Our multipliers are better adapted to the specific commutation rules on the Lie algebra of the given group

SENSITIVE TO THE GROUP STRUCTURE

PAWE L G LOWACKI

Abstract. We propose new sufficient conditions for L^p-multipliers on homogeneous nilpotent groups. The multipliers generalise the flag mul- tipliers of Nagel-Ricci-Stein-Wainger, but the approach and the tech- niques applied are entirely different. Our multipliers are better adapted to the specific commutation rules on the Lie algebra of the given group.

The proofs are based on a new symbolic calculus fashioned after H¨ormander.

We also take advantage of Cotlar-Stein lemma, and Littlewood-Paley theory in the spirit of Duoandikoetxea-Rubio de Francia.

Contents

1. Introduction 1

2. Basic Setup 4

3. Metrics 6

4. A partition of unity 10

5. The Melin operator 12

6. Estimates for the basic metric 16

7. Twisted multiplication and L²-multipliers 20

8. Convolution of N-kernels 23

9. L^p-Multipliers 29

10. Appendix. Convolution of distributions. 32

Acknowledgements 33

References 33

1. Introduction

Classical multiplier operators are convolution linear operators of the form S(R^N) 3 f 7→ TKf = f ? K ∈ C^∞(R^N),

where

f ? K(x) = Z

R^N

f (x − y)K(y)dy = hK, fxi, fx(y) = f (x − y).

Here K is a tempered distribution on R^N such that m(ξ) = bK(ξ) is a locally integrable function. The function m is then called a multiplier. As the space of test functions f one may adopt the Schwartz class S(R^N) of smooth

2010 Mathematics Subject Classification. 42B15 42B20 42B25.

Key words and phrases. homogeneous groups, L^p-multipliers, Fourier transform, sym- bolic calculus, H¨ormander metrics, singular integrals, flag kernels, Littlewood-Paley theory.

1

functions which decay rapidly at infinity along with all their derivatives.

The primary goal of the study is the boundedness of such operators in terms of the function m. We say that a linear operator T on S(R^N) is bounded in the Lebesgue norm k · k_L^p, where 1 < p < ∞, if there exists a constant C > 0 such that

kT f k_L^p ≤ Ckf k_L^p, f ∈ S(R^N).

There are numerous classical results which give (mostly) sufficient conditions on the function m that assure the boundedness of the multiplier operator.

Let us mention just two such results relevant to the subject of this paper.

By the Plancherel theorem we know that the multiplier operator is boun- ded on L²(R^N) if and only if the symbol m is essentially bounded. The other result is known as the Marcinkiewicz theorem and gives a sufficient condition for the boundednes on the L^p(R^N)-spaces, for all 1 < p < ∞. One of the versions of this theorem requires that m is differentiable everywhere except where ξ_k = 0, for some 1 ≤ k ≤ N , and satisfies the estimates

|D^αm(ξ)| ≤ C

N

Y

k=1

|ξ_k|^−αk,

for a constant C > 0 and all multiindices α = (α₁, α₂, . . . , α_N), where α_k= 0 or 1.

Similar questions have been dealt with in the context of a nilpotent ho- mogeneous group. Instead of the usual addition of vectors (x, y) 7→ x + y in R^N we consider a group multiplication

(x, y) 7→ xy = x + y + P (x, y), where P : R^N× R^N → R^N is a polynomial mapping

P (x, y) =

P₁(x, y), P₂(x, y), . . . , P_N(x, y)

with terms of order at least two. For every j, the polynomial P_j depends only on the variables xk, yk, where 1 ≤ k < j. Furthermore,

P (x, 0) = P (0, x) = P (x, −x) = 0,

for every x ∈ R^N. Additionally, we assume that our group is homogeneous, that is, there exist numbers 1 ≤ p1 ≤ p₂ ≤ · · · ≤ p_N such that the mappings

δtx =



t^p1x1, t^p2x2, . . . , t^pNxN



are group automorphisms called dilations. Such a group is necessarily con- nected, simply connected, and nilpotent. The simplest and best known noncommutative group of this type is the Heisenberg group, where the poly- nomial multiplication is defined on R³ in the following way

xy =



x1+ y1, x2+ y2, x3+ y3+1

2(x1y2− x₂y1)

 . The mappings

δ_tx =

t^p1x₁, t^p2x₂, t^p3x₃

, t > 0, are automorphic dilations if and only if p1+ p2 = p3.

The multiplier operators on (R^N, P ) are convolution operators S(R^N) 3 f 7→ f ? K ∈ C^∞(R^N),

where

f ? K(x) = Z

R^N

f (xy⁻¹)K(y)dy = hK, fxi, fx(y) = f (xy⁻¹), for a tempered distribution K on R^N. Again, the question is: What are the conditions we may impose on the Fourier transform bK that are sufficient for the operator f 7→ f ? K to be bounded on L^p(R^N)? Various examples in the theory of pseudodifferential operators which is intimately connected to the analysis on the higher-dimensional Heisenberg groups show that the conditions we know from the classical theory of multipliers have to be sub- stantially strengthened. For instance, the boundedness of bK is no longer sufficient for the boundedness of TK on L²(R^N, P ). The weakest sufficient conditions for the L^p-boundedness have been obtained by Nagel-Ricci-Stein- Wainger [16]. Here the smoothness of bK is assumed where ξN 6= 0 and it is required that

|D^αK(ξ)| ≤ Cb α N

Y

k=1

 X

j≥k

|ξ_j|^1/pj−p_kαk

,

for |α| ≤ M , where M is sufficiently large. (Actually, this is a somewhat simplified version of the condition. For a full version, see Example 8.10 below.) The operator norm of the multiplier operator depends on a finite number of constants Cα.

Let us pause here to reflect on the occurrence of the variables with higher indexes in the above estimates. This is a consequence of the noncommuta- tivity of the group and is closely related to the fact that the polynomials Pk defining the multiplication depend on the “earlier” variables. Turning the matter around we may say that each variable x_j or y_j occurs only in polynomials P_k, for k > j. This suggests the idea of a new order in the set

N = {k ∈ N : 1 ≤ k ≤ N }.

Let us write k ≺ j if P_j really depends on x_k or y_k. This relation can be extended to a partial order in N . Examples show that the order may be much coarser than the natural linear order in N .

In this paper we wish to offer an improvement on the described above results. The condition for the L²-boundedness is weakened to

(1.1) |D^αK(ξ)| ≤ Cb _α Y

k∈N \Nmax

 X

jk

|ξ_j|^1/pj−p_kα_k

,

where N_maxconsists of the maximal elements of N with regard to the partial order ≺. The condition for the L^p-boundedness takes the form

(1.2) |D^αK(ξ)| ≤ Cb α

Y

k∈N

 X

jk

|ξ_j|^1/pj−p_kαk

.

In (1.1) as well as in (1.2) it is assumed that ξj 6= 0 if j ∈ N_max. As before, the norms of the multiplier operators depend on a finite number of constants Cα. The flag kernels of Nagel-Ricci-Stein-Wainger [16] satisfy (1.2).

These conditions are not only weaker but also seem much better adjusted to the individual group structure. For example, if the group is Abelian, the conditions (1.2) easily reduce to the classical Marcinkiewicz C^∞-conditions.

Our approach is based on a new symbolic calculus, where the typical estimates look like

(1.3) |D^αa(ξ)| ≤ C_α Y

k∈N

g_k(ξ)^−pkαk, ξ ∈ (R^N)^?.

In the most interesting cases the weight functions g_k do not depend on the independent variables, that is the variables ξ_j, where k 6 j. If, for instance,

g_k(ξ) = 1 +X

jk

|ξ_j|^1/pj,

then by differentiating with respect to the variable ξ_k no gain is produced in the independent directions.

The idea of the calculus goes back to H¨ormander [12] and Melin [14] (see, Manchon [13] or G lowacki [8]). Central to the method is the formula

(f ? g)^∧(ξ) = Z Z

R^N×R^N

e^−ihx+y,ξie−ihP (x,y),ξif (x)g(y)dxdy,

for Schwartz functions f, g. This calculus seems to be well suited to the approximation of “symbols” like those in (1.2) by symbols (1.3) of the cal- culus and transferring, perhaps in a weaker form, some of the properties of the latter. In particular, it is shown that if distributions K1, K2 satisfy (1.2), then they are convolvable (see Definition 10.3) and the convolution K₁ ? K₂ also satifies (1.2). This is proved without using the fact that the corresponding operators are bounded on L² and extends to similar classes of distributions of order different from zero.

This paper is heavily influenced by Duoandikoetxea-Rubio de Francia [4], H¨ormander [12], and Melin [14]. The idea of N-kernels has been inspired by the concept of flag kernels of Nagel-Ricci-Stein-Wainger [16].

2. Basic Setup

Let X be a real N -dimensional vector space with a fixed linear basis {e_k}^N_k=1. Accordingly, each element x ∈ X has a representation as

x = (x₁, x₂, . . . , x_N).

Occasionally, it will be more convenient to write

x = (x_k)_k∈N, N = {1, 2, . . . , N }.

The space X is assumed to be homogeneous, that is, endowed with a family of dilations {δ_t}. The vectors in the basis are supposed to be invariant under dilations:

δ_te_k= t^pke_k, t > 0, k ∈ N , where 1 ≤ p1 ≤ p₂ ≤ · · · ≤ p_N. The number Q = PN

k=1pk is called the homogeneous dimension of X. We have

dδtx = t^Qdx, t > 0.

The function

ρ(x) =

N

X

k=1

|x_k|^1/pk, x ∈ X,

will play the role of a homogeneous norm on X. We also choose and fix the l¹-norm

(2.1) kxk = k

N

X

k=1

x_ke_kk =

N

X

k=1

|x_k|.

By α, β, . . . , A, B, . . . , a, b, · · · we shall denote multiindices in A_N = N^N, where N stands for the set of all nonnegative integers. Let

|α| =

N

X

k=1

α_k. p(α) =

N

X

k=1

p_kα_k.

The set of multiinices M = (m1, m2, . . . , mN), where mk ∈ R will be de- noted by A_R. Here R denotes the field of real numbers.

We shall adopt the following notation for partial derivatives:

D_k = D_xk = ∂

∂xk

, D^α =Y

k

D_k^αk. For a function f on X we shall write

f (x) = f (−x),e x ∈ X.

The Schwartz space of smooth functions which vanish rapidly at infinity along with all their derivatives will be denoted by S(X). The seminorms

f → max

|α|+|β|≤msup

x∈X

|x^αD^βf (x)|, m ∈ N ,

form a complete set of seminorms in S(X) giving it a structure of a lo- cally convex Fr´echet space. Needless to say that the seminorms are actually norms. The subspace C_c^∞(X) of functions with compact support is dense in S(X). By L¹(X) and L²(X) we denote the usual Lebesgue spaces. S(X) is a dense subspace of the Lebesgue spaces.

Analogous notation will be applied to the objects on the dual space X^? with the dual basis {e^?_k}_k∈N and the dual dilations still denoted by {δt}_t>0. The Fourier transforms are denoted by f 7→ f^∧ and g 7→ g^∨. We choose Lebesgue measures dx in X and dξ in X^? so that

f^∧(ξ) = bf (ξ) = Z

X

f (x)e^−ihx,ξidx, g^∨(x) = Z

X^?

g(ξ)e^ihx,ξidξ, where f ∈ S(X), g ∈ S(X^?) and

hx, ξi =

N

X

k=1

xkξk

is the duality of vector spaces. If P is a polynomial on X, then P f
∧

= P (iD) bf , for f ∈ S(X).

By |M | we denote the cardinality of a finite set M .

Let A, B be positive quantities. We shall write A ∼ B to say that there^<

exists a constant c > 0 whose precise value is irrelevant such that A ≤ cB.

Some more notation is explained in Appendix (Section 10) or in the cur- rent text of the paper.

3. Metrics

Let ≺ be a partial order in N = {1, 2, . . . , N } such that k ≺ j implies k < j. Instead of k ≺ j we shall also write j  k. The expressions k  j and j  k mean that k ≺ j or k = j.

Another basic structure in N which is going to be intrumental throughout the paper is filtration. A family N = {N_k}_k∈N of subsets of N is called a filtration if for every k ∈ N and every j ∈ N ,

a) j  k implies j ∈ N_k. b) j ∈ N_k implies N_j ⊆ N_k.

If, for every k ∈ N , k ∈ Nk, we say that the filtration is closed.

Any filtration N = {N_k}_k∈N determines a set of partial homogeneous norms

(3.1) N_k(ξ) = X

j∈Nk

|ξ_j|^1/pj, ξ ∈ X^?.

We shall consider metrics, that is, families of norms on X^? of the form

(3.2) g_ξ(η) =

N

X

k=1

|η_k|

g_k(ξ)^pk, ξ ∈ X^?, η ∈ X^?,

where gk are continuous strictly positive functions. Every filtration N de- termines a metric g = g_N, where

g_k(ξ) = 1 + N_k(ξ), ξ, η ∈ X^?. Definition 3.3. This class of metrics will be denoted by G.

With few exceptions, these are the metrics we are going to consider here (see Remark 3.19 below). The metric corresponding to the filtration {N_k}, where

N_k= {j ∈ N : j  k}

is special. We will denote it by q and will refer to it as the basic metric on X^?. We have

(3.4) q_ξ(η) =

N

X

k=1

|η_k| q_k(ξ)^pk, where

q_k(ξ) = 1 +X

jk

|ξ_j|^1/pj, k ∈ N , ξ ∈ X^?.

The following proposition says that the metric q is self-tempered.

Proposition 3.5. There exist C₀, T > 0 such that, for all ξ, η, ζ ∈ X^?,

 q_ξ(ζ) qη(ζ)

±1

≤ C₀ 1 + qξ(ξ − η)
T

.

Proof. It is sufficient to show that, for every k ∈ N , there exists T_k> 0 such that

(3.6)  q_k(ξ)

q_k(η)

±1

∼<



1 +X

jk

|ξ_j− η_j| q_j(ξ)^pj





Tk

, for ξ, η, ζ ∈ X^?. Let us prove (3.6). We have

q_k(ξ) q_k(η)

∼ 1 + q< k(η)⁻¹X

jk

|ξ_j− η_j|^1/pj

∼ 1 +< X

jk

|ξ_j− η_j|^1/pj qj(η)

∼ 1 +< X

jk

|ξ_j− η_j| qj(η)^pj .

The other part is proved by reverse induction. If k is a maximal element with respect to ≺, then

q_k(ξ) = q_k(η) = 1.

If not, let us assume that

(3.7) q_j(ξ)

qj(η)

∼<



1 +X

lj

|ξ_l− η_l| q_l(ξ)^pl ,





Tj

for j  k. As in the first part of the proof, q_k(ξ)

qk(η)

∼ 1 +< X

jk

|ξ_j − η_j|

qj(ξ)^pj · q_j(ξ) qk(η)

pj

, so that, by (3.7) and q_j(η) ≤ q_k(η),

q_k(ξ) qk(η)

∼<



1 +X

jk

|ξ_j− η_j| qj(ξ)^pj







1 +X

lj

|ξ_l− η_l| ql(ξ)^pl





pNTj

∼<



1 +X

jk

|ξ_j− η_j| q_j(ξ)^pj





Tk

, (3.8)

where T_k = p_Nmax_jkT_j+ 1, which implies (3.6).  Corollary 3.9. The metric q is slowly varying, that is, if q_ξ(ξ − η) < 1, then

(3.10)  q_ξ(ζ)

qη(ζ)

±1

≤ C₁, for some C₁ ≥ 1.

Corollary 3.11. Let qe_k(ξ) = |ξ_k|^1/pk + q_k(ξ), k ∈ N , ξ ∈ X^?. For every k ∈ N , there exists Rk> 0 such that

 qe_k(ξ) qe_k(η)

±1

∼<



1 +X

jk

|ξ_j− η_j| q_j(ξ)^pj





R_k

, The metrics g ∈ G are q-tempered:

Proposition 3.12. Let g ∈ G. Then, for every k ∈ N , q_k ≤ g_k, and, for all ξ, η, ζ ∈ X^?,

 gξ(ζ) g_η(ζ)

±1

∼ 1 + q< ξ(ξ − η)
T

, where T > 0 is as in Proposition 3.5.

Proof. We note first that q_k ≤ g_k, since the sets N_k for the basic metric q are, by definition, minimal. Now, observe that, for every k ∈ N ,

g_k(ξ)

gk(η) ≤ 1 +g_k(η − ξ) gk(η) .

If j ∈ Nk, then by the filtration property, Nj ⊆ N_k, hence qj ≤ g_j ≤ g_k. Therefore,

1 +gk(η − ξ)

g_k(η) ≤ 1 + X

j∈Nk

|η_j− ξ_j|^1/pj qj(η)

∼ 1 +<

X

j∈N

|η_j − ξ_j|

q_j(η)^pj = 1 + q_η(η − ξ), which implies

g_ξ(ζ) g_η(ζ)

∼ 1 + q< ξ(ξ − η).

By the above and Proposition 3.5, g_η(ζ)

gξ(ζ)

∼ 1 + q< η(η − ξ)∼ 1 + q^< ξ(ξ − η)
T

,

which completes our proof. 

Let g be a metric (not necessarily in G) and let m be a strictly positive function on X^?. For a smooth function f on X^?, ξ ∈ X^? and s ≥ 0, let (3.13) |f |^m,g_s (ξ) = m(ξ)⁻¹ max

p(α)≤sg^α(ξ)|D^αf (ξ)|, where

(3.14) g^α(ξ) =

N

Y

k=1

gk(ξ)^pkαk. Let also

|f |^m,g_s = sup

ξ∈X^?

|f |^m,g_s (ξ).

Instead of | · |^1,g_s we shall write | · |^g_s. Let

S(m, g) = {f ∈ C^∞(X^?) : ∀s≥0 |f |^m,g_s < ∞}.

The space S(m, g) with the seminorms | · |^m,gs is a Fr´echet space.

Let g₁, g₂ be metrics and m₁, m₂ strictly positive functions. A linear mapping

(3.15) T : S(m₁, g₁) → S(m₂, g₂)

is Fr´echet continuous, if for every n, there exist k such that

|T f |_n^m2,g2 ∼ |f |^< _k^m1,g1, f ∈ S(m1, g1).

The product S(m₁, g) × S(m₂, g) is a Fr´echet space with the product topol- ogy. We shall also consider Fr´echet continuous bilinear mappings

(3.16) T : S(m₁, g) × S(m₂, g) → S(m₃, g).

We need, however, still another concept of convergence. A sequence fn ∈ S(m, g) is said to be weakly convergent if it is bounded and point- wise convergent (see H¨ormander [12], page 369). Note that, by the Ascoli theorem, for a bounded sequence, pointwise convergence is equivalent to the convergence in C^∞-topology: for each α, the sequence D^αfn is uniformly convergent on compact subsets of X^?.

A linear mapping (3.15) is said to be weakly continuous if the weak con- vergence f_n → f in S(m₁, g₁) implies the weak convergence T f_n → T f in S(m2, g2). A bilinear mapping (3.16) is said to be weakly continuous if the weak convergence fn→ f in S(m₁, g) together with the weak convergence g_n → g in S(m₂, g) imply the weak convergence T (f_n, g_n) → T (f, g) in S(m3, g2).

Proposition 3.17. Let g be a metric and m a strictly positive function on X^?. The subspace C_c^∞(X^?) is weakly dense in S(m, g).

Proof. Let ψ ∈ C_c^∞(X^?) be equal to 1 in a neighbourhood of the origin.

Let ψn(ξ) = ψ(δ_1/nξ). Let f ∈ S(m, g). Of course, the sequence ψnf is pointwise convergent to f , so we only have to show that it is bounded in S(m, g). In fact, if p(α) ≤ s, then

m(ξ)⁻¹g^α(ξ)|D^α(ψnf )(ξ)|

∼<

X

β+γ=α

(1 + ρ(ξ))^p(β)n^−p(β)m(ξ)⁻¹g^γ(ξ)|D^γf (ξ)|

≤ |f |^m,g_s

since ρ(ξ) ≈ n on the support of D^βψ_n, if β 6= 0. If β = 0, the estimate is

obvious. 

We denote by M(q), the class of strictly positive functions m on X^? such that there exists T₁> 0 such that, for all ξ, η ∈ X^?,

 m(ξ) m(η)

±1

∼ 1 + q< ξ(ξ − η)
T1

.

We may express this property, by saying that the functions m are q-tempered.

The elements of M(q) will be called q-weights or simply weights. Observe also that if m₁, m₂ ∈ M(q), then m₁m₂ ∈ M(q) and m^θ₁ ∈ M(q), for every θ ∈ R.

Proposition 3.18. Let g ∈ G. Let M = (m₁, m₂, . . . , m_N) ∈ A_R and let mM(ξ) =

N

Y

k=1

gk(ξ)^mk,

Then, m_M is a q-weight. In particular, for every α ∈ A_N, g^α∈ M(q).

Proof. It follows from Proposition 3.12 that functions g_k are g-weights.

Thus, by the preceding remark mM ∈ M(g), for every M ∈ A_R. 

Let g ∈ G. The metric g ⊕ g on X^?× X^? is defined by

(g ⊕ g)_(ξ1,ξ2)(η₁, η₂) = g_ξ1(η₁) + g_ξ2(η₂), ξ₁, ξ₂, η₁, η₂ ∈ X^?. This metric no longer belongs to G. For f ∈ S(X^?× X^?) and a strictly positive function m on X^?× X^?, let

|f |^m,g⊕g_s = sup

ξ,η

sup

p(α)+p(β)≤s

m(ξ, η)⁻¹g^α(ξ)g^β(η)|D_ξ^αD_η^βf (ξ, η)|.

Remark 3.19. Througout the paper, the metric g ⊕ g, where g ∈ G, is the only instance of a metric which is not in G. In addition, it will only play an auxiliary role.

4. A partition of unity Let v be a norm on X^?. Denote

B_v(a, r) = {ξ ∈ X^? : v(ξ − a) < r},

for a ∈ X^?, r > 0. The following is a simplified lemma of H¨ormander ([12], Lemma 2.5). We adapt the original proof to our needs.

Proposition 4.1. There exists a discrete set B = {bν : ν ∈ N } ⊆ X^? such that the family of the balls A_ν = B_qbν(b_ν, 1/2) is a covering of X^?, and no point ξ ∈ X^? can belong to more than (4C₁³+ 1)^N larger balls Bν = B_qbν(b_ν, 1). There also exists a sequence of functions ϕ_ν ∈ C_c^∞(B_ν) bounded in S(1, q) and such that for every ξ ∈ X^?

X

ν

ϕ_ν(ξ) = 1.

Furthermore, there exist constants m, M > 0 such that

(4.2) X

ν∈N



1 + q_bν(ξ − b_ν)−m

≤ M, for every ξ ∈ X^?.

Proof. Let C1 be as in (3.10). Let {bν} be a maximal sequence of points in X^? such that

(4.3) q_bν(bµ− b_ν) ≥ 1

2C₁, µ 6= ν.

Let ξ ∈ X^?. Note that

qξ(ξ − bν) < 1/2C1 implies qbν(ξ − bν) < 1/2.

Therefore, either q_bν(ξ − b_ν) < 1/2 for some ν, or q_ξ(ξ − b_ν) ≥ 1

2C₁ and q_bν(ξ − b_ν) ≥ 1/2 ≥ 1 2C₁.

The latter contradicts the maximality of our sequence. The former implies that {A_ν}_ν∈N is a covering, which proves the first statement of the propo- sition.

To show that the covering {B_ν}_ν∈N is uniformly locally finite take a ξ ∈ X^? and let

M (ξ) = {ν : ξ ∈ Bν}.

If ν ∈ M (ξ), then q_bν(ξ − b_ν) < 1, which implies q_ξ(ξ − b_ν) < C₁. On the other hand

q_ξ(bµ− b_ν) ≥ q_bν(bν− b_µ)/C1 ≥ 1/2C₁², for µ 6= ν. Thus, for every ν ∈ M (ξ),

B_qξ(b_ν, 1/4C₁²) ⊆ B_qξ(ξ, C₁+ 1/4C₁²), and the balls B_qξ(b_ν, 1/4C₁²} are pairwise disjoint, so

|M (ξ)| ≤ C₁+ 1/4C₁² 1/4C₁²

N

= (4C₁³+ 1)^N, as claimed.

Let ψ ∈ C_c^∞



Bk·k(0, 1)



be equal to 1 on the smaller ball Bk·k(0, 1/2).

Let

ψ_ν(ξ) = ψ ∆⁻¹_ν (ξ − b_ν)
, where

(4.4) ∆νξ =



q_k(bν)^pkξ_k



k∈N. By (3.10),

q^α(ξ)|D^αψ_ν(ξ)| = q^α(ξ)q^−α(b_ν)|D^αψ(∆⁻¹_ν (ξ − b_ν)|

≤ C₁^p(α) sup

kηk≤1

|D^αψ(η)| = Cα,

for all ν and all α, which shows that ψ_ν ∈ S(1, q) with uniform bounds.

Since {Aν} is a covering, P

µψµ(ξ) ≥ 1 for every ξ ∈ X^?, and it is not hard to see that the sequence

ϕν(ξ) = ψ_ν(ξ) P

µψµ(ξ) is a partition of unity.

It remains to prove (4.2). Given ξ ∈ X^? and k ∈ N , let M_k(ξ) = {ν : q_bν(ξ − bν) < k}.

For every ν ∈ M_k(ξ),

q_ξ(ξ − bν) ≤ C0k(1 + k)^T ≤ C₀(1 + k)^{T +1},

wher C₀ is as in Proposition 3.12. Furthermore, for ν, µ ∈ M_k(ξ) such that b_ν 6= b_µ,

q_ξ(bν − b_µ) > C₀⁻¹q_bν(bν− b_µ)(1 + k)^−T > 1

2C₀C₁(1 + k)^T, so the balls

Bqξ



bν, 1 4C0C1(1 + k)^T



are mutually disjoint and contained in the ball B_qξ(ξ, 2C₀(1+k)^{T +1}). There- fore,

|M_k(ξ)| ≤ (8C₀²C1(1 + k)^{2T +1})^{N <}∼ (1 + k)^{(2T +1)N}.

Finally, X

ν∈N



1 + q_bν(ξ − bν)−m

≤ |M₁(ξ)| +

∞

X

k=1

(1 + k)^−m  

M_k+1(ξ) \ M_k(ξ)   

∼ 1 +<

∞

X

k=1

(1 + k)−m+(2T +1)N ≤ M,

if m > (2T + 1)N + 1. 

Let

dν(ξ) = 1 + qbν(bν− ξ), ν ∈ N , ξ ∈ X^?. Proposition 4.5. We have



1 + q_ξ(ξ − b_ν)_{T +1}¹

∼ d< ν(ξ) ∼^<



1 + q_ξ(ξ − b_ν)T +1

uniformly in ξ and ν.

Proof. In fact,

1 + qξ(ξ − bν) ∼ 1 + q^< bν(bν− ξ)

1 + qbν(bν − ξ)T

≤ d_ν(ξ)^{T +1}, for ξ ∈ X^?. The other inequality is proved in a similar way. 

Let

dνµ= max n

dν(bµ), dµ(bν) o

. Corollary 4.6. We have

d_νµ ∼ d^< ν(ξ)^{2T +1}d_µ(ξ)^{2T +1} uniformly for ν, µ and ξ.

Proof. In fact,

(4.7) 1 + q_ξ(b_ν − b_µ) ≤

1 + q_ξ(ξ − b_ν)

1 + q_ξ(ξ − b_µ) , so, by (4.7) and Proposition 4.5,

d_ν(b_µ) = 1 + q_bν(b_ν− b_µ) ∼^<



1 + q_ξ(b_ν− b_µ)

1 + q_bν(b_ν− ξ)T

∼<



1 + q_ξ(ξ − b_ν)

1 + q_ξ(ξ − b_µ) d_ν(ξ)^T

∼ d< ν(ξ)^{T +1}d_µ(ξ)^{T +1}d_ν(ξ)^{T <}∼ dν(ξ)^{2T +1}d_µ(ξ)^{2T +1}.

Our claim follows by symmetry. 

5. The Melin operator

We specify the abstract structure of X. From now on X = g is a homoge- neous Lie algebra with the Campbell-Hausdorff multiplication (see Corwin- Greenleaf [3])

(5.1) xy = x + y + P (x, y),

where P is a polynomial mapping with non-zero terms of order at least 2.

We assume there is a fixed family of automorphic dilations on g. By definition, dilations of a Lie algebra are automorphisms of the form

δ_t= t^P, t > 0,

where P : g → g is a diagonalizable linear operator with positive eigenvalues 0 < p1≤ p₂≤ · · · ≤ p_N,

listed along with their multiplicities. We assume that p₁ ≥ 1. If D = {p_kj} is a complete set of pairwise different eigenvalues, then

g=M

j

g_(k

j), where

[g_(ki), g_(kj)] ⊆

(g_(ks), if pki+ pkj = pks, {0}, if p_ki+ p_kj 6∈ D.

Let {e_k}^N_k=1be a basis of eigenvectors for P. Of course, there is some freedom in the choice of such a basis. However, once we have chosen and fixed one, we may apply the notation of Section 2.

The mapping P commutes with the dilations, that is, (5.2) P (δ_tx, δ_ty) = δ_tP (x, y),

so the dilations are also automorhisms of the group. We also have (5.3) DxkPj(x, y) = DykPj(x, y) = 0, j ≤ k,

where Pj(x, y) = hP (x, y), e^?_ji. From

−x − y − P (x, y) = −xy = (xy)⁻¹ = y⁻¹x⁻¹

= −y − x + P (−y, −x), one gets P (x, y) = −P (−y, −x), which in turn implies (5.4) D_xkP_j 6= 0 iff D_ykP_j 6= 0.

Definition 5.5. We define ≺ as the smallest order in N such that DxkPj 6= 0 or, equivalently, Dy_kPj 6= 0 implies k ≺ j (cf. (5.4)).

Remark 5.6. The partial order ≺ is determined by our choice of basis and is by no means canonical.

Lemma 5.7. An integer k ∈ N is maximal if and only if the basis vector e_k is central in g.

Proof. In fact, suppose that k is maximal and let x ∈ g. Since Dx_kPj(x, y) = D_ykP_j(x, y) = 0, for j > k, it follows that P_j(x, e_k) = P_j(e_k, x) = 0, for every j ∈ N . Therefore, xe_k = x + e_k = e_kx. The converse implication is

trivial. 

Lemma 5.8. Let N = {N_k}_k∈N be a filtration in N . Let g_k be the linear subspace of g generated by the vectors ej, j ∈ Nk. Then, gk is an ideal in g.

Proof. Let j ∈ N_k. By the Campbell-Hausdorff formula, for any l, s ∈ N , λs= h[el, ej], e^?_si = 2D_xlDyjPs(0, 0).

If λs 6= 0, then s  j, hence s ∈ N_k. Therefore, [el, ej] = X

s∈Nk

λses∈ g_k.

 Example 5.9. Let g be the 6-dimensional Lie algebra with a basis {e_k}⁶_k=1 and the nonzero commutators

[e₁, e₂] = e₄, [e₁, e₅] = e₆, [e₂, e₃] = e₅, [e₃, e₄] = −e₆. As automorphic dilations one can take

δt(x) = (tx1, tx2, tx3, t²x4, t²x5, t³x6).

If

X =

6

X

j=1

xjej, Y =

6

X

k=1

ykek, then

[X, Y ] = (x1y2− x₂y1)e4+ (x2y3− x₃y2)e5

+ (x1y5− x₅y1− x₃y4+ x4y3)e6, and

[X, [X, Y ]] = (x1x2y3− x₁x3y2− x₃x1y2+ x3x2y1)e6, so, by the Campbell-Hausdorff formula,

P (x, y) = 1

2[X, Y ] + 1 12



[X, [X, Y ]] + [Y, [Y, X]]



=

6

X

j=1

P_j(x, y)e_j, where

P1(x, y) = P2(x, y) = P3(x, y) = 0, P₄(x, y) = 1

2(x₁y₂− x₂y₁), P5(x, y) = 1

2(x2y3− x₃y2), P6(x, y) = 1

2(x1y5− x₅y1− x₃y4+ x4y3) + 1

12(x₁x₂y₃− x₁x₃y₂− x₃x₁y₂+ x₃x₂y₁ + y1y2x3− y₁y3x2− y₃y1x2+ y3y2x1).

Then,

q1(ξ) = 1 + |ξ4|^1/2+ |ξ6|^1/3, q2(ξ) = 1 + |ξ4|^1/2+ |ξ5|^1/2+ |ξ6|^1/3, q3(ξ) = 1 + |ξ5|^1/2+ |ξ6|^1/3, q4(ξ) = q5(ξ) = 1 + |ξ6|^1/3, q6(ξ) = 1.



Definition 5.10. We define the Melin operator on g^?× g^? by (5.11) Uf (ξ) =

Z Z

g×g

e^−ihx+y,ξie−ihP (x,y),ξif^∨(x, y)dxdy, for f ∈ S(g^?× g^?).

By a linear differential operator on g^? with polynomial coefficients we understand a differential operator of the form

Lf (ξ) =X

α

p_α(ξ)D^αf (ξ), f ∈ C^∞(g^?),

where p_α are polynomials and the sum is finite. If f is a differentiable function on g^?× g^? we write Dξf = D1f and Dηf = D2f for the partial derivatives with respect to the variable (ξ, η) ∈ g^?× g^?.

Lemma 5.12. For every linear differential operator L with polynomial co- efficients on g^?, there exists a finite number of operators L_k of the same type on g^?× g^? such that

|LUf (ξ)| ∼^<

X

k

kL_kf k_A(g^?_×g^?₎, f ∈ S(g^?× g^?), ξ ∈ g^?, where

kf k_A(g?×g^?)= Z Z

g×g

|f^∨(x, y)|dxdy.

Consequently, the mapping U : S(g^?× g^?) → S(g^?) is continuous.

Proof. Let

P_j,k¹ (x, y) = Dx_kPj(x, y), P_j,k² (x, y) = Dy_kPj(x, y) and

D_j,k¹ = P_j,k¹ (iDξ, iDη) D_j,k² = P_j,k² (iDξ, iDη).

Directly from (5.11) one obtains

ξkU(f )(ξ) = U(T_k¹f )(ξ) −X

jk

ξjU

 D¹_j,kf

 (ξ) (5.13)

= U(T_k²f )(ξ) −X

jk

ξjU

 D²_j,kf

 (ξ), where

T_k¹f (ξ, η) = ξ_kf (ξ, η), T_k²f (ξ, η) = η_kf (ξ, η).

We also have

(5.14) D_kUf (ξ) = U(D_kf )(ξ), where

D_k = D_ξk+ Dη_k− iP_k(iD_ξ, iDη).

By iteration of (5.13) and (5.14), for every linear differential operator L with polynomial coefficients, there exist operators L_k of the same type such that

|LU(f )(ξ)| ≤X

k

|U(L_kf )(ξ)|.

By (5.11),

|U(f )(ξ)| ≤ Z Z

g×g

|f^∨(x, y)|dxdy = kf k_A(g^?_×g^?₎, for f ∈ S(g^?× g^?) and ξ ∈ g^?. Therefore,

|LU(f )(ξ)| ≤X

k

kL_kf k_A(g^?_×g^?₎.

 Lemma 5.15. Let g ∈ G. For every α ∈ A_N,

D^α = X

p(A)+p(B)=p(α) g^α≤g^Ag^B

cABD_ξ^AD^B_η,

where

D^α= D^α₁¹D₂^α2. . . D^α_N^N, and the symbols g^α have been defined in (3.13).

Proof. It is not hard to see that if the assertion holds for α and β, then it holds for α + β as well. Therefore, it is sufficient to prove it for single derivatives D_k. We have

D_k = X

p(A)+p(B)=pk

c_ABD^A_ξD_β^B. (5.16)

If c_AB 6= 0, then A_j = Bj = 0 unless j  k. The sequence gj is decreasing, so

g_k^pk = g^p(A)_k g^p(B)_k ≤ g^Ag^B.

 6. Estimates for the basic metric

In this section we only consider the basic metric q.

Lemma 6.1. Let {Bν}_ν∈N be the covering of Proposition 4.1. Then,

|U(f )(ξ)|∼ |f |^{< q⊕q}N +1, f ∈ C_c^∞(B_ν× B_µ), uniformly in ν, µ.

Proof. We have

|U(f )(ξ)|∼^<

Z Z

g×g

|f^∨(x, y)|dxdy = Z Z

g×g

|F^∨(x, y)|dxdy, where

F (ξ, η) = f (b_ν + ∆_νξ, b_µ+ ∆_µη) and ∆_ν is as in (4.4). Note that

q_bν(∆_νξ) = kξk, hence F is supported in the product K × K, where

K = {ξ ∈ g^? : kξk < 1}.

By the Sobolev inequality (10.1),

|U(f )(ξ)| ≤ max

|α|+|β|≤N +1kD^α_ξD^β_ηF k_L2(g^?×g^?),

where

kD^α_ξD^β_ηF k²_L2(g^?×g^?)= Z

K

Z

K

|D^α_ξD^β_ηF (ξ, η)|²dξdη

= Z

K

Z

K

q_α(b_ν)²q_β(b_µ)²|D_ξ^αD_η^βf (b_ν + ∆_νξ, b_µ+ ∆_µη)|²dξdη.

Recall that, by (3.10),

qα(bν) ≈ qα(bν+ ∆νξ), for b_ν+ ∆_νξ ∈ B_ν. Thus,

kD_ξ^αD_η^βF k²_L2(g^?×g^?) <

∼ Z

K

Z

K

|f |^q⊕q_{N +1}(b_ν+ ∆_νξ, b_µ+ ∆_µη)²dξ dη

∼<

|f |^q⊕q_{N +1}2

.

 Let

d_ν,k(ξ) = 1 +X

jk

|ξ_j− (b_ν)j|

q_j(b_ν) , ξ ∈ X^?. By Corollary 3.11, there exists R > 0 such that, for every k ∈ N , (6.2)

 eqk(ξ) qe_k(b_ν)

±1

∼ d< ν,k(ξ)^R.

Recall that the functions qek have been defined in Corollary 3.11.

Proposition 6.3. For every L ∈ N , there exists s₀ > 0 such that, for all ν, µ,

|Uf (ξ)|∼ d^< ν(ξ)^−Ld_µ(ξ)^−L|f |^q⊕q_s0 , ξ ∈ g^?, if f ∈ C_c^∞(Bν × B_µ).

Proof. For the sake of the proof we refine our claim:

For every L ∈ N and every 1 ≤ k ≤ N , there exists s_k> 0 such that, for all ν, µ,

(6.4) |Uf (ξ)| ∼ d^< ν,k(ξ)^−Ldµ,k(ξ)^−L|f |^q⊕q_s

k .

Once we prove (6.4) for all minimal k ∈ N , our claim will follow. We proceed by induction starting with maximal k. If k is maximal in N , then, by (5.13),

ξ_k− (b_ν)_k

q_k(b_ν)^pk Uf (ξ) = UT_k¹− (b_ν)_k q_k(b_ν)^pk f

(ξ), where

T_k¹− (b_ν)_k q_k(b_ν)^pk f

≤ |f |, f ∈ C_c^∞(Bν× B_µ),

so our claim is reduced to that of Lemma 6.1. Otherwise, assume that, for any j  k, any L₁, L₂ ∈ N , and some s > 0,

(6.5) |Uf (ξ)| ∼ C(ξ)|f |^{< q⊕q}s ,

where C(ξ) =



1 +|ξ_k− (b_ν)_k| q_k(b_ν)

−M1

1 +|ξ_k− (b_µ)_k| q_k(b_µ)

−M2

d_ν,j(ξ)^−L1d_µ,j(ξ)^−L2. The inductive step consists in showing that (6.5) implies that, for every j  k, there exists s⁰ such that

(6.6) |ξ_k− (b_ν)_k|

qk(bν) |Uf (ξ)| ∼ C^< 1(ξ)|f |^q⊕q_s0

and

(6.7) |ξ_k− (b_µ)_k|

q_k(b_µ) |Uf (ξ)| ∼ C^< 1(ξ)|f |^q⊕q_s0 , where

C₁(ξ) = d_ν,j(ξ)^2Rd_µ,j(ξ)^2RC(ξ),

for some m1, R > 0. Since the cases (6.6) and (6.7) are almost identical, let us only consider the first one.

By (5.13),

ξk− (b_ν)k

q_k(bν)^pk Uf (ξ) = U

T_k¹− (b_ν)_k q_k(bν)^pk f

 (ξ) (6.8)

−X

jk

ξj

q_k(b_ν)^pkU

 D¹_j,kf

 (ξ) (6.9)

= U(f_ν,k)(ξ) −X

jk

ξj

q_k(b_ν)^pkU(D¹_j,kf )(ξ).

(6.10)

Note that, by (6.5),

|Uf_ν,k(ξ)| ∼ C(ξ)^<

T_k− (b_ν)_k q_k(b_ν)^pk f

q⊕q

s⁰

∼ C(ξ)|f |< ^q⊕qs .

To prove (6.6), it is, therefore, sufficient to estimate each of the terms U_k,j(ξ) = ξ_j

q_k(b_ν)^pkU(D_k,j¹ f )(ξ).

Note that, by (5.16),

D¹_j,k = X

p(α)+p(β)=pj−p_k

c_αβD_ξ^αD_η^β. Therefore, by (6.5), there exists s⁰ > 0 such that

|U(D_j,k¹ f )(ξ)|∼ |D^{< 1}j,kf |^q⊕q_s

∼ C(ξ)|f |< ^q⊕q_s⁰ X

p(α)+p(β)=pj−p_k

qα(bν)⁻¹qβ(bµ)⁻¹, where αr = βr = 0 unless r ≺ j. Therefore,

qα(bν)^{−1 <}∼ eqj(bν)^−p(α), qβ(bµ)^{−1 <}∼qej(bµ)^−p(β). By (6.2),

|U(D¹_j,kf )(ξ)| ∼ C(ξ)|f |^{< q⊕q}_s⁰ qe_j(ξ)^pk−pjd_ν,j(ξ)^R/2d_µ,j(ξ)^R,