IC/96/218

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUASI-INVARIANT MEASURESON GROUPS OF DIFFEOMORPHISMS

OF REAL BANACH MANIFOLDS

Sergey V. Liidkovsky1

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Groups of diffeomorphisms of real Banach manifolds are defined. Quasi-invariant

measures on them, relative to dense subgroups, are constructed. Applications of measures

are discussed for studying unitary representations including irreducible.

MIRAMARE - TRIESTE

October 1996

Permanent address: Theoretical Department, Institute of General Physics, Str. Vav-ilov 38, Moscow, 117942, Russian Federation.

1 Introduction.

In the papers [?, ?, ?] quasi-invariant measures (QIM) on groups of diffeomorphisms

(GD) of manifolds modelled on real Hilbert or non-Archimedean Banach spaces (BS)

were constructed. Also the regular representations associated with QIM and irreducible

representations of GD were investigated [?, ?]. Let Dif ft(M) be a group of diffeomor-

phisms with compact supports over a locally compact Riemannian manifold M over the

field R, where t correspponds to a class of smoothness Ct. In [?] the idea of the construc-

tion of Gaussian QIM on Dif ft(M) with values in R for finite noninteger t bounded from

below by a constant depending on a dimension dimRM = n was given. The presented

formulas were suitable for flat manifolds (see IV.2 and V.4 in [?]). QIM on Difft(M)

give the possibility for constructions of unitary representations [?, ?]. Particularly, on

Diff+1(S1) QIM was constructed in [?].

The problem about the construction of QIM was formulated by I.M. Gelfand in sixties

in connection with represenatation theories of groups. On the other hand, measurability of

representations of Difft(M), Banach-Lie groups and locally compact groups was studied

t(This article is devoted to construction of QIM on Difft(M) for (i) real Banach man-

ifolds M; (ii) all nonnegative t, (iii) classes of smoothness C°° and Can, (iv) classes of

weighted Holder spaces (HS) and Schwarz spaces and the corresponding groups of diffeo-

morphisms, (v) in particular, on groups of homeomorphisms Hom(M). It is necessary to

note that QIM on G = Difft(M) may be only relative to (dense) subgroups G' G,

since G is not locally compact and because of the theorem IV.5.3 in [?] and the classical

theorem of A. Weil [?].

In §2 we present definitions and notations. Differentiability of measures is defined

for topological groups here as in 2.6 [?] and differs from that of the linear topological

spaces or measurable manifolds along vector fields [?, ?]. In §3 we give some specific

isomorphisms of BS appearing as closed subspaces of HS. This is necessary for finding

greater G' relative to which QIM exists. The results of §3 sum up the known theorems

about elliptic differential equations in HS and weighted HS for manifolds that are compact

or Euclidean at infinity. In [?] the case of elliptic equations in a dual space L* to a Schwarz

space L was considered. In [?, ?, ?] cases of weighted HS and Sobolev spaces on manifolds

Euclidean at infinity were considered. The aim of §3 is the use of pseudo-differential

operators (PDO, see for example, [?, ?, ?, ?, ?, ?]). With the help of PDO we give

continuous extensions L of (b + A6), where A is a Beltrami-Laplace operator on HS Ct

for M with a metric g, s E N and t < 2s. This is used for a more simple construction

of QIM than in [?]. In §4 we construct QIM \i on G relative to G' for Riemannian

manifolds M with arbitrary connections and briefly discuss their applications. Here G'

are dense in G and differentiability of JJL is studied, that also improves the results of [?],

where G' were not dense in G in some cases. The results given below in the particular

case of a Hilbert manifold N impose weaker conditions on At(N) than in [?] and G' are

larger. The cases of Schwarz and analytic classes were not considered previously, but are

important in mathematical physics. The new technique for this is created. It is proved

that G does not have measures left and right quasi-invariant simultaneously, or relative

to inner automorphisms αg(h) = ghg~1, g E G', h E G. Finally, unitary representations

are discussed. The main results are theorems 4.2-4.4, 4.6

2 Notations, definitions and preliminary results.

2.1. Definition. Let M be a complete connected Riemannian manifold fulfilling one of

the following conditions:

(i) Euclidean at infinity [?] with dM = 0 or

(ii) compact; assume also that M is with a metric g and a finite atlas At(M) =

{(Ui,φi) : i = 1,...,k} of class Cl with charts (Ui,φi), where l = 1,2,... or l = oo [?], Ui

are homeomorphic to R n or R^ := (x E R n : x = (x1 ,...,xn),x1 > 0) such that M is

admissible in (i,ii) (analogously for finite l) [?];

(iii) for l = (an, r(2)) let M be compact, dimRM = n and there is an embedding

M ^ B(R 2 n + 1 ,0 , r (1)) , φi(Ui) are bounded in R n , cl(Int φi(Ui)) = cl(φi(Ui)), φi(Ui) C

B(R n ,0,r(0)),if dM ^ 0 and Ui n dM ^ 0 then φi(Ui n dM) C S(n-l,r(0)) := {x E

R n : |x| = r(0)}, c/>j φ (f)jl are analytic with radius of convergence greater than r(2) (for

decompositions into series with centres y for each y E Ui n Uj =£ 0), where 0 < r(0)

< r(1) < r(2) < oo, B(X, x, r) := {y E X : \\x — y\\x < r} for a normed space X.

2.2. Definition. For integer s > 0 and l > s let Cs(TM) denote a comple-

tion of a space of s times continuously differentiable sections f of a vector Riemannian

(tangent) bundle TM with f o (f>~1 E L(Rn) or L(R") for corresponding Ui (and fi-

nite linear combinations of such fi, initially may be taken fi with supp(fi) C Ui) with

||/||c«(TAf) := sup{E[||V6/(^)| | : b = 0,1,..., s, x E M}, where V is a covariant differential,

n = dimRM = dimRTxM, L(N) is a Schwarz space of functions on N with values in R n

[?]. For t = s + q with an integer s and 0 < q < 1 weighted HS Cβt(TM) ( C0t =: Ct) were

defined in [?, ?] with a weight factor < x > β , β E R + , < x > = (1 + |x | 2 ) 1 / 2 , |x| := d(x, 0),

0 is a fixed point in M, d(x,y) is a Riemannian distance along a geodesic joining x and

y E M. For β < 0 let Cβt(TM) be a linear space of sections f of the tangent bundle TM

such that for each compact set V C M (with V = cl(Int V)), f|V E Ct are satisfied the

following conditions:

(1) WfWc-iTM) := H/llc^d(x,x') < ρ(x), x E M} < oo,

(2) lim{| |/ |Mfl | | c* ( T M i 0 : R -+ } = 0, where MR := {x : x E M, d(x,0) > R}, ρ, d,

σ(x) and T(X,X') are as in [?], 0 is a fixed point in M.

Let Cβt(M, N) denote a space of Cβt mappings f : M —> N topogolized with the help of

a metric d(f,g) = J2ij=i \\fij — Sfij||c*(Rn,Rm)) where fi,j = faoffa1 and it is implied that

(for fi,j) it is defined on U n i o(f)jl([Jj) if it is nonvoid and fi,j is zero otherwise, (Uj, φj)

is an atlas of N, dimRN = m, dimRM = n, analogously for Ct(M, N) with β = 0, M and

N are Cβl-manifolds with l > t + 1. We suppose that Cβt(Rn,Rm) are HS of mappings

/ : R n —> R m , t G [0, 00), M and N fulfil (i), (ii) or (iii) in the corresponding cases.

Let for M as in (iii) Can,r(TM) be BS of real analytic sections f with \\f\\c^,r(TM)

1,2,...} < 00, where 4>jof4>jl(y) = £ f c fj,kyk, (Uj, φj) are charts ofM,ye B(Rn, 0, r(0))n

φj(Uj) =: E(j), x = (p-\y) G M, r(2) > r > 4r(1), yk = y1k(1)...ynk(n). Then Can,r(M,M)

denotes a space of C a n , r mappings f : M —>• M topogolized with the help of a metric

d(f,g) = E i j \\fi,j ~ 9i,j\\can^{TE{j)), where fi,j = φi o f o 0J1 as above for t G R+.

In view of the fundamental theorem of Riemannian geometry on M exists the unique

Levi-Civita connection w = wg for g. We assume that g is of the same class of smoothness

as M.

2.3. Definition. Thereafter, Diffβt(M) (or Difft(M), or Diffan,r(M) denotes a

group of homeomorphisms f of M with (fi,j — idi,j) belonging to Cβt(Rn,Rn) (or Ct,

or Can,r) topologized by the corresponding left-invariant metric ρ(f,g) := d(g~l o f,id),

where id(x) = x for each x G M, t G R+, β > 0, r > 0. Henceforward, dM denotes a

boundary of M and A denotes a Beltrami-Laplace operator for (M, g) (see Appendix 14

in [?]), γ0f denotes f\dM for f G Ct.

Henceforth, Cf{TM) := n{C^(TM) : t G N}, Difff(M) := n{Diffp(M) : t G N},

L(TM) := n { C ; ( T M ) : β G N}, L(M,N) := n{C^°(M,^) :/?GiV}, Diff(L,M) :=

(~\{Diffβt(M) : β G N} in topologies given by countable families of norms and metrics

correspondingly (for a C°°-manifold or C2?-manifold with (31 > β or L-manifold M in the

corresponding cases, for β = 0 we drop β), Ct(TM; h) = {f G Ct(TM) : γ0AJ'/ = 0 for

j = 0,..., h}, A0 := I is the identity operator, Ct,q(TM) : = { f : f G Ct(TM), 7 o V 6 / = 0

for b = 0,...,q}, Dt(M;h) := {f G Difft(M) : γ0V6/ = 7 o V 6 id for b = 0, ...,h},

Dif\h'-M) := {f G Difft(M) : 7 o V 6 / = 70V6«i for b = 0,...,fr'}, where h G Z,

t>2h>0,tieZ,ti<t,M fulfils (ii) and is a Cl-manifold with l > t + 1 or (iii) in the

corresponding cases with (an,r) instead oft, C\TM) := {f G Ct(TM) : πν(f(x)) = 0

for each x G 5M} for dM ^ 0, where πν denotes the projection on a normal to dM in

each x G <9M.

We also consider

(i) an infinite countable At(M) such that M has a sequence {UE n : n G N} of

submanifolds satisfying the conditions (2.1.i) or (2.1 .ii) imposed above with \Jn UEn = M,

4

where UE := Uj ee Uj, E E Σ, Σ is a family of all finite subsets of N. Then for t = (an, r)

we define the corresponding spaces with infinite At(M) with the help of the strict inductive

limits as in [?].

2.4. Definitions. 1. Let X be separable BS over R. Suppose that Fn C Fn+1 C ...X,

dimRFn = n, is a sequence of finite-dimensional subspaces. Let {zn : n E N} be a

sequence of linearly independent vectors in X with ||.zn||x = 1, SPR{Z1, ...,zn} = Fn for

each n. For open U and V in X we consider a space of all infinitely many times Frechet

differentiable functions f,g : U —> V fulfilling (i, ii) in 2.1 [?] and with ρβ,γt(f,h) < oo,

where h : U —> V is some fixed smooth (of class C°°) mapping h : U ^ V, Dxα for α =

(α 1,..., αn) is the operator of differentiation by (x1, ...,xn) E Fn, but with URc := {x E U :

| |x| |x > R} and < x >= (1 + ||x||^-)1//2 apart from [?]. We denote by E^r/ the completion

of such metric space and consider ET[U, V) as in [?].

2. Let M be a paracompact separable metrizable manifold modelled on X [?] and

fulfilling (i, ii) below:

(i) an atlas At(M) = [(Uj,φj) : 1 < j < k + 1] is finite, k E N (or countable k = ω0),

φj : Uj —> X are homeomorphisms of Uj onto φj(Uj) 3 0, Uj and φj(Uj) are open in M

and X respectively, (φj o <f>~1 - id) E E™s((f)j(Ui n Uj),X) for each Ui n Uj ^ 0, where

CJ > 0, γ > 0, id(x) = x is the identity mapping, Ω0 is the initial number of cardinality

No [?];

(ii) M contains a sequence of Mk and Lk submanifolds. They are of class E^°7 with

dimRMk = k for Mk and codimRLk = k for Lk, k = k(n) E N, k(n) < k(n + 1) for each

n, Mk C Ml and Lk D U for each k < l, M = MkU Lk, Mkn Lk = dMk n 5Lfc for each k

such that Ufc Mk is dense in M. Moreover, M and At(M) are foliated. That is, they fulfil

(a) (j)itj : foo^lfaiUinUj) -»• X are of the form (j)itj((xl : ! G N ) ) = (^j.fc^1, - , ^ f c ) ,

γi,j,k((xl : l > k))) for each n E N, k = k(n), when M is without a boundary, <9M = 0. If

<9M 0 then:

(β) for each boundary component M0 of M and C/jflMo =£ 0 we have φi : C/jflMo - - l ,

where Hl = {x E X : xl > 0}, xl = Pzl(x) is the projector of X onto Rzl along X 0 Rzl

(see [?] ).

3. Analogously to definition 2.3 [?] we consider spaces Eβ,γt,θ(M, M) and Diffβ,γt(M) for

M and M as in 2.4.2. Then Diff^(M) is defined as f\l&i Difffi(M) and Diff^(M) =

C\t€N Dif fβ,γt(M) with the corresponding standard topologies of projective limits [?, ?].

Evidently, Diffβ,γt(M) is the separable topological group. It is metrizable with a

left-invariant metric d, when At(M) is finite (see the proof of theorem 3.1 [?]).

2.5. Lemma. Let M be a E^-domain in X. Then there exists a Hilbert space Y

such that Y C X, Y is dense in X, ||x||y > \\x\\x for each x e Y and Dif f^, ,(N) is

a dense subgroup in Diffβ,γt(M), where N = M n Y, oo > t > 0, t' > t, oo t f > 1,

P' > β > 0, i > γ + 2, to β P', δ > 7'.Proof. In view of theorem I.4.4 [?] for BS X there exists a Hilbert space Y, Y C X,

\\X\\Y > ll^llx for each x G X. We take {Fn : n G N} in X and an orthonormal base

{en : n G N} in Y with e1 = z1, ei = 2}=i ^J-^J a r e chosen by induction, bi,i ^ 0. Since

Z ^ i ( Z ^ = n T O < i ) < °° for e a c h d < ~2, then there is a Hilbert space Y0 with an injection

T : Y0 -^ X being a nuclear operator [?, ?], Tx = 52ili(x,yi)Yo

zi, where x G Y0, (*, *)y0

is an inner product in Y0, {yi} is a base in Y0 such that S £ i |z/i|y0 < °°- Moreover, we

can choose e = 6^^^. Let 7 0 C F C I , IMIyo ^ ll^lly — ll^llx f° r e a c h x G lo- Then

from definition 2.1[?] of p^ and l2,γ, also from the consideration of multipliers naj, nnaj,

it follows that each g G Diff^l/yl(N) belongs to Hom(M), since Fn cY C X, t' > 1,

< x > y > < x >X for each i e F . Therefore, g has the unique continuous extension g on

M such that g G Diffβ,γt(M), since N is dense in M and we can choose for each 0 < γ

the space Y0 with |yi| < i~2~e for each i e N .

2.6. Definition. Let M be a E^-manifold as in 2.4. Henceforward, we suppose that

there exists E^,-submanifold N in M; N is modelled on a Hilbert space Y, where Y is as

in 2.5 with Diff™s,(Y) C Diff™s(X) for the corresponding 5' > δ, where M and N are

separable. Also let N satisfy conditions in 2.2 and 2.4 [?] such that Mk C N, Nk C N,

Nk is dense in Lk for each k G N.

2.7. Corollary. Let M be a Banach E1^-manifold and N be a Hilbert E^s,-manifold

such that they satisfy 2.6. Then Diffp,(N) is a dense subgroup of Diffβ,γt(M), if

£' > δ > y > γ + 2, t > 1, oo t H > t > 0 and ω > β.

Proof. For charts (Vj,ψj) of N with Vj n Vi =£ 0 a mapping ψ o -0"1 is in the class

of smoothness E^,. In view of definitions 2.4, 2.6 and lemma 2.5 Dif ft ,(N) is a dense

subgroup of Diffβ,γt(M).

2.8. Note. From the definitions of groups of diffeomorphisms for a separable Banach

manifold M, it follows, in particular, that when dimRM < oo, a group Diffβ,t0(M)

coincides with Dif fβt(M) from definition 2.3, Diff£>0(M) = Diff(L, M), Diff£0(M) =

Diff^(M). When (i) At(M) consists of charts (Uj,φj) with φj(Uj) C B(X,xj,rj),

xj G X, 0 < rj < oo for each j , we drop the index β from Diffβ,γt(M) and Eβ,γ

2.9. Definition. Let M be a Riemannian locally compact Cγl-manifold, where l > 1.

Suppose P is PDO corresponding to (1 + A) and P€, Pl~€ are PDO such that (P€Pl~€ -

P) G OPS11,0 (see theorems II.3.8 and II.4.4 [?]). For compact M, l > n + 1, 0 < δ < 1,

0 < γ < 1, 0 < δ + γ < 1we denote by C~n-e's(TM) 3 f BS that is, a completion

of space P^n+€+&)/2Cl{TM) 3 g relative to a norm | | / | | C -™-^(TM) := \\g\\c^{TM) for each

/ = p(n+t+&)ng_ F o r a locally compact Cγl-manifold M Euclidean at infinity with l>n+l,

7 > 0 let C~n-e's(TM) 3 f be BS equal to the completion of a space p ( » + ^ ) / 2 ^ ( T M )

with UJ G R, uj = uj' + n + t + 8, relative to a norm | | / | | C -«-E,«( T M N = ||sf||c5

/(TM) for

6

2.10. Lemma. Let C1-n-"'s(TM) =: Xi and C^-^^TM) =: Xi,ω be given by 2.9

with γ1 + δ1 = γ2 + δ2. Then X1 (or X1,ω) is isomorphic with X2 (or X2,ω) respectively.

Proof. If Q e OPS10,0, then Q : (7£(rM) -> C e(TM) for compact M and Q : Cω ->

C^(TM) for noncompact M is a continuous linear operator due to theorems XI.2.1, 2.2 in

[?] and [?, ?]. In fact, Pe is given by theorem XII. 1.3 as a function of a self-adjoint elliptic

operator. For each PDO p(e+<5)/2 corresponding to the degree of the elliptic operator

(1 + A)(£+<5)/2 there exists a decomposition for δ1 < δ2: p^+W* = P(e1+s1)/2p(s2-s1)/2

(mod OPSei+S2~1) [?, ?, ?]. But Pα : Cγβ(TM) -> C^(TM) is the isomorphism for

7 > 0 , R \ N 9 o ; > 0 , R \ N 9 / 3 > a ; f o r M Euclidean at infinity. Also Pα : Cβ(TM) ->

CP~a(TM) is continuous onto a closed subspace for compact M with β > α > 0 (see

theorem XI.2.5 [?] and [?] ).

2.11. In view of lemma 2.10 we denote C~n-e's(TM) and C~n-e's(TM) simply by

C~n-e(TM) and C~n-e(TM) correspondingly.

3 Some specific isomorphisms of Banach spaces.

3.1. Theorem.Let M be a Cl-manifold fulfilling (2.1.ii), dM ^ 0, s E Z, s > 0,

0<q< 1,t = s + q, l>2j > s, j G Z, andA|Cl(TM) = (b + Aj + Ao,-fO,loV, •-,7o^"1),

where A0|Cl(TM) is a differential operator with an order ord(A0) < 2j, A0|C

l(TM) =

E c ( i r ; p ( l ) ! . . . ] j ( n ) ) V f ) . . . V f ) ! 0 < p(1) + ... + p(n) = p < 2j, c(x;p(1), ...,p(n)) E

CZ(T*M) with z = l — 2j + p, b E R, η = δ or η = Vv, ν is a normal to dM for each

x E dM, γ0f = f\dM. Then there exists b E R and a continuous extension of A onto

Ct(TM) and a Banach subspace YAt(TM) of C1t-2j(TM)®{d C1t-hi(TdM)}, such that

A : Ct(TM) —Y At(TM) is an isomorphism, where h = ord(η).

Proof. In view of the results in [?, ?, ?] we have for l > t' > 2j, 0 < q < 1, that there

exists b E R \ {0} such that

(1) A : Ct(TM) -> C*-2i(TM) 0 {<8)to (f~hi(TdM)} is an isomorphism, where

t' = s' + q. Indeed, this is true for j = 1 and by induction for (b' + A) J instead of b + AJ,

Ct"(TM) C C1t'(TM) is a compact operator of embedding fort" > f [?], (6'+A) J -6-A^ =

n = i ( i ) ( ^ ) J " i A i , b = (P')j and we may use theorem I.12.4 in [?].

Let at first M be a C°°-manifold. As it follows from theorems IV. 14,16 (and exer.

21,24,25 of Ch. V) in [?] and chapter 3 in [?] for J in a topologically dual space (Cr(TM))*,

r E N there exist Radon (Baire) measures // for |β | < r such that they are with values

in R and of finite variations on M:

(2) J(f) = J2\f3\=rJM(^rf(x))(Xi(i)>-->Xi(r))Vi3(dx), where e1,.. .,e n is the standard

base in R n , β = ei(1) + ... + ei(r), Xi, Zi are continuous vector fields on M such that

spR{Z1(x),...,Zn(x)} = TxM for each x E M and 0 < inf{gx(Xi(x),Xi(x)) : x E M,

i = 1,...,n} < sup{gx(Xi(x),Xj(x)) : x, i and j = 1,...,n} < 1, n = dimRM, since

4>~l 1 (pj are in Cl for Ui fl j ^ 0 (III.8 in [?] ); analogously for dM instead of M

and V G (Cr(TdM))*, when <9M ^ 0. We have also (Cr)* C (C*)* C (C r + 1 )* for

0 < r < i < r + l , r e Z .

Now let a(x,ξ) G S™(T<9M) be a symbol of PDO corresponding to A [?]. For

a6(a;,f) = < f >^ a , b G R and 0 < α < n the Fourier transform F ξ in L*(TRn) by ξ

is defined,

(3) afc(x, k) = F?(a6(x, ξ ))(k) G L™/(ra"a) and for b G R\ {0} it decreases exponentially

by k G R n (exer. IX.50 and for α G R, b G R \ {0} by theorem IX.46 in [?]), where

< ξ >b:= (b + | ξ | 2 ) 1 / 2 , < ξ >:=< ξ >1, |ξ | := gx(ξx,ξx). In view of theorem XI.2.5 [?] :

(4) \\OP(< £ > ? ) / | | L 2 < C x | |/ | |C9(TV). Therefore, from the Holder conditions and

(4) using principal symbols of PDO we have:

(5) \\OP(p)f\\ < C x \\f\\c\TM), where C" is a constant, p G S\T*M) (see also §V.4

Using theorems about compositions of PDO and convolutions of generalised functions

and functions we have, that for each α G R and u = f o (f)~l G L(TRn) or L(TR") in

dependence of Ui is defined OP(< £ >^a)u} 6 ^ 0 (see also [?] ).

Now let L be an elliptic operator of order 2m, m G N, Bj with j = 1,..., m be operators

of orders kj corresponding to a boundary problem and fulfilling conditions as in [?], then

we denote their symbols as a(x,ξ) and bj(x,ξ) respectively (see below). At first we may

consider a dense subspace P(TM) of Ct(TM) consisting of f with f o 0" 1 G D(TRn)

or D(TR") (for example, for functions f with supp(f) C Ui and compact supp(f o (f)~l)

and then their finite linear combinations). Indeed, M is compact and there exists a finite

family (j>il{y{i,j)) that is a covering of M and V(i,j) = cl(Int(V(i,j)) are compact

i n R n o r R ^ [?]. F o r A = ( L , γ 0 B 1 , ...,γ0Bm) a n d f , g G P ( T M ) , gj G P(TdM) for

kj > t and gj G C^^'iTdM) for t > kj with gj G ^BjPiTM) for each kj and t we have

(Af,g) = J(g) + JM + ... + J m (g m ) , where J = Lf = OP(a)f + L0f, Jj = γ0Bjf =

γ0OP(bj)f + γ0Bj,0f, g= (g,g1, ...,gm),

(6) J(g) = JM(OP(a)f)(x)g(x)f,(dx), Jj(gj) = ^{OP^f^g^^dx'), mea-

sures /i(dx) and /i'(dx') correspond to the Riemannian (n and (n — 1) forms of) volume

elements on (M, g) and {dM, g') respectively, where g' is induced by g. If 2m > t or kj > t

we may use compositions of PDO with symbols a1 G Sf^{T*M) and a2 G S*2™"* {TdM),

bj,iSlQ{T*M) and bj,2 G S^(TdM) such that

(7) a - a1a2 =: α G Sl^~l(T*M) and &,- b j,1bj,2 =: ζ e S'fjT^^M) by 18.1.3,8 in

[?], where 0 < |t - t" | < 1/4 such that 2m - t" > 0, t" > 0.

Indeed, in view of proposition 18.1.19 [?] there exist a G S2m(T*M) and bj G SkJ(T*M)

such that L = a(x,D) + L0, Bj = bj(x,D) + Bj,0 with kernels (of integral form) of

operators L0 and Bj,0 belonging to C°°(M x M). In view of theorem 18.1.23 in [?] L =

OP(a1)OP(a2) G Ψ2m(M), Bj = OP(bj,1)OP(bj,2) G ^(M) and the principal symbols

of a(*) and bj(*) are equal to the products of the corresponding principal symbols of

a1, a2, bj,1, bj,2 respectively. T h e t e r m s a — a1a2, bj — bj,1bj,2 for j = 1,...,m do n o t

influence the conclusion about isomorphisms due to theorem 18.1.9 in [?] and formulas

in [?, ?], where the elliptic problem was reformulated in terms of PDO. Then a(x,ξ) ~

J2<jCLo(ij(x,$,), a,j(x,£) are homogeneous of degrees (2m — j) by £, dj(x,^) ~ < ξ >2m

aAxi!i)i < ? > < s > < s > ) < s > — sl ^j=o ^ j Jlsl r o r Is I > x) wnere

aj(x, ξ) are homogeneous of degrees (—j). This factorization may be accomplished further

for α and each ^ and so on by induction.

The operator (L,γ0B1, ...,γ0Bm) corresponds to PDO A = P M

T

+ G initially defined

on C°°(TM), where PM, G, T are PDO given in [? ,? ,? ] , A is of order 2m and class

r = m a x ( f c l r . , t ) + 1. Here PM = OP(p(x,ξ)) V e SJ™, T = (T1,...,Tm), (Tbu)(xf) =

OPT(n(x>, 0)u = (2TT)-» / R n -i e-'«' /0°° T3(X>, O(e+u(O)AdCndC, n(x>, 0 = E ^ b stf, O

r ' ^ ' ^ ) , where r ' G H: 1 ; s^x',^') G ^ " ^ A ^ R 1 1 " 1 ) , M' is open in R 1 1" 1 (initially for

domains in R n and R " " 1 then for charts and manifolds), G = YIjZl Kjlj + G', G'u(x) =

are Poisson operators of orders (2m — j), (Kj(f)j)(x) = (2vr) n / R n etx^K,(x',(

K, G H+ as a function of ξn, φj = γ0OP(ξnj)u =: γju.

For p(x,ξ) = a1(x,ξ)a2(x,ξ) = P(Ξ) (that are symbols independent from x G M),

f,g G D(TRn) we have JK4OP(a1)f](y)[OP(a2)g](y)dy = / R ^ F " 1 ^ ! F(f))](y)

[F-l(a2(F(g))](y)dy = / R n a i ( £ ) / ( £ ) a 2 ( £ M R = Jn4OP(aia2)f](y)g(y)dy, since the

Fourier transforms F and F~l are unitary on L2, where g(£) = F(g)(ξ), (g)v(x) =

[F~l(g)](x), x,y,ξ G R n . Therefore,

(8) JM[OP(ala2)f](y)g(y)fi(dy) = JM[OP(al)f](y)[OP(a2)g](y)fi(dy) is the bilinear

functional by f, g G P(TM) having the continuous extension on Ct” (TM) ® C2m-fr (TM)

due to theorem II.4.4 [?], the Lebesgue-Fubini theorem and lemma 2.5.

At first we treat operators L and γ0Bj with constant coefficients and the corresponding

A, then with variable coefficients approximating symbols p(x, ξ) by linear combinations of

cj(x)dj(ξ) with cj(x) G Cl(M, R) and symbols di(ξ) independent from x G M using this

approach for p(x,ξ), Sj(x',$t'), g'(*) and so on. This consideration may be accomplished

for a parametrix B such that AB - / and BA - / G S-°°(T*M). In general we have

A = ( P + ^ K ) : (P(TM) ® P(TdM)m)+ -> (P(TM) ® P(TdM)m)+, where in a Green

operator A PDO P, G and T are respective to the elliptic problem with (L, γ0B1,..., γ0 Bm),

P is PDO in M satisfying the uniform transmission condition, G is a singular Green

operator, K is a Poisson operator, T is a trace operator, S is PDO in dM, the superscript

+ denotes the transposition operation. There exists a Green operator B such that BA — /

and BA — / I 5*"°°. In view of theorem 5.1 in [?] and [?] there are Green operators B',

A(1), A(2) such that A - A(1)A(2) G S2m~l and BA(1) - A(1)B' G 5"*"-1, A(2) is in St”,

9

A(1) is in S2m * (with the corresponding orders, that may be integers, of P(j), G(j),T(j), K(j) a n d S(j) f o r j = 1,2). Hence (A(1)B')A(2)(*) has the continuous extension

on (C*(TM)®{0})+into {Cl-2m{TM) ® {®?=1 Ct~k^TdM))+ =: Z by g E Ct(TM) (see

also §2.4, theorem 2.7.9 and corollary in [?] ).

Using this principle and equations (3,5,6,8), theorem II.4.4 [?] analogously, for M and

bj,1, bj,2, j = 1, ...,m, PM, G, T we have the following inequality

| ( A / , <?)| < C\\f\\ct"(TM)(\\g\\c2™-t"(TM) + Cl | |S ' l | |c f c i-*"(T6iM) + ••• + Cm\\gm\\Ckm-t" (TdM)),

since the principal symbols of PDO and their adjoint are complex conjugated (theorem

18.1.34,12 in [?]), where C, C1,..., Cm are positive constants depending on (L, B1 ,... ,Bm),

but they do not depend on f and g,gj. Therefore, there exists A : Ct(TM) -> C^^iTM)®

{<8>™i C*~fcj(T<9M)}+ =: Z and it is the linear continuous operator with

(9) ||A/|U < C(1 + d + ... + C m ) | | / | | c t ( T M )

Therefore, A : (Ct(TM) ® {0})+ -> YAt(TM) is the isomorphism, where YAt(TM) is

the closure of A((C*(TM) 0 {0})+) in Z. In view of theorems I.3.4, 4.1 and results of

chapters II, III in [?] and formulas (4-9) above, the operator B : A(C*'(TM)) -> C*(TM)

is continuous and has a continuous extension on A(Ct(TM)), since

where L and Bj are, in general, with nonconstant coefficients, C, C'j are constants. There-

fore, (L,γ0B1, ...,γ0Bm) = A has the continuous extension on Ct(TM) and due to (9,10)

is the isomorphism onto Y = A(C t(TM)) = c/A(P(TM)), that is, the Banach subspace of

CP-^iTM) ® {<S>T=i C^^iTdM)} in view of the Banach theorem. From theorem I.12.4

in [?] and its version for PDO [?] the statement of theorem 3.1 for the operator A and the

C°°-manifold M follows. Using the embedding of M into R 2 n + 1 and the approximation

of the Cl-manifold M by C°°-manifolds Mi we obtain the statement of this theorem.

3.2. Theorem.Let M be a Cl-manifold fulfilling (2.1.i) with regularly asymptotically

Euclidean metric g, s E Z, s > 0, 0 < q < 1, t = s + q, l > 2j > s, β > 0, j E Z,

2(β+j) >n,n = dimRM and Aβ|Cβl(TM) = (b+Aj + A0), where b E R and A0|Cβl(TM)

is as in theorem 3.1. Then there are a linear operator A, b E R and a Banach subspace

Y/3+2j(TM) ofCC^_2j(TM), such that Aβ : Cβt(TM) —Y Y^2j(TM) is an isomorphism.

Proof. In view of the results in [?, ?, ?, ?] there exists b E R \ {0} such that

(11) Aβ : C%(TM) -> C%~2j(TM) is the isomorphism, where t' = s' + q, l > t! > 2j,

0 < q < 1, since the proof in [?] may be generalized for each t' > 2j or using results of [?, ?]

for weighted Sobolev spaces Hm,α and the following embeddings: Hm,α(TM) D Cp(TM)

for β > α + n/2, f > m E N, β E R, t! > 0, t' i Z; H [ t ] + [ ( n + 1 ) / 2 ] + n + 1 , β + [ ( n + 1 ) / 2 ] + 1 ( T M )

C Cβt(TM) for β E R and 0 < t E R \ Z (see Appendix to §V.3 in [?] and [?]).

Let at first M be a C°°-manifold. Also let a1 E Sr(T*M), a2 E S2j-f'(T*M), a(*) be

a symbol of (b + Aj) and a - a1a2 E S2m-\T*M), then (8) is valid for f,gE Cf{TM),

since 2(β + j) > n and JM < x >-2(J+/3) jj,{dx) < oo. That is,

10

(12) JM[OP(al)f](y)[OP(a2)g](y)ii(dy) = JM[OP(ala2)f](y)g(y)ii(dy), where ai(x,ξ) =ai(ξ). Then we take into account PDO dependent on x (see the proof of 3.1), consequently,

Aβ defined by (11) has extension onto Cβt(TM) and

(13) Aβ : Cβt(TM) -> Y!~2UTM) is continuous, since \(kpf,g)\ C

ft' > n-2j - β, where (*,*) denotes the scalar product on L2(M,JJ), Y^(TM) =

d(A,,(C;(TM)) C Cj~|-(TM), Aβ = OP(a1a2), C = const. There exists continuous

Bβ = (A^IC^(TM))- 1 on H := A ^ C ^ T M ) ) C C^+2j(TM) such that

(14) ll/llc*'(TM) — ll^llc'^^fTM) ^ o r f = ^z3^' h E H, as follows from the Oskolkov-

Tarasov theorem.

The following subspace E := {f|f E C°°(TM), there exists R = R(f) such that

supp(f) C M n BR} is dense in Cβt(TM) for β > 0, since each g e Cβt(TM) fulfils

(2.1,2) and for each b > 0 there is R = R(g) > 0 for which ||<7|Mfl||ct(TM) < b/3, so

there is f e H 0 with supp(f) C M n ^ ^ such that ||(# - f ) | M n 5^11^ ( T M ) < b/3 and

||/|Mfl| |c* < b/3, hence ||^ - / | | C | ( T M ) < b, where BR := {x|x G M;dM(x,0) < R}. Then

A β ( E ) =: H 0 is dense in H, consequently, for h E H there is a sequence {hk : k} C H0

with l i m k ||/i — hk\\ct-2j(TM) = 0. In view of theorem 3.1, formulas (11) and (14) for

2j = t' — t the following sequence {fk : fk = A~lhk,k E N} is the Cauchy sequence

in Cβt(TM), since for each k there exists C°°-submanifold Nk C MR(k) \ M2R(k) with

dimRNk = n — 1 such that fk G Ct(TMk;j), supp(fk) C M (~\ B2R(k), where Mk are

compact C°°-submanifolds, dimRMk = n, dMk = Nk, Mk C M n BR(k). Therefore,

/ = limfc^oofk G Cβt(TM) and Aβf = h, consequently, AβBβ = I and BβAβ = I on

C^iTM) and HB^/iUc^y^-) < C x ||/i||ct-2j/TMw where C = const. Then from (13) and

the last inequality analogously to the last part of the proof of 3.1 we obtain the statement

of theorem 3.2.

4 Quasi-invariant measures.

4.1. Theorem. Let M be a locally compact Cl-manifold fulfilling (2.1.i) or (2.1.ii) or

(2.1.iii), l > t + 1 (l = oo for t = oo; l = (an, r') for t = (an, r) with r' > r). Let G be a

group of diffeomorphisms defined in 2.3. Then

(i)for each Eβt(M, TM)-vector field V its flow ηt

is a one-parameter subgroup of Dif fβt(M), the curve t —η t is of class C1, the mapping

Exp : TeDif fβt(M) —> Diffβt(M), V —η 1 is continuous and defined on a neighbourhood

of the zero section in TeDif fβt(M)

(ii) TfDiffβt(M) = {VE βt(M, TM)|π o V = f};

(iii) (V,W)= /M

11

is a weak Riemannian structure on a manifold Diffβt(M), where JJL is a measure induced

on M by φj and a Lebesgue measure on R n , 0 < /i(M) < oo;

(iv) the Levi-Civita connection V induces the Levi-Civita connection

V on Diffβt(M);

(v) E : TDiffβt(M) -> Diffβt(M) is defined by

EV(V) = expη(x) o Vη on a neighbourhood of the zero section in TηDiffβt(M) and is a

E^ mapping by V onto a neighbourhood Wη = Wi(i η of η E Diffβt(M); E is the

uniform isomorphism of uniform spaces V and W. Analogously for GD in the analytic

and Schwarz cases.

Proof. Each geodesic τ = xt on M with initial conditions (x,X) is a solution of

the following system of differential equations d2xi/dt2 + Y,j,k Γij,k (dxj / dt) (dxk / dt) = 0,

i = 1,..., n in local coordinates {xi : i = 1,..., n}, hence dτ/dt is in the class of smoothness

ci-i (^oo o r C(an,r') for t = oo or t = (an, r)), where

(!) rlfc = gk'l(dgiJ/dxi + dgiti/dxj -dgitj/dxl)/2 e Cl~l. This induces the exponential

mapping exp such that for each x E M a neighbourhood TxM D Nx 3 (x,0) exists and

exp : Nx ->• Ux is Cl~l (or C°° or C^'^) diffeomorphism, since TM is the Cl~l (or

C°° or C'(ara'r"/) respectively; r' indicates only on the radius of convergence, but not on the

region, where exp is nondegenerate) manifold, where M D Ux 3 x is a neighbourhood

(Prop. III.8.2 in [?] ). Therefore, for finite At(M) there exists 1 > b > 0 such that for

each z E Ct(TM) with ||.Z||CI(TM) < b we have: f(x) = exp(z(x)) is in Difft(M) due

to the theorem about inverse transformation, where ρ(f, id) < C x | |Z||C*(TM), C = const

depends on M and t only for t E [1, oo). Then analogously to the proof of theorem 3.4 [?]

due to the theorem about flows on manifolds [?] (see also [?] ).

4.2. Theorem. Let M be a Cl -manifold satisfying (2.1.i) or (2.1.ii) with dM ^ =

or (2.3.i) and G be a group (1) Diffβt(M) or (2) Dift(q; M), (3) Dt(M; h). Then there

exist dense subgroups G' equal to (1’) Diff^(M) or (2') Dif(q;M), (3') D*(M;h)

respectively and G" (k) (for l > t' + k + 1 and k E N, l = oo for k = ooj such that

there exists QIM /j, that is quasi-invariant relative to G' and Ck-differentiable relative to

G"(k) C G', where l > t' + 1, t E [0, oo), t > q > 4mn+1, t > 2h > 4m - 2, t > 2h > 0,

t > q > 0, t' > t + 2m + ζ for t > 1 and t' > t + 2m + 1 + ζ for 0 < t < 1, ζ = 0 for

t E (0, oo) \ Z and 0 < ζ < 1 for t E Z, t > 0, β > 0, /?' > β + 2m, m = [(n + 1)/2] + 1,

n = dimRM, [d] is the integral part of d ([d] < d).

Proof. (A). Let at first £ > l , £ e R \ Z and At(M) be finite. Suppose A is the same

as An;m(n) in 4.11 [?]. Therefore, in view of lemma 4.12 [?] and theorem 4.1 we have that

A(φ of)- A(f) E Cj+2

2|+^(TM) for each φ E Dif fβt++zz(M) and f E Diffβt(M). Let us

choose z = 2m and 2ξ > max(t, 4mn), ξ E N.

12

(A.1). Let at first M fulfil (2.1.i). In view of theorem 4.1, theorem 3.2 and the theorem

about inverse transformation there are open U0 3 id, U C Diffβt(M) and an open ball

V0 9 0, V0 C Yp~ll(TM) such that A : U 0• A(id) + (V0/2) is the uniform isomorphism

for suitable k R , since 2(β + ξ) > n. There are open U, W 3 id, U C Diffβt(M) and

W C Diffβt++z

z(M) such that U C U0 and WU C U0, so for V = A(U) the following

mapping Sφ : V —> V0, Sφ(v) = A((f)(A~l(v))) — v for each φ e W may be defined. Then

Sφ(v) : W x V —> V0 is in C°° by φ G W, since A(φ o f) is the polynomial by φ, f and

by their covariant derivatives, Lφf := φ o f, dLφ/dφ = I, φ G Ct+z and appearing Va(f)

and corresponding PDO in Sφ(f) have orders a < 2ξ — 1, the convolution of h £ Cβ1(TM)

with g G L2(M, TM; dx) n C^iTM) \sh*geCf.

We have the embedding (Cb^(TM))* C H{_b]t_^_[n/2](TM) for b = t - 2ξ < 0, 1 < t E

R\Z,/3>0,/3 ' = β + 2ξ, consequently, Cb^(TM) D H1+[b]^,+n/2(TM) =: H, since b G R\

Z and -[-&] = 1 + [b], whence Y%,(TM) D Z := C%(TM)nH. There exists C = const > 0

such that ||/1|cb,(TM) < C x II/IIH f° r e a c h f ^ H, hence Z is the Hilbert subspace of

H, since Yt(TM) is BS. We may take T : H —> H satisfying the following conditions

(a-c): (a) T(H) D Hi+[b}+2m,f3'+2m+n/2(TM); (b) T is self-adjoint injective positive of trace

class together with T|Z : Z —> Z, T(Z) C Z; (c) T|Z has continuous injective self-adjoint

positive of trace class extension T : H_2£,f3'(TM) —H i7_2^)/g/(TM), T(H-2t,/3'(TM)) D

H2m-2t,,i3'+2m{TM). The correspondence between (C^/)*-cylinder subsets (that are given

with the help of finite families of continuous linear functional on C%(TM)) in H and Cβ

produces the Gaussian measure on Cp,(TM).

Therefore, T induces the Guassian measure with the correlation operator T and zero

mean value, its restriction on Vo = V0 D Z is denoted by /2. Let X G Bf(V0), so there

are V1 C V0, W1 C W, W1 3 id, W1 C Di//*+ 2

2^(M), ^ = A " 1 ^ ) and W1U1 C U0,

/i(X n V0) > 0, consequently, p,<f>(X) := p,(A o i ^ o A - 1 (X)) > 0 and p,<f>(X) is equivalent

to p,(X) due to theorem 26.2 in [?]. Therefore, p, induces QIM JJL on Bf(U1) with the

continuous QI factor q{4>, f) by {4>,f) G Wi x U\, /i(L^-idf) = n((f)df) = q((f), f)/i(df).

Further, for l > t' + k + 1, k G N let us take a family J of one-parameter subgroups in

Diffβt”” (M) (see theorem 4.1) for t" = t' + k (l = t" = oo for k = oo), β" = β + 2m + k, a

subgroup G" C Dif fβt””(M), such that G" is the minimal subgroup generated by J U {gi :

z = 0,1, 2,...}. Hence \i is QIM relative to G" =: G"(k), where {gi : i} is a sequence of

elements in Diffβt”” (M) such that g0 = id, {giWi : i} is a locally finite open covering of

G, Wi C W for each i. The rest of the proof is analogous to that of theorem 4.5 [?].

(B.1). Now let 0 < t(1) < t and {z(i) : i = 1,2,...} be dense in M. We may de-

fine the following subsets of W and W(1) C Diffβt(1)(M) =: G(1), W(1) C G =: W ,

W(k,t(1),c;f) := {g G W(1) : ρ(k;g,f) < c}, W(k,t,c;f) := { J G W : ρ(k;g,f) < c},

where oo > c > 0 , k N , / e W, the mappings ρ(k, k';g, f) := Ea) fcsup{[cr(x)] i+/3 |V i((/-1o

y)o,6(a:)-<6(a;))| : j = 0,1,..., s(1), x G

13

ida,b(x)) -T(x,y)V<l\(f-1 o g)a,b(y) - ida,b(y))|]/[d(x,y)]q(1) : d(x,y) < ρ(x); (x,y) e

F2(k,k') and for (x,y) exists a chart Ui 3 x, Ui 3 y, x ^ y} are uniformly continuous

on G(1) relative to ρG(1) (g,f), F(k,k') := {z(k),..., z(k')} for each fc' > k; ρ(k;g,f) :=

ρ(1,k;f,g), t(1) = s(1) + q(1), 0 < s(1) G Z, 0 < q(1) < 1, a{x) := min[σ(x), σ(y)] for a

pair (x,y), ha,b = φa o h o ^ l as in §2, so W(k + 1,t(1), c; f) C W(k,t(1), c; f) for each

fceN.(A.2). Now let M fulfil (2.1.ii) with dM ^ 0. In view of theorem 3.1 for 2ξ > t > 1,

£ G R \ Z and t > q > ξ — 1 (t > 2q > 2ξ — 2) the operator A given above induces

QIM on Dift(q;M) (on Dt(M;q) respectively) , since for q > ξ: Ct(q;TM) is closed in

Ct(ξ - 1;TM) and in view of the Banach theorem A' : Ct,q(TM) -> ^ ( ^ ' " ( T M ) ) is

the isomorphism onto the Banach subspace in C^^iTM). Hence we obtain QIM (Ck-

differentiable) for each t > 0 analogously to the case of Diffβt(M). Then we use the

same procedures as in [?] (1) for infinite At(M), (2) for the consideration of differentiable

measures and (3) for induced measures from dense subgroups, when 0 < t < 1.

4.3. Theorem. Let M be a E™>s-manifold as in §2. Then Diffβ,γt(M) has QIM (and

k times differentiable) JJL relative to dense subgroups G' (and G" respectively, G" C G'),

where δ > γ + 2, γ > 0.

Proof. In view of corollary 2.7 there exists a dense subgroup Diffp,(N) in Diffβ,γt(M)

with topologies r ' and τ respectively such that τ|Dif fptJ>(N) C r', since δ > γ + 2 . There-

fore, QI (and k times differentiable) λ on Diffp,(N) relative to G' (and G" respectively)

induces QI (and k times differentiable) JJL on Diffβ,γt(M) analogously to the proof of the-

orem 4.5 [?]. But we simplify below the construction of λ on Diff^y(N).

In view of lemma 4.12 [?] and theorem 3.2 there are BS E^)r/,(TN) such that the

operator Ak;m(k) from Y ® (^)7,(TiV))®4m(fc)fc into E^)tY(N,TN) is a continuously dif-

ferentiable operator by V G Y, where t(k) = t - s(k), β(k) = β + s(k), s(k) =

4m(k)k, k = k(n), n E N , Y is an open neighbourhood of id in TidDif'f^,(N). Let

A{ip) •= J2n=iBnAk{n)Mk{n))(ij)e'n e 0 ~ = i ( 4 ( ( S , y ( T ^ ) 0 e'™) = : Z ( s e e a b o u t a d i "

rect sum of spaces in [?] ). Then there are Bn such that A(ψ) is well defined on Y,

where {e'n : n E N} is the standard orthonormal base in l2. For G' := Diff™^,(N)

the operator S^V) := ^[^(A"^!^))] — V is nuclear on Y with values in H, when

2m(n) > n for each n and m / — liniyj oo m(n)/n = c > 1 (see also 4.12-4.15 [?]), where

# : = {ξ = (e'nBnA'k{n)Mk{n))( :neN)\(E ^(TN)} is B S w i t h t h e following n o r m

\\£\\H := \\CWE* ,(TN)

a n d Bn are chosen such that A : Y —H i7 is the local uniform isomor-

phism, At(N) is finite, γ" > 7 '+ 2. Therefore, Z is dense in H. Then a Gaussian measure

v on H induces a QI (and k times differentiable) measure λ on Diffp,(N) relative to

G' = G". Then as in [?] we consider infinite At(N). Hence fj, on Diffβ,γt(M) is QI (and k

times differentiable) relative to the dense subgroup G', since G' is dense in Dif fβ ,(N).

Particularly, when N satisfies the conditions of §4 [?] the group G' = Diff™^, (N) con-

14

tains as a subgroup Di{{lγ}},δ(N) from theorem 4.5 [?] (see also the estimates given in 4.4 [?]

). Evidently, this theorem also is true, when the conclusion of lemma 2.7 is accomplished

with δ > 7' > γ.

4.4. Theorem. Let a manifold M fulfil (2.1.iii) and G(j) be a group of diffeo-

morphisms (1) G(1) := Diff(an,r)(M), or (2) G(2) := D(an, r)(M;h), or (3) G(3) :=

Dif(an,r)(q;M) with h > 0, q > 0, h and q G Z. Then there exists QIM ν on G(j)

relative to (1’) G'(l) := Dif f(an,R)(M), or (2') G'(2) := D(an,R)(M;h), or (3) G'(3) :=

Dif(an,R)(q; M) respectively and infinitely differentiable relative to a dense subgroup G" (j),

where r(2) > R > r > 4r(1) > 1. Moreover, if a manifold M' fulfils (2.1.i) and is diffeo-

morphic to M\dM, f : M' -> M\<9M is a diffeomorphism, f G Cf{M', M) or L(M', M)

and M fulfils (2.1.iii) with dM ^ 0, then there exists QIM ν on (4) G(4) := Diff^(M'),

or (5) G(5) := Dif f(L, M') relative to dense subgroups G'(4) or G'(5) and infinitely dif-

ferentiable relative to G"(4) or G"(5) respectively, where G"(j) C G'(j) C G(j), G"(j)

are dense in G(j) ,β > 0.

Proof. In view of theorem 4.1, E1^ : TidG(j) —> G(j) is the local uniform isomorphism,

i.e. for open subsets V C TidG(j) and i ( ie W e G(j), where V is a neighbourhood of

the zero section in TidG(j). The spaces C(an,r)(TM) and C{an'l){TM') are isomorphic

if an isomorphism f : M" -> M is in C( a n , r )(M", M) and M" C B ( R 2 n + 1 , 0 , r '(l)),

0 < r'(l) < 1, so we may consider r = 1. Let E(an,s)(TM) : = { f : f is a real analytic

section of TM, | | / | | B ( o n , . ) ( r M ) := E[|/,,fc|/(l + (k)q) : |k| = 0,1, 2,...; j = 1, 2,...] < 00} be

BS with s = qn, n = dimRM, qeN, (k)q := (k(1))q...(k(n))q, (c)q := (c+1)(c+2)...(c+q)

for c G Z. Therefore, there exists b G R such that (b + As) : C(an,1)(TM;s - ! ) - » •

E(an>8\TM) and (6 + A s) : G ^ ' ^ ' ^ ^ T M ) -> £ W ( r M ) are isomorphisms of these

Banach spaces, where s = 2n2, m = n. For M with <9M ^ f) an analytic manifold

M" exists such that M is the submanifold of M" with r"(2) = r(2), r"(i) > r(i) for

z = 0,1, dimRM" = dimRM, dM" ^ 0, 5M" n M = 0, consequently, in view of

the Stone-Weierstrass theorem C(an<r'"><8-1(TM") and C(an,r")(TM"; s - 1) are dense in

C(an,r)(TM), where r" > r. Hence a Gaussian measure on ^ " ^ " ' ( T M " ) induces a

Gaussian measure on C(an,r)(TM) and C(m- r»(TM), moreover, Dif(an,r”)(s - 1;M") is

dense in Diff(an,r)(M). Whence (infinitely differentiable) QIM ν on Dif^an'r"\q'-M")

[D(an<r")(Mn;h')\ induces (infinitely differentiable) QIM on G(j) with j = 1,2,3 and

g' > q, h' > h, q' > s - 1, h' > s - 1.

Let A be of the same type operator as in the proof of theorem 4.2 with its degree

2ξ = 4n2, hence ALφf - Af e E^'8-2^(TM), evidently, E^'3-2'^(TM) is the dense

Indeed, there exist Gaussian measures on E(an,s)(TM) =: X(j) that induce ν on

Bf(W) such that they are quasi-invariant on Bf(V) relative to the left action of G'(j) D

V 3 V, where L^{ji{E)) := fji(ipE) for E G Bf(W), V 3 id, V is open in G'(j),

15

V'~l = V, V'V C W, V'V C W, V 3 id, W and V are open in G(j). Since, in view of

theorem I.4.4 in [?], for X(j) exists the abstract Wiener space (S,H,X(j)), where H is

the separable Hilbert space, S : H —> X(j) is the injective embedding, S(H) is dense in

X(j), H+ C H C E- is the rigged Hilbert space, H_ = X(j) [?].

For S G σ22(X(j)) (the Hilbert-Schmidt operator) there is the canonical Gaussian

measure \i on X(j) with the correlation operator T = SS* and zero mean value. Its

quasi-invariance follows from §26 in [?], since 2[fc|=o 1/(1 + (k)2) < °°- I n view of theorem

4.1 analogously to the proof of theorem 4.2 we obtain infinitely differentiable QIM ν on

Bf(G(j)) forj=1,2,3. Then f~l ogof is in G(j) for each g G Dif(an,r)(0; M), where j = 4

for f G ^ ( M ' , M) and j = 5 for f G L(M', M), hence f~l o Dif(an,r)(0; M) o f =: G

is dense in G(j) and the topology of G induced from G(1) is stronger than that of G(j).

Taking G'(j) = f~l ° G'(3) o f and G"(j) = Z" 1 o G"(3) o f for j = 4,5 with q = 0we

prove other statements of theorem 4.4 in analogy with the proof of theorem 4.2.

4.5. Note. Let Inv(g) := g~l for g G G(j) and w(E) := (Inv(E)) for each E G

Bf(W) with W~l = W, consequently, R^*(w(E)) := w(Eψ) = u^E-1) and w is

QIM and infinitely differentiable from the right (with Rψ instead of Lψ, Rψf := f o ψ,

Lψf = ψ ° f), since Inv is the continuous automorphism of G(j). For example, M' = R n

fulfils the conditions of theorem 4.4.

On topological groups of diffeomorphisms Ds defined and topogolized with the help

of Sobolev spaces H2t(TM) for compact M and s > n/2 + 2 [?] (differentiable) QIM

relative to D2m+s may be constructed using the proof of theorem 4.2 and the corresponding

isomorphism A as in theorem 3.1 A : H2t(TM) -> Hl~2j(TM) ® {(g)^1 Bl~1/2~hi(TdM)},

here H2t and B2t are the Bessel-potential and Besov spaces respectively [?, ?, ?].

4.6. Theorem. Let (1) G be a group of diffeomorphisms of a manifold M defined as

in §2 above or as in [?, ?, ?], or (2) G be an infinite-dimensional over the corresponding

field (R or K, Q p C K C Cp) Banach-Lie group such that for its Banach-Lie algebra

g there is not any dense subalgebra g' such that ad(h) are nuclear in the case of R or

compact in the case ofK operators for each h G g'. Assume that G' is a dense subgroup

of G. Then it does not have non-trivial QIM /i with values in R or F which is QI relative

to (a) Lψ and Rψ simultaneously or (b) relative to inner automorphisms αh(f) := h~1fh

for each h G G', where F is a local field.

Proof. For a Banach-Lie group there exists the exponential mapping exp : V —> W

which is the local isomorphism of open V and W, where V C g and e G W C G.

Therefore, ln(W n G') =: V is dense in V (see the Hausdorff series, §§II.6-8, ch. II,

§VII.3 in [?] ).

For a group of diffeomorphisms there exists a refinement At” (M) of At(M) such

that At"(M) provides a locally finite covering of M by charts U"j. Therefore, U :=

U1 \ (Ui/i U" i) is open M [?]. Let GU denote a subgroup of G consisting of f G G with

16

supp(f) C U. The set M \ U is closed in M, hence GU is closed in G. From the definition

of topology in G it follows that G'C\Gu =: G'u is dense in GU. In view of [?, ?] a retraction

r exists of G onto GU. Hence QIM p n G relative to G' induces QIM ν(S) = fi(r~lS) for

each S G Bf(GU) relative to G'u- But U is the flat manifold and expx is trivial for each

x G U, consequently, E is trivial on a sufficiently small neighbourhood W of id G GU.

We denote ν and GU again by \i and G.

To each local one-parameter subgroup of G corresponds a vector field on U for the

group of diffeomorphisms or an element of g for the Banach-Lie group. If G is the non-

Archimedean group of diffeomorphisms then ad(h) is not a compact operator for each

smooth (i.e. of class G°°) h in E-\W), where W = W n G'. For the real group

of diffeomorphisms it is not a nuclear operator for each such h. This follows from the

consideration of the algebra of smooth vector fields on U and the fact that the group of

diffeomorphisms is simple and perfect.

If /j, is QIM on G and fulfils (a) or (b) then In or E~l induce a measure λ on V that

is QI relative to ad(h) for each h G V, V = ln{W) or V = E-\W) respectively. But

this contradicts theorems 2.31, 3.12, lemma 3.26 [?] in the non-Archimedean cases, the

Minlos-Sazonov theorem and theorem 19.1 [?] in the cases of the Banach-Lie group or M

over R for the group of diffeomorphisms.

4.7. Note . As in [?] QIM \i on GD of a real Banach manifold induces the strongly

continuous regular unitary representation. In view of corollary 2.7 each strongly contin-

uous irreducible unitary representation (IUR) of Diffp,(N) has a continuous extension

on Diffβ,γt(M). Each QIM on N relative to Diffβt””,γ”(N) induces QIM on M relative to

the same group. Theorefore, IUR constructed in [?] induces IUR for groups of diffeomor-

phisms of infinite-dimensional Banach manifolds M.

References

[1] S. Agmon, A. Douglis, L. Nirenberg. "Estimates near the boundary for solutions of

elliptic partial differential equations satisfying general boundary conditions I, II",

Comm. Pure Appl. Math. 12(1959), 623-727; 17(1964),35-92.

[2] C. Amrouche, V. Girault, J. Giroire. "Weighted Sobolev spaces for Laplace's equation

in R n " , J. Math. Pures et Appl. 73(1994), 579-606.

[3] Ya. I. Belopolskaya, Yu. L. Dalecky. "Stochastic equations and differential geometry"

(Kluwer, Dodrdrecht, 1990).

[4] V.I. Bogachev, O.G. Smolyanov. "Analytical properties of infinite-dimensional dis-

tributions", Uspekhi Mat. Nauk 45(1990), 3-83.

17

[5] N. Bourbaki. "Lie groups and algebras". (Mir, Moscow, 1976).

[6] A. Chaljub-Simon, Y. Choquet-Bruhat. "Problemes elliptiques du second ordre sur

une variete Euclidienne a l'infini", Ann. Fac. Sci. Toulouse Math. Ser. 5. 1(1979),

9-25.

[7] Y. Choquet-Bruhat, D. Christodoulou. "Elliptic systems in Hs,δ spaces on manifolds

that are Euclidean at infinity", Acta Math. 146(1981), 129-150.

[8] Yu.L. Dalecky, S.V. Fomin. "Measures and differential equations in infinite-

dimensional spaces" (Kluwer Acad. Publ., Dordrecht, 1991).

[9] Yu. A. Dubinski. "Sobolev spaces of infinite order", Uspekhi Mat. Nauk 46(1991),97-

131 (N 6).

[10] D.G. Ebin, J. Marsden. "Groups of diffeomorphisms and the motion of incompressible

fluid", Ann. of Math. 92(1970),102-163.

[11] R. Engelking. "General topology" (Mir, Moscow, 1986).

[12] Federer H. "Geometric measure theory."(Springer-Verlag, Berlin, 1969).

[13] P. Flashel, W. Klingenberg. "Riemannsche Hilbertmannigfaltigkeiten. Periodische

Geodatische." Lect. Notes in Math. 282(Springer-Verlag, Berlin, 1972).

[14] I.M. Gelfand, N.Ya. Vilenkin. "Generalized functions" v. 4 (Fiz.-Mat. Lit., Moscow,

1961).

[15] G. Grubb, N.J. Kokholm. "A global calculus of parameter-dependent pseudo-

differential boundary problems in Lp Sobolev spaces", Acta Math. 171(1993), 165-

229.

[16] G. Grubb. "Parabolic pseudo-differential boundary problems and applications", in:

Lecture Notes in Math.(Springer, Berlin, 1991) 1495, 46-117.

[17] G. Grubb. "Functional calculus of pseudo-differential boundary problems"

(Birkhauser, Boston, 1986).

[18] E. Hewitt, K.A. Ross. "Abstract harmonic analysis" v. 1 (Springer, Berlin, 1979).

[19] L. Hormander. "The analysis of linear partial differential operators" (Mir,

Moscow, 1986).

J.R. Isbell. "Uniform neighborhood retracts", Pacif. J. Math. 11(1961), 609-648.

18

[21] S. Kobayashi, K. Nomizu. "Foundations of differential geometry" v.1 and 2 (Nauka,

Moscow, 1981).

[22] A.V. Kosyak. "Irreducible Gaussian representations of the group of the interval and

circle diffeomorphisms", J. Funct. Anal. 125(1994), 493-547.

[23] H.-H. Kuo. "Gaussian measures in Banach spaces" (Springer, Berlin, 1975).

[24] S. Lang. "Differential manifolds." (Springer-Verlag, Berlin, 1985).

[25] G.P. Lopushanskaya. "About solution of boundary problems for elliptic systems in a

space of generalized functions", Differential Equations 28(1992), 1401-1410.

[26] S.V. Liidkovsky. "Measurability of representations of locally compact groups", Mat.

Sb. 186(1995), N 2, 83-92.

[27] S.V. Liidkovsky. "Classification of some types of locally compact groups by their

unitary representations", Uspekhi Mat. Nauk 47(1992), N 5, 185-186.

[28] S.V. Liidkovsky. "Measurability of infinite-dimensional groups representations", Us-

pekhi Mat. Nauk 51(1996), N 3.

[29] S.V. Liidkovsky " Measures on groups of diffeomorphisms of non-Archimedean Banach

manifolds", Uspekhi Mat. Nauk 51 (1996), 169-170 (N 2).

[30] S.V. Liidkovsky. "Representations and structure of groups of diffeomorphisms of non-

Archimedean Banach manifolds", parts 1 and 2, Preprints No IC/96 /180 and No

IC/96 /181 of Intern. Centre for Theoret. Phys., Trieste, Italy, September, 1996.

[31] S.V. Liidkovsky. " Quasi-invariant measures on a group of diffeomorphisms of an

infinite-dimensional Hilbert manifold and its representations. Intern. Centre for The-

oret. Phys, Trieste, Italy, Preprint No IC/96/202 , October 1996.

[32] S.V. Liidkovsky. "Quasi-invariant and pseudo-differentiable measures on a non-

Archimedean Banach space". Intern. Centre for Theoret. Phys., Trisete, Italy,

Preprint No IC/96/210 , October 1996.

[33] S.V. Liidkovsky. "Quasi-invariant measures on a non-Archimedean group of diffeo-

morphisms and on a Banach manifold". Intern. Centre for Theor. Phys. Trieste, Italy,

Preprint IC/96 /215 , October, 1996.

[34] M.P. Malliavin, P. Malliavin. "Measures quasi invariantes sur certain groupes de

dimension infini", C.R. Acad. Sci. Paris Ser. 1 311 (1990), 765-768.

19

[35] Yu.A. Neretin. "Some remarks on quasi-invariant actions of loop groups and the

group of diffeomorphisms of the circle", Comm. Math. Phys. 164(1994), 599-626.

[36] A. Pietsch. "Nuclear locally convex spaces" (Springer, Berlin, 1972).

[37] M. Reed, B. Simon. "Methods of modern mathematical physics" v. 1, 2 (Acad. Press,

New York, 1972, 1975).

[38] R. Seeley. "The resolvent of an elliptic boundary problem", Amer. J. Math.

91(1969),889-920.

[39] R. Seeley. "Singular integrals and boundary value problems", Amer. J. Math.

88(1966), 781-809.

[40] H.H. Schaefer. "Topological vector spaces." (Mir, Moscow, 1971).

[41] E.T. Shavgulidze. "On a measure that is quasi-invarint with respect to the action

of a group of diffemorphisms of finite-dimensional manifolds", Soviet Math.Dokl.

38(1989), 622-625.

[42] A.V. Skorohod. "Integration in Hilbert space" (Springer- Verlag, Berlin, 1974).

[43] E.M. Stein. "Singular integrals and differentiability properties of functions" (Princ.

Univ. Press, New Jersey, 1970).

[44] M.E. Taylor. "Pseudo-differential operators". (Princ. Univ. Press, New Jersey, 1981).

[45] H. Triebel. "Interpolation theory. Function spaces. Differential operators" (Deutsche

Verlag, Berlin, 1978).

[46] V.S. Vladimirov. "Uravneniya Matematicheskoy Fiziki" (Nauka, Moscow, 1971).

[47] A. Weil. "L'integrtion dans les groupes topologiques et ses applications", Actual.

Scient. et Ind. N 869 (Herman, Paris, 1940).

20