PSEUDODIFFERENTIAL CALCULUS ON COMPACT LIE GROUPS

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1 Helsinki University of Technology Institute of Mathematics Research Reports Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja Espoo 2001 A431 PSEUDODIFFERENTIAL CALCULUS ON COMPACT LIE ROUPS Ville Turunen

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3 Helsinki University of Technology Institute of Mathematics Research Reports Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja Espoo 2001 A431 PSEUDODIFFERENTIAL CALCULUS ON COMPACT LIE ROUPS Ville Turunen Helsinki University of Technology Department of Engineering Physics and Mathematics Institute of Mathematics

4 Ville Turunen: Pseudodierential calculus on compact Lie groups ; Helsinki University of Technology Institute of Mathematics Research Reports A431 (2001). Abstract: A continuous linear operator acting on the test function space on a compact Lie group is presented by a family of convolution operators (i.e. by a convolution operator -valued mapping on the group) arising naturally from the Schwartz kernel of the operator. This family, called the symbol of the operator, is closely related to the symbol of a pseudodierential operator on a Euclidean space. We characterize (1, 0)-type pseudodierential operators on compact Lie groups by symbol inequalities, that is operator norm inequalities for operator-valued symbols. Furthermore, we study associated symbol calculus, giving asymptotic expansions for amplitude operators, adjoints, transposes, compositions and parametrices. Main tools are the Schwartz kernel theorem and a non-commutative operator-valued Fourier transform of distributions on a group. We also present a commutator characterization of pseudodierential operators on closed manifolds. AMS subject classications: 35S05, 22E30, 43A77, 46A32, 46E40, 58J40, 47B47, 4730 Keywords: pseudodierential operators, Lie groups, non-commutative Fourier transform, operator-valued functions, symbol calculus, Schwartz kernels, asymptotic expansions, parametrices, commutators ISBN ISSN Otamedia Oy, Espoo, 2001 Helsinki University of Technology Department of Engineering Physics and Mathematics Institute of Mathematics P.O. Box 1100, HUT, Finland

5 1 Introduction In this paper we present a symbolic calculus and a characterization of the Hörmander (1, 0)-class of pseudodierential operators on compact Lie groups. The operator classes are characterized by operator norm inequalities on a family of convolution operators parametrized by the group, and these inequalities are analogous to the traditional symbol inequalities of pseudodierential calculus on the Euclidean spaces. As a special case, the denition of periodic pseudodierential operators (i.e. operators on the torus T n ) given by Agranovich in 1979 follows. The genesis of pseudodierential theory can perhaps be traced back to the studies of singular integral operators by S.. Mikhlin in the 1940s and later by A. P. Calderón and A. Zygmund in the 1950s. There one can see the idea of presenting an operator A by a family of convolution operators, i.e. (Af)(x) = (s A (x) f)(x) = (s A (x))(x y) f(y) dy. R n In 1965 J. J. Kohn and L. Nirenberg [9] created the modern pseudodierential theory (soon generalized by L. Hörmander [8]), where the main emphasis is on certain weighed inverse Fourier transforms. More precisely, (Af)(x) = σ A (x, ξ) ˆf(ξ) e i2πx ξ dξ, R n where the symbol function (x, ξ) σ A (x, ξ) is the object of interest. These approaches are connected formally via the Fourier transform, σ A (x, ξ) = ŝa(x)(ξ). Our treatise draws inspiration from the schools of both MikhlinCalderón Zygmund and KohnNirenbergHörmander, and we considerably utilize the work of M. E. Taylor [15]. In other words, starting from the (left) regular representation π L : L(L 2 ()) of a compact Lie group we introduce the Fourier transform f π(f) of distributions on by π(f)g = f g; i.e. we map distributions to corresponding convolution operators. Then we present a pseudodierential operator A Ψ m () as a family of convolution operators π(s A (x)) = σ A (x) (where s A : D (), s A (x)(y) = K A (x, y 1 x)) so that (Af)(x) = (σ A (x)f)(x) = (s A (x) f)(x) = Tr (σ A (x) π(f) π L (x) ). This trace formula is important from the application point of view; however, we shall not examine aspects of numerical analysis in this paper. 2 Non-commutative Fourier analysis This section deals with harmonic analysis of distributions on compact Lie groups. We introduce the concept of the operator-valued symbol of a continuous linear operator acting on distributions, and given a symbol with suitable 3

6 properties, we study Sobolev space boundedness of the corresponding operator. Throughout this treatise, the space of continuous linear operators between topological vector spaces X, Y is denoted by L(X, Y ), and L(X) := L(X, X). All the manifolds are (real) C -smooth and without a boundary. For basic facts about harmonic analysis on compact Lie groups we refer to [21]. Notions C and R stand for the elds of complex and real numbers, respectively, and Z, N 0 and N stand for integers, non-negative integers and positive integers, respectively. Let be a compact Lie group and µ its normalized Haar measure, i.e. the unique regular Borel probability measure which is left translation invariant: f(x) dµ (x) = f(yx) dµ (x) for every f C() and y. Then also f(x) dµ (x) = f(xy) dµ (x) = f(x 1 ) dµ (x). The convolution f g L 2 () of f, g L 2 () is dened (almost everywhere) by (f g)(x) = f(y) g(y 1 x) dµ (y). (1) Let π L : L(L 2 ()) be the left regular representation of, that is (π L (y)g)(x) = g(y 1 x) (2) for almost every x. The right regular representation π R : L(L 2 ()) is dened by (π R (y)g)(x) = g(xy) (3) for almost every x. The Fourier transform (or the global Fourier transform, see [12]) of f L 2 () is by our denition the left convolution operator π(f) L(L 2 ()), denoted also by π(f) = π(f)g = f g, (4) f(y) π L (y) dµ (y). The inverse Fourier transform is then given in L 2 ()-sense by f(x) = Tr (π(f) π L (x) ), (5) where Tr is the trace functional; notice that Tr(AB) = Tr(BA). By choosing an orthonormal basis of representative functions (trigonometric polynomials) provided by the PeterWeyl theorem we can realize the operators π L (x) and π(f) as block diagonal matrices with nite-dimensional blocks, each block 4

7 corresponding to some irreducible unitary representation of. Each matricial entry of the matrix form of π L (x) is then a C -smooth function on, whose translates span a nite-dimensional subspace of L 2 (); moreover, the set of these matricial entries contains an orthogonal basis of L 2 (). Let D() be the set C () equipped with the usual Fréchet space structure, and D () = L(D(), C) its dual, i.e. the set of distributions with D() as the test function space. We equip the space of distributions with the weak -topology. The Fourier transform of a distribution f D () is dened to be the left convolution operator π(f) L(D()), that is π(f)φ := f φ. The duality D() D () C is denoted by φ, f := f(φ), (6) where φ D() and f D (), and an embedding D() D (), ψ f ψ = ψ is given by φ, ψ = φ(x) ψ(x) dµ (y). (7) The transpose of A L(D()) is A t L(D ()) dened by and the adjoint A L(D ()) is given by Aφ, f = φ, A t f, (8) (Aφ, f) = (φ, A f), (9) where (φ, f) = φ, f, f(x) := f(x). Notice that π(f) = π( f) where f(x) = f(x 1 ), and π(f) t = π( ˇf) where ˇf(x) = f(x 1 ). In the sequel, we denote π(a)π(f) := π(af) if A L(D()) and f D(); i.e. π(a) is just A put through the Fourier transform. Stay alert: in this notation, if φ D(), f D (), then π(φ f) = π(φ)π(f), but π(φf) = π(m φ )π(f) where M φ is the multiplication operator f φf. Let D() D () denote the complete locally convex tensor product of the nuclear spaces D() and D () (see [11]). Then K D() D () denes a linear operator A L(D()) by Aφ, f := K, f φ. (10) In fact, the Schwartz kernel theorem states that L(D()) and D() D () are isomorphic: for every A L(D()) there exists a unique K A D() D () such that the duality (10) is satised with K = K A, which is called the Schwartz kernel of A. Duality (10) gives us also the interpretation for Lemma 1. (Aφ)(x) = Let A L(D()), and let K A (x, y) φ(y) dµ (y). (11) s A (x, y) := K A (x, y 1 x) (12) in the sense of distributions. Then s A D() D (). 5

8 Notice that D() D() = D( ). Let us dene the multipli- m : D() D() D(), m(f g)(x) := f(x)g(x), Proof. cation the co-multiplication : D() D() D(), ( f)(x, y) := f(xy), and the antipode S : D() D(), (Sf)(x) := f(x 1 ). These mappings are a part of the (nuclear Fréchet) Hopf algebra structure of D(), see e.g. [1] or [14]. The mappings are all continuous and linear. The convolution of operators A, B L(D()) is said to be the operator A B := m(a B) L(D()); it is easy to calculate the Schwartz kernel or K A B (x, y) = K A (x, yz 1 ) K B (x, z) dµ (z), K A B = (m t )(id τ id)(k A K B ), where τ : D () D() D() D (), τ(f φ) := φ f, and t : D () D () D () is the transpose of the co-multiplication, t (f g) = f g (i.e. t extends the convolution of distributions). Now (A S)S L(D()), hence K (A S)S D() D () by the Schwartz kernel theorem, and K (A S)S (x, y) = K A S (x, y 1 ) = K A (x, y 1 z 1 ) K S (x, z) dµ (z) = K A (x, y 1 x) = s A (x, y) Any distribution s D() D () can be considered as a mapping s : D (), x s(x), where s(x)(y) := s(x, y). If D L(D()) and M L(D ()) then (D M)s D() D (). For instance, D could be a partial dierential operator, and M a multiplication; these examples will be of great relevance for us in the sequel. 6

9 Denition. The symbol of A L(D()) is the mapping σ A : L(D()), x π(s A (x)), (13) where s A (x)(y) = K A (x, y 1 x) as in Lemma 1. Now (Af)(x) = = K A (x, y) f(y) dµ (y) s A (x)(xy 1 ) f(y) dµ (y) = (σ A (x)f)(x) = Tr (σ A (x) π(f) π L (x) ). Hence we can consider the symbol σ A as a family of convolution operators obtained from A by freezing it at points x. Moreover, if A L(D()) is a right-invariant operator, i.e. Aπ R (x) = π R (x)a for every x, then its symbol is the constant mapping x σ A (x) A (A is a left convolution operator). Conversely, let us be given a function σ : L(D ()) such that σ(x) = π(s(x)), where s D() D (). Then we can dene Op(σ) L(D()) by (Op(σ)f)(x) = (σ(x)f)(x). (14) Let us dene derivations of symbols of operators A L(D()) in the following way: Let D L(D()) be a partial dierential operator, and (Dσ A )(x) = Dσ A (x) := σ B (x), where s B = (D id)s A. Then B = Op(Dσ A ) L(D()), because D L(D()), id L(D ()). The next theorem is an adaptation of M. E. Taylor's result (see [15]). Theorem 2. Let be a compact Lie group of dimension n and σ C k (, L(L 2 ())) with k > n/2. Then Op(σ) L(L 2 ()). Proof. We clearly have Op(σ)f 2 L 2 () = (σ(x)f)(x) 2 dµ (x) sup (σ(y)f)(x) 2 dµ (x), y and by an application of Sobolev embedding theorem we get sup (σ(y)f)(x) 2 C k (( y α σ)(y)f)(x) 2 dµ (y). y α k 7

10 Therefore using Fubini theorem to change the order of integration, we obtain Op(σ)f 2 L 2 () C k (( y α σ)(y)f)(x) 2 dµ (x) dµ (y) α k C k α k = C k α k C k α k sup y (( α σ)(y)f)(x) 2 dµ (x) sup ( α σ)(y)f 2 L 2 () y sup ( α σ)(y) 2 L(L 2 ()) f 2 L 2 (). y The proof is complete, because is compact and σ C k (, L(L 2 ())) Let be the bi-invariant (right- and left-invariant) Laplacian of, i.e. the Laplace operator corresponding to the bi-invariant Riemannian metric of (see [17]). The Laplacian is symmetric, I is positive; let Ξ σ (I ) 1/2(x) = (I ) 1/2. Then Ξ is bi-invariant, so that it commutes with every convolution operator. Moreover, Ξ s L(D()) and Ξ s L(D ()) for every s R. Let us dene (f, g) H s () = (Ξ s f, Ξ s g) L 2 () (f, g D()). The completion of D() with respect to the norm f f H s () = (f, f) 1/2 H s () is called the Sobolev space H s () of order s R. Thereby H 0 () = L 2 (). This denition of H s () coincides with the denition obtained using any smooth partition of unity on the compact manifold. Notice that σ AΞ r(x) = Ξ r σ A (x) = σ A (x)ξ r (A D()), and that Ξ r is a Sobolev space isomorphism H s () H s r () for every r, s R. Therefore Theorem 2 yields a simple consequence: Corollary 3. If is a compact Lie group, A L(D()) and Ξ m σ A C (, L(L 2 ())) (or equivalently σ A C (, L(H m (), H 0 ()))) then A L(H m (), H 0 ()). Convention: Let D be a right-invariant vector eld on. Then σ D (x) = D for every x, so that the reader must be cautious! In the sequel Dσ A (x) refers to the derivative (Dσ A )(x) of the operator-valued function σ A, whereas σ D σ A (x) means the composition of two convolution operators, namely σ D and σ A (x). More precisely, Dσ A (x) = σ B (x) with s B = (D id)s A, and σ D σ A (x) = σ C (x) where s C (x) = s D (x) s A (x). Let D = xj be a rst order right-invariant partial dierential operator with constant symbol σ D (x) D. Then for A L(D()) we have DA L(D()), and it is clear that x σ D σ A (x) is a symbol of an operator belonging to L(D()). Then σ DA (x) = ( xj σ A )(x) + σ D σ A (x). 8

11 Lie algebra: Traditionally the Lie algebra g of a Lie group is the set of the left-invariant vector elds on. In this work, however, we dene the Lie algebra g of a Lie group to be the set of the right-invariant vector elds on. This non-conventional practice simplies our notations; recall that rightinvariant operators are left convolution operators. Hence our Lie algebra g can be identied with rst order scalar right-invariant partial dierential operators on. Of course, g = T e () = R n, where T e () is the tangent space of at e, n = dim(). In the sequel, { j = xj 1 j n} denotes a vector space basis of g. 3 Pseudodierential operators on compact manifolds In this paper all the pseudodierential operators are of the Hörmander type (ρ, δ) = (1, 0). Recall the appearance of the pseudodierential operators on the Euclidean spaces, (Af)(x) = σ A (x, ξ) ˆf(ξ) e i2πx ξ dξ, (15) R n where f belongs to the Schwartz space S(R n ), and the Fourier transform ˆf S(R n ) is dened by ˆf(ξ) = f(x) e i2πx ξ dx; R n the symbol σ A C (R n R n ) is said to belong to S m (R n ), the set of pseudodierential symbols of order m R, if α ξ β x σ A (x, ξ) CAαβm (1 + ξ ) m α, (16) uniformly in x R n for every α, β N n, and then A is called a pseudodierential operator of order m, denoted by A OpS m (R n ). Imitating the analysis on compact groups (see the previous section), we can denote by σ A (x 0 ) (x 0 R n ) the convolution operator with the symbol (x, ξ) σ A (x 0, ξ), and then (Af)(x) = (σ A (x)f)(x). Now A L(S(R n )), A L(S (R n )), and σ A (x, ξ) = e ξ (x) 1 (Ae ξ )(x), where e ξ (x) = e i2πx ξ. On a closed manifold M the set Ψ m (M) of (1, 0)-type pseudodierential operators of order m R is usually dened using locally the denition for the Euclidean case, see [8], [16] or [17], and one denes the Sobolev spaces H m (M) using any smooth partition of unity on M. On the torus M = T n = R n /Z n, however, we can apply the Fourier series, as rst suggested by Agranovich [3]. More precisely, for A L(D(T n )) we can write (Af)(x) = ξ Z n σ A (x, ξ) ˆf(ξ) e i2πx ξ, (17) 9

12 where the Fourier transform ˆf of f D(T n ) is dened by ˆf(ξ) = f(x) e i2πx ξ dx; T n the symbol σ A C (T n Z n ) is said to belong to S m (T n ), the set of pseudodierential symbols of order m R, if α ξ β x σ A (x, ξ) CAαβm (1 + ξ ) m α, (18) uniformly in x T n for every α, β N n, and then A is called a periodic pseudodierential operator of order m, A OpS m (T n ). Here α ξ is a dierence operator, α ξ σ(ξ) = γ α ( ) α ( 1) α γ σ(ξ + γ). γ Again, σ A (x, ξ) = e ξ (x) 1 (Ae ξ )(x). The fact OpS m (T n ) = Ψ m (T n ) was proven by Agranovich [2], and the result was generalized by McLean [10] for more general (ρ, δ)-classes on the tori. Beals discovered in [4] that pseudodierential operators on the Euclidean spaces can be characterized by studying Sobolev space boundedness of iterated commutators by dierential operators. Related studies (also on closed manifolds) were pursued by Dunau [7], Coifman and Meyer [5] and Cordes [6]. In [18] another variant of commutator characterization was given on closed manifolds (i.e. compact manifolds without boundary), equivalent to the following denition: Denition. Let M be a closed smooth orientable manifold. An operator A L(D(M)) is a pseudodierential operator of (1, 0)-type and of order m R, denoted by A Ψ m (M), if and only if A k L(H m+d C,k (M), H 0 (M)) for every k N 0 and for every sequence C = (D k ) k=1 L(D(M)) of smooth vector elds, where A 0 = A and A k+1 = [D k+1, A k ], d C,0 = 0 and d C,k+1 = k+1 j=1 (1 deg(d j)). Here deg(d j ) is the order of the partial dierential operator D j, i.e. deg(m φ ) = 0 for a multiplication operator M φ, and deg( β ) = β. A variant of this denition was applied in proving that periodic pseudodierential operators are traditional pseudodierential operators [18]. Let us give another equivalent criterion for pseudodierential operators, simpler in formulation: Theorem 4. Let M be a closed smooth orientable manifold. An operator A L(D(M)) belongs to Ψ m (M) if and only if (A k ) k=0 L(Hm (M), H 0 (M)) for every sequence (D k ) k=1 L(D(M)) of smooth vector elds (i.e. continuous derivations of the function algebra D(M)), where A 0 = A and A k+1 = [D k+1, A k ]. 10

13 Proof. Let Op m denote the set of operators satisfying the the property expressed by the iterated commutators in the claim. In fact, Ψ m (M) L(H s (M), H s m (M)) for every s, m R, and there is the nice commutator property [Ψ m 1 (M), Ψ m 2 (M)] Ψ m 1+m 2 1 (M), and hence it is clear that Ψ m (M) Op m. Now suppose that A Op m. First, Op r L(H r (M), H 0 (M)) for every r R. Hence in order to prove the theorem it suces to verify that [M φ, A] Op m 1 for every φ D(M). Let g be some Riemannian metric for M, and let g be the corresponding Laplacian (see [17]). Then I g is a Sobolev space isomorphism H s (M) H s m (M) for every s, m R. We write the second order partial dierential operator I g in the form i,j E if j + F + M ψ, where F, E i, F j L(D(M)) are some smooth vector elds and ψ D(M). Now [M φ, A] = [M φ, A](I g )(I g ) 1 ( ) = ([M φ E i, A] M φ [E i, A])F j + [M φ, A](F + M ψ ) (I g ) 1. i,j Here M φ L(H s (M)) for every s R, and A L(H m (M), H 0 (M)), so that [M φ, A] L(H m (M), H 0 (M)), and moreover (I g ) 1 L(H m 1 (M), H m+1 (M)), F j, F + M ψ L(H m+1 (M), H m (M)), ([M φ E i, A] M φ [E i, A]) L(H m (M), H 0 (M)). This yields [M φ, A] L(H m 1 (M), H 0 (M)). For a smooth vector eld D we clearly have [D, [M φ, A]] = [M φ, [D, A]] + [[D, M φ ], A], so that also [D, [M φ, A]] L(H m 1 (M), H 0 (M)); notice that [D, A] Op m. Then assume that we know [D k, [D k 1, [D 1, [M φ, A]] ]] = γ J k [M φγ, A γ ], where J k is some nite index set, φ γ D(M), A γ Op m, D j (1 j k) smooth vector elds. Then [D k+1, [D k, [D k 1, [D 1, [M φ, A]] ]]] = γ J k ( [Mφγ, [D k+1, A γ ]] + [[D k+1, M φγ ], A γ ] ) = γ J k+1 [M φγ, A γ ] for a nite index set J k+1, φ γ D(M), A γ Op m when γ J k+1. Hence [M φ, A] Op m 1 (M), so that Op m Ψ m (M) 11

14 Let M be a closed smooth manifold. As it is well-known, for any given sequence of pseudodierential operators A α Ψ mα (M) (α N n 0) with m α as α, there exists a pseudodierential operator A Ψ max{mα} (M) satisfying the following: for every r R there exists N r N such that A α <N A α Ψ r (M) whenever N > N r. Then the formal sum α A α is called an asymptotic expansion of A, denoted by A α 0 A α ; if M = is a compact Lie group, we also write σ A (x) α 0 σ Aα (x). Notice that this asymptotic expansion denes the operator only modulo Ψ (M) := r R Ψ r (M) = {B L(D(M)) B(D (M)) D(M)}; then the asymptotic expansions endow the set Ψ (M) = r R Ψ r (M) with a non-hausdor topology, but for Ψ (M)/Ψ (M) we obtain a complete metrizable topology, where addition and composition of equivalence classes of operators are continuous, i.e. we have a complete metrizable topological ring. This division into classes modulo innitely smoothing operators is reasonable, if we are only interested in information related to the singularities of distributions. 4 Operator norm symbol inequalities for pseudodierential operators In this section we provide operator norm inequalities for symbols characterizing the classes Ψ m () for compact Lie groups ; the main result is Theorem 9, in the end of the section. In the sequel, n = dim(). Taylor expansion: By identifying the Lie algebra g of with R n, a function f C (g) can be presented as a TaylorMaclaurin expansion, f(x) = 1 α! xα ( α f)(0) + R N (x) (19) α <N where the remainder term R N is of the form R N (x) = 1 α! xα ( α f)(θ x x) (20) α =N with some θ x (0, 1). Let V, W g be open balls centered at 0 g, V = B(0, r V ) W = B(0, r W ), such that the exponential mapping exp : g is injective on W. Thereby there exist functions q α C () satisfying q α (exp(x)) = 1 α! xα 12

15 when x B(0, r V + ε) g with some ε > 0; the smooth functions q α have zeros of order α at the point e. It is enough to construct functions q α C () for α = 1, and then dene higher order polynomials recursively by q α+β := q α q β for every α, β N n 0. Furthermore, for every x \ {e} we must have q α (x) 0 for some α = α x N n 0 with α = 1; this condition will be explained right after the denition of the symbol class S0 m (). It is easy to see that this condition can be fullled: For instance, let α = 1, and let ψ : [0, r W ] [0, 1] be an increasing C -smooth function such that ψ [0,rV +ε] = 0 and ψ [rw ε,r W ] = 1 where 0 < ε < (r W r V )/2, and set q α (exp(x)) := iψ( x ) + (1 ψ( x )) 1 α! xα when x W, and set q α (y) := i when y \ exp(w ). Consequently a function φ C () can be presented by φ(x) = q α (x) ( α φ)(e) + R N (x), (21) α <N when x exp(v ), with the remainder R N (x) = q α (x) ( α φ)(ψ(x)), (22) α =N where ψ(x) exp(v ), and x ( α φ)(ψ(x)) is C -smooth; in the sequel we abbreviate this by φ(x) q α (x) ( α φ)(e). α Then let us dene a quasi-dierence Q α = π(mˇqα ), that is, Q α σ A (x) = π(y ˇq α (y)s A (x)(y)). (23) The idea of this denition stems from the Euclidean Fourier transform F, which obeys (F(x x α ˆf(x)))(ξ) = ( α F(f))(ξ); a multiplication by a polynomial turns to a dierentiation when put through the Fourier transform. Let V = B(0, r V ), W = B(0, r W ) be as above, and let U = B(0, r U ) g with 0 < r U < r V. Using a partition of unity we may present an operator A L(D()) in a form A = A 0 + A 1 where A 0, A 1 L(D()), such that K A0 (x, y) = s A0 (x)(xy 1 ) = 0 when xy 1 exp(v ), and K A0 (x, y) = s A1 (x)(xy 1 ) = 0 when xy 1 exp(u). Suppose A is a pseudodierential operator; we know that the Schwartz kernel of a pseudodierential operator is C -smooth outside the diagonal of, and consequently A 0 is the interesting part of A. Let {x j } N j=1 such that {U j := x j exp(u)} N j=1 is an open cover for, and let {(φ j, U j )} N j=1 be a partition of unity subordinate to this cover. Let σ A0,j (x) := φ j (x j x)σ A0 (x j x); (24) then σ A0 (x) = N j=1 σ A 0,j (x 1 j x), and supp(s A0,j (x)) exp(v ) = V for each x exp(u) = U and j {1,..., N}. Notice that σ A0,j (x) = 0 when x 13

16 exp(u). Hence we can interpret s A0,j to be a distribution supported in a small neighbourhood of the origin of the Euclidean space R n R n. Let (B j f)(x) = s A0,j (x)(x y) f(y) dy, R n (25) i.e. σ Bj (x, ξ) = s A0,j (x)(ξ) (x, ξ R n ). Notice that α ξ β x σ Bj (x, ξ) = s Bjαβ (x)(ξ) (x, ξ R n ), (26) where σ Bjαβ (x) = Q α β x σ A0,j (x) (x ). Taylor ([15], Propositions 1.1 and 1.4) states essentially the following: Proposition 5 (Taylor's characterization of Ψ m ()). be as above. Then A Ψ m () if and only if K A1 C ( ) and {σ Bj } N j=1 S m (R n ). Let A L(D()) It is noteworthy that Taylor deals with not only compact Lie groups, but locally compact ones; in Proposition 5 we presented just a special statement suitable for our purposes. In the course of proving a generalization of Proposition 5, Taylor obtains also the following useful results (see [15], Propositions ): Lemma 6. Let A Ψ m A () and B Ψ m B (). Then Op(σ A σ B ) Ψ m A+m B () and Op(σ A ) Ψm A (), where (σ A σ B )(x) = σ A (x)σ B (x) and σ A (x) = σ A(x). Moreover, Op(Q α β x σ A ) Ψ m A α () for every α, β N n 0 and σ A C (, L(H m A (), H 0 ())) Relation (26) motivates the denition of S m 0 (): Denition. σ A S m 0 (), if Ξ α m Q α β x σ A (x) L(L 2 ()) C Aαβm (27) uniformly in x, for every α, β N n 0. Remark: The L(L 2 ())-norm of a convolution operator σ A (x) is just the supremum of the l 2 -operator norms of the nite-dimensional blocks of the canonical matrix representation of σ A (x) (see [15], p. 33). By Corollary 3, OpS0 m () L(H m (), H 0 ()). Now we are going to recursively dene symbol classes Sk m() such that Sm k+1 () Sm k (), and we shall prove that Ψ m () = OpS m () with S m () = k=0 Sm k (). 14

17 Remark: Recall that the functions q α, α = 1, vanish simultaneously only at the neutral element e, and that q α+β = q α q β for every α, β N n 0. This implies that if σ A S0 m (), then (x, y) s A (x)(y) is C -smooth function on ( \ {e}), i.e. (x, y) K A (x, y) may have singularities only at the (x = y)-diagonal. Moreover, notice that f H m () π(f) L(H s (), H s m ()), (28) π(g) L(H m (), H 0 ()) r > n/2 : g H (m+r) (). (29) On a non-commutative group the multiplication of operator-valued symbols is not usually commutative, and this causes some complications in symbolic calculus. Recall Theorem 4; we now nd additional requirements for the symbols by studying the commutator characterization of pseudodierential operators. Let A Ψ m (), and let D = M φ j be a derivation on D(). By Taylor [15] σ [D,A] (x) = φ(x) [σ j, σ A (x)] + ( ( ) ) φ(x) Q γ σ j γ x σ A (x) ( xφ(x)) γ (Q γ σ A (x)) σ j 0< γ <N + γ =N R γ (x) φ(x) [σ j, σ A (x)] + ( ( ) ) φ(x) Q γ σ j γ x σ A (x) ( xφ(x)) γ (Q γ σ A (x)) σ j γ >0 is an asymptotic expansion of the symbol of [D, A], where R γ Ψ m N (). This expansion inspires new demands on symbols: Denition. σ A S m k+1 (), if σa S m k (), [σ j, σ A ] S m k (), (Q γ σ j )σ A S m+1 γ k () and (Q γ σ A )σ j S m+1 γ k () for every j {1,..., n} and γ N n 0 with γ > 0. Denition. S m () = k=0s m k (). Lemma 7. Let A, A j L(D()) such that σ Aj S m () and K A Aj C j ( ) for every j N. Then σ A S m (). Moreover, ψσ A S m () for every ψ C (). 15

18 Proof. For any multi-indices α, β N n 0 we can evidently choose j αβ N so large that Ξ α m Q α β x σ A Ajαβ (x) L(L 2 ()) C (A Ajαβ )m, and hence σ A S0 m (); the rst step of induction is established. Next, suppose we have proven σ B Sk r() whenever there exists (σ B j ) j=1 S r () such that K B Bj C j ( ) (for every r R). Now [σ i, σ A ] = [σ i, σ Aj ] + [σ i, σ A Aj ]; we know that [σ i, σ Aj ] S m () for every j N, and that the Schwartz kernel corresponding to [σ i, σ A Aj ] gets arbitrarily smooth when j tends to innity. Thereby [σ i, σ A ] S m k (); similarly one proves that and (Q γ σ i )σ A S m+1 γ k () (Q γ σ A )σ i S m+1 γ k (). Thus σ A Sk+1 m (). Consequently we have proven that σ A S m (), the rst claim of Lemma 7. By induction we will get the result C ()Sk m() Sm k (). Indeed, for k = 0 this is trivial. Assuming C ()S m k () S m k () for every m R, let ψ C () and σ B S m k+1 (). Then ψσ B S m k (), and [σ j, ψσ B ] = [σ j, ψ]σ B + ψ[σ j, σ B ] S m k (), (Q γ σ j )(ψσ B ) = ψ(q γ σ j )σ B S m+1 γ k () (Q γ (ψσ B ))σ j = ψ(q γ σ B )σ j S m+1 γ k (). Thereby C ()S m k+1 () Sm k+1 (), so that C ()S m () S m () Lemma 8. Let σ A S m A (), σ B S m B (). Then σ AB (x) = (Q α σ A (x)) x α σ B (x) α <N + σ RN,α (x), α =N where for every j N there exists N j N such that K RN,α C j ( ) when α = N N j. 16

19 Proof. Observe that (ABf)(x) = (s A (x) (Bf))(x) = s A (x)(xz 1 ) (σ B (z)f)(z) dµ (y) = s A (x)(xz 1 ) s B (z)(zy 1 ) f(y) dµ (y) dµ (z) = s A (x)(xz 1 ) s B (z)(zy 1 ) dµ (z) f(y) dµ (y) = s A (x)(xz 1 ) ˇq α (xz 1 ) x α s B (x)(zy 1 ) dµ (z) α <N f(y) dµ (y) + (s A (x))(xz 1 ) ˇq α (xz 1 ) r B,N,α (x, z)(zy 1 ) dµ (z) α =N f(y) dµ (y), where s B (z)(y) = ˇq α (xz 1 ) x α s B (x)(y) α <N + ˇq α (xz 1 ) r B,N,α (x, z)(y), α =N with the remainder terms (x, z) r B,N,α (x, z) in C (, H m B r ()) for every r > n/2. Hence where K RN,α (x, y) = σ AB (x) = (Q α σ A (x)) x α σ B (x) α <N + α =N σ RN,α (x), s A (x)(xz 1 ) ˇq α (xz 1 ) r B,N,α (x, z)(zy 1 ) dµ (z). Notice that (x, y) š A (x)(y)ˇq α (y) gets arbitrarily smooth as N = α grows (cf. the behaviour of x Q α σ A (x)), so that the kernel K RN,α gets indeed smoother with growing N Theorem 9. Ψ m () = OpS m (). 17

20 Proof. Let A Ψ m (). By Lemma 6 we have σ A C (, L(H m (), H 0 ())), and Op(Q α x β σ A ) Ψ m α (). Thus Ψ m () OpS0 m (). We shall proceed by induction: Assume that Ψ m () OpSk m m R. By Lemma 6, we have () for every Moreover, Op((Q γ σ j )σ A ), Op((Q γ σ A )σ j ) Ψ m+1 γ (). Op([σ j, σ A ]) Ψ m (), because [σ j, σ A (x)] = σ [ j,a](x) ( j σ A )(x). Hence Ψ m () OpS m k+1 (), so that we have proven Ψ m () OpS m (). Now we are going to prove that [D, OpS m ()] OpS m () for a partial dierential operator D = M φ j Ψ 1 (); by Theorem 4 this would mean OpS m () Ψ m (), because OpS m () L(H m (), H 0 ()) due to Corollary 3. Let σ A S m (). Lemma 8 yields σ [D,A] (x) = φ(x)[σ j, σ A (x)] + ( ( ) ) φ(x) Q γ σ j γ x σ A (x) ( xφ(x)) γ (Q γ σ A (x)) σ j 0< γ <N + γ =N R N,γ (x), where the remainder terms R N,γ behave nicely enough: the corresponding Schwartz kernels K RN,γ become arbitrarily smooth as N grows (see Lemma 8). Thereby due to Lemma 7 we get σ [D,A] S m (); combining this with OpS m () L(H m (), H 0 ()) we have proven that OpS m () Ψ m () 5 Symbolic calculus In this section we present some elementary symbol calculus on compact Lie groups, and the results resemble closely the Euclidean case (see e.g. [8], [16] or [17]) and the torus case (see [19]). Strichartz constructed a calculus for left-invariant pseudodierential operators on Lie groups in [13], and Taylor for the class Ψ m () in [15]. We review some of Taylor's results. We also get 18

21 some new results: we dene and study a class of amplitudes in analogy to the Euclidean case, and present an asymptotic expansion for transposed operators. The culmination is the nal Proposition 14 yielding another asymptotic expansion for a parametrix of an elliptic operator. So far we have studied a pseudodierential operator A as a family x σ A (x) = π(s A (x)) of convolution operators, where (Af)(x) = (σ A (x)f)(x) = s A (x)(xy 1 ) f(y) dµ (y). Suppose we are given a two-argument family (x, y) a(x, y) = π(t a (x, y)) of convolution operators, and formally dene an operator Op(a) by (Op(a)f)(x) = t a (x, y)(xy 1 ) f(y) dµ (y). (30) More rigorously: Denition. A convolution operator-valued function a : L(D()) is called an amplitude of order m R, denoted by a A m (), if the mapping (x, y) Ξ m a(x, y) belongs to C (, L(L 2 ())) and the mapping x x β y γ a(x, y) y=x belongs to S m () for every β, γ N n 0. Example. Clearly S m () A m (), and if σ S m () then also (x, y) σ(y) belongs to A m (). Proposition 10 reveals us that Op(a) belongs to OpS m () whenever a A m (): Proposition 10. If a A m () then Op(a) = A OpS m () and the symbol has an asymptotic expansion σ A (x) α 0 Q α α z a(x, z) z=x. Proof. Examining (30), rst we develop t a (x, y) into Taylor series in variable y at x: t a (x, y)(xy 1 ) = α <N + α =N ˇq α (zy 1 ) α z t a (x, z)(xy 1 ) z=x ˇq α (zy 1 ) r a,n,α (x, z, xy 1 ) z=x, where (x, z, y) ˇq α (y)r a,n,α (x, z, y) gets arbitrarily smooth on as N grows ( α = N). Lemma 7 then yields the conclusion 19

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