let / C S'(l,^),\f{x)\ >_ A for x ^ K, and assume that (3.2.6) holds, so that

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), withfo^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) - f^ #/(^)).j v ^\J -tt-JK \^) ~ fK1

But since / and f^1 are ^-locally-constant, we have f # f^1

/ p 1 #f = f ' /K1^ an^ we have proved the following

THEOREM (3.2.9). — For global p, every elliptic f e S(m, g) defines aFredholm 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) = <S(V), for any temperate

7>0

weight m. Let / e S(m,g) be a real valued elliptic symbol. By (0.1.13),p(f) is formally self-adjoint. From (2.3.15) we know that p(f) is boundedas an operator from L^ô) to L^(Vo). Let us denote by A the unboundedoperator on L^(Vo) defined by p(f), with domain L^^Vo).

Note that we can write f^#f= I-RQ, with RQ <E S(V) as in (3.2.5),hence if (p e L^ (Vb) is any element such that p(f)y e ^2(^0), then

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 ofp(f) in ^'(Vb), we deduce that the operator A is closed. We claim that A isself-adjoint, and to prove it we shall show that A equals the closure of itsrestriction to <S(Vo). Let ^k denote the partial sums of the characteristicfunctions of the sets 2L, corresponding to the metric-covering g. The <I>âre a bounded sequence in S(l,g) covering to 1 in C^(V), and by (2.3.11)we have for any <p e L^(Vo), p{^>k)^ -^ ^ in ^(Vb); moreover, since^k ^ S(V) the operators p(^/c) are smoothening, and so p(^/c)^ e ^(Vb).For^G47 n )(yo)wehave

Ap(^ = p(f #^k# f^W) + P(f # # -Ro) .

The / # k # fK\ and / # # J?o, are bounded in 5'(1, ), and convergeto f#f~K1^ and to /ô, in Coo(y), respectively, hence by (2.3.11) wehave

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 hencefor some real number Ao, {A - \o) is invertible. Its inverse (A - \o)~1 isa self-adjoint injective operator with image L^^Vo). Assume now thatm~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 orthonormalbasis {(^} for L^Vo) consisting of eigenfunctions for (A - Ao)~1 , with realnon-zero eigenvalues converging to 0. Hence the 0/s are eigenfunctions


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 theabove 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 anessentially self-adjoint operator p(f). Denoting by A the unique self-adjointextension of p(f), Domain(A) = L^^Vo), and iff>.0 is positive, A isbounded from below. If moreover, m~1 e Co(V) vanishes at infinity, then Ahas 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 \jconverging to +00.

To study the asymptotic behaviour of the eigenvalues \j, we mayassume that f(x) > c • m(x) > e > 0, and moreover that minAj > e, sincewe 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 righthand 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

(3.3.5)

' sup | (/-A)-1 (a;) | <, |c-mj-A|-1 <. |e-A|-1,X^j

sup g,{x, - a;2)-Q|(/ - A)-^^!) - (/ - A)-^)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^llôoC^) ^ c^^ and similarlyô.o.t) Ûl^llB^^^-y^) ^C^.

From (3.3.6) and (3.3.7), combined with (2.3.8), and with (2.3.15), wehave as |A[ —> oo, Re(A) ^ î,

fllw-ArîLôdAi-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 dA2^ ^,-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^/ ^(/-^-Âe^m^)^).-" l' J e^—ioo


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 remarksafter (1.3.3).

Combining (3.3.10), (3.3.11), (3.3.12) with (3.3.4) we conclude thatfor Re(s) < 0

f A8 = p(f^8) with f^8 = /s +r<^ € ^(m1^,^,(3.3.13) {

[r^ C<?(V).

Since for 5 an integer (3.3.13) holds with f : ^ s = f # - • ' ^ f , s times, wesee that (3.3.13) actually holds for all s € C. Thus we have proved thefollowing

THEOREM (3.3.14). — For a global certain temperate metriccovering (7, a temperate weight m with m~1 € Co(V), and a positiveelliptic symbol f € 5(m, 5'), denoting by A the unique self-adjoint extensionofp(/), and assuming that A is positive^ A generates a holomorphic groupof 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 functionof 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 therecursion 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 symbolsin S^m"5,^) correspond to trace-class operators for s > 2d/OQ. Thus wecan 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 , < A } ,

Nf{\)=vo\{x\f(x)<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 weobtain 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 theself-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 ofparagraph 3.1. Using the notation of the same paragraph, / is ^-locally


constant and the operator A = p(f) is explicitly diagonalizable; denotingby \j its eigenvalues, we have {\j} = f{V), so that for any positive real Awe have an equality # {Xj ; |A^| < A} = yo\{x ; \f{x)\ < X} if finite, andmore precisely we have an equality of measures

(3.3.22) ^^.=/.(d.r).3

Here there are no ellipticity or positivity assumptions on /. If howeverthe assumptions of Theorem (3.3.21) are satisfied, then / is also positiveelliptic, and so for any e > 0, combining (3.3.21) with (3.1.2) we have asA —> +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 Toeplitzsymbols defined in (1.4.12).

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1052 SHAIHARAN

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Manuscrit recu Ie 24 aout 1992,revise Ie 3 mai 1993.

Shai HARAN,Dept of MathematicsTechnion Institue of TechnologyTechnion City32000 Haifa (Israel).

ANNALES DE L INSTITUT OURIER - Centre MersenneThere are thus no differential operators over Qp. But...

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Transcript of ANNALES DE L INSTITUT OURIER - Centre MersenneThere are thus no differential operators over Qp. But...