Optimal boundedness of central oscillating multipliers on compact Lie groups

BLAISE PASCAL

Jiecheng Chen & Dashan Fan

Optimal boundedness of central oscillating multipliers on compact Lie groups

Volume 19, n^o1 (2012), p. 123-145.

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Optimal boundedness of central oscillating multipliers on compact Lie groups

Jiecheng Chen Dashan Fan

Abstract

Fefferman-Stein, Wainger and Sjölin proved optimalH^pboundedness for cer- tain oscillating multipliers on R^d. In this article, we prove an analogue of their result on a compact Lie group.

1. Introduction

Let G be a connected, simply connected, compact semisimple Lie group of dimension n. In this paper, we will study theH^p(G) boundedness for the oscillating multiplier operator

T_γ,β(f)(x) =K_γ,β∗f(x), γ≥0 and 0< β <1.

Here K_γ,β is a central kernel defined by K_Ω,γ,β(y) = ^X

λ+δ∈Λ\{0}

e^ikλ+δkβ

kλ+δk^γd_λχ_λ(ξ),

where y is conjugate to the element expξ in a fixed maximal torus of G (the detailed definition can be found in the second section). Thus the operatorT_γ,β has the Fourier expansion

Tγ,β(f)(x) = ^X

λ+δ∈Λ\{0}

e^ikλ+δkβ

kλ+δk^γdλχλ∗f(x) for anyf ∈C^∞(G).

Keywords:Oscillating multiplier,H^pspaces, Compact Lie groups, Fourier series.

Math. classification:43A22, 43A32, 43B25, 42B25.

The formulation of T_γ,β is an analogue of the oscillating multiplier op- erator Sγ,β(f) on R^d:

S_α,β(f)(x) = Z

R^d

fb(ξ)Ψ(|ξ|)m(ξ)e^ihx,ξidξ, where

m(ξ) = e^i|ξ|β

|ξ|^γ

and Ψ(|ξ|)∈ C^∞(R^d), satisfying Ψ(|ξ|) = 0 for |ξ|<1/2 and Ψ(|ξ|) = 1 for |ξ|>1.

It is well known that the operatorS_γ,βis bounded onH^p(R^d) if and only if|1/2−1/p| ≤γ/(dβ) for all 0< p <∞ (see [14, 19, 20, 22]). We notice that when 1 < p < ∞, the boundedness of Sγ,β has been generalized to many different settings of Lie groups and manifolds (see [1, 5, 15, 18]). In a recent paper [5], we established the following optimal L^p(G) boundedness of T_γ,β on a compact Lie group.

Theorem 1.1. LetGbe a connected, simply connected, compact semisim- ple Lie group of dimension n. For0< β <1, the operator T_γ,β is bounded on L^p(G) if and only if | ¹₂− ¹_p |≤ _nβ^γ for all 1< p <∞.

In [5], we are able to extend Theorem 1.1 toH^p(G) for 0< p0< p≤1.

However, the method in [5] only allows us to obtain the result when p₀ is close to 1. Thus, the aim of this paper is to give a complete solution of Theorem 1.1 by establishing the following optimalH^p boundedness for all p >0.

Theorem 1.2. LetGbe a connected, simply connected, compact semisim- ple Lie group of dimension nand0< β <1. The operatorT_γ,β is bounded on H^p(G) if and only if |¹₂ −¹_p |≤ _nβ^γ for all 0< p <∞.

We want to point out that the extension of Theorem 1.1 to all 0< p <

∞ is not trivial and actually it is quite involved, due to the structure of a semi-simple Lie group. Our proof will use powerful results of Clerc [8], in which the Weyl denominator and its derivatives are carefully estimated based on the classification of the root system. This allows us to obtain sharp estimates on the kernel and its derivatives. This same method was recently also used in [6] to study the wave problem (β= 1).

In order to prove the main theorem, we use a standard analytic inter- polation argument (see [2]). Define an analytic family of operators

T_z,β(f)(x) = ^X

λ+δ∈Λ\{0}

e^ikλ+δkβ

kλ+δ k^zd_λχ_λ∗f(x), z∈C.

By the Plancherel theorem, it is easy to see that kT_z,β(f)k_L2(G) kfk_L2(G)

if <(z) = 0. Thus, to complete the proof of the sufficiency part of The- orem 1.2, by the analytic interpolation theorem it suffices to show the following endpoint estimate.

Proposition 1.3. Let p < _β+2^2β andγ_p =nβ(¹_p −¹₂). If <(z) =γ_p, then kT_z,β(f)k_Hp(G) kfk_Hp(G).

The plan of this paper is as follows: in Section 2, we will recall some necessary notation and known results on a compact Lie group; the kernel K_z,β and its derivatives will be carefully estimated in Section 3; we will prove Proposition 1.3 in Section 4.

In this paper, we use the notationABto mean that there is a positive constant C independent of all essential variables such that A≤CB. We use the notation A'B to mean that there are two positive constantsc₁ and c₂ independent of all essential variables such thatc₁A≤B≤c₂A.

Acknowledgments. The first author is supported by the NSFC Grant 10931001, 10871173.

2. Notation and known results

2.1. Some definitions

Let G be a connected, simply connected, compact semisimple Lie group of dimensionn. Let g be the Lie algebra of G and τ the Lie algebra of a fixed maximal torus T inGof dimensionm. LetAbe a system of positive roots for (g, τ) so Card(A) = ^n−m₂ , and let δ= ¹₂^P_a∈Aa.

Let |. |be the norm of g induced by the negative of the Killing form B on g^C, the complexification of g. The norm |.| induces a bi-invariant metric d on G. Furthermore, since B|_τC×τ^C is nondegenerate, given λ ∈

hom_C(τ^C,C), there is a unique H_λ in τ^C such that λ(H) = B(H, H_λ) for each H ∈ τ^C. We let h. , .i and k . k denote the inner product and norm transferred from τ to hom_C(τ, iR) by means of this canonical iso- morphism.

LetN={H∈τ : expH =I}, whereI is the identity inG. The weight lattice P is defined by P = {λ∈ τ :hλ, ni ∈ 2πZ for any n ∈ N} with dominant weights defined by Λ = {λ ∈ P : hλ, ai ≥ 0 for any a ∈ A}.

Λ provides a full set of parameters for the equivalence classes of unitary irreducible representation of G: for λ∈Λ, the representation Uλ has di- mension

d_λ = ^Y

a∈A

hλ+δ, ai hδ, ai and its associated character is

χλ(ξ) = P

w∈W(w) eihw(λ+δ),ξi

P

w∈We^ihwδ,ξi

where ξ ∈τ, W is the Weyl group, and (w) is the signature of w ∈W. Any function f ∈L¹(G) has the Fourier series

X

λ∈Λ

d_λχ_λ∗f(x).

The oscillating multiplier

T_z,β(f)(x) = ^X

λ+δ∈Λ\{0}

e^ikλ+δkβ

kλ+δk^zd_λχ_λ∗f(x)

is initially defined on all f ∈C^∞. Thus T_z,β is a convolution operator Tz,β(f)(x) =Kz,β∗f(x),

where K_z,β is a central kernel defined by K_z,β(y) = ^X

λ+δ∈Λ\{0}

e^ikλ+δkβ

kλ+δ k^zd_λχ_λ(ξ),

and expξ ∈T is conjugate to y. Let Qbe a fixed fundamental domain of T and

Q_ν ={ξ+ν : ξ ∈Q}, ν ∈N.

Up to sets of measure zero,{Q_ν}is a sequence of mutually disjoint subsets in the Lie algebra τ.

Any element y ∈Gis conjugate to exactly one element in exp(Q). We denote y∼expξ ify is conjugate to expξ.

2.2. Hardy spaces H^p(G)

There are many equivalent definitions of the Hardy space H^p. The reader can see [3, 4, 7, 8, 9, 10, 11, 13, 16] and the references therein for more details and background on the Hardy space. Below, we briefly review the definition of H^p by using the heat kernel, and the atomic characterization of Hardy spaces.

For the heat kernel (see [21]) Wt= ^X

λ+δ∈Λ

e^{−t{kλ+δk2−kδk2}}dλχλ,

the Hardy space H^p(G), 0 < p <∞, is the collection of all distributions f ∈S⁰(G) such that

kfk_Hp(G)=ksup

t>0

{|W_t∗f|}k_Lp(G)<∞.

An exceptional atom is anL^∞function bounded by 1. In order to define a regular atom, one considers a faithful unitary representation Π ofGsuch that Π(G)≈U(L,C). ThenGcan be identified as a submanifold in a real vector space V underlying End(C^L). A regularp-atom for 0< p≤1 is a function a(x) supported in a ballB(x₀, ρ) such that

kak_L2(G)≤ρ^{−n(1p−12)}, Z

G

a(x)℘(Π(x))dx= 0,

where ℘ is any polynomial onV of degree less than or equal to` for any integer `≤[n(¹_p −1)], the integer part of n(¹_p−1).

The space H_a^p(G), 0 < p≤1, is the space of all f ∈S⁰(G) having the form

f =^Xc_ka_k with ^X|c_k|^p <∞,

where each a(x) is either a regular p-atom, or an exceptional atom. The

“norm” kfk_H^p

a is the infimum of all expressions (^P|c_k|^p)^1/p for which we have a representation f =^Pc_ka_k. As we discussed in [5] (see also [3], [4]), to show that the operator T_z,β is bounded onH^p(G), it suffices to prove

kT_z,β(a)k_Lp(G)1

uniformly for any regular p-atom a(x) with support in B(I, ρ), where 0< ρ < r and r is a fixed sufficiently small number.

2.3. Decomposing the Weyl denominator Denote

D(θ) = ^X

w∈W

ei<wδ,θ>. D(θ) is called the Weyl denominator and it satisfies

D(θ) = ^Y

a∈A

sinα(θ) 2 , where

α(θ) =hα, θi.

In this section, we introduce some notation in [8]. LetB_s be the set of all simple roots in A and let BL be the set of all largest roots. For θ ∈ Q, introduce the following sets

I =I_θ={α∈B_s:α(θ)≤ 1 R} J =J_θ={β∈B_L:β(θ)≥2π− 1

R}.

When R is a large number, elements in Iθ and Jθ are independent. We define two facets

FI,J ={ξ ∈Q:α(ξ) = 0 for α∈I, β(ξ) = 2π forβ ∈J, 0< α(ξ)<2π forα∈Bs\I, 0< β(ξ)<2π forβ ∈BL\J} and let F_I,J be the affine subspace generated by F_I,J so that

F_I,J ={ξ∈τ :α(ξ) = 0 for α∈I and β(ξ) = 2π forβ ∈J}.

A positive root γ is R−singular of type I atθ, if the following equivalent conditions are satisfied:

(i) γ{F_I,J}={0}, (ii) γ{F_I,J}={0}, (iii) γ can be written as

γ=^X

α∈I

nαα, nα ∈N.

A positive root γ is R-singular of type II atθ, if the following equivalent conditions are satisfied:

(i) γ{F_I,J}={2π}, (ii) γ{F_I,J}={2π},

(iii) γ can be written as, for someβ ∈J γ =β−^X

α∈I

n_αα, n_α ∈N.

Both R−singular roots of types I and II are called singular roots. By this definition, it is easy to see that ifγ is a positive non-singular root, then

1

R < γ(θ)<2π− 1 R.

For a singular root α of type I, let Sα be the orthogonal symmetry with respect to the hyperplane α = 0. For a singular root β of type II, let Seβ

be the orthogonal symmetry with respect to the hyperplane α= 2π. Let WI,J be the group generated by

{S_α}_α∈I∪ {Seβ}_β∈J.

This group is a finite subgroup of the affine Weyl group. Now we define Γ_θ = Γ^(R)_θ = convex hull of {wθ}w∈W_I,J.

Also, we write the root system

A=A_s∪A_ns

where As is the set of all singular (R−singular) roots and Ans is the set of all non-singular roots. Denote by µ^R the number of singular roots and denote

D^R(θ) = ^Y

a∈Ans

sin(hα, θi 2 ).

Thus

D(θ) =D^R(θ) ^Y

a∈As

sin(hα, θi 2 ).

The above definition of D(θ) and Γ^(R)_θ can be defined on the torus T itself. In fact, if x∈T, thenx= expθ for someθ inQ. We define

d(expθ) =D(θ), and Γ^(R)_x = exp Γ^(R)_θ .

Some properties of the domain Γ^(R)_x can be found in [8]. In particular, it is known from Lemma 2.9 in [8] that there exists a constant c such that

D^R(ξ)≥cD^R(θ) (2.1) for all ξ in Γ^(R)_θ .

2.4. Derivatives on central functions.

Fixing a vector basis of g^C, say Y1, Y2, . . . , Yn, we denote the element Y₁^j1Y₂^j2· · ·Y_n^jn byY^J, whereJ = (j₁, . . . , j_n). AsJ varies over all possible n−tuples, the {Y^J} forms a basis of the complex universal enveloping algebra U(g) ofg. Similarly, we fix a basis Θ1, . . . ,Θ_m of τ^C and use the notation Θ^I for Θⁱ₁¹Θⁱ₂²· · ·Θⁱ_m^m, with I = (i1, . . . , im). We can find the following two theorems in [8].

Theorem 2.1. Letp, qbe positive integers. Assume thatf is aC^∞central function. Then for each I with |I| ≤ p and J with |J| ≤ q, there exists a constant C such that

Θ^IY^Jf(y)C

p+q

X

j=0

R^{j X}

|K|≤p+q−j

sup

v∈Γ^(R)_y

Θ^Kf(v).

Theorem 2.2. Letpbe a positive integer. Assumef(y) =d(y)⁻¹g(y) and thatgis aC^∞central function which is skew-invariant by the Weyl group.

For each I with |I| ≤p, there exists a constant C such that

Θ^If(y)Cd^R(y)⁻¹

p

X

j=0

R^{j X}

|K|≤p+µ^R−j

sup

v∈Γ^(R)y

Θ^Kg(v).

3. Estimates on the kernel K_z,β, <(z)> n

In this section, we denoteγ =<(z) and assumeγ > n. First, we recall the following lemma, which is an easy modification of results in [12] or [17].

Lemma 3.1. Let A ⊆ R^m denote an open set and Φ ∈C₀^∞(A). If Ψ ∈ C^∞(A) satisfies

|det(∂²/∂xi∂xjΨ(x))| ≥C >0

for all x∈supp(Φ), then for large |λ|,

Z

A

ei(λΨ(x)+hξ,xi)Φ(x)dx |λ|^−m/2kΦk_C2m.

By the definition, it is easy to see

K_z,β(y) = P

λ+δ∈Λ\{0}

e^ikλ+δkβ kλ+δk^z ( ^Q

a∈A

hδ+λ, ai)^P_w∈W(w) eihw(λ+δ),ξi

D(ξ) ^Q

α∈A

hα, δi

= ( ^Q

a∈A

∂

∂α) ^P

λ∈P\{0}

e^ikλkβ kλk^z e^ihλ,ξi D(ξ) ^Q

a∈A

ha, δi , where y is conjugate to expξ ∈T.

Choose a C^∞ radial functionψ on R^m such that

ψ(t) = 0 if t∈(0, c1) and ψ(t) = 1 ift∈(c2,∞), where c₁ and c₂ are two fixed positive numbers such that

K_z,β(y) = ( ^Q

a∈A

∂

∂α) ^P

µ∈P e^ikµkβ

kµk^z ψ(kµk)e^ihµ,ξi D(ξ) ^Q

a∈A

ha, δi .

Since <(z) = γ > n, the above summation is absolutely convergent. By the Poisson summation formula (see [11]) we obtain

K_z,β(y)' P

ν∈N

( ^Q

a∈A

∂

∂α)<_ν(ξ) D(ξ) ^Q

a∈A

ha, δi , where

<₀(ξ) = Z

R^m

e^ikHkβ

kHk^z ψ(kHk)e^−ihξ,HidH, and

<_ν(ξ) =<₀(ξ+ν).

For simplicity, we write

<(ξ) =<₀(ξ) and Θ(H⁰) = (^Y

a∈A

hα, H kHki).

Also, without loss of generality, we may use the Euclidean norm | . | instead of k.k. Thus, forν∈N,

(^Y

a∈A

∂

∂α)<_ν(ξ) = (^Y

a∈A

∂

∂α)<(ξ+ν)' Z

R^m

e^iΦ(H,ξ+ν)

|H|^z−n−m2

Θ(H⁰)ψ(|H|)dH, where

Φ(H, ξ+ν) =|H|^β− hξ+ν, Hi is the phase function.

Fix a small ρ >0. Let ϕ(t) be a C^∞ function on (0,∞) that satisfies ϕ(t) = 0 if t <0.05 , ϕ(t) = 1 ift >0.1,

and let

ϕ∞(t) =ϕ(ρt), ϕ0(t) = 1−ϕ(ρt).

For each ν∈N, we write ( ^Y

a∈A

∂

∂α)<_ν(ξ) = (^Y

a∈A

∂

∂α)<_ν,0(ξ) + (^Y

a∈A

∂

∂α) <_ν,∞(ξ), where

(^Y

a∈A

∂

∂α) <_ν,0(ξ) = Z

R^m

e^iΦ(H,ξ+ν)

|H|^z−n−m2 Θ(H⁰)ψ(|H|)ϕ0(|H|)dH and

(^Y

a∈A

∂

∂α) <_ν,∞(ξ) = Z

R^m

e^iΦ(H,ξ+ν)

|H|^z−n−m2

Θ(y⁰)ψ(|H|)ϕ_∞(|H|)dH.

Furthermore, for y∼expξ, we define four central kernels:

=_∞,0(y) = P

ν∈N\{0}

( ^Q

a∈A

∂

∂α)<_ν,0(ξ) D(ξ) ^Q

a∈A

ha, δi , =_∞,∞(y) = P

ν∈N\{0}

( ^Q

a∈A

∂

∂α)<_ν,∞(ξ) D(ξ) ^Q

a∈A

ha, δi , and

=_0,0(y) = ( ^Q

a∈A

∂

∂α)<_0,0(ξ) D(ξ) ^Q

a∈A

ha, δi , =_0,∞(y) = ( ^Q

a∈A

∂

∂α)<_0,∞(ξ) D(ξ) ^Q

a∈A

ha, δi . Then, we decompose the kernel K_z,β by

K_z,β(y)'=_0,∞(ξ) +=_∞,∞(ξ) +=_0,0(ξ) +=_∞,0(ξ).

We will give different estimates on these kernels.

Lemma 3.2. Fix a small positiveρsuch that ρ≤1/c₁. Let ξ∈Qν∩ {ξ :

|ξ|>1000nρ^1−β}. For any positive integerL and N =γ− ^n+m₂ +L, we have

|=0,∞(ξ)| |ξ|^−Lρ^N

|D(ξ) ^Q

a∈A

ha, δi |, (3.1)

and

k=∞,∞k_L1(G) ρ^N.

Proof. For|ξ| ≥1000nρ^1−β,there is at least one coordinateξj satisfying

|ξ_j| ≥1000ρ^1−β.

On the other hand, if H lies on the support ofϕ∞ we have

|H|^β−1 ≤100ρ^1−β. Thus we have

∂

∂yj

Φ(H, ξ)

≥ |ξ_j| −β|H |^β−1 |ξ_j| ' |ξ|.

By this observation, we perform integration by parts on H_j−variable to obtain

Z

R^m

e^iΦ(H,ξ)

|H|^z−n−m2 Θ(H⁰)ϕ∞(|H|)dH

|ξ|^−Lρ^{γ−n+m2 +L}. This shows (3.1). As a consequence, we obtain

Z

R^m

e^iΦ(H,ξ+ν)

|H |^z−n−m2 Θ(H⁰)ϕ∞(|H |)dH

|ξ+ν|^−Lρ^{γ−n+m2 +L}. Thus, by the Weyl integral formula we have

k=_∞,∞k_L1(G) ^X

ν∈N\{0}

ρ^{γ−n+m2 +L} Z

Q

|ξ+ν|^−L|D(ξ)|dξρ^N after choosing a suitably large Land letting γ−^n+m₂ +L=N. Lemma 3.3. Assume |ξ|> 1000nc^β−1₁ . We have, for any multi-index J and any positive integer L,

( ∂

∂ξ)^J Z

R^m

e^iΦ(H,ξ)

|H|^z−n−m2 Θ(H⁰)ϕ₀(|H |)ψ(|H|)dH

(1 +|ξ|)^−L.

Proof. It is easy to see that ( ∂

∂ξ)^J Z

R^m

e^iΦ(H,ξ)

|H|^z−n−m2 Θ(H⁰)ϕ0(kHk)ψ(|H|)dH

=

|J|

X

k=1

Z

R^m

e^iΦ(H,ξ)

|H|^z−n−m2 P_k(H)Θ(H⁰)ϕ₀(|H|)ψ(|H|)dH =

|J|

X

k=1

L_k(ξ), where P_k is a homogeneous polynomial of degree k. Without loss of gen- erality, we may write, for eachk,

L_k(ξ) = Z

R^m

e^iΦ(H,ξ)

|y|^{z−n−m2 −k}Θ(H⁰)ψ(|H|)ϕ₀(|H|)dH.

The support condition of ψ implies that the phase function satisfies

n

X

j=1

∂

∂H_jΦ(H, ξ)

|ξ|.

Thus the lemma follows by integration by parts N times for a suitably

large N.

Lemma 3.4. Assume 1000nc^β−1₁ ≥ |ξ|>1000ρ^1−β.For any multi-index J, if

(α−ⁿ₂ − |J|)

1−β −m

2 <0, then we have

( ∂

∂ξ)^J Z

R^m

e^iΦ(H,ξ)

|H|^z−n−m2 Θ(H⁰)ϕ₀(|H|)ψ(|H|)dH

|ξ|

(γ−n 2−|J|) 1−β −^m₂

. Proof. As in the previous lemma, we need to show for each k,

|L_k(ξ)|= Z

R^m

e^iΦ(H,ξ)

|H|^{z−n−m2 −k}Θ(H⁰)ψ(|H|)ϕ₀(|H|)dH

|ξ |

(γ−n 2−k)

1−β −^m

2 +1.

Choose C^∞ functions Γ_j(t),j= 1,2,3 such that

Γ₁(t) = 1 ifβt^β−1 >3 and Γ₁(t) = 0 if βt^β−1 ≤2, Γ₂(t) = 1 if βt^β−1< 0.1

n and Γ₂(t) = 0 if βt^β−1 ≥0.2.

Let

Γ₃(t) = 1−Γ₁(t)−Γ₂(t), and

Γ_j(t, ξ) = Γ_j(|ξ|^1−β1 t), j = 1,2,3.

Therefore, L_k(ξ) =

3

X

j=1

Z

R^m

e^iΦ(H,ξ)

|H|^{z−n−m2 −k}Γ_j(|ξ|^1−β1 |H|)Θ(H⁰)ψ(|H|)ϕ₀(|H|)dH

=

3

X

j=1

L_k,j(ξ).

By polar coordinates, L_k,1(ξ)'

Z

S^m−1

Θ(H⁰)

×ⁿ Z ∞

0

e^ig(t) t^{z−n−m2 −k−m+1}

Γ₁(t|ξ|^1−β1 )ψ(t)ϕ₀(t)dt^odσ(H⁰), whereS^m−1 is the unit sphere inR^m with the induced Lebesgue measure dσ(H⁰), and the phase function

g(t) =t^β−ithξ, H⁰i satisfies

d dtg(t)

t^β−1, and d dt

1

g⁰(t) = β(1−β)t^β−2

g⁰(t)² =O1 t^β

,

iftlies in the support of Γ₁(t|ξ|^1−β1 ). By this observation, it is easy to see that after using integration by parts N times for a sufficiently large N, we have

|L_k,1(ξ)| 1. (3.2)

For the term L_k,2(ξ), it is easy to check that, in some direction Hj,

∂

∂H_jΦ(H, ξ)

≥ |ξ| |H|

ifH lies in the support of Γ₂(H|ξ|^1−β1 ). Again, performing integration by parts in the H_j direction for sufficiently many times, we obtain

|L_k,2(ξ)| 1. (3.3)

For the term L_k,3(ξ), changing variables we have L_k,3(ξ)'

|ξ|−i=(z)/(β−1)

|ξ|

(γ−n+m 2 −k) β−1

Z

R^m

e^iΨ(H,ξ)

|H|^{z−n−m2 −k}Γ₃(|H|)Θ(H⁰)(ψϕ₀)(|H| |ξ|^β−11 )dH, where

Ψ(H, ξ) =|ξ|^β−1β |H|^β− |ξ|^β−11 hξ⁰, Hi.

Recalling that the support of Γ₃ lies in the set S=ⁿy: (0.1

βn)^1−β ≤ |H| ≤(3 β)^1−βo,

without loss of generality, we may also assume that |ξ|is small such that

|H| |ξ|^β−11 > c₂ for allH ∈S.

We now have

L_k,3(ξ)' |ξ|−i=(z)/(β−1)

|ξ|

(γ−n+m 2 −k) β−1

Z

R^m

e^iΨ(H,ξ)Γ^f₃(|H|)Θ(H⁰)ϕ0(ρ|H| |ξ|^β−11 )dH, where

Γf₃(|H|) = Γ₃(|H|)

|H|^{z−n−m2 −k}

is a C^∞function supported in the set S. By Lemma 3.1 we have

|L_k,3(ξ)| |ξ|

(γ−n+m 2 −k)

1−β |ξ|

mβ

2(1−β) =|ξ|

(γ−n 2−k) 1−β −^m₂.

(3.4) We now obtain the lemma by combining (3.2),(3.3) (3.4).

Next, we fix a number η such that d(u, I) ≤ 2η implies α($) ≤ π for all α∈A, whereu∼exp$.

Lemma 3.5. Assume d(u, I) ≤ η. For any non-negative integer L and any multi-index M with|M|=q, we have

Y^M=_∞,0(u)R^q|d^R(u)|^{−1 X}

ν6=0

|ν|^−L.

Proof. By [8], d(u)=_∞,0(u) is skew-invariant by the Weyl group. Thus we invoke Theorems 2.1 and 2.2 and (2.1) to obtain

Y^M=∞,0(u) ^X

|J|≤q

sup

x∈Γ^(R)_u

Θ^J =∞,0(x)R^q|d^R(u)|⁻¹

× ^X

|J|≤q

X

|I|≤|J|+µ^R

sup

x∈Γ^(R)_u

sup

y∈Γ^(R)_x

( ∂

∂ξ)^{I X}

ν∈N\{0}

( ^Y

a∈A

∂

∂α)<_ν,0(ξ) , where expξ∼y and

( ^Y

a∈A

∂

∂α)<_ν,0(ξ)'( ∂

∂ξ)^I Z

R^m

e^iΦ(H,ξ+ν)

|H|^z−n−m2 Θ(H⁰)ϕ₀(|H |)ψ(|H|)dH . Observing that there is a σ >0 such that for allξ ∈Q,

|ξ+ν| ≥σ ifν 6= 0, and

|ξ+ν| ' |ν| if |ν| is large,

we obtain Lemma 3.5 from Lemma 3.3 and 3.4.

Now we turn to estimate Y^I=_0,0(u).We have the following estimate.

Lemma 3.6. Let the number η be the same as in Lemma 3.5. For any multi-index I with |I|=q, if

d(u, I)≤η and N >−γ+n−1 2 +q, then we have

Y^I=_0,0(u) sup

y∈Γ^(R)_u

(|ξ|⁻

(γ−n 2)−q

β−1 |ξ |⁻ⁿ²) +|ξ|^−N−n+12 ), where y∼expξ.

Proof. By Theorem 2.1, without loss of generality, we may write

Y^I=_0,0(u) ^X

|J|=q

sup

y∈Γ^(R)_u

( ∂

∂ξ)^J=_0,0(y) By [22, Ch.4] (or see [8]), we have

<_0,0(ξ)' Z ∞

0

e^itβ ψ(t)ϕ₀(t)V^m−2

2 (t|ξ|)t^−z+m−1dt,

where

V^m−2

2 (t) = J^m−2

2 (t) t^m−22 , and J^m−2

2 (t) is the Bessel function of order ^m−2₂ . An easy computation shows (see [5]),

( ^Y

a∈A

∂

∂α)V^m−2

2 (t|ξ |)'( ^Y

a∈A

hα, ξi)Vⁿ⁻²

2 (t|ξ|)t^n−m. Thus, we obtain

=_0,0(y)' ( ^Q

a∈A

hα, ξi)

( ^Q

a∈A

sin^hα,ξi₂ ) Z ∞

0

e^itβψ(t) ϕ0(t)Vⁿ⁻²

2 (t|ξ|)t^−z+n−1dt.

By choosing a sufficiently large R, without loss of generality, we may assume α(ξ)≤π for all α∈A. Thus, taking the advantage that

( ^Q

a∈A

hα, ξi)

( ^Q

a∈A

sin^hα,ξi₂ ) is an analytic function, we may assume

( ∂

∂ξ)^J=_0,0(expξ)

( ∂

∂ξ)^J Z ∞

0

e^itβψ(t) ϕ₀(t)Vⁿ⁻²

2

(t|ξ |)t^−z+n−1dt . Using a derivative formula for the Bessel function (see [23]) we have

( ∂

∂ξ)^J=_0,0(expξ)

≤

q

X

k=0

|℘_k(ξ)|

where

℘_k(ξ) =

q

X

k=0

P_k(ξ)

|ξ |^{n−22 +k} Z ∞

0

e^itβϕ₀(t)ψ(t)Jⁿ⁻²

2 +k(t|ξ|)t^−z+n2+kdt, and Pk is a homogeneous polynomial of degreek.

For each k, using the asymptotic expansion of the Bessel function (see [23]), for any positive integer N we have

℘_k(ξ) =

N

X

µ=0

aµ

P_k(ξ)

|ξ|^{n−12 +µ+k} Z ∞

0

e^itβ e^−it|ξ| ϕ0(t)ψ(t)t^{−z+n−12 −µ+k}dt

+

N

X

µ=0

bµ

Pk(ξ)

|ξ |^{n−12 +k+µ} Z ∞

0

e^itβ e^it|ξ|ϕ0(t)ψ(t)t^{−z+n−12 −µ+k}dt

+O P_k(ξ)

|ξ|^{n+12 +N+k} Z ∞

0

ϕ₀(t)ψ(t) dt t^{z−n−12 +N+1−k}

N

X

µ=0

| F_k,µ(ξ)|+O 1

|ξ|^{m+12 +N} , where

F_k,µ(ξ) = 1

|ξ |^{n−12 +µ} Z ∞

0

e^itβ e^±it|ξ| ϕ0(t)ψ(t)t^{−z+n−12 −µ+k}dt.

Let Γ_j,j= 1,2,3 be defined in Lemma 3.4. We write F_k,µ(ξ) =

3

X

j=1

F_k,µ,j(ξ), where, for simplicity of notation, we write

F_k,µ,j(ξ) = 1

|ξ |^{n−12 +µ} Z ∞

0

e^itβ e^±it|ξ|ϕ0(t)Γj(t|ξ|^1−β1 )ψ(t)t^{−z+n−12 −µ+k}dt for j= 1,2,3.

By the same argument as we estimate L_k,1and L_k,2 in Lemma 3.4, we have

|F_k,µ,1(ξ)|+|F_k,µ,2(ξ)| 1.

By changing variables we have that

F_k,µ,3(ξ) = |ξ|

−γ+n−1 2 −µ+k+1

β−1 |ξ|−i=(z)/(β−1)

|ξ|^{n−12 +µ}

× Z ∞

0

e^iφ±(t,ξ) (ϕ₀ψ)(|ξ|^β−11 t)Γ₃(t)t^{−z+n−12 −µ+k}dt, where

φ±(t, ξ) =|ξ|^β−1β t^β±t.

Thus, by Lemma 3.1 and an easy computation we have

|F_k,µ,3(ξ)| |ξ|

−γ+n−1 2 +k+1 β−1

|ξ |ⁿ⁻¹²

|ξ|^2(1−β)β |ξ|^1−βµ

|ξ |^µ . Therefore,

q

X

k=0

|℘_k(ξ)|

q

X

k=0 N

X

µ=0 3

X

j=1

|F_k,µ,j(ξ)|+O( 1

|ξ |^{m+12 +N} )

q

X

k=0 N

X

µ=0

|ξ|

−γ+n−1 2 +k+1 β−1

|ξ |ⁿ⁻¹²

|ξ|^2(1−β)β |ξ|^1−βµ

|ξ|^µ +O( 1

|ξ |^{m+12 +N} )

|ξ|

−γ+n 2

β−1 |ξ|^β−1q |ξ|⁻ⁿ² +O( 1

|ξ |^{m+12 +N} ).

This completes the proof of the lemma.

Lemma 3.7. Let the number η be the same as in Lemma 3.5. Ifd(u, I)>

η

100 then for any multi-index M with|M|=q and any L >0,

|Y^M(=_0,0(u) +=_∞,0(u))|R^q |d^R(u)|⁻¹







1 +^X

ν6=0

|ν|^−L





 . Proof. Observe that (=_0,0(u)+=_∞,0(u))d(u) is skew-invariant by the Weyl group. The proof is the same as that of Lemma 3.5.

4. Proofs of the theorem

4.1. H^p Boundedness Of T_z,β

We will prove Proposition 1.3 in this section. Precisely, we will prove the H^p(G) boundedness of the operator Tz,β with

<(z) =γp=nβ(1 p− 1

2).

Observing that the assumption

p < 2β β+ 2

implies γp > n, we can use the estimates of K_z,β obtained in Section 3.

Moreover, as mentioned in Section 2, to prove theH^pboundedness ofT_z,β it suffices to prove

kT_z,β(a)k_Lp(G)1,

for anyp-atoma(x) of support inB(I, ρ) with sufficiently small ρ.

Let ς = 1−β. We have kT_z,β(a)k^p_Lp=

Z

d(x,I)≤10⁴nρ^ς

|T_z,β(a)(x)|^p dx

+ Z

d(x,I)>10⁴nρ^ς

|T_z,β(a)(x)|^p dx=I₁+I₂. By Hölder’s inequality, we have

I1kT_z,β(a)k^p_L2(G) ρ^{2−p2 nς} kak^p

L²_−ap(G) ρ^{2−p2 nς} kak^p_Lr(G) ρ^(1−2p)nς, 1/2 = 1/r−αp/n.

Note ¹_p −¹₂ = _nβ^γp and ς = 1−β. An easy computation shows I1 ρ^{pγp−n(1−p/2)}ρ^{2−p2 ςn} 1.

By Hölder’s inequality and Lemma 3.2, we have Z

d(x,I)>10⁴nρ^ς

Z

G

a(y)=∞,∞(xy⁻¹)dy

p

dx kak_L1(G)k =_∞,∞k_L1(G)1.

Using Hölder’s inequality and Lemma 3.2 again , we have Z

d(x,I)>10⁴nρ^ς

Z

G

a(y)=_0,∞(xy⁻¹)dy

p

dx kak_L1(G)

Z

d(x,I)>8⁴nρ^ς

| =_0,∞(x)|dx ρ^{(γp−n+m2 +L)}ρ^−np+n

Z

{|θ|≥10nρ^ς}∩Q

|D(θ)| |θ|^−Ldθ1 if we choose a large positive integer L.

It remains to show Z

d(x,I)>10⁴nρ^ς

| Z

G

a(y)(=_∞,0(xy⁻¹) +=_0,0(xy⁻¹))dy|^p dx1.