Microfunctions at the Boundary and Mild Microfunctions
Microfunctions at the Boundary and Mild Microfunctions
复制标题
边界微函数和轻度微函数
DOI：
10.2977/prims/1195174864
复制
发表时间：
1988
影响因子：
1.2
通讯作者：
G. Zampieri
中科院分区：
文献类型：
--
作者：
P. Schapira;G. Zampieri
设 X 为实流形，且为 D(X} 的对象，即 X 上交换群滑轮有界复形范畴的派生范畴。[3] 中定义的函子 //hom(• , • ) 似乎是一个有用的工具，特别是在偏微分方程的边值问题理论中。本文的目的是计算 Rrzfj.hom(Ns, IF) 的茎，当 Q 为凸（最多微分同胚）和 X 的闭子流形 M 的开子集，Z 是 T*X 的闭凸真锥。作为一个应用，我们展示了如何以简短且函数的方式恢复 Kataoka 的温和微函数理论 [5]。设 X 为 C° 类实流形，T*X 为 X 的余切丛，K : T*X-*X 为 X 上的投影，orx 为 X 上的方向束。 X 的子流形 1 表示 X 中 M 的共正交束，特别是 1 表示 T*X 的零截面。对于 X 的两个子集 S 和 V，1 表示 S 沿 V 的法向圆锥（参见 [2]、[3]）。表示 X 上交换群的滑轮范畴的派生范畴，并令 D(X) 为由有界上同调的复形组成的完整子范畴，对于 ^^Ob(D(X)) 和 AdX 局部闭，可将 ^ 沿 A 的微局域化关联起来： ( l .D ^(^)=jMhom(ZA,JO ，其中 ZA 是 X\A 中为零的滑轮，且常数滑轮与A 上的茎 Z，其中 /jhom( • , • ) 是由 M. Kashiwara 于 1987 年 7 月 27 日发布的双函子。 * Universite Paris-Nord, 93430 Villetaneuse, France ** Dip. Mat., Universita, Via Belzoni, 7, 35131 Padova, Italy 496 PIERRE SCHAPIRA AND。 GIUSEPPE ZAMPIERI 在 [3] 中，这是 Dco,ic(T*X) 的一个对象，D(X) 的完整子类别由其上同调对象在 T*X 中的 R 轨道上恒定的复合体组成。此外：d.2) (1.3) (1.4)（其中 SS(J) 是 & 的微支撑，如 [3] 中定义）令 M 为 X 的实子流形，余维数为 n。并令 Q 为 M 的开 C°-凸子集（即：在任何 x&X 处，存在 Q 为凸的局部图）。我们首先注意到 ZQ 是上同调构造的（参见 [3]），并且： (1.5) R^n(ZQ,Z^=RrQ(Zx)^Z-Q®orM/x{_-n-\。因此应用 [3] 的命题 5. 6. 3，我们得到： (1.6) RrT^x(/2Q(^))=^®orM/xl-nJ( = X，它在 D*(X) 中给出一个可区分的三角形： (1. 7) &z®orM/x\_-n-\ »RrQ(&} *R 回想一下，如果向量丛 E 的圆锥子集不包含直线，则称为真圆锥子集。如果 Z 是这样的集合，我们用 Z° 表示对偶向量丛中的极（闭）锥体定理 1.1 设 Z 为包含 Q X TX X 的 T* X 的闭凸真圆锥，并令 x^X，则对于 X 中 x x 的合适开邻域 X'，我们有： (1.8) H'z(T*X':fi0(& ')®orM/x)=lun H'-*(U;&) u > 其中 U 的范围为 X' 的开子集族z=&. 证明，我们从一开始就假设 X 是 X 的向量子空间，Q 是 M 的凸子集；我们记得 /^(<F) = p.A(R&^(q2 ZQ, g{^))3，其中 A 是 XxX 的对角线，qii XxX-^X, (i = l ,2 ) 是第 i 投影，其中 X 被标识为 A ，而 T*X 为T^(XxX) 通过 T*XXT*X 上的 MlCROFUNCTIONS 和 MILD MlCROFUNCTIONS 497 使用 [3] 命题 2. 3. 2，我们有： (1.9) = lim H W 对于 X X X 的 W 开子集： (1.10) C*(XxX\W)n IntZ=0 我们也有： 我们声称我们可以为 (1. 10) 中的 W 族选择一个基本系统，使得： (1. 11) q?(x) 0 ((XxO) fW) 对于所有 x^X 都是凸的。事实上，对于向量空间 £" 和线性子空间 MdE，让我们采用具有凸纤维的开锥体 ?dTME，并选择线性投影 q:E-*M。然后根据 [1] 引理 1，我们可以找到开集的基本系统。 WdE，其中 CM(E\W) 0^= 0 ，使得 q\w 具有凸纤维。我们将上述说明与 E = XXX, M = 4 f = Int Z, q = ql9 一起应用，然后得到 (1.11) (1. 12) (RquZwmx^x = Rrc(qr W H (WH (XxQ»;ZXxX\^lw), x^X)。 xGq^Wft (XxQ))，我们在 (1. 12) 中找到 0。否则 gf W n (WH (XxQ}} =Wn ({x} X/3) 是 M 的非空开子集。如果我们另外假设这是凸的，则 (1. 12) 右侧的项同构于： RFC (M; ZM) =Z\_ dim M~\ （用于选择Af，即同构 orM=ZM~)。因此，我们证明，如果 (1. 11) 满足，则： (1. 13) Rqii(Zwn(x*sn) Zql(wn(xxQ»®orM\_ — l + n'])。我们总结到目前为止所建立的结果： Ifor W 满足 (1.10) 和 (1.11)。498 PIERRE SCHAPIRA 和 GIUSEPPE ZAMPIERI 我们现在需要两个引理。对于任何子集 AdXxX 和 BdX，我们有：（根据 T*XxT*X 上定义的第一个投影来标识 T*X=T%(XxX)）。设 0^C(X\q1((XxB) fU),S)。 + 这样：我们选择一个序列 zn »x, zn^B 使得：
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: