ABSTRACT Title of Dissertation: Microfunctions for Sheaves of Holomorphic Functions with Growth Conditions Sin-Chnuah Cheah, Doctor of Philosophy, 1994 Dissertation directed by: Carlos A. Berenstein, Professor Department of Mathematics Mikio Sato devised microfunctions as a means of measuring the singularities of hyperfunctions. In 1970, Icawai and Sato introduced Fourier hyperfunctions in their study of partial differential operators. The class of Fourier hyperfunctions has been generalized by Saburi, Nagamachi, and Kaneko, among others, and most recently by Berenstein and Struppa. Berenstein and Struppa introduced Fourier p-hyperfunctions, where p is a plurisubharmonic function satisfying certain smoothness and growth conditions. p(z) = IzlS, s 2 1 are the cases studied by Sato, Kawai, Nagamachi, and Kaneko. Following the methods of Sato, Kawai and Icashiwara, this dissertation intro- duces Fourier p-microfunctions functorially, though under very severe conditions on p. These restrictions on p are satisfied when, for instance, p(z) = log+ If 1 where f is a product of 1 variable holomorphic functions with zeroes uniformly bounded away from the real axis. Kaneko has introduced Fourier microfunc- tions for p(z) = I%ezls, s > 0, using tubes. When s < 1 these p's are not plurisubharmonic. Thus the results here complement his. MICROFUNCTIONS FOR SHEAVES OF HOLOMORPHIC FUNCTIONS WITH GROWTH CONDITIONS Dissertation submitted to the Faculty of the Graduate School of The University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1994 Advisory Committee: Professor Carlos A. Berenstein, Chairman/Advisor Associate Professor Daniel I. Five1 Associate Professor Mark A. Shayman Professor Daniele C. Struppa Professor Scott A. Wolpert PREFACE In their work on Dirichlet series, Berenstein and Struppa [1988], introduced a new theory of interpolation for Ap,@(I') (definition 1.2. I), the space holomorphic functions in an open cone r E Cn satisfying growth conditions depending upon a plurisubharmonic function p, and for its dual Ap,O (I?)'. They noticed that the proofs of these interpolation theorems, and some theorems on mean periodic functions amounted to theorems on the vanishing of cohomology groups. In the spirit of Kawai [1970], they [preprint] thus introduced sheaves of holomorphic functions with growth conditions. These sheaves are defined on the radial compactification IDn of Rn, or its h corresponding "complexification7' Cn = IDn + GRn. Both compactifications were introduced by Sato and Icawai in Kawai [1970]. Sato and Kawai defined the sheaves, 8, of slowly increasing holomorphic functions and, 0, of rapidly de- N creasing holomorphic functions. Then they defined the Fourier hyperfunctions in the same manner that hyperfunctions are defined, namely as := RnrW (6'). The sheaf of Fourier hyperfunctions on IDn. The sheaves Berenstein and Struppa [preprint] defined were the sheaf of holo- morphic functions of minimal type p, W, where the plurisubharmonic function p satisfies, among other things, Hormander7s condition (definition 1.2.1(4) (ii)' ), and the sheaf of rapidly decreasing functions of type p, As in Kawai [I9701 Hormander [1967]. they introduced the sheaf of Fourier p-hyperfunctions, here denoted %. When p(z) = I . 1, these are the Fourier hyperfunctions of Kawai and Sato. Saburi [197812 introduced Fourier hyperfunctions using a radial compactification of (Cn , and Kaneko [I9851 has introduced Fourier hyperfunctions when p is the (not necessarily plurisubharmonic) function I%ez IS (s > 0). As one of their interest lay in the singularities of Dirichlet series, Beren- stein & Struppa asked what would correspond to microfunctions for Fourier p-hyperfunctions. Microfunctions (for ordinary hyperfunctions), it should be re- called, were introduced by Sato [1970], and defined functorially in Sato, Kawai & Kashiwara 119731. They measure the extent to which hyperfunctions fail to be real analytic, thus measuring the singularities of hyperfunctions. By using tubes, Kaneko [I9851 has introduced microfunctions for the Fourier hyperfunctions he defined. Following Sato, Kawai & Ibshiwara [1973], this paper introduces Fourier p-microfunctions. The results here complement Kaneko's. Eventhough the con- ditions to be imposed on the plurisubharmonic function p in chapter 2 will turn out to be rather severe, they allow for functions not considered by Kaneko. However, the results here do not include Kaneko's, since p(z) = I%ezls is not plurisubharmonic when s < 1. It is shown in chapter 4 that Fourier p-microfunctions defined here are con- centrated in one degree. More specifically it is shown (theorem 4.2.18) that S*R is purely n-codimensional with respect to T-'W in analogy to the case of (ordinary) microfunctions. It should be noted here that 6 and IDn are mani- folds with boundary. Thus this result contrasts with microfunctions up to the boundary studied by Schapira3, where the microfunctions are not in general 2For references to Saburi, the reader should also refer to Saburi [I9821 and [1985]. 3See for instance Schapira [I9881 52. concentrated in one degree. On the other hand, Lieutenant [I9861 showed that h S*O is purely n-codimensional with respect to 7r-'i,@, where i : Cn -+ Cn is the inclusion. Thus the result here is similar to his. As a more tenuous justification for studying such microfunctions, one might note that Fourier hyperfunctions have appeared in quantum field theory as a means of enlarging the space of states4. 4See Briining & Nagamachi [I9891 and references contained therein. DEDICATION Of Course for who have known for many years what the Old Masters knew well and who are Old Masters themselves ACKNOWLEDGMENTS This has been a long and oftentimes arduous journey. Along the way, I was aided by my advisor, Professor Carlos Berenstein, who suggested this problem, and Professor Daniele Struppa. They shared with me a preliminary version of their paper for which I am much obliged. They awaited my arrival with great patience, and made suggestions on avenues to take when I found myself at a dead end. Dr. Bao-Qin Li shared some of his thoughts with me. Professors Daniel Fivel, Mark Shayman, and Scott Wolpert served on my dissertation commit tee. I am thankful for everyone's help. . As the reader will notice, this work owes much to the English and french works of many authors on hyperfunctions and microfunctions. Those works provided the maps without which I would have been hopelessly lost. It should be mentioned that this paper was typeset with AMS-TEX, the T@ macro system of the American Mathematical Society, and w-pic, K. H. Rose's 'IJjX macro for typesetting diagrams. Finally, I am, most of all, grateful (and much indebted) to Poh and Tom Willson for the shelter and sustenance they provided during thunderstorms. While they may find it hard to believe since I kept raiding their refrigerator, for a while they allowed me to think more about mathematics than food or fleas. This work would have been impossible without that assistance. TABLE OF CONTENTS Section Chapter I. Introduction and Basic Triangles 51.1 Introduction 5 1.2 Review of Results 51.3 The Basic Triangles 51.4 Computation of Terms of the Triangles Chapter 11. Theorems of Kawai 52.1 Conditions on the Plurisubharmonic Functions p 52.2 Spaces of Holomorphic Functions with Growth Conditions 52.3 Kawai's Approximation Theorem 52.4 A Vanishing Theorem Chapter 111. Topological Lemmata 53.1 Exhaustion Functions $3.2 General Lemmas 53.3 Lemmas on Traces Chapter IV. Theorems on Pure Codirnensionality and Fundamental Exact Sequences 54.1 Computation of X& (rV1W) 54.2 Computation of X&n(~-lW) 54.3 Fundamental Exact Sequences References vii CHAPTER I INTRODUCTIOb1 AND THE BASIC TRIANGLES David Harum says, "A reasonable amount of fleas is good for a dog. They keep him from broodin' on bein' a dog." A goodly supply of fleas might likewise keep man from brooding over anything deeper than the presence of these fleas, but in some cases this in itself is a rather serious thing to brood over. -Asa C. Chandler, I~~iroduction to Parasitology [1944]. To define the Fourier p microf~~pctions, I have basically followed the results of Sato, Kawai & Kashiwara [I9731 for (ordinary) hyperfunctions. More specifi- cally, in this chapter, we will note that all the basj c triangles for hyperfunctions remain true without modification on open subsets of IDn, and that most of the terms of these triangles can be computed in exactly the same manner. These results do not depend on any assumptions on p. In the following chapters we proceed to compute the other' terms of these triangles. Again as in Sato, Kawai & Kashiwara, these terms are the two van- ishing theorems in chapter 4. The preliminaries needed to prove one of these vanishing theorems (4.2.18), are laid out in chaptlers 2 and 3. As in the case of hyperfunctions, proposition 1.4.12 below reduces one of the vanishing theorems (4.2.7) to a computation of H~,~(V; W), where G is a wedge and V an open set. Then following Kashiwara, Kawai & Kimura [I9061 we show in chapter 3 and h 54.1 that V n G can be written as I<' - K for suitable compact subsets of Cn. The techniques involved are elementary if tedious. Theorem 2.4.8 in chapter 2, then shows Hfc,-K(Cn; W) = 0 for k # n. To prove theorem 2.4.8 is the goal of chapter 2. This theorem generalizes proposition 2.2.2 of Kashiwara, Kawai & Kimu:ra [I9861 to certain compact subsets of 6, and by remark 2.4.9, to compact subsets of Cn equal to their plurisubharmonic hull. This result is however essentially contained in Kawai [1970], and I have closely followed his ideas. Horrnander's L2 methods provide the main tools. Crucial to this goal is Kawai's a:pproximation theorem, which is noted to hold not only for subsets of Dn, as stated in his paper, but also for h compact subsets of Cn that are in some sense equal their plurisubharmonic hull. It is in chapter 2 during the course of proving theorem 2.4.8 that restrictions are placed on the plurisubharmonic function p. These restrictions are the prop- erty (Pp) introduced by Berenstein & Struppa (in analogy with a condition in Meril [1983]), and the existence of holomorphic fun.ctions of "controlled growth", which is implicit in Kawai and Saburi's work. As a philosophical point, one might note that such "controlled growth" func- tions will play only a catalytic role in the proofs. They are used locally only at h points at infinity (of R, or, more precisely, of some Cn neighbourhood of R), and then only to bring functions from one space to another and then back again, by first multiplying and then dividing by the functioin of "controlled growth". Ex- amples, although admittedly scant,-of plurisubharmonic functions p that satisfy these conditions are given in chapter 2. The other vanishing theorem (4.1.5) requires sufficiently many W-pseudocon- vex sets. Unlike the case of Cn, where several characterizations of pseudoconvex sets are known, little more than the definition characterizes W-pseudoconvex sets.' As a consequence to produce an 49'-pseudoconvex set almost certainly requires the exhibition of an exhaustion function. This indeed was one of the problems that necessitated the explicit calculatio~is to prove the previous van- ishing theorem. In this case, the Grauert tubulas theorem is used to exhibit the W-pseudoconvex sets and exhaustion functions. This theorem is proven by Kawai [1970], and Berenstein & Struppa [preprint] for open subsets of Dn, and in detail by Saburi [I9851 for open subsets of Dn in a different compactification of Cn. A proof following Harvey & Wells' 119721 proof of Grauert7s original theorem (for real analytic submanifolds of comp1e:x manifolds) is supplied here for the reader's convenience. After a proposition on smoothing p1urisubhar:monic functions, proposition 4.1.4 shows that points at infinity on -9~1 have sufficiently many neighbour- h hoods whose projection on Cn - Dn is W-pseudoc~onvex. The classical proof of the vanishing theorem can then be used to show theorem 4.1.5 with no modifi- cation. In summary, all the main ideas in this work are due to Sato, Kawai & Kashi- wara [1973], and Kawai [1970]. Here, only a few ~a~lculations are added to their already extensive and formidable work. $1.2 Review of Results Listed below are some of the main definitions of Berenstein and Struppa [preprint]. As an important remark, the properties listed below that the plu- risubharmonic function p are to satisfy form the ildeal case. In actuality more severe restrictions shall have to be made; this is done in the following chapters. The results in ??3 and 4 hold regardless. For instance convex sets are pseudoconvex. This provides an abundance of albeit unin- teresting pseudoconvex sets. (1) B is the sheaf of holomorphic functions on Cn; (2) IP is the radial compactification of Rn, viz. IDn := Rn U SF-1, being the n - 1 sphere at infinity, which is identified with Rn - {o)/R+; (3) 6 := ID^ + GR~; (4) p is a smooth plurisubharmonic function on Cn satisfying: (i) P(.) 2 0, log(l + IzI) = O(P(~>>, (ii) there are const ants Kl , I{2, K3, I<4, such that Izl - z2 I < - e~p(-I{~p(z~) - IC2) implies p(zl) 5 1{3p(z2) + I(4; and (iii) p is Cm and convex; (5) For a pseudoconvex region, U, in Cn, Ap(U) is the set { f E B(U) : 3 positive constants A and B s. t. If (z) 1 < Ae BP(~)) (6) For U 2 6, open, W(U) is the set of all holomorphic functions f E B(U n Cn) such that, for any e > 0 and any compact set I{ C U, sup ~f(z)e-'~(')l < m. r ? lr'nCn These W(U) form a sheaf, denoted W'; (7) For U 6, open, B(U) is the set of all holomorphic functions f E B(U n Cn) such that, for any compact set I< C U, 36 > 0 such that sup [f(z)e6p(')~ < 00. r E KnCn These W(U) form a sheaf, denoted W (8) BDn, or simply B, denotes the sheaf of Fourier hyperfunctions on IDn; this by definition is the sheaf RnFm (W). 0 REMARK 1.2.2. Kawai [I9701 uses 9 to denote the Fourier hyperfunctions on W; this corresponds to the case p(z) = lzl in Berenstein and Struppa [preprint] Fourier p-hyperfunctions on IDn, p9. D Instead of the notation above for Fourier hyperfunctions, this paper will use DEFINITION 1.2.3. Let S1 be an open subset of IDn. Define the sheaf of Fourier p-hyperfunctions on S2 to be Rnrn (49). This sheaf will be denoted by @R or simply %. 0 DEFINITION 1.2.4~. An open set V C Cn satisfies property (Pp) if (PP 1 34 E @(V) such that VM > 0, sup(-%e+(z) + Mp(z)) < oo. v REMARK 1.2.5. Clearly if V' _> V satisfies property (Pp) then so does V. D REMARK 1.2.6. Any V cc Cn satisfies (Pq) for q plurisubharmonic and merely upper semicontinous on Cn. Take 4 - 0. D DEFINITION 1 .2.73. An open set U C 6 is is-pseudoconvex if (1) U n Cn satisfies property (Pp); (2) There is a C2 plurisubharmonic function B on U n Cn such that (i) Vc E R, {z : O(z) < c) cc U; (ii) VI< C U, co~pact, 3M1< such that suPKncn O(Z) < MK. 0 THEOREM 1 .2.84. Let U G 6 be is-pseudoconvex. Then 2Meril [1983], Berenstein & Struppa [preprint]. 3Kawai [1970], Saburi [1978], Nagamachi [1981], Berenstein & Struppa [preprint]. 4Kawai, Meril, Saburi, Nagamachi, I 0, R, := cv(R U {&-v1ej : ej is the jth unit vector in Wn, j = 1,. . . , n; 0 < v' < v)); This is a complexification of 52. (2) F := c~R and Fu := c16R,. (3) $a:= R x Sn-1; SF := F x These are the sphere bundles. (4) S*R := R x SL-,; S*F := F x Sk-,. The dual sphere bundles. (5) 6, := (R, - R) U SR; F, := (Fu - F) U SF. The real monoidal transforms. (6) 8: := (R, - a) U S*R; F; := (F, - F) IJ S*F. The real comonoidal transforms. (7) DR := {(x, t, q) E R x 9,-1 x Sk-l : (t, 7) L 0). (8) DF := {(x,t,q) E F x %-I x 9;-1 : (I,q) 5 0); .--v - (9) DO: := (R, - 0) u DO; DF$ := (F, - F) U DF; (11 : 6; c,: F,-F-+ F,. 0 REMARK 1.3.8. To a map f : X -+ Y is associated the mapping cone triangle: where 9' E K+(Y), and CO(T) is the mapping cone of the canonical adjunction 7. Twisting and translating this triangle produces REMARK 1.3.10. gist, can be considered a functor from K+(Y) to KS(Y). However Co(d* + 9) is not a functor nor does it normally give rise to a derived f~nctor.~ D As in Sato, Kawai & Kashiwara [1973], many of the triangles in the sequel will take the form We now use (3-1) in the following situation. Consider the inclusions and projections in the following diagram: P,, - ao,, J R,-,R- *fi,,_L_t.R,, .1 J J. F,, - F-P" * F,, For 9' E K+(R") or S* f K+(F,,) there are triangles These triangles form an octahedron, and the octahedral axiom provides the dashed arrows: See Komatsu [I9711 $8. The triangle with dashed arrows gives rise to a triangle in the derived category D+(Y): Note that R 9istTi(9') = Rrn (2") and RT* R 9isti(~-'s') = RT* RI'M ~-'9', so (3-8) gives: 51.4 Computation of Terms of tihe Triangles The proofs given here are, with little or no moclification, due to Sato, Kawai & Kashiwara [1971]. PROOF'. The long exact sequence from the triangle (3-5) gives and the equality R~ 9istT(9) = R~-'T,T-'B, fork 3 2. cf. Prop. 2.3.3 of Kawai, Kashiwara & IGmura [19SG]. Since T is a closed map and R, is metrizable, one has, for x E R,,7 R* Jist,(S) = H*(T-'{x} + x; gz) This proves the lemma. - LEMMA 1.4.2. Let T : DR: -+ 0, be the canonical projection. Then there is an isomorphism PROOF. Let Y* E K+(!?,). There is a composition of canonical maps Consequently for 9 E ~+(fi,), this induces, in the derived category, a map When g* = T-~S' this is the map claimed in the lemma. That this map is an isomorphism is proven in Lieutenant [198618. LEMMA 1.4.3. Consider the following diagram and maps where the left arrows are inclusionsg: 7Kawai, Kashiwara & Kimura [1986], proposition 2.3.6. 8Page 105 equation (1). 'After Sato, Kawai & Kashiwara [I9731 There is an isomorphism (4-3) RT*T-' Rrm (T-' 9') e ~rg* 0 T-' 9'. PROOF. The triangle (3-1) gives rise to the following triangle in the derived category: Applying Rrs* 0 produces the triangle By (4-1) the last term of this triangle is From the proof of Lemma 3.9 in Lieutenant [1986, 19881, R Oist,(n-'9') = 0, SO Rrs* 0 n-' 9' -=+ RT* n-' ~r~ (T-I 9'). PROOF. From the proof of lemma 1.4.2 we see that (4-2) gives the quasi- isomorphism Taking derived functors gives and using the Vietoris-Begle isomorphism g' -+ Rrr,rr-lC4' (which is possible by lemma 1.4.5 below) gives the isomorphism For '3' = ~-'9' this isomorphism produces: CV LEMMA 1.4.5. rr : DSt+ + a, is proper and has contractible fibres. THEOREM 1.4.6. There is a triangle (4-4) + Rh (3') [n] -+ RT. R~S* (n-' ~*)[n] 3 . PROOF. Substitute the terms computed in lemma 1.4.1 and proposition 1.4.4 into the triangle obtained from the octahedral axiom (3-9) and translate by [nl. (1) Pd~ := (w)n, (2) pZ2n : = Rrm (T -'W) [+ I], (3) WO := Rrs* ~1 (rr-lW)[nIa, where a is the antipodal map. 0 (4-3) gives PROPOSITION 1.4.8. Rr,r-'p%2 = W[l - nIa. Theorem 1.4.6 gives PROPOSITION 1.4.9. There is a triangle PROPOSITION 1.4.10'~. Let S be a sheaf on a,,, and let xo + itow E S*Q. Then there is an isomorphism ~2~ (T-' 9),,+ib, lim ----+ H'$,~(v; 9), V3xo G where V runs through neighbourhoods of zo in Q,, and G through the following Grn,[o := Q + i {Y : (ylt) < O,v(I E ~rn,[o} PROOF. To be explicit, let V, (E > 0) be the intersection of a basis of W- pseudoconvex neighbourhoods of xo with 0, that decreases to xo as E decreases to 0. Let Urn,, be the neighbourhoods of xo + it0co in defined by -1 urn,, :=r K n ((Q + iym,z is an isomorphism, it follows that for k = 0 lirn H*(v,; 91ny) = + m>c for k # 0, and {)+i fk=O lirn H* (urn,,; ~~'9) = -----f m,c for k # 0. Thus (4-9) provides o----tlim +m,e H8m,c0nK (v,; 5) - sX,, (4- 10) J. *m,c 1 -, lirn H:* anurn,. (Urn.,; ~-'9) + sx,, 4 lim HO(% - Gmlc0; 9) -lim H&m,,on~c +m,e +m,~ (K;9) -0 J, 1 + , lirn HO(Um,, - $*St; ~-'9) + lirn Hs. nnum,. (Urn,,; n-l9) -+ 0, dm,? +m,e and for k > 2, (4- 1 1) 0-lim Hk--l +m1c (K - Gmlto; 5) -1im H&m,conve(x; 9) -0 +m,c 4 k 4 0 + lim H*-'(U~~, - S*Q; ~-'9) + lim Hs*nnum,c (Urn,.; ~~'9) + 0 -m,c +m,c The vertical maps, being induced from isomorphisms, are themselves isomor- phisms. It follows from the five lemma applied to (4-lo), and from (4-11) that This proves the theorem. REMARK 1.4.11. Clearly the proposition holds for 9 defined on F, mutatis mutandis. D The theory for these sheaves is however not complete at present. CHAPTER I1 THEOREMS OF KAWAI Balances are delicate and easily tipped. The social status of a word, its force, its length, its history of use: anything can do it. Syntax sets up the scale, but semantics puts the weights in the pans. The follow- ing are out of balance: (1) "the bandit shot my son, stabbed me in the arm, and called me names," (2) "what bitter things both life and aspirin are!," (3) "I have boated everywhere--on the Po and on Paw- tucket Creek," (4) "you say your marriage suffers from coital insuffi- . . ciency and greasy fries?," (5) "yeah, my wife kisses her customers and brings their bad breath to bed." -William H. Gass, 'And' in Habitations of the Word [1985]. This chapter presents some restatements of Kawai7s [I9701 results, especially his theorems 2.2.1, 3.2.1 and 3.2.2. There is essentially nothing new here. The thrust of the effort has been to distill the essence of Kawai7s results, to make sure that his results hold for these slightly more general, plurisubharmonic functions. This has been carried out the way a janitor might go about making sure things are in order. $2.1 Conditions on the Plurisubl~arrnonic Function p DEFINITION 2.1.1. Let V Cn and p(z) a plurisubharmonic function defined on Cn . A holomorphic function $ E O(V) is controlled exponential type (K, p(.)) if 3~', 0 < K' < K, 3AK > 0, B, > 0 such that B~"P(') < I+(z)l < A~"P(') r E V. A An open set U E Cn is said to have a function of controlled exponential type (~,p) if there is function of controlled exponential type (~,p) on U n Cn. 0 REMARK 2.1.2. Suppose U _> U' has a function of controlled exponential type (~,p) then clearly so does U'. D REMARK 2.1.3. Suppose V cc Cn, and q is a continuous plurisubharmonic function. Then there are holomorphic functions of type (K, q) on V for every K > 0. Take II, E 1 in (1- 1) and note that q attains its maximum and minimum on clQlnV. D DEFINITION 2.1.4. In addition to the assumptions made in 51.1, we shall im- pose more restrictive conditions on the plurisubharmonic growth function p. Explicitly: (1) p 2 0, p E Cm is convex. h (2) For every compact I< C Cn, log (1 + 121) = o(p(z)) as z ---+ co, z E I< n cn; (3) 3A, B > 0 such that lz - C1-e 1 p(C) .< Ap(z) + B;' (4) For sufficiently small v every point of 52, - Cn has a neighbourhood with functions of controlled type (~,p) for every K > 0. 0 EXAMPLES 2.1.5. (1) p(z) = (1 + lr12)S/2 or p(z) = 12Is, s > 0; when s = 1 this is the case considered by Kawai [1970], and Meril [1983]. (2) p(z) = I%ezls, s 2 1; Kaneko [I9851 considers the case s > 0. For s < 1 these p7s are not plurisubharmonic, so the methods here are not applicable to his case. This condition goes back to Berenstein and Taylor. See references in Struppa [I9831 (3) ~(z) = log+ If (z) 1 where f (z) = n; fj (zj), with fj entire and uniformly bounded away from 0 in a 6 neighbourhood of W. In this case (1-1) will be satisfied. D Recall DEFINITION 2.1 .62. An open set V 2 6 is Saburi type (1) if for some a > O Here 13mzl = Cjyjv2 and Ifllezl = JF. 0 F EXAMPLE 2.1.8. Suppose V C 6 is Saburi type (1). Let p(z) = lzl, the case considered by Kawai and Saburi. Then $,(z) := cosh (tcfiJv) is a function of controlled exponential type tc, I . / for V. PROOF. First note from the series expansion that $, is entire. For computational purposes let c := 1 JmzI < 1, n(z) := I SUPvne lRezl+ a \/xjzjl2, or(z) :=Wtn(z), and ai(z) := 3ma(z). Note that 1 cash fitc. 1 = cash 2 + cos 2 Gnoi. Let zj = zj + iyj = rjeaj. Then g(z) = JC jr? e2"j . ~efine r and o by r2e2i6' .- .- C .r2e2"j. Then 3 3 2 r = \/(xjr? cos 20j)2 + (Cjr? sin 20i)2, C. r2 cos 20j g(z) = reie = r cos 8 + i sin 8, and cos 28 = ' 3r2 . So up to sign Hence r2 + x jr: cos 20) r2 - jr: cos 20, Similarly, using "cartesian" coordinates one gets Then Cjxj,2 - j,2 cos 29 = Y r2 In terms of these coordinates Thus up to sign 1) Upper bound. BY Now estimate a, using the "polar" coordinates (1-3). Since By choosing A sufficiently large, there is an upper bound 2) Lower bound. For the lower bound the case when cosh = 0 is dispensed with and then a asymptotic growth is obtained. a) cosh fi~~ = 0 if and only if 2fi~0, = 0 and 2fi~0i = (2k + 1)rr k E Hence This would be impossible if it fails to satisfy Saburi7s type (1) inequality (1-2). SO cosh fiKo # o if Simplifying and completing the square gives 2 Since (1 - c2) (JF - A) > 0, if tc can be chosen so that the other 1-c2 - two terms of the last line are greater than zero, (1-2) will be false; i. e. K has to be chosen so that Equivalently cLa' -- - for k E Z. 1 -c2' This has to be true for all k E Z, so K has to be chosen such that This will clearly hold for K < KO for some small KO. In summary when 0 < K < KO, lcosh fi~cl > 0. b) Now provide a lower bound for the asymptotic behaviour. Assume first of all that lzl > 1, and note that . 24; ru, -24$KU, e (1-6) COS~ 26~~. = + 2 e > -e 2 1 2fir1u.i Now estimate larl in (1-6). From (1-4) (1-2) implies that for sufficiently large I Jxjxj121, xj yj12 < Cjxjf2; SO in this case (1-4) and (1-5) give = (1 - c)\/~~xji~ - ca. It follows that (1-7) On the other hand, and Together with (1-7), this yields 1 2fi.1uc(z)1 > ~e-2fi"C"e-2fi'~cae2fiK~P(%) ?ie 2 I for large Jw. Since cosh zlca > 0 from step (Za), this lower asymptotic bound shows I PI that Choosing B sufficiently small gives the lower bound 52.2 Spaces of Holomorphic Functions with Growth Conditions The topologies of the spaces involved are first recalled from Saburi [1978], Nagamachi [1981], Meril [I9831 and Berenstein & Struppa [preprint]. h DEFINITION 2.2.1. Let U C @In be open and I{;, li'; CC int@ I{;+, be an h exhaustion of U by compact subsets of Cn. pxO(u) := lirn L~ (int@ IC; n Cn ; ep(z)) ; t j,c'ho J,&t (u) : = lim L~ (int@ I(: n Cn ; - cp(z)) ; ---+ j,c'hO P@(U) := lim Lm (int6 I{; n Cn; ep(z)) n 6' (id,-- K; n a~") ; t j,c'hO p@(U) := lim lim p&dd (int6 I{; n @" ; -sp(z)) . t+ j ALo h For K compact in Cn and intG Icj a basis of compact neighbourhoods of I{, let p6'(K) := lim pB~dd (int6 n Cn; -~P(z)) , j ,GO where pOBdd(Ic;4) := {f E B(Kn@") : sup lfle-' =: Ilfll~ < m), and, IinCn LEMMA 2.2.23. Let L$ = intG Iq n Cn, Ki increasing as in definition 2.2.1. Let m > 0. Then lim B LJ; - $p(z) - 2m log(l + lz 12)) = lim B (LJ; - fp(z)) ( as TVS. - 3 - 3 PROOF. Clearly B L;; -fp(z) - 2m log(I. + 1zl2)) + lim B (LJ; -ip(z)) ( - 3 is a continuous inclusion. Hence lim B LJ; - f p(z) - 2m log(l + lz 12)) + lirn B (LJ; - fp(z)) - 3 ( --+ 3 is a continuous injection. On the other hand, let f E B (LJ; -fp(z)), and choose 6 > 0 such that 4m6 - A < 0. Since log(1 + lzl) = o (p(z)) as z + oo there is an R such that 3(3+1) log(1 + lzl) < dp(z) for lzl > R. Thus It follows that 1 < M / 1 f 12e?P(z)dh. L:: +I So the map induced by restriction lim B LJ; -fp(r) - 2m log(1 + 1zl2)) +- lim B (L;; - fp(z)) - 3 ( - 3 is continuous. This proves the lemma. PROPOSITION 2.2.34. Let Li = inte ICj n Cn, ICj decreasing as in definition 2.2.1. Then 1 pB (ints - fp(z)) = lim + B L j ; - f P(z)) as TVS. antG Kj cjK L j PROOF. First note that the map L is well-defined and continuous because if f E pB(intGI(,) then sup^^ 1 f leSP < oo. Thus 1 - -P However log(1 + lzl) = o(p(z)) as z --+ oo, so e j dX < oo. Thus =j is continuous. Next we show lirn . ,8 ints Kj; - f p(z)) 4 lirn B (L~; - f p(z)) is surjec- -3 ( --+j tive. Consider the map given by restriction: where the brackets [.I here denote the greatest integer, and the constant A comes from definition 2.1.4(3) Choose r so that B(z, r) Lj for all z E Lj+l. Then But from definition 2.1.4(3) p(C) > (p(z) - B)/A, so ~hus p@ ( int~I{[2j~+l]; --P(z)) + B (L j; - fp(z)) is well-defined. This proves the surjectivity. Since the preimages of barrels are barrels, lim d Lj; -fp(z)) is barreled. --fj ( Moreover as the direct limit of injective5 weakly compact6 maps it is a DFS* space, and thus Hausdorff. lim ,B (idF I<,; -fP(z)) = pB(IC) is a DFS space, and the strong dual of -i a Frkchet space, thus it is fully complete7. Thus lim intF I$; - f p(z)) -+ lim B (L j ; - $p(z)) is open8. It is clear- -j ( + j ly 1-1. limp@ Lj; - f p(z)) = lim lim B (Lj; -Jp(z)) + ( +--f Similarly for B replaced with p@ and the weight -Ap replaced with --Ap - 3 3 2mlog(l + 1zI2). This follows because limits commute and from "diagram chasing". D 5Each component of int@Kj intersect Ii' by assumption. 6The spaces 6 (L ; - f P(z)) are Hilbert spaces; Aloaglu- Bourbaki theorem. 'page, W. [I9881 theorem 21.3(ii). 8Page, W. [I9881 corollary 21.9. 26 DEFINITION 2.2.5'. Let X (Lj; -Jp(z)) denote the closure of @(Lj; -2Sp(z)) in L2 (Lj; -Jp(z)). Similarly for X (L~; -Jp(z) - 2m log(1 + lzI2)). 0 REMARK 2.2.6. X (L j; -JP(z)) G - 0 (L j ; -Jp(z)), since @ (Lj ; -JP(~)) being the kernel of -8, is closed in L2 (Lj; -Jp(z)). D lim X (L~; -6p(z) - 2mlog(l + (z12)) = lim X (Lj; -Jp(z)) . + + ah0 ah0 PROOF. The proof is essentially the same as in lemma 2.2.2. First consider the map This is well-defined and continuous since If fk E B(~j;-2Jp(z)-2mlog(l+lz/~)), and fk i f in L~ (L~;-~P(z) -2mlog(l + lzI2)), then fk E B(Lj;-2Jp(z)) and fk + f in L2 (Lj; -6p(~)). On the other hand, for S < S', 'Kawai [1970]. lo Kawai [1970]. is well-defined since Again if fk E @(Lj;-2S1p(z)), fk -+ f E L2 (Lj;-S1p(z)), then fk E @ (Lj; -2dp(z) - 2m Iog(1 + lzI2)) and fk + f in L2(Lj; -Sp(z) - 2mlog(l + 1zI2)). LEMMA 2.2.8". Let I 0, ),-;+1 = ,-Wl*/k < (,supn-We4 l/k < (esuI'n -Be4+fp 'Ik < m, - - 1 Vb. So Lebesgue's dominated convergence theorem gives By the Hahn-Banach theorem, 6' (L~; -26p(z) - 2m log(l + Iz12)) is dense in @(~~;-d~1(~)-2mlog(1+1~1~)). DEFINITION 2.2.9. For an open set U 6 and a family of increasing compact sets Kc, c E R, Kc U, Kc c c ints Kc! for c < c', define ( Cn) := 1 lim B (Kc n C; -S1p(z) - 2mlog(l + 1.~1~)) , t + cpco 6'\0 p@(U) := lim lim @(I(,;-d'p(~) -2mlog(l+Iz12))- 0 t 4 c/o0 6!\0 LEMMA 2.2.10. For U and Kc as in definition 2.2.9, O2 (U n Cn ) = p@(U) as sets. PROOF. The proof follows that of Proposition 2.2.3. Let p@(U) -+ 02(U n Cn ) be the "identity" : f H f. To show this is well- defined, let I< be a compact subset of U. Without loss of generality, I{ can be taken to be Kc for some c. By definition 3S>O suchthat sup If1 e6~(z)+2mlog(l+lz12) < zcC nu? Thus To show that the inverse #(U) t B2(U n Cn) is well defined, let I< be a compact subset of U, and f E @2 (U n Cn ) . Then K cc intG Kc for some c. Choose r > 0 so that B(z, r) c II, for all z E K n Cn. By definition there is a S > 0 such that Following the argument in proposition 2.2.3, we have for z E I<, Thus So suplcnc,, ~f (z)le6"p(z)+2m'og(1+1z12) < m. This proves the lemma. LEMMA 2.2.1 112. Suppose ISj is a decreasing sequence of compact neighbour- hoods of a compact set I< 5 U and that U satisfies (Pp). Then for S < 6' there is a dense inclusion In fact the closure of the image in L2 (Lj; -Sp(z) - 2mlog(l + 1zl2)) is 8 (L~; -Stp(*) - 2mlog(l + lzI2)). . PROOF. Recall that 8 (Lj; -Sp(z) - 2m log(1 + /*I2)) is a closed subspace of L2 (Lj; -Stp - 2m log(1 + [*I2)). Follow the proof of lemma 2.2.8. has dense image. PROOF. This follows from general definitions of direct limits. Let p be a con- tinuous linear functional and suppose each fj has dense image: Supposepf = 0. Thenpfpj = 0 =pp;fj, and henceppj = 0 Vj. Thisimplies thatp=O. $2.3 Kawai's Approximation Theorem In this section we note that Kawai's approximation theorem remains true for sets not necessarily in IDn 13. LEMMA 2.3.1. Consider the inductive system {A,) in an abelian category. (For simplicity assume this category is concrete.) 13See also Saburi [I9851 s2.3. Given morphisms f and f' consider the pull- backs p: f and p: f' The following are equivalent: (1 :AB pf =O + pft=0; (2) p,:A, -t B; V p,p,f =O =+ p,p,fl=O. PROOF. l4 Given p,, p exists from the definition of direct limits. Let a E A. Then there are an e and an a' E A such that p,(af) = a. If moreover a = f (l), then (I, a') E p:L. SO p f (a) = pp,(p: f)(l, a') = 0. By hyphothesis this implies that p f' = 0. So p?(p:f) = 0 ve. Suppose p is given such that p f = 0. Let p, := pp,. Then pf = 0 ==+ p,(p: f) = 0 Ve. By hypothesis this implies p,(p: f ') = 0 YE. Let I' E L'. 3a1 E A,, for some E, such that f1(l') = p,(al). So (It, a') E p:L1. But then pf'(l')=pp,(p:f')(l',a')=O;i.e. pfl=O. LEMMA 2.3.2. Suppose w E L2 (U; dp(z) -I- 2m log(l lzI2)) and Id1 > on U. Then W - E L~ (u; -ep(r) - 2mlog(l + lz12)) , for e < 2~' - 6. II, 2 Now note that e(~+6-2"')~(z)+4mlog(l+I~l ) E La when e < 2~' - 6, and 1~12~-6~(z)-~mlog(l+lzl') E ~1 . 14This should be true without assuming that the objects are sets. LEMMA 2.3.3. Suppose v E L2 (U; -Sp(z) - 2m 1og(l+ Iz12)) and < AeKP on U. Then vd E L~ (u; ep(z) + 2m log(l+ lzI2)) , for 26 - s 5 E. Note that e(2~-~-6)I'-4m log(1-k I "I2) E Lm when 26 - 6 5 E. REMARK 2.3.4. Note that p@(I 0. Then p@'(U) --+ p@(I{O) has dense image in the topology induced by lim L2(~,;-~p(z)-2mlog(l+/z12)). -+rho PROOF. Note first that p@(U) injects into pO(I{o). Its image will again be denoted pB(U). The Hahn-Banach theorem will be applied to show lim L~ (LC; -~p(z) - 2m log(1 + IzI2)) and ,-L (#(u)) = 0 implies p (p@(I E; i. e.F = 0 on K,C fl (Cn Let be the densely defined operator T = 8. According to proposition 2.2.1 of Hormander [1965], 9(p,q) is dense in graph norm in Dom(T). It follows that aw - =~c~~F~~A, for wtDom(T). In paticular this is true for w E 6' (L,; [p(z) + 2m log(1 + lz12)) since w can be extended by 0 to all of U, and since F and u both vanish outside L,. Such w are thus in Dom(T). The formula above shows that Recall that (2-8) Hence p vanishes on '6 (L,; [p(z) + 2m log(l 4- lzI2))- BY 1emma 2-33 *I Since E < 2K1 - [ 5 2~ - [ =: ?11, Thus p vanishes on B(L,; -~"p(z) - 2mlog(l + lzI2)). By lemma 2.2.11 d (L,; -E"P(z) - 2m log(1 + lzI2)) is dense in 0' (L,; -~p(z) - 2m log(1 + lzI2)). Thus p vanishes on d (L,; -~p(z) - 2mlog(l + (zI2)). But note that P;l (,@(KO)) = B (L,; -cp(z) - 2m log(1-t lzI2 1) . So the proposition is proven. LEMMA 2.3.6. Let I be a directed index set. Then lim A. - lim lim Aj. j '--k--tj>k PROOF. (Here j > k means j 2 k and j # k. Suppose given fj : Aj + B. Consider the diagram The maps into l~l,p, Al and l~l,j,,J A1 are well defined, and a unique dashed arrow exists. REMARK 2.3.7. The lemma above is applied to the theorem below in step 2 withindexset I={IcV :KCC I 0. Then pB(U) -, p8(IC) has dense image. 16Kawai [1970]. cf. Hijrmander [I9901 theorem 4.3.2. Kawai states his result only for subsets in W , eventhough it is applicable without this restriction. p(I<) = lim limp&dd (V; -Sp(z)) + + U>>V_>K 6 = lim lim limp@Bdd (W; -Sp(r)) , (Kv := cZ,-- {OV < 0)) , + ++ Kv>KW>Kv 6 = lim p@'(IK By proposition 2.3.5, pB(U) is dense in p@(I 0 one of these neighbourhoods has a function of controlled exponential type (~,p). Then the sequence is exact. h PROOF^^. Let f E &(K) satisfy 8f = 0. Since I< is compact and P is Haus- dorff, ,X(K) = .X(V) where V may be assumed to be relatively com- pact W-pseudoconvex neighbourhoods of I<. The representative off in &(,,,) (V) for some V satisfies F 18Berenstein & Struppa [preprint]. lgKawai [1970], Saburi [1978], Nagamachi [1981], Meril [1983], Berenstein & Struppa [preprint]. 'O~awai [1970], Saburi [1978], Nagamachi [1981], Berenstein & Struppa [preprint]. ~aburi [1978]. By choosing K. sufficiently small, and restricting f to a smaller W-pseudoconvex neighbourhood if necessary, we may suppose that T), f E PX(p,q)(V), where $, is a function of controlled exponential type (~,p). Since a($, f) = 0 there is a g E PX(,,,-l)(V) such that ag = $d, by proposition 2.4.3 and lemma 2.3.2. Then 32 = f and 2 E J(p,,-l)(V). COROLLARY 2.4.522. There is an exact sequence PROOF. This follows from the assumption that points at infinity (Q, -(Cn) have a basis of neighbourhoods having functions of controlled exponential growth p for every K.. COROLLARY 2.4.6. Let I< be a compact subset of 6 satisfying the conditions of proposition 2.4.4. Then Recall the following theorem from I 0. Then H~-(u;~B)=O, fork#% and H( 0) 2 p(I<)'. These results together yield the main theorem of the chapter. 22Kawai, Saburi, Nagamachi, Berenstein & Struppa. THEOREM 2.4.823. Let K C I<' G 6 be two compact subsets of 6 satisfying (1) K' and I< have fundamental systems of 49'-pseudoconvex neighbour- hoods; (2) there is an open W-pseudoconvex neighbourhood U of I<' having a holo- morphic function of controlled exponential type (~,p) for any K > 0; (3) there is a function Bv for every 6 neighbourhood V of I< satisfying the conditions of theorem 2.3.8. Then H$,-,(6;W') = 0 for k # n. PROOF. Recall that rz(X; 9) = FZ(V; 9), where Z is locally closed and V is an open set containing Z as a closed subset. Thus for the situation here H; (u; W) = Hfi (6 ; 49') = and similarly for I('. Now consider the long exact sequence h Since H&, (P; W) = Hk(6; W) = 0, for k # n by corollary 2.4.6 and theorem 2.4.7, H!~,-~(~;PB) = 0, fork # n- l,n. 23 cf. Kawai, Kashiwara & Kimura [I9861 proposition 2.2.2. For k = n - 1, n there is the exact sequence A By corollary 2.3.9 pb(Ir')' 4 pb(K')' is injective. Hence H;I,;:,((C~ ; W) = 0; 1. e. H,(;~)=O, forkin. A REMARK 2.4.9. If K and I<' are compact in Cn satisfying Ir' = I{;, the -P plurisubharmonic hull of K, and I<' = KtU, then the conditions of the the- orem are automatically satisfied by remarks 1.2.6 and 2.1.3 above, and theorem 2.6.11 in Hormander [1990]. (See also scholium 4.3.1 below.) Thus the theo- rem generalizes proposition 2.2.2 of Kawai, Kashiwara & Kimura [1990], which states that H;~.-~(C~; 0) = 0 for k # n when I< and I" are compact analytic polyhedra. D CHAPTER I11 TOPOLOGICAL LEMMATA Who, if I cried out, would hear me among the angels' hierarchies? and even if one of them pressed me suddenly against his heart: I would be consumed in that overwhelming existence. For beauty is nothing but the beginning of terror, which we still are just able to endure, and we are so awed because it serenely disdains to annihilate us. Every angel is terrifying. -Rainer M. Rilke, Duino Elegies [1923].' The purpose of this chapter is to show that the traces at infinity (definition 1.3.1) of certain neighbourhoods are well behaved. The method used is simply to look at the asymptotic expansions of the functions that define these neigh- bourhoods. These calculations are simple and terrifying, but, unfortunately, not beautiful. $3.1 Exliaustion functions The following functions will be crucial in this and the next chapter. While they play an important role, their importance is merely technical in that they serve only to make the machinery work. CONVENTION 3.1.1. In this and the following chapter, sums over k run from 2,. . . , n, while sums over j run from 1,. . . , n (n being as usual the n in (Cn). A 'Translated by Stephen Mitchell. (1) ,oa(z) := Cklzk12 +ylv2 + xjlY j2+ I 1 where x(xl - 1/~) x(xl - l/a)' For simplicity x will not be explicitly written in most cases. Instead pa shall be written as NOTATION 3.1.3. To simplify notation let x := x1 - l/a when dealing with pa7 and x := x1 - l/e when dealing with p'. No confusion should arise from this imprecision. Superscripts are used to denote coordinates, and this necessitated the more perverse notation x12 (etc.) for (xl - 11~~)~. A 3~he definition of pa is essentially due to Nagamachi [1981]. The idea for the function yl comes from a similar function in Kawai Kashiwara, & Kimura [1986]. Thus REMARK 3.1.5. (1) $, > 0 if and only if NOTATION 3.1.6. For the rest of this chapter let xo := (170,. . . ,0) E Dn - IWn. A (1) pa is Cm strictly plurisubharmonic where it is defined; (2) Let S, := {z E Cn : pe(z) < ?1, and let 3, = int6c16S,. Then is a fundamental system of neigl~bourhoods of xo + iO. (3) SE is W-pseudoconvex, having as exhaustion function and exhaustion sets. PROOF. (1) Since the last two terms of pa7 Cj yjt2 and l/xt2 are Cm plurisubharmonic where ever pa is defined, it is sufficient to show likewise for the first term of pa7 Compute the Levi form: Thus the matrix of the Levi form is This is positive definite. (2) Let and let N := int6 clF N' C 6. Then N is a neighbourhood of xom + iO. Let z E N'. Then x1 > $ implies that . since E < 1. Thus So z E S,. Thus N C 3'. Clearly given any "conic" basic neighbourhood of xo + iO, there is an E such that 3, is contained in that neighbourhood. (3) t H I/(& - t) is a convex increasing function for t < E. Hence q'(z) := 1/ (E - pC(z)) is Coo strictly plurisubharmonic. The corresponding exhaustion sets are {q' < c}, for < c < m, or since q' < c if and only if p' < E - f, these sets can be rewritten as {p' < p}, for 0 < ,b' < E. 53.2 General Lemmas LEMMA 3.2.1. If A is open in a topological space X then clxintxclxA = clxA. PROOF. A C intxclxA so clA E clxintxclxA. If C is closed and C _> A then C 2 intx clxA. So C _> clxintxclxA. Thus clA 2 clxintx clxA. LEMMA 3.2.2. Let X be a topological space and let U be an open subset of X. For A c X, (clxA) n U = clU(A n U). PROOF. (clxA) n U is a closed subset of U containing An U. So (clxA) n U _> clu(An U). Now let x E (clxA) n U. Then x E U and every neighbourhood N of x meets A. Since U is open, N n U is a neighbourhood of x. So N n A n U # 0. This implies that (N n U) n (A n U) # 0. Thus x E clu(A n U). LEMMA 3.2.3. Let X, U, A be as in the previous lemma. Then (intx A) n U = intv(A n U). PROOF. (intxA) n U is open in U and is contained in An U. So (intxA) n U 2 intu(A n U). On the other hand, intu(A n U) is open in X since U is open. Thus (intxA) n U 2 intu(A U U). LEMMA 3.2.4. Let A be an open convex subset of a TVS X. Then intxclxA = A. PROOF. Clearly A C intxclxA. Suppose now that zo E intxclxA. By transla- tion, zo can be taken to be 0. Since 0 E intx clxA, there is a neighbourhood, N, of 0 such that N C clxA. By considering N n -N we may suppose N = -N. Since 0 is a limit point of A, and A is open, there exist a and an open set V such that a E V G A n N. Then -V G N. Now -V must contain a point of A, for otherwise, -a E N is not in the closure of A, contradicting N C A. So there exists b E V n N c A such that -b E A n N. Since A is convex, 0 = '6 2 + f (-b) E A. Thus intxclxA A. LEMMA 3.2.5. Let A C Cn . Suppose that inb c&p A = A. Then int6 c16 A = A U tr,A. PROOF. Clearly ints cl6 A _> A U tr,A. Suppose 20 E int,-- cl,--A. If zo E @", then ro E (id6 elF A) n Cn = intQln ch A = A. h If zo f Cn - Cn , then there is a neighbourhood r of zo such that I' C cl3 A. Hence I'n Cn G c16A n Cn = cbA. Since I" n Cn is open in Cn, I'n Cn C intQln c&p A = A. By definition, zo E tr,A. Conversely there is LEMMA 3.2.6. Suppose A S Cn is open. Then int@ c16 A = A U tr, A implies intQln cb A = A. COROLLARY 3.2.7. IfA & Cn is convex then intGclQIA = AU tr,A. LEMMA 3.2.8. (1) inb ckn {pa < c) = {pa < c), (c > 0); (2) inb ~kn {$a > 0) = {$a > 0). PROOF. (1) Since {pa < c) is open, inb cb {pa < c) _> {pa < c). Now let z = (xl, yl,. . . , xn, yn) E inb cb {pa < C) inb {pa 5 c). Since a neighbour- hood of z in Cn must project to a neighbourhood of (yl,.. . , yn), and since z is an interior point of {pa 5 c) we cannot have because increasing the values of yj's will increase the value of pa. Thus z E {P" < 4. (2) AS above inb ckn {$a > 0) _> {$a > 0), and inb C& {$a > 0) inkn {$, > 0). Let B 2 in& clp {$a > 0) be a neighbourhood of z and suppose that $,(z) = 0. Since $, is harmonic and $,(z) 2 0 for z E B, the minimum principle implies that $, - 0 on B. Rut $, is real analytic when x > O(x = x1 - llcu), so $, EE 0. This is clearly impossible. Thus $,(z) > 0; i. e. in& ckn {$, > 0) {$, > 0). (1) inte el(@ {pa < c) = {pa < c) U tr,{pO < c); (2) inte el(@ {$a > 0) = {$, > 0) U tr,{$, > 0). (1) cb{p" < c) = {pa _< c); (2) clp (4, > 0) = {$, 2 01, for a outside a set of measure 0; (3) ink- {pa I c) = {pa < c); (4) inb {$a 2 0) = {$a > 01, for a outside a set of measure O. PROOF. (1) Suppose z = (xl + iyl,. . . ,xn + iyn) satisfies ~~(z) = c. Then zt := (xl + ityl,. . . , xn + ityn) 0 5 t < 1 satisfies and zt -, z ast f 1. (2) This is a consequence of Sard's theorem. Recall (remark 3.1.5) that $, = 0 if and only if $0 = -a. Since $0 is Cm when x > 0, ($0 = -a) is a Cm hypersurface in EX2n when a is outside a set of measure 0. Suppose z E Cn satisfy $,(z) = 0. Since {$, = 0) is a (smooth) submanifold of there is a sequence z, E {$, > 0) that tends to zlO. (Take for instance z, to be a sequence along the normal.) So z E clp {$, > 0). (3) is a corollary of (1) and lemma 3.2.8. (4) is a corollary of (2) and lemma 3.2.8. $3.3 Lemmas on Traces PROOF. It is clear that tr,{pa < c) _> Uo 0 such that t 2 T zt E {pa < c). Let Then < 0, since x: > 0. SO pa(zt) < p(zT) < c for t > T. Thus z, E clG {pa < pa(zT) =: c'). Next it is shown that z, E tr,{pa < c'). Let I := ] - 1,1[ . Let N, be the basic neighbourhood of z, defined by Claim: For sufficiently small e, N, n Cn 2 {pa < c'). Proof. A sketch of the proof is given. Let z: := TxO/e + SX' + iy' E N, n Cn. By drawing a picture, it is seen that zI, E {pa ,< c') for small E: let z" = I1 (xll" + iyll", . . . , any + iyn*"). Then Let z" = z~,~ = Tx,/E + iy,. Then the inequality (3-1) will remain true for y E ?In for all s when E is small. Hence zI, E {pa < c'). This proves the claim and the lemma. LEMMA 3.3.2. Let V cc U 6. Suppose U intG cl3 {pa < c). Then 3cr, 0 < c' < c such that V C intG cl~ {pa < c'). = U inte^clG{pa < d). O c} = Uw>d>c t~m{$~ > d}. PROOF. Clearly trw{$a > c) _> Uoo>d>c trw{$a > d}. SO we show Let z, := x, + iy, E tr,{$, > c) where x, E = IDn - Rn. Define zt := tx, + iy, for t > 0, and let N, be the basis of neighbourhoods of z, defined by N, := {x,/E + sx + iy, + ie2y E Cn : s > 0, x ? .BR~ (x*, e2) fl sn-1, y E In} U{xm + iy, + s2iy : x E BR~ (x,, e2) U Sn-l,y E In}, where I = 1- 1,1[ . By definition of tr,, 3eo > 0 such that 0 < e < ?0 + N, U Cn C {$a > c}. Let &.(s) := $,(re,,), where z,,, := x./e + sx + iy, + isZy E N.Cn. This of course depends on x and y. Explicitly xi/? + sxl - l/a (3-2) 4, (s) = a - (X~E + sx1 -- 1/a)2 + (yi + e2y1)2 + Cj(pjy* + e2pjy) Now examine the asymptotics of this function when 0 < E << ?0. Assume first that xi # 0. Consider each of the terms above separately. For convenience set xj = xi + SEX^, for j = 1,. . . , n. (1) 2nd term of (3-2): (2) 4th term numerator of (3-2): (XC,X*'~ - ~e~C,(yf + e2y*)2 - e2Cj(pjy* + e2p'y)) 2 1 212 x ((XI - - e (y, + e y ) - 2e2 (y: + e2y1)) 2e2 (~xc~x~~: + sZyk) + CI.pkx + P'X - ?/a) x (X1 - e/a)(y; + e2y1 + 1) I (3) 4th term denominator of (3-2): (4) 4th term of (3-2): Putting the numerator and denominator calculated above gives (5) Hence s . XC,Xk*2 &(s) = a - + xjy: - (XI - ?/a)2 -+ o(s) > c, for small e. (XI - ?/a) If e (small) is decreased, 4, will decrease because of the second term. Now s occurs, if at all, only in the denominators of each term, including the o(e)- term. Moreover as s increases, the second term decreases in size, so that 4, has its minimum at finite s. Thus one sees that reducing 6 to say 6' provides the estimate inf $,,(s) > C. s,xEBR~(x*;E~) Let d = inf,,,EsRn(r*;r~) $.~(s), then N,, n Cn G tr,{$. < d); i. e. z. E trm {$a < d). (6) Consider now the case x: = 0. As before let z,,, := x,/e + sx + iy, + ie2y E N, n Cn . For notational simplicity, let y" = y, + e2 y. When xi = 0, (3-2) reduces kk 2XCk(x:/f f Sx )c + xb(~k~*/~ + spkx) +. [' + P'x*/? + sp1x - l/a x (sx' - 1/a)(g1 + 1) L y . ((sx' - 1/a)2 - -It2 - 2ijq2 + 4(sx1 - l/a)2(g' + By assumption 3e0 such that 0 < E < ?0 N, 17 Cn C {+a > c); i. e. 44s) > c for z,,, E N, n CY This inequality must remain true for x' = 0. In this case only the 4th term depends on s. So consider its behaviour when s is large. As the denominator will in this case be independent of s, only the numerator will be significant. (6;) 4th term numerator of (3-4): Let Ax; := xi - xi. Then the above numerator can be rewritten as 2XCk ((s + $)xt + SAX:) ek a (el + 1) + C' ((s + $)pix* + SP'AX*) - l/a For s >> 1 and 0 < E<< ?0, this gives - X(zk ((s + f)2 xt92 + 2 (s + j) SX~AX~ + s2~s:72))(l/a2 - elf2 - 2c1) + 0(s2) This estimate provides an upperbound on 4, as follows. Recall that c is small but fixed; that 1 = c jx:'2 = ~~x:'~ since x: = 0; and that x E Barn (x, ; e2), so that < r2. Then Thus, since r is small, (64 The next goal is to show provided 0 < E 5 r' for some r' which can be taken to be less than ?0. Let 0 < e" < E'. Once this is proven the proof of the lemma is completed by noting that (3-7) d' : = inf +.(x* /st' + i y, + id"' y) ~?1" Then choose d such that c < d < dl. For z,~,, E N,I n Cn; z,I,, := x,/d + sx + iy, + ii92y, $a (~?1 ,s ) > $a (~?1 ,o) > min $.(x,/el + iy, + ie~~~) yEIn = dl > d. Thus Net n Cn 5: {$, > d); i. e. z, = x,m + iy, E tr,{$, > d). (6iii) Differentiating (3-4) yields As in 6;) above, let y" = y, + e2y. When x: = 0, (3-8) simplifies to (3-9) (6iv) First 2 terms of (3-8): But xi = 0 and x E Bw (x* ; e2), implies la1 1 < c2. Thus the first 2 terms is o(e). (6v) Numerator of the 3rd and 4th terms of (3-8). First consider the numerator of the 3rd term of (3-8): (numerator of the 3rd term of (3-8)) x (((sxl - ~/a)~ - sly2 - 2~~)~ + 4(sx1 - ~/a)~(sl + I)~) 1 The numerator of the 4th term of (3-8) is With common denominator the numerator of the 3rd and 4th terms of (3-9) combined is Note that the o(eO) term cannot be disregarded because it contains s. This term will be studied separately in each of the two cases below. (64 Case 1: x1 = 0. In this case the denominator (3-10) is independent of s while the numerator (3-11) including the o(cO) term explicitly written out reduces to In this case the o(eO) term is independent of s, so, for small E, the dominating terms are Thus putting (3-10) and (3-13) together yields Now >(I + SE) - (1 + 2s~) Jc~A~~~~Jc~~:~~ - SEE 4 >O for e sufficient1.y small. (s > 0). Together with (3-5), this shows that (3-14), the 3rd and 4th terms of (3-8) combined, is greater than 0 for small e and all s > 0. (6vii) Case 2: x1 # 0. In this case the largest power of s in the denominator (3-10) is 8 while it is at most 7 in the numerator (3-ll)(this includes the o(rO) term). Thus the o(rO) term in (3-11) can be estimated by bounds independent of s, x1 being estimated Since Ixl 1 < r2, the terms in the numerator involving x1 are o(r2), and can be grouped with the o(rO) term; thus the sum of the 3rd and 4th terms of (3-8) As in the previous case Again the terms containing AX; are o(el) and are thus o(rO). Thus (3-15) reduces to - -2X(1+ sr) ((sxl - l/a)2 - jjli" gl) + 0 (E-~) ?2 ((sxl - l/a)2 - - gq2 + 4r2(sx1 -- l/a)2(Sl + 1)2 > 0, for sufficiently small E. independent of s. (6;s) Together with the first two terms computed in (6iv) it follows that d4e -(s) > 0, for sufficiently small E and any s > 0. ds This proves the lemma. COROLLARY 3.3.5. For a outside a set of measurle 0, LEMMA 3.3.6. Let V CC U C 6. Suppose U c int6 cle {$. > c). Then 3c1, co > c' > c such that V C int6 cl6 {$, > c'). PROOF. clFV is compact and cl6V C Um>d..,c ~nt~cl~{+~ > d). LEMMA 3.3.7. For c' > c, c& {$. > c'} idf; elG {$, > c}, when a, c, c' are outside a set of measure 0. PROOF. As in lemma 3.3.3, this is equivalent to proving cl~tr,{+~ > c'} G tr,{$, > c}. Let z, = x,co + iy, E c16tr,{$, > c1),x, E So there is a sequence 2, = XmW + i~m + z*, zm E trm{$a > c'}. (1) Case 1: x: # 0. Since x, + x, it can be assumed that x& # 0. Then from (3-3) in the proof of lemma 3.3.4: where the ?,/a terms are collected with the om(?,) term. Since x in (3-3) is here chosen to be x,, AX^ = 0. Moreover, choose 6, -, 0 as m -+ oo. By assumption $a(x*/?m + SmXm + iym) > c'? VSm > O when em is suffi- ciently small. On the other hand, for x E BWn (x,; e2) n and y E In, + om (em) + o(E). Clearly the 2nd, 3rd, and 4th terms can be made arbitrarily small for small e and large rn. The 5th and 6th terms combined give: This can be made arbitrarily small since since (X~~X:'~ - ~2~x1'~) -3 0 as m + m, and the other terms in the numerator contain AX:-terms which are small for 0 < E << 1. As in lemma 3.3.4, the power of s in the numerator is no greater than that in the denominator. Thus by choosing all the terms except the first term to be less than S in absolute value, $a (x+ /E + SX + iy* + ie2y) 2 $a (xm/% + SmXm + iyrn + iym) - 6 > c'-Szc, for O c). (2) Case 2: xi = 0. In this case pick zm --+ 2,. We can assume zm # z, and since tr,{$, > c') is open choose z, so that xk # 0 for all m. Consider Ga(xt/c + sx + iy, + ic2y) = $a(xm/cm + smxm + iy,) ($a(x*/c + sx + iy, + ic2 y) - $J~(x,/E, + s,xm + iym)) . By choosing E, sufficiently small, the first term on the right is greater than c' for all sm 2 0. For simplicity, let The next step is to estimate T, - Tm. From (3-4) and (3-16) -xCk(x: + + xE~c~~~'~ + E2~k~ky*) x ((sx' - I/Q)~ - 8:~~ - 2~:) kk 2XCk(xt + sex )g* + Ck(pkx, + SLP~X) I' + P1x, + seP1x - ?/a x (sx' - l/a)(g: + 1) The first term is bounded. The 2nd term can be made arbitrarily small by choosing m sufficiently large so that em is small, and then fixing m and choosing sm large. Since ym i y,, Cj(I'jy, - Pjym) > -6 for an arbitrary S > 0 by choosing m large. To estimate the 5th term, consider the following two cases. (3) Case 2a: (xi = 0 and) x' = 0. In this case the 5th term reduces to Estimate the numerator The first term in square brackets is greater than 0 by (3-5). Thus choosing e << 1 makes the 5th term as large as needed. (4) Case 2b: (xt = 0 and) x1 # 0. Because the power of s in the numerator is less than that in the denominator, the dominating term in the numerator, for small e, will be since lxl J = Is1 - x: 1 < e2. Again by (3-5), this is greater than 0. Thus the 5th term can be made as large as needed. In conclusion T, - Tm > -6 for S > 0 by suitably choosing e << 1. Thus T* = Tm + (T* - Tm) > C' - (c' - C) = c on N,. SO Z* E tr,{$, > c). This proves the lemma. CHAPTER IV THEOREMS ON PURE CODKMENSIONALITY AND FUNDAMENTAL EXACT SEQUENCES Whoso has sixpence is sovereign (to the length of sixpence) over all men; commands cooks to feed him, philosophers to teach him, kings to mount guard over him,-ta the length of sixpence. -Thomas Carlyle, Sartor Resartus [1833]. In this chapter we show that SO is pure 1-codimensional with respect to r-'49 (i. e. X&(r-lW) = 0 for k # 1); and S*R is pure n-codimensional with respect to T-~W. Since the Fourier-Sato transform works just as well on $2 C Dn as on a real analytic manifold, many of the usual results for microfunctions on a real analytic manifold are seen to remain true for Fourier p-microfunctions. Specifically one has the usual short exact sequences on the sphere and cosphere bundles, SR and S*R respectively. These are stated in 54.3. Some preliminaries are needed to begin. Proposition 4.1.2 allows one to smooth plurisubharmonic exhaustion. It is modelled after a classical result. Next we recall the Grauertl tubular neighbourhood theorem in the form Kawai l Grauert [1958], $3. used for open subsets of IDn. The proof given here is almost exactly Harvey and Wells7 [I9721 proof that dispenses with Grauert's original cone construction. Finally essentially by intersecting the Grauert tubular neighbourhood with a wedge, we show that every point of GSfl has a basis of neighourhoods whose h projection on (Cn - IDn is W-pseudoconvex. (T-'@) can then be calculated using the classical proof. LEMMA 4.1.1. Let (X, U) be a uniform space, and let f : X -+ R and g : X + R be uniformly continuous. Then f V g := maz( f, g) is uniformly continuous. PROPOSITION 4.1 .22. Let U be an open subset of 6, and Ino a compact subset of U. Suppose q is a continuous plurisubharmonic function such that (1) {q < C) cc U, c E R; (2) suP~<~n~ q < 0; (3) for every compact subset K E U, suplincn q < m; and (4) for every compact subset Ii' C_ U, q is uniformly continuously on I< n Cn . Then 3i E Cm(U n Cn) strictly plurisubharmonic, ij > q, satisfying (I), (Z), (3) and (4). PROOF. Let := iq < j), and where y is a Friedrich mollifier, and Sj is chosen so small that vo < 0, and suP~,,-,~ v1 < 0. This is possible because of condition (2) in the statement and because for a compact set Ino, sup^^ 13m2I2 < 00. Moreover uniform conti- nuity of q, condition (4), shows that for j = 2,3, . . . the Sj 's can be chosen so =This is essentially the second part of Hormander [I9901 theorem 2.6.11. The same proof goes through with these new hypotheses. that vj < q + 1 on Vj since < 1, for small Jj. Thus there is a 6-neighbowhood, fi of c&p 4, such that on n Cn, vj is strictly plurisubharmonic (because of the 13mz12 term) and is > q. Moreover note that vj vanishes outside a Gj-neighbourhood of Vj+1. Let ~(t) be a convex Cm function that is 0 when t 5 0, and > 0 when t > 0, such that X' > 0 when t > 0. h Then x (vj + $ - j) is strictly plurisubharmonic in a C? neighbourhood of tip Vj - VjVl (intersected with C) since d2vj 2 x1($)gtmw7 for z outside Vj-5. Next inductively choose constants aj and define urn by h so that urn is strictly plurisubharmonic on a Cn n.eighbourhood of C& Vm, and Urn > q. urn can be chosen strictly plurisubharmonic since v,-~ vanishes outside a Sm-1 neighbourhood of Vm g.iving Thus a similar calculation as in (1-1) shows that a2x (urn-, + $ -- j) wt ---W > - -Crn-lIwI , outside Vrn-1. 2 azaz Choosing a, sufficiently large thus makes urn strictly plurisubharmonic. urn can be chosen > q since on Vm - Vrn-1 can be chosen greater than m! + 1, the maximum of q there. Let 6 := limrn+, urn. 6 .is Coo and strictly plurisubharmonic on U n Cn. This is uniformly continuous on I< n Cn since each vj is. ij satisfies the other requirements of the proposition. Recall the following Grau.ert tubular neighbourhood theorem from Kawai [197013. THEOREM 4.1.3. Let 0 be an open subset ofDn, and U a complex neighbour- hood of 0 such that U n Dn := 0. There is an W-,pseudoconvex neighbourhood WofOsuchthatO~W~lJandW~Dn=O. Moreover a strictly plurisubharmonic exhaustion function, q, of V can be chosen to satisfy (1) 4 2 0; 3See also Saburi [1985]. The proof given here follows Harvey & Wells [1972]. 77 (2) q is Cm on W (W considered as a manifold with boundary); (3) For every compact subset I< C W there is a constant XK such that the Levi form of q, Lq, satisfies Lq(z)(w, w) 2 Xalw12, for z E I<. PROOF. AS in Saburi [1985], let w be the Cm diffeomorphism of 5 onto B(0; 1) + iRn given by - + iy, if x = X'OO E Sn-lm = Dn - Rn, w(x + iy) = x +iy, if x +iy E Cn. Let I 0, 42) - e(a(z)) will be strictly plurisubharmonic on U fl Cn. This follows by directly computing the Levi form on each a-I (Kk) n Cn and using Lemma 2.2.1 of Saburi [I9851 (with his notation): for z E I c > 0 for some constant c, we must have d(yk, BC) + 0. Thus yk + Oor lfi-vol + e. But sinceyk +yo # Oand yo E B, thisisa contradiction. Thus suP~~@n - log d(z, Vt) < 03 Thus if I< is compact in U:, sup 8<00. ItnCn Moreover q is uniformly continuous on K n Cn. Next we show the same is true for -log d(z, Vt), and hence for 8. Let z,zl E K nCn. Then I - logd(z,Vz) + logd(zl,V,C)I = log 1 :::;cc; 1 But by what was proven earlier, l/d(z, V,C) < Mo on I< n Cn. Since d(-, V:) is uniformly continuous on Cn , it follows that - log d(., V,C) is uniformly continuous on I 1. Consider first the morphism T-~W' -+ j, j-lr-lW' = j,Wlnu-n in (1-2). It shall be shown that this map is injective. This map is obtained as follows. Let 0 C flu be a neighbourhood of zo E 0,. The map above is the direct limit as 0 runs through a basis of neighbourhoods zo of Recall that r-'W'(O) consists of sections a' o T, where a' is continuous and Thus since 0 n (R, - R) = 0 - SR the map above is a' o T c a'lii-sQ. - Suppose that a'lo-sQ = 0. If zo E R, - 52 = R, - SO, then for sufficiently small TO = 0 R, - 52, then a' = 0 and hence a = 0. On the other hand, if zo = xo + ivoO E SR where vo = 1 then we can take 0 to be the sets O6 defined as follows. If xo E Rn define If xo E IDn - Rn say xo = x'oa where IIx'II = 1, define { x r, := x E Rn - (0) : - E BRn (x', E) U {XCO E Sn-l~ : x E BRn (x', E)} llxll I A, := (r, + xo/e) + i{v : - v0l < E, ll~lil = 110 E SSt. In both cases define B, := {x + itv : 0 < t < e,x + ivO E A,, llvll = 1) 0, := (A? u B?) n 6, Then note that in either case 7.0, is open in R,. Hence a' is a section of PB, a' E W(T &). Thus a' E B(T 0, ir Cn ) is an analytic function and by the - uniqueness of analytic continuation, a'lir.-sQ = 0 implies a' 0 on TU.. This proves that T-~W' ---+ j, j-lr-'W = j,Wln, -n is injective. It follows from (1-2) that %&(T-'W') = 0. Now consider the isomorphisms (1-3). As before Rkj*j-l~-lW' = Rk j, (Win, -n). This is the sheaf associated to the presheaf For z E flu, (1-4) ~*j,(p@l~,-~)~ = 15 H~(O n (Q, - R);PB). 03z If z E flu - SR = Q, - R, (1-4) becomes the direct limit over neighbourhoods If on the other hand z E 90, z = xo + ivoO, with xo E R, and llvoll = 1, take 8 = as in proposition 4.1.4, with oc forming a basis of neighbourhoods of z Then since Oc n (52, - R) is W-pseudoconvex, So in either case the direct limit vanishes. This proves the claim and the theorem. From the proof of the lemma one has COROLLARY 4.1.6. There is short exact sequence 84.2 Computation of R&n(n-f~) REMARK 4.2.2. As in chapter 3, we set so := (1,0,. . . ,0)m E llY - Rn. These sets Wa will be a basis of neighbourhoods of xo, as is seen below. We will then write these sets as a difference of compact sets, IC1 and I<:. D LEMMA 4.2.3. {Wa) form a basis of neighbourhoods of xo + iO E 6 for a > 0. PROOF. By definition Wa C intG c16 {pa < 2a). It remains to show that for sufficiently small E, {pa < E) C {$J, > 0). Suppose pa(z) < E, where as usual z = (xf + iyl,. . . , xn + iyn). Next each of the terms of $I, is estimated. Write = a - TI - T2 - T3 in (111.1-2). Since pff < e, it follows from (111.1-1) that Now examine the third term: T3 : = 1 + 2s (2XC,x'yX + Ckpkx + Px) (yl + I)( (xy2 - Y - 2yf)2 + 4xJ(yf + 2)2 The denominator of these two terms simplifies: Note that y112 + 2y1 + 2 > 0, so the 2nd summand of T3 can be estimated as follows Similarly for the first summand of T3, Thus TI + T2 + T3 < a, for t: > 0 sufficiently small. That is, $,(z) > 0 when t: is sufficiently small. This proves the lemma. LEMMA 4.2.5. For t: > 0 clew,-, 5 Wa. PROOF. Recall that W, := int6 ele ({$a > 0) n {pa < 2a)). Thus the lemma is a corollary of corollary 3.2.9, and lemmas 3.3.2, 3.3.6, 3.3.7, since $,-, > 0 if and only if $, > t:. ASSUMPTION 4.2.6. From now on assume that cr is very small, and in particular smaller than This is used in lemma 4.2.9 below. (1) G := {(zl ,... ,zn) : Ply 5 0 ,... ,Pny 5 0.); (2) K1 := ele (Gn {pa 5 F}); (3) K,Z := clG (K1 n {$, 5 0)). PROOF. From corollary 3.2.9, Wa fl G n Cn = {$a > 0) n {pa < 2a) fl G; i. e. 'the "ints cle" operator does not add points of @" to {$a > 0) n {pa < 2a).' Similarly (K1 - I{:) n Cn = G n {pa < :) n {$a > 0). Since 0 < a < :, 2a < f. So WanGna7. 2 K1-I<:. Let z E (WanG)-@". Thenbythe lemmas 3.3.2 and 3.3.7 (since z E Wa - Cn) there is a (conic) neighbourhood of z, say I?, such that I' n Cn E Was So $a(I' n Cn) 2 S > 0. Hence z $ CI@{$~ < 0). Moreover z E K1 since {$a > 0) n {pa < 2a) n G 2 K1. So WanG&K1-K:. PROOF. First we show I{' - Kz n Cn C {$, > 0) n {pa < 2a). From (111.1-3) (2-1) x $,>O@a> X'2 + y1,2 - CjpjY + (Xp" - Xp") As in the proof of the previous lemma (I<' - I-) n Cn = G n {pa 5 ;) n {$, > 0) C {pa 5 :) n {$, > 0). As usual let z = (xl + iyl,. . . ,xn + iyn). We shall show that if Pjy < 0 then $.(z) > 0 + pa(z) < 2a when z E {pa < P). To this end, we shall show that the first three summands together are nonnegative when piy 5 0 1) The first summand of (2-1) -Cjp'y - XC 3 .yi,2 = CjlpjYl - Xllyll: 2 Y - Y since ll~lla < 1, 2 KIIY112 - XIIY112 2 0, for X < I<. 2) The second summand of (2-1). (111.3-1) gives Moreover by assumption 4.2.6 Hence 3) Now estimate the third summand of (2-1) We shall examine the numerator of (2-2) by collecting powers of x. Let c4 denote the coefficient of xi4, co the coefficient of xy2, and let co be the last term of (2-4). By assuming a is sufficiently small, a computation shows a) co L 0 b) c2 > 0 since lyl 1 < fi when pa(z) < a/2. c) On the other hand c4 may be less than 0. However together with the term computed in "(2) above, we see that for small a this term is o(l/x). Thus the 2nd and 3rd summands of (2-1) together are greater than 0 for small a. Hence if t,ba(z) > 0 then This shows Now let z, E (K1 -I<:) - Cn. 3zn E Gn {pa 2; ;} - Kz such that zn -t z, in 5. By corollary 3.2.9, z, $ Ic implies there is a (conic) neighbourhood r 3 z, such that $,(I' n Cn) > 0. Then for z, cf I" cc I?, 3e > 0 such that $,-,(I" fl Cn) > 0 (lemma 3.3.6). Now zn E I" for large n, and by (2-5), lemmas 3.3.3 and 3.3.6, is contained ,I, {$a-, > 0) n {pa < ~(u-E)) = el,- int,- clG {$,-, > 0) CI {pa < 2(a - e)) = c1,- wa-c C Wa (lemmas 3.3.3, 3.3.6) So K1 - Kz C Wa; and since K1 - I<: G, I<' -- 11; C_ Wa n G. This proves the lemma. El The two previous lemmas show COROLLARY 4.2.10. I<' -I<: = Wa fI G, forsmall a > 0. REMARK 4.2.11. (111.1-3) shows that if b < 0 then intsc16{$s < 0) is a 6 neighbourhood of K1. D DEFINITION 4.2.12. Suppose a and a are given. Let 0' := rnax(P$,, pa - z, P1 y,. . . , Pn y), where p > 0 is chosen so small that a- $ < 0. Let 8" := max(pa - f, Ply,. . . , Pny). Let Uo := int6 cl6 {Of < $1. Let G:=int~cl,-{z~C~:~~~~0,j=1, ..., n). 0 REMARK 4.2.13. By remark 4.2.11, Uo is a neighlbourhood of K1. D REMARK 4.2.14. 0' is plurisubharmonic, and (0' < f) & {pa < a}. The same is true for 8". D LEMMA 4.2.15. 0' and # are uniformly continuous on {pa < ra). PROOF. Clearly each y I+ pj LJ is uniformly continuous on {pa < ra) . By taking the derivatives of pa - 5, and showing that each of the partials is bounded on {pa < 7-4, one concludes that pa - is uniformly continuous on {pa < ra}. Similarly the techniques of the previous chapter and those of lemma 3.3.4 in particular show that 2da has bounded partials on {pa < ra) . LEMMA 4.2.16. int@ cl@ {O' < e)o<, 0 since = (2 inte ~1~ {$a > d)) , by lemma 3.3.5 In particular one has Similarly Moreover each of the sets in the intersection on the right hand side is a neigh- bourhood of the corresponding set on the left. Note that is a neighbourhood of Ki. In fact these form a fund.amenta1 system of neighbour- hoods of KZ since K; is compact and 6 is metrizable. These sets are relatively compact for each E and tend to I{: as E tends to 0. Moreover these sets are W- pseudoconvex: by lemma 4.2.15, 8' is uniformly continuous on compact subsets; then consider (E - el)-'; finally smooth these according to proposition 4.1.2. This proves the lemma. Similarly one has LEMMA 4.2.17. intF; ~1~ 16 < e)o<,cl is a basis of W-pseudoconvex neigh- bourhoods of K1. THEOREM 4.2.18. X&n(x-lW) = 0, fork # n. PROOF. By proposition 1.4.12 this is equivalent to showing lim HSnGt (V; '6) = 0, + for k # n. V3x0 G ' If xo E Cn then this reduces the usual result about microfunctions on Cn (scholium 4.3.2 below). Suppose first that xo = (1,O,. . . , O)m E IDn - Rn, and let G' = G (definition h 4.2.12). Since {Wa)o 0, f > EV such that I{: 2 int6 ck {0' - ry < 0) Now let xv(-) be a convex increasing function sucln that (1) xv is uniformly continuous on {t : t 5 d) for every d E R; (2) limt-+ xv(t) = m; (3) xv (7) < 0; and (4) xv 0 8' > 0 on VC. Then xv o 8' is a plurisubharmonic exhaustion function of Uo satisfying the conditions of proposition 4.1.2. Thus it can be smoothed to produce an W- pseudoconvex exhaustion function, Bv, of Uo. 8v satisfies the requirements of theorem 2.3.8. For general xo E Dn - Rn, take a unitary transformation R mapping xo to (1,O, . . . , 0)oo. Modify the functions pa and $, as follows: Ckl(Rz)'12 + (Ry)'12 + Cjlyj12 + 1 pa(.> = ((Rx)' - ~/a)~ ((Rx)' - l/a)' ' . XiC,(~z)~i' - Ckpk(~z) - (P1(Rr) - l/a) Qa(z) = ia+ CjPJz - ((Rz)' - 110)~ + %i((Rz)l - l/a) =a- (Rx)' - l/a ((Rx)' - I/CX)~ + (Ry)lJ + Cjp's Calculations similar to those in chapter 3 and in lemma 4.2.9 show that lemma 4.2.10 still holds. This proves the theorem. $4.3 Fundamental Exact Sequences SCHOLIUM 4.3.1. Let U be an open subset of a topological space X. Let Z be a locally closed subset of a topological space U and V an open subset of U containing Z as a closed set. For a.sheaf 9 on X, H~(v; 9) = H~(v; 91~). PROOF. In fact take a flabby resolution 2' of 9. Then since U is open 9' Iu is a flabby resolution of 91~. Moreover rz(V; 9) = rz(V; Flu) and similarly rz(v; 3') = rZ(v; 2.1~). PROOF. This follows from the previous scholium since ?%a (U) = HEnU(U; p8), and WIc = 8. SCHOLIUM 4.3.3. l~nnllg")-tmsn-l~ = ennRn . PROOF. Let x, + i[,m E (a n Rn) + flSn-lm. Then (xp)+ lim -----+ H:,~(V; PB), But since x, E Rn, V runs through bounded of Cn. Thus again by scholium 4.3.1, (4-1) becomes the usual limit for microfunctions on U Rn. We have now computed all the terms in the triangle (1.3-5) of proposition 1.3.9. We suppose as always the conditions on the plurisubharmonic function p stated in Chapter 11. Recall that Rrn(W)[n] vanishes except in degree 0; it is the sheaf of Fourier p-hyperfunctions, ?%a, and Pd vanishes except in degree 0. From the long exact sequence associated to the triangle in proposition 1.4.9 it follows that Rjx,WR = 0, for j # 0. From theorem 4.2.7 we now have the stronger result that W is concentrated in degree 0. Thus there is THEOREM 4.3.4. Let 52 be an open subset ofDn. There there is a short exact sequence 0-Pdo -??afi ---t~*%~ -0 PROOF. Take the long exact sequence of triangle (1.3-5), and use theorem 4.2.7. Similarly one can produce the other short exact sequences involving Wn as in Sato, Kawai & Kashiwara [1973]. References Berenstein, C. A. & Struppa, D. C. [preprint]. 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