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Let X be a real manifold, & an object of D(X}, the derived category of the category of bounded complexes of sheaves of abelian groups on X. The functor //hom( • , • ), denned in [3], appears to be a useful tool especially in the theory of boundary value problems for partial differential equations. The aim of the present paper is to calculate the stalk of Rrzfj.hom(Ns, IF), when Q is a convex (up to diffeomorphism) and open subset of a closed submanifold M of X, and Z is a closed convex proper cone of T*X. As an application we show how to recover in a short and functorial way, the theory of mild microfunctions by Kataoka [5]. § 1. Let X be a real manifold of class C°°, T*X the cotangent bundle to X, K : T*X-*X the projection, orx the orientation sheaf on X. If M is a closed submanifold of X one denotes by T^X the conormal bundle to M in X and by orM/x the relative orientation sheaf. In particular one denotes by TXX the zero section of T*X. One sets f*X=T*X\T%X,n = n ^x. For two subsets S and V of X, one denotes by C(S, V) the normal cone of S along V (cf. [2], [3]). Let D(X) denote the derived category of the category of complexes of sheaves of abelian groups on X, and let D(X) be the full subcategory consisting of complexes with bounded cohomology. To ^^Ob(D(X)) and AdX locally closed, one associates the microlocalization of ^ along A: ( l .D ^(^)=jMhom(ZA,JO where ZA is the sheaf which is zero in X\A and the constant sheaf with stalk Z on A, and where /jhom( • , • ) is the bifunctor defined Communicated by M. Kashiwara, July 27, 1987. * Universite Paris-Nord, 93430 Villetaneuse, France. ** Dip. Mat., Universita, Via Belzoni, 7, 35131 Padova, Italy. 496 PIERRE SCHAPIRA AND GIUSEPPE ZAMPIERI in [3], This is an object of Dco,ic(T*X), the full subcategory of D(X) consisting of complexes whose cohomology objects are constant on the orbits of R in T*X. Moreover : d.2) (1.3) (1.4) (where SS(J) is the microsupport of & as defined in [3]). Let M be a real submanifold of X with codimension n and let Q be an open C°°-convex subset of M(i.e. : at any x&X there is a local chart in which Q is convex). We first note that ZQ is cohomologically constructive (cf. [3]) and that: (1.5) R^n(ZQ,Z^=RrQ(Zx)^Z-Q®orM/x{_-n-\. Thus applying Proposition 5. 6. 3 of [3] we obtain : (1.6) RrT^x(/2Q(^))=^®orM/xl-nJ( = X which gives a distinguished triangle in D*(X): (1. 7) &z®orM/x\_-n-\ »RrQ(&} *R Recall that a conic subset of a vector bundle E is called proper if it contains no line. If Z is such a set we denote by Z° the polar (closed) cone in the dual vector bundle and we set Z=~Z°. Theorem 1.1. Let Z be a closed convex proper cone of T* X containing Q X TX X and let x^X, Then for a suitable open neighborhood X' of x x in X, we have : (1.8) H'z(T*X':fi0(& ')®orM/x)=lun H'-*(U;&) u > where U ranges through the family of open subsets of X' such that z=&. Proof, We assume from the beginning that X is a vector space, M a vector subspace of X, Q a convex subset of M ; set l = dimX. We recall that /^(<F) = p.A(R&^(q2 ZQ, g{^))3 where A is the diagonal of XxX, qii XxX-^X, (i = l ,2 ) , is the i projection, and where X is identified to A and T*X to T^(XxX) by the first projection on T*XXT*X. MlCROFUNCTIONS AT THE BOUNDARY AND MILD MlCROFUNCTIONS 497 Using [3] Proposition 2. 3. 2, we have : (1.9) = lim H W for W open subset of X X X with : (1.10) C*(XxX\W)n IntZ=0. We also have : We claim that we can choose a fundamental system for the family of W in (1. 10) such that: (1. 11) q?(x) 0 ((XxO) fW) is convex for all x^X. In fact for a vector space £", and a linear subspace MdE, let us take an open cone ?dTME with convex fibers, and choose a linear projection q:E-*M. Then according to [1] Lemma 1, we can find a fundamental system of open sets WdE, with CM(E\W) 0^= 0 , such that q\w has convex fibers. We apply the above remark with E = XXX, M = 4 f = Int Z, q = ql9 and then get (1.11). Now note that : (1. 12) (RquZwmx^x = Rrc(qr W H (WH (XxQ»;ZXxX\^lw), x^X. If xGq^Wft (XxQ)), we find 0 in (1. 12). Otherwise gf W n (WH (XxQ}} =Wn ({x} X/3) is a non-empty open subset of M. If we assume in addition that this is convex then the term in the right side of (1. 12) is isomorphic to: RFC (M; ZM) =Z\_ dim M~\ (for a choice of an orientation on Af, i.e. an isomorphism orM=ZM~). We have therefore proved that if (1. 11) is fulfilled, then: (1. 13) Rqii(Zwn(x*sn) Zql(wn(xxQ»®orM\_ — l + n']. We summarize up the results established until now by : Ifor W satisfying (1.10) and (1.11). 498 PIERRE SCHAPIRA AND GIUSEPPE ZAMPIERI We need now two lemmas. Lemma 1.2. For any subsets AdXxX and BdX, we have: (under the identification T*X=T%(XxX) by the first projection defined on T*XxT*X). Proof. Let 0^C(X\q1((XxB) fU),S). There exist sequences {(*„, , and [cn}ndR + such that: We choose a sequence zn »x, zn^B such that:

Microfunctions at the boundary and mild microfunctions