JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES ANDERS MELIN On the construction of fundmentl solutions for differentil opertors on nilpotent groups Journées Équtions ux dérivées prtielles (1981), p. 1-5 <http://www.numdm.org/item?id=jedp_1981 A15_0> Journées Équtions ux dérivées prtielles, 1981, tous droits réservés. L ccès ux rchives de l revue «Journées Équtions ux dérivées prtielles» (http://www. mth.sciences.univ-nntes.fr/edp/) implique l ccord vec les conditions générles d utilistion (http://www.numdm.org/legl.php). Toute utilistion commercile ou impression systémtique est constitutive d une infrction pénle. Toute copie ou impression de ce fichier doit contenir l présente mention de copyright. Article numérisé dns le cdre du progrmme Numéristion de documents nciens mthémtiques http://www.numdm.org/
Conference n 15 ON THE CONSTRUCTION OF FUNDAMENTAL SOLUTIONS FOR DIFFERENTIAL OPERATORS ON NILPOTENT GROUPS by A. MELIN I shll here outline some steps towrds the construction of prmetrices ( nd locl fundmentl solutions) for some clsses of differentil opertors on nilpotent groups. Some of the results re s yet only estblished for 2-step groups, but it seems plusble tht the technique developed is pplicble to ny group which is grded nd nilpotent. (For other tretments of the rnk 2 cse we refer to Geller [l] nd Miller [3,4]). 1. Generl considertions Let Q. be grded nilpotent Lie lgebr of step r. This mens tht ^= ^^... ^3^ direct sum of vector spces, where \3., ^.] c: ^.. (=o if i+j > r). We consider ^ x.y = x + y +j[x,y] +^ (dx) y + s Lie group t the sme time with multipliction (dy) x +... given by the Bker-Cmpbell Husdorff formul. Let ^ be the subspce of <?' (<?) consisting of distributions which re rpidly decresing t oo nd smooth outside the origin. Then ^ is closed under group convolution nd if T is unitry irreductible representtion of ^ then T(u) is defined on the Schwrtz spce T(Uk V) = TUo TV. ^?(T) of T when u ^.W e hve The lgebr D(^) of right invrint differentil opertors on CL is vi the mp P -* P(S identified with the sub lgebr of (^,*) consisting of elements supported by the origin : If P D(^) nd u C W we hve Pu = Pfi * u. Thus (J 0 (7 to find fundmentl solution (or prmetrix) for P mounts to finding n inverse for Pfi in ^ (modulo the idel <^(4) ). We denote the Eucliden Fourier trnsform of u by Fu = u nd ^ is the vector spce dul of L. The symbol of P 0 /\ * ^ is by definition the polynomil p ) = (P6 ) ( ;) on 4 Since Q is grded there re nturl definitions of qusi-homogeneity for opertors nd functions on j2 ^ etc.. We set Pol"^ ^ ) = ^ Pol (4 ), (7 * ' ' 0<j<m j where Pol.(^ ) is the set of polynomils on 4 which re qusi-homogeneous of degree j. If Ot = (,..., ) is sequence of multi indices corresponding to r 3k choice of bses for the Q. then we set II ll = S j.1. For ; = ^ +...+ ^ ^ o - J i J l r (7
2 j^ 3 we let 11^11 = Z ^. be the homogeneous norm. Then S"^ ) is the set of ll I D U 1 oo * C (4 ) for which for ll. sup (1 + 1101 )" " " m ld (i;) < oo ^ Definition 1.1 : There is self djoint differentil opertor $ = $(^n, D,D ) = Z $ (^,T1.,D,D,...,D.,D ) on ^* x O* - > 11 J J - ] y ^l " i -j-i "j-.i such tht (u*v)^) =^#<?(^ ^ (e^u-^^x^ when u,v -S (^). Moreover, $ is liner in ;,n nd qusi homogeneous of degree 0. Proposition 1.2 : The multipliction # extends to S (<2 ) = U S" 1^*) nd # b S" 11^" (^*) if S" 11 (^) nd b S" 1 " (^*). m 3R Definition 1.3 : is invertible in S (<7 )/-d(^). We sy tht p Pol" 1^ ) is elliptic w.r.t # if its principl prt Recll Rocklnds conditions for P nd P : (Ro) * V T(P ) nd T (P ) re injective on -0 m m T when T is non trivil unitry - irreductible representtion of ^L. We wnt to prove the following (which is still unproved for r > 2). (C) p is elliptic if F^p stisfies (Ro). Remrk 1.4 ; One would insted consider right nd left inverses. Under the ssumptions bove it follows from the clculus for # tht it suffices to consider the cse when p is qusi-homogeneous. Non elliptic polynomils p might be invertible w.r.t. # fter dding lower order terms to p.
2. The induction step dimensions. Assume tht the result (C) is lredy proved for ll groups of lower Let p C Pol (^ ) stisfy (Ro). Choose vector e ^ 0 in 01 nd consider m o ^r ^ the quotient group ^ = ^ /SRe. The imge P of P under the projection / ^' ^ y^ will then stisfy (Ro) nd the symbol of P is the restriction of p to W : <^,e> =0. This implies tht one cn find q S (^ ) so tht p # q = 1+b with b in S 0 '^ ) so tht S^(^ ) is the set of ll b S^ (^ ) so tht s \i + iigi 7 Sup f 1^ ^'Fd ^ 11^11 ^ " - k! D^) I < oo ^ 0,1 /^* 0, * for ll. The clculus then llows us to replce S ( 6? ) by S (^ ). Set S 1^00^ ) == {b S 3^00^); b = 0 for < ;, > smll or negtive} nd S (~SR ; ^0( W)) the spce of <?(W) -vlued order k symbols on 3R vnishing for smll t. Considering t = <E,,e> s prmeter we obtin nturl isomorphism S^(f) -S^IR^W)). In the right hnd side we my view <>(W) s the restriction of ^(^') to <^,e> = 1 nd the restriction of # to this spce is well defined. The multipliction # will then be respected by the isomorphism bove in view of the homogeneity of $. By single prtition of unity w.r.t. the vribles ^ we my lso ssume tht hs its support in smll conic neighborhood of vector e*. This will imply tht I ; I < Cst. in the support of the imge 3 of under the isomorphism considered bove. 3. The rnk 2 cse Let (V,,g) be symplectic vector spce with positively qudrtic * * form g on V with dul form ^ on V. We shll ssume tht we hve bound (3.1) 0(x,y) 2 < C^g(x)g(y) with C fixed ll the time. If u e o 0 * 0(V ) we set u,(^) = Z mx lu^1 (^.,n-...,ti.) /glen,) 1/2...g*(n.) 1/2. D 1 : 1 3 j<k n * * Note tht defines differentil opertor D on (V x V/) nd n ssocitive multipliction # on >5(V ) is defined by
id /2 (u # v) (^) = e u u(^) v(n)/ S== U. After quntiztion every u -o(v ^ *) cn be viewed s the symbol of pseudo 2 differentil opertor nd its opertor L -norm is independent of choice of quntiztion s well s its Hilbert-Schmidt norm. We denote these by Hull nd H u l l -. Set Hull, = mx lu!( ^). rn grk k 0,L ^ Lemm 3.1 : There is positive C = C(n), n = dim V so tht the following Jt * ^ * holds : If u G "0(V ) nd Hull n < c then there is unique v in -<?(v ) so tht fl (1-u) # (1-v) = (1-v) # (1-u) = 1. There re lso mps k -> k', C' depending on n nd C so tht K. 0 llvll, < c'd + Hull.J^ Hull -. grk k g,k' g,k' By using (Ro) nd Lemm 3.1 with 0 = B>. (x,y), x,y Q /Rd B,- (see lso -' 2 '2 the cse Q =the Heisenberg group treted in Melin [2]) one cn lwys modify so tht S( = considered s n element in S (3R, -^o(w)) vnishes long some orbit 0(^) for the co-djoint representtion with» = rj = fixed element in + 2. 2. CL. The norm IIXIL 3 is not chnged much when one pss to nerby orbits. This llows one (by nrtition of unity rgument) to find with supll'sii ^ smll when ^ = r\ e,l 2 nd n ppliction of Lemm 3.1 gives then n vnishing identiclly for ^ = F). Finlly one hs to consider derivtion w.r.t. ^ for ^ in ^ z. 2. compct set. These derivtives must be smooth where B >- hs mximl rnk nd the 2 estimtes we obtin for then re uniform. REFERENCES [1] D. Geller : Locl solvbility nd homogeneous distributions on the Heisenberg group, Comm. Prtil Differentil Equtions, 5 ( 5 ), (1980) 475-560. [2] A. Melin : Prmetrix constructions for some clsses of right invrint differentil opertors on the Heisenberg group. To pper in Comm. Prtil Differentil Equtions. [3] K. G. Miller : Hypoellipticity on the Heisenberg group. J. Funct. Anl. 31 (1979) 306-320.
[4] K. G. Miller : Prmetrices for hypoelliptic opertors on step two nilpotent Lie groups. Coimn. Prtil Differentil Equtions, 5 (11) (1980) 1153-1184. *