The Melin calculus for general homogeneous groups

c 2007 by Institut Mittag-Leffler. All rights reserved

The Melin calculus for general homogeneous groups

Pawel Glowacki

Abstract.The purpose of this note is to give an extension of the symbolic calculus of Melin for convolution operators on nilpotent Lie groups with dilations. Whereas the calculus of Melin is restricted to stratified nilpotent groups, the extension offered here is valid for general homogeneous groups. Another improvement concerns theL²-boundedness theorem, where our assumptions on the symbol are relaxed. The zero-class conditions that we require are of the type

|D^αa(ξ)| ≤Cα

R j=1

ρj(ξ)^−|αj|,

whereρj are “partial homogeneous norms” depending on the variablesξkfork>j in the natural grading of the Lie algebra (and its dual) determined by dilations. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.

The purpose of this note is to give an extension of the symbolic calculus of Melin [7] for convolution operators on nilpotent Lie groups with dilations. The calculus can be viewed as a higher order generalization of the Weyl calculus for pseudodifferential operators of H¨ormander [3]. In fact, the idea of such a calculus is very similar. It consists in describing the product

a#b= (a^∨b^∨)^∧, a, b∈C_c^∞(g),

on a homogeneous Lie group G, where f^∧ and f^∨ denote the Abelian Fourier transforms on the Lie algebra g and its dual g, and its continuity in terms of suitable norms similar to those used in the theory of pseudodifferential operators.

An integral part of the calculus is an L²-boundedness theorem of the Calder´on–

Vaillancourt type.

This has been done by Melin whose starting point was the following formula a#b(ξ) =U(a⊗b)F(ξ, ξ),

where

U(F)^∨(x, y) =F^∨

x−y+xy

2 ,y−x+xy 2

, x, y∈g.

Melin shows that the unitary operator U can be imbedded in a one-parameter unitary groupU_twith the infinitesimal generator Γ which is a differential operator ong×gwith polynomial coefficients, and he thoroughly investigates the properties of Γ under the assumption that G is a homogeneous stratified group. From the continuity of U he derives a composition formula for classes of symbols satisfying the estimates

|D^αa(ξ)| ≤C_α(1+|ξ|)^m−|α|, (0.1)

where | · | is the homogeneous norm on g and |α| is the homogeneous length of a multiindex α. He also proves anL²-boundedness theorem for symbols satisfying (0.1) withm=0.

Our extension goes in various directions. First of all the calculus of Melin is restricted to stratified nilpotent groups, whereas the extension offered here is valid for general homogeneous groups. Another improvement concerns the L²-bound- edness theorem, where our assumptions on the symbol are less restrictive. The zero-class conditions that we require are

|D^αa(ξ)| ≤C_α R

j=1

ρ_j(ξ)^−|αj|,

where ρ_j are “partial homogeneous norms” depending on the variablesξ_k fork>j in the natural grading of the Lie algebra (and its dual) determined by dilations, and α=(α₁, α₂, ..., α_R) is the corresponding representation of the multiindexαrelative to the grading. This direction of generalization of the boundedness theorem had been suggested by Howe [5] even before the Melin calculus was created. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.

Most of the techniques applied here have been already developed in a very similar context of [2]. They heavily rely on the methods of the Weyl calculus of H¨ormander [3]. We take this opportunity to clarify some technical points which remained somewhat obscure in [2]. One major mistake is also corrected. Some repetition is therefore unavoidable. In [2] the reader will also find more on the background and history of various symbolic calculi on nilpotent Lie groups.

1. Preliminaries

LetX be a finite-dimensional Euclidean space. Denote by ·,· and · the scalar product and the corresponding Euclidean norm. These are fixed through- out the paper. Whenever we identify X with X, it is by means of the duality determined by the scalar product. LetX=_R

j=1X_j be an orthogonal sum and δ_tx_j=t^dj, x_j∈X_j,

be a family of dilations with eigenvaluesD={d_j}^R_j=1, where 1 =d₁< d₂< ... < d_R.

Let

|x|= R

j=1

x_j^2/dj _1/2

be the corresponding homogeneous norm and ρ(x) = 1+|x|. Let us define a family of Euclidean norms

p_x(z)²= R j=1

z_j²

ρ(x)^2dj, x∈X.

Lemma 1.1. We have 1 2≤ρ(x)

ρ(y)≤2 if p_x(x−y)<

1 2√ R

_dR (1.2)

and

ρ(x)≤√ R+1

ρ(y)(1+p_y(x−y)).

(1.3)

Proof. Observe thatp_x(x−y)<

1/

2√ R_dR

yields x_j−y_j^1/dj≤ρ(x)

2R , 1≤j≤R, so

|x−y| ≤¹₂ρ(x),

and consequently

ρ(x)≤ρ(y)+|x−y| ≤ρ(y)+¹₂ρ(x), ρ(y)≤ρ(x)+|x−y| ≤³₂ρ(x), which implies

1 2≤ρ(x)

ρ(y)≤2.

We also have

x_j−y_j^1/dj

ρ(y) ≤x_j−y_j ρ(y)^dj +1, so

|x−y| ρ(y) ≤√

R+p_y(x−y), and finally

ρ(x)≤ρ(y)+|x−y|=ρ(y)

1+|x−y| ρ(y)

≤√ R+1

ρ(y)(1+p_y(x−y)), which completes the proof.

For 0≤j≤R, let

ρ_j(x) =ρ(x^j) =ρ R

k=j⁺1

x_k

and let

q_x(z)²= R

j=1

z_j²

ρ_j(x)^2dj, x∈X, be another family of norms onX determined byρ.

Lemma 1.4. There exist constantsC, M >0such that ρ_j(x)≤Cρ_j(y)(1+q_y(x−y)) (1.5)

and

ρ_j(x)≤Cρ_j(y)(1+q_x(x−y))^M. (1.6)

Proof. Inequality (1.5) is proved in the same way as (1.3). The second in- equality is proved by induction. In fact, ifj=R, there is nothing to prove. Assume

that (1.6) holds fork>j with 1≤C=C_k≤C_j+1andM=M_k≤M_j+1. Then ρ_j(x)≤ρ_j(y)+|(x−y)^j| ≤ρ_j(y)

1+|(x−y)^j| ρ_j(y)

≤ρ_j(y)

1+

^R

k=j⁺1

x_k−y_k^2/dk ρ_k(y)²

≤

R−jρ_j(y)

1+

^R

k=j+1

x_k−y_k² ρ_k(y)^2dk

≤C_jρ_j(y)

1+

^R

k=j⁺1

x_k−y_k² ρ_k(x)^2dk

(1+q_x(x−y))^dj+1Mj+1

≤C_jρ_j(y)(1+q_x(x−y))^Mj, which shows that (1.6) holds also forj with

C_j=

R−jC_j^d₊^j+1₁ ≥C_j+1≥1, M_j=d_j+1M_j+1+1≥M_j+1.

A family of Euclidean norms (a metric) g={g_x}x∈X on X is called slowly varyingif there exists 0<γ≤1 such that

γ≤g_y g_x≤1

γ, if g_x(x−y)< γ.

(1.7)

A metricgonX is calledtempered with respect to another metricG, or briefly G-tempered, if there existC, M >0 such that

g_x g_y

±1

≤C(1+G_x(x−y))^M, g_x≤G_x. Note that a self-tempered metric is automatically slowly varying.

Lemma 1.8. Ifgis a self-tempered family of norms with constantsCandM, then for every x, y, z∈X,

1+g_x(x−y)≤C(1+g_y(y−x))^M+1,

1+g_x(x−y)≤C²(1+g_x(x−z))^M(1+g_y(y−z))^M+1. Proof. In fact,

1+g_x(x−y)≤1+Cg_y(y−x)(1+g_y(y−x))^M≤C(1+g_y(y−x))^M+1,

as required. Moreover, by the first inequality, 1+g_x(x−y)≤1+g_x(x−z)+g_x(z−y)

≤1+g_x(x−z)+Cg_z(z−y)(1+g_x(x−z))^M

≤(1+g_x(x−z))^M(1+Cg_z(z−y))

≤C²(1+g_x(x−z))^M(1+g_y(y−z))^M+1, which completes the proof.

Corollary 1.9. The metricspandq are slowly varying andq-tempered.

Proof. This follows immediately from Lemmas 1.1 and 1.4.

A strictly positive function m on X is a weight on X with respect to the G-tempered metricg, if it satisfies the conditions

m(x) m(y)

±1

≤C ifg_x(x−y)≤γ (1.10)

and

m(x) m(y)

±1

≤C(1+G(x−y))^M (1.11)

for some C, M >0. The weights form a group under multiplication. A typical example of a weight forporqism(x)=ρ(x). A universal example is

m(x) =g_x(x−x₀), (1.12)

where x₀ is fixed.

Letmbe a weight with respect to a metricg. Forf∈C^∞(X) let

|f|^m_(k)(g) = sup

x∈X

g_x(D^kf(x)) m(x) , and

|f|^m_k (g) = k

j=0

|f|^m_(j)(g),

where Dstands for the Fr´echet derivative, and g_x(D^kf(x)) = sup

g_x(yj)≤1|D^kf(x)(y₁, y₂, ..., y_k)|.

Let

S^m(X,g) ={a∈C^∞(X) :|a|^m_k (g)<∞for allk∈N}.

S^m(X,g) is a Fr´echet space with the family of seminorms| · |^m_k (g). Thusf∈C^∞(X) belongs toS^m(X,p) if and only if it satisfies the estimates

|D^αf(x)| ≤C_αm(x)ρ(x)^−|α|,

where |α| is the homogeneous length of a multiindex α. The estimates for f∈ S^m(X,q) are

|D^αf(x)| ≤C_αm(x) R

j=1

ρ_j(x)^−djαj.

Note that everyp-weightmis also aq-weight andS^m(X,p)⊂S^m(X,q). Moreover for everyk,

| · |^m_k (q)≤ | · |^m_k (p) (1.13)

so the inclusion is continuous.

Apart from the Fr´echet topology in the spacesS^mit is convenient to introduce aweak topology of theC^∞-convergence on Fr´echet bounded subsets. By the Ascoli theorem, this is equivalent to the pointwise convergence of bounded sequences in S^m. Following Manchon [6] we call a mappingT:S^m1!S^m2 double-continuous, if it is both Fr´echet continuous and weakly continuous. Moreover,C_c^∞(X) is weakly dense inS^m(X,g). The last assertion is a consequence of Proposition 2.1 (b) below.

2. The method of H¨ormander

The following construction of a partition of unity is due to H¨ormander [3]. Also the lemma that follows is an important principle of the H¨ormander theory. For the convenience of the reader we include the proofs here.

Proposition 2.1. Letg be a metric onX.

(a) For every0<r<γ there exists a sequencex_ν∈X such that X is the union of the balls

B_ν=B_ν(r) ={x∈X:g_xν(x−x_ν)< r}

and no pointx∈X belongs to more than N balls, whereN does not depend on x.

(b) There exists a family of functions φ_ν∈C_c^∞(B_ν)bounded in S¹(X,g) and such that

ν

φ_ν(x) = 1, x∈X.

(c) Forx∈X let

d_ν(x) =g_xν(x−z).

If the metric gis self-tempered, then there exist constantsM, C₀>0such that

ν

(1+d_ν(x))^−M≤C₀, x∈X.

All the estimates in the construction depend just on the constant γin(1.7)and the choice of r.

Proof. (a) Let 0<r<γ. Let{x_ν}ν be a maximal sequence of points inX such that

g_xν(x_µ−x_ν)≥γr, µ=ν.

Letx∈X. Note that

g_x(x−x_ν)< γr implies g_xν(x−x_ν)< r.

Therefore, either g_xν(x−x_ν)<rfor someν, or

g_x(x−x_ν)≥γr and g_xν(x−x_ν)≥r≥γr.

The latter contradicts the maximality of our sequence. The former implies that X⊂

νB_ν.

To show that the covering is uniformly locally finite suppose thatx∈B_ν. Then g_xν(x−x_ν)<r, which impliesg_x(x−x_ν)<r/γ <1. On the other hand

g_x(x_µ−x_ν)≥γr forµ=ν.

The number of points from a uniformly discrete set in a unit ball is bounded inde- pendently of the given normg_x so we are done.

(b) Let 0<r<r₁<γ. Letψ∈C_c^∞(−r₁², r²₁) be equal to 1 on the smaller interval [−r², r²]. If

ψ_ν(x) =ψ(g_xν(x−x_ν)²),

then, by part (a),

µψ_µ(x)≥1 for everyx∈X, and it is not hard to see that φ_ν(x) = ψ_ν(x)

µψ_µ has all the required properties.

(c) Letr<γ. Letx∈X. Fork∈N let

M_k={ν:d_ν(x)< k}.

It is sufficient to show that the number |M_k| of elements in M_k is bounded by a polynomial ink. Letν∈M_k and let

V_ν={z∈X:g_x(z−x_ν)< r_k}, where

r_k= r C(1+k)^M.

Observe thatV_ν is contained both inB_ν (see part (a)) and in the ball V ={z∈X:g_x(z−x)< R_k},

whereR_k=r_k+C(1+k)^M+1. In fact, ifg_x(z−x_ν)<r_k, then g_xν(z−x_ν)≤Cg_x(z−x_ν)(1+k)^M< r, and

g_x(z−x)≤g_x(z−x_ν)+g_x(x_ν−x)< r_k+Cg_xν(x_ν−x)(1+g_xν(x_ν−x))^M

< r_k+C(1+1+g_xν(x_ν−x))^M+1< r_k+C(1+k)^M+1 Hence

C₁|M_k|r_k^dimX≤

ν∈Mk

|V_ν| ≤N

ν∈Mk

V_ν

≤N|V| ≤C₁NR_k^dimX,

which immediately implies the desired estimate

|M_k| ≤N

1+R_k r_k

_dimX

.

Lemma 2.2. LetX be a finite-dimensional vector space with a Euclidean norm · . Letr₁>r>0. LetL be an affine function such that L(x)=0for x∈B(x₀, r₁).

Then for every k∈N, D^k1

L(x)

≤ k!r₁

(r₁−r)^k+1|L(x₀)|, x∈B(x₀, r).

The estimate does not depend on the choice the norm.

Proof. We may assume that x₀=0 andL(0)=1. Let ξ be a linear functional onX such thatL(x)=ξ, x+1. Since

L(x) =ξ, x+1>0, x< r₁, it follows that ξ≤1/r₁and

L(x)≥1−r

r₁, x∈B(0, r).

Consequently,

D^k1 L(x)

≤ k!ξ^k

|L(x)|^k+1≤ k!(1/r₁)^k (r₁−r/r₁)^k+1

≤ k!r₁ (r₁−r)^k+1 forxinB(0, r).

For the general theory of slowly varying metrics and its applications to the theory of pseudodifferential calculus the reader is referred to H¨ormander [4], vol. I and III.

3. The Melin operator U

Letgbe a nilpotent Lie algebra with a fixed scalar product. The dual vector space g will be identified with g by means of the scalar product. We shall also regardgas a Lie group with the Campbell–Hausdorff multiplication

x₁x₂=x₁+x₂+r(x₁, x₂), where

r(x₁, x₂) =¹₂[x₁, x₂]+₁₂¹([x₁,[x₁, x₂]]+[x₂,[x₂, x₁]])+₂₄¹[x₂,[x₁,[x₂, x₁]]]+...

is the (finite) sum of terms of order at least 2 in the Campbell–Hausdorff series forg. Let{δ_t}t>0, be a family of group dilations ongand let

g_j=

x∈g:δ_tx=t^djx

, 1≤j≤R,

where 1=d₁<d₂≤...<d_R. Then

g=g₁⊕g₂⊕...⊕g_R (3.1)

and

[g_i,g_j]⊂

g_k, ifd_i+d_j=d_k, {0}, ifd_i+d_j∈D/ ,

where D={d_j:1≤j≤R}. Letx!|x| be the homogeneous norm ongas defined in Section 1.

For a functionf∈C_c^∞(g×g) let Uf(y) =

g×g

e^−ix,yf^∨(x)e^−ir(x),˜ydx,

wherex=(x₁, x₂),y=(y₁, y₂)∈g×g, and ˜y=(y₁+y₂)/2. We shall refer toUas the Melin operator ong. The importance ofUconsists in

f g(y) =U( ˆf⊗ˆg)(y, y), y∈g, (3.2)

which is checked directly.

Let

g=g₁⊕g₂⊕...⊕g_R−1. (3.3)

The commutator

g×g(x₁, x₂)−![x₁, x₂]∈g,

where stands for the orthogonal projection onto g, makes g into a Lie algebra isomorphic tog/g_Rwithx!xplaying the role of the canonical quotient homomor- phism. The group multiplication ing is

x₁x₂=x₁+x₂+r(x₁, x₂). Proposition 3.4. Forf∈C_c^∞(g×g),

Uf(y, λ) =U(P_λf(·, λ))(y), y∈g, λ∈g_R, (3.5)

where

P_λg(y) =

g×g

e^−ix,yg^∨(x)e^{−ir(x),λ˜} dx, g∈C_c^∞(g×g),

is an integral operator on C_c^∞(g) invariant under Abelian translations, and U stands for the Melin operator on g.

Proof. In fact, Uf(y, λ) =

g×g

e^−ix,ye^−it,λf^∨(x, t)e^−ir(x),˜ye^{−ir(x),˜λ}dxdt

=

g×g

e^−ix,y(f(x^∨, λ)e^{−ir(x),λ˜} )e^−ir(x),˜ydx

=

g×g

e^−x,y(P_λf(·, λ))^∨(x)e^−ir(x),˜ydx

=U(P_λf(·, λ))(y)

for allf∈C_c^∞(g×g),y∈g×g,λ∈g_R×g_R.

The reader who is familiar with our previous paper may be surprised that (3.5) differs from an analogous formula in [2] by the order of the operators P_λ and U. As a matter of fact, (3.4) in [2] is false and we take this opportunity to correct it. Fortunately only the proof of Proposition 5.1 in [2] requires some obvious corrections. This mistake has been brought to my attention by Jacek Dziuba´nski.

For the background on homogeneous groups we recommend Folland–Stein [1].

4. The inductive step

In what follows we apply the notation of Section 1 among others to X=g andX=g×g. In the latter case we employ the product normx²=x₁²+x₂², the product dilations δ_tx=δ_tx₁+δ_tx₂, and the product homogeneous norm |x|²=

|x₁|²+|x₂|². In addition,ρ(x)=1+|x|.

From now on,galways denotes a metric of the form g_x(z)²=

R j=1

z_j² g_j(x)^2dj,

where g_j(x)≥ρ_j(x). We also assume that g is q-tempered. Of course, what we focus on isg=porg=q.

Letλ∈g_R×g_R(see (3.1) and (3.3)). It is easily seen that the family of metrics g_x^λ(y) =g_(x,λ)(y,0), x,y∈g,

is uniformly slowly varying and uniformly q^λ-tempered. In particular q^λ are uni- formly slowly varying. Letγbe a joint constant for all the metricsg^λ andq^λ. Let B_ν=B_ν(x_ν, r)⊂g×g be the common covering of Lemma 2.1 for all these metrics.

Let

d^λ_ν(y) =q^λ_x

ν(y−x_ν)

(cf. Proposition 2.1). Letρ^λ(x)=ρ(x, λ) andg^λ_j(x)=g_j(x, λ).

Here comes the crucial step in our argument.

Lemma 4.1. For everyN, there existC andksuch that

|P_λf(y)| ≤C|f|¹_k(g^λ)(1+d^λ_ν(y))^−N (4.2)

for f∈C_c^∞(B_ν) uniformly inλ∈g_R×g_R andν∈N.

Proof. Letf∈C_c^∞(g×g) be supported inB_ν. There existC andksuch that

|P_λf(y)| ≤

g×g|f^∨(x)|dx=f_A(g×g)=f_λA(g×g)≤C|f|¹_k(g^λ), (4.3)

where

f_λ(y) =f _R−1

j=1

g^λ_j(x_ν)^djy_j

,

and · A(g×g)stands for the Fourier algebra norm. The last inequality is achieved by the Sobolev inequality

f_A(g×g)≤C(s)

|α|≤s

D^αf2

applied tof_λwhich is supported in a ball of radius 1 with respect to the norm · . Assume now that (4.2) is true for someN. Letd^λ_ν(y)=a>1. Note that other- wise the estimate is easy. Therefore there exists ξ∈(g×g) of unit length with respect to the norm dual to q^λ_x

ν such that ξ(y−x)≥a for x∈B(x_ν, r₁), where 0<r<r₁<γ. The norm one condition reads

1 = (q^λ_x

ν)(ξ)²=

1≤j≤R⁻1

(ρ_j)_λ(x_ν)^2djξ_j²≥

1≤j≤R⁻1

(1+λ^1/dR)^2djξ_j². ThenL(x)=x−y, ξdoes not vanish onB(x_ν, r₁) so, by Lemma 2.2,

g^λ_x

ν

D^k1

L(x)

≤C_k(r, r₁)

a , x∈B(x_ν, r).

Note thatL(y)=0. Therefore, P_λ(Lf)(y) = [P_λ, L]f(y) =

R−1

j=1

ξ_j(1+λ^1/dR)^djP_λ(f_λ,j)(y),

where

f_λ,j(z) =1

i(1+λ^1/dR)^dR−dj

r_j(iD),

λ˜ (1+λ^1/dR)^dR

f(z), and

r_j(x, λ) =D_(x1)jr(x, λ)+D_(x2)jr(x, λ) is a homogeneous polynomial of degreed_R−d_j. It follows that

|P_λ(f)(y)| ≤ R−1

j=1

P_λ 1

Lf

λ,j

(y)

²_1/2 ,

and consequently, by Lemma 2.2 and the induction hypothesis,

|P_λf(y)| ≤C_k(r, r₁)

a |f|¹_k(g^λ)(1+d_ν(y))^−N≤C_k(r, r₁)|f|¹_k(g^λ)(1+d_ν(y))^−N−1, which completes the proof of (4.2).

Letmbe ag-weight. Thenm_λ(x)=m(x, λ) is a weight ong×gwith respect to g^λ (which is q^λ-tempered), and the family of weights is uniform in λ. Let φ^λ_ν∈C_c^∞(B_ν) be the partition of unity ong×g forg^λ. By Proposition 2.1, φ^λ_ν are bounded inS¹(g×g,g^λ) uniformly in ν andλ.

Observe that

m_λ(y)≤C₁m_λ(x_ν)(1+q^λ_x

ν(y−x_ν))^M≤C₁m_λ(x_ν)(1+d^λ_ν(y))^M. (4.4)

Proposition 4.5. For everyλthere exists a unique double-continuous exten- sion of P_λ to a mapping

P_λ:S^mλ(g×g,g^λ)−!S^mλ(g×g,g^λ).

All the estimates hold uniformly in λ.

Proof. By Lemma 4.1 and (4.4),

|P_λ(φ^λ_νf)(y)| ≤C₁|φ^λ_νf|¹_k(g^λ)(1+d^λ_ν(y))^−N and

m_λ(y)⁻¹|P_λ(φ^λ_νf)(y)| ≤C₂m_λ(x_ν)⁻¹(1+d^λ_ν(y))^M|P_λ(φ^λ_νf)(y)|

≤C₃|f|^m_k^λ(1+d^λ_ν(y))^−N+M.

LetN be so large that

ν

(1+d^λ_ν(y))^−N+M<∞,

see Proposition 2.1 (c). Then our estimate which remains valid forf in a bounded subset of S^mλ(g×g,g^λ) without any restriction on the support implies that for everyy∈g,

f−!

ν

P_λ(φ^λ_νf)(y)

defines a weakly continuous linear form onS^m(g×g,g^λ). Consequently,P_λadmits a (unique) weakly continuous extension to the whole ofS^m(g×g,g^λ), and

|P_λ(f)(y)|=

ν

P_λ φ^λ_νf

(y)≤C₅|f|^m_k^λm_λ(y).

The estimates for the derivatives of P_λf follow from the fact that P_λ commutes with translations, and hence with differentiations.

5. Continuity of U and symbolic calculus

Recall from Section 4 that the Melin operator U has been defined for f∈C_c^∞(g×g).

Theorem 5.1. Letmbe ag-weight ong×g. There exists a double-continuous extension of the Melin operator to

U:S^m(g×g,g)−!S^m(g×g,g).

Proof. Suppose that gis as in (3.1) and proceed by induction. If R=1, gis Abelian and U=I so the assertion is obvious. Assume that our theorem is true for g as in (3.3) and U=U. For λ∈g_R and f∈S^m(g×g,g) let f_λ(y)=f(y, λ), (ρ_λ)(y)=ρ(y, λ), and m_λ(y)=m(y, λ).

By hypothesis, f_λ=f(·, λ)∈S^mλ(g×g,g_λ) uniformly in λ (cf. the previous section). Now Proposition 4.5 yields

P_λf_λ∈S^mλ(g×g,g^λ)

uniformly inλso, by the induction hypothesis, UP_λf_λ∈S^mλ(g×g,g^λ)

uniformly in λ. The same holds true for the derivatives (∂/∂λ)^jUP_λf_λ which is checked directly. Thus, by (3.5), we get the desired estimate: There exists C >0 such that for everyk₁∈N, there existsk₂∈N such that

|Uf|^m_k1(g)≤C|f|^m_k2(g), f∈S^m(g×g,g).

This proves our assertion.

Corollary 5.2. Let m₁ andm₂ beg-weights ong. Then C_c^∞(g)×C_c^∞(g)(a, b)−!a#b= (a^∨b^∨)^∧∈ S(g) extends uniquely to a double-continuous mapping

S^m1(g,g)×S^m2(g,g)−!S^m1m2(g,g).

Proof. This is a straightforward consequence of (3.2) and Theorem 5.1 applied to the metric g⊕gong×g.

Letφ_ν be the standard partition of unity for the metricq ong. Let Φ_µν(x)=

φ_µ(x₁)φ_ν(x₂), wherex=(x₁, x₂)∈g×g. LetQ=q⊕q. Note that, by Lemma 1.8, 1+q_xν(x_ν−x_µ)≤C²(1+q_xµ(x_µ−y))^M(1+q_xν(x_ν−y))^M+1

(5.3)

for everyy∈gand everyµandν.

Corollary 5.4. Let f∈S¹(g×g,Q). Let f_µν(y) =U(Φ_µνf)(y, y)

be a function on g. Then, for every N, there exists a norm | · |¹(Q) in S¹(g,Q) such that for every µ andν,

f_µνA(g)≤ |f|¹(Q)(1+q_xν(x_µ−x_ν))^−N. Proof. If

m_µν(y) = (1+q_xν(x_µ−y₁))^−N(M+1)(1+q_xν(x_ν−y₂))^{−N M}, where y=(y₁, y₂), then, of course,m_µν is a weight (see (1.12)), and

Φ_µνf∈S^mµν(g×g,Q)

uniformly so, by Proposition 5.1,

U(Φ_µνf)∈S^mµν(g×g,Q) uniformly inµandν. Hence, by (5.3),

|D_y^αf_µν(y)| ≤ |f|^m_k^µνm_µν(y, y)≤ |f|¹_k(1+q_xν(x_µ−x_ν))^−N

for|α|≤k. Ifkis large enough, our assertion follows by the Sobolev inequality.

We conclude with the L²-boundedness theorem for a∈S¹(g,q). Recall that S¹(g,p)⊂S¹(g,q) and, by (1.13), the inclusion is continuous.

Theorem 5.5. Let a∈S¹(g,q). The linear operator f!Af=f a^∨ defined initially on the dense subspace C_c^∞(g) of L²(g) extends to a bounded mapping of L²(g). To be more specific, there exists a norm| · |¹(q)inS¹(g,q)such that

AfL²(g)≤ |a|¹fL²(g), f∈C_c^∞(g).

Proof. Let

A_vf=f (φ_va)^∨, f∈L²(g).

Since φ_v∈C_c^∞(g), the operators A_v are bounded. Furthermore, by (3.2) with the notation of Corollary 5.4,

A_uA_vf(y) = (a⊗a)^∨_u,vf, A_uA_vf(y) = (a⊗a)^∨_u,vf, so that, by Corollary 5.4,

A_uA_v+A_uA_v ≤(|a|¹(q))²(1+g_xν(x_ν−x_µ))^−N,

where N can be taken as large, as we wish, and | · |¹(q) is a norm in S¹(g,q) depending only onN.

On the other hand,

a=

u

φ_ua,

where the series is weakly convergent in S¹(g,q) so that

Af=

u

A_uf, f∈C_c^∞(g).

Thus, the sequence of operatorsA_usatisfies the hypothesis of Cotlar’s Lemma (see e.g. Stein [8]), and therefore the series

uA_uis strongly convergent to the extension of our operatorAwhose norm is bounded byC₀|a|¹(see Proposition 2.1).

Acknowledgements. The author wishes to express his deep gratitude to W. Czaja, J. Dziuba´nski, and B. Trojan for their critical reading of the manuscript and many helpful comments.

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Math.32(1979), 359–443.

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Pawel Glowacki

Institute of Mathematics University of Wroclaw pl. Grunwaldzki 2/4 PL-50-384 Wroclaw Poland

glowacki@math.uni.wroc.pl Received October 17, 2005

published online February 10, 2007