Optimal boundedness of central oscillating multipliers on compact Lie groups

Fefferman-Stein, Wainger and Sjölin proved optimal H boundedness for certain oscillating multipliers on R. In this article, we prove an analogue of their result on a compact Lie group.


Introduction
Let G be a connected, simply connected, compact semisimple Lie group of dimension n.In this paper, we will study the H 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 where y is conjugate to the element exp ξ in a fixed maximal torus of G (the detailed definition can be found in the second section).Thus the operator T γ,β has the Fourier expansion The formulation of T γ,β is an analogue of the oscillating multiplier operator S γ,β (f ) on R d : It is well known that the operator S γ,β is bounded on H 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 S γ,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.Let G be a connected, simply connected, compact semisimple Lie group of dimension n.For 0 < β < 1, the operator T γ,β is bounded on L p (G) if and only if | 1  2 − 1 p |≤ γ nβ for all 1 < p < ∞.
In [5], we are able to extend Theorem 1.1 to H p (G) for 0 < p 0 < p ≤ 1.However, the method in [5] only allows us to obtain the result when p 0 is close to 1. Thus, the aim of this paper is to give a complete solution of Theorem 1.1 by establishing the following optimal H p boundedness for all p > 0.
Theorem 1.2.Let G be a connected, simply connected, compact semisimple Lie group of dimension n and 0 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 interpolation argument (see [2]).Define an analytic family of operators By the Plancherel theorem, it is easy to see that if (z) = 0. Thus, to complete the proof of the sufficiency part of Theorem 1.2, by the analytic interpolation theorem it suffices to show the following endpoint estimate.
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 notation A B to 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 constants c 1 and c 2 independent of all essential variables such that Acknowledgments.The first author is supported by the NSFC Grant 10931001, 10871173.

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 in G of dimension m.Let A be a system of positive roots for (g, τ ) so Card(A) = n−m 2 , and let δ = 1 2 a∈A a.Let | .| be the norm of g induced by the negative of the Killing form B on g C , the complexification of g.
for each H ∈ τ C .We let ., .and .denote the inner product and norm transferred from τ to hom C (τ, iR) by means of this canonical isomorphism.
Let N = {H ∈ τ : exp H = I}, where I is the identity in G.The weight lattice P is defined by P = {λ ∈ τ : λ, n ∈ 2πZ for any n ∈ N} with dominant weights defined by Λ = {λ ∈ P : λ, a ≥ 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 U λ has dimension and its associated character is χ λ (ξ) = w∈W (w) e i w(λ+δ),ξ w∈W e i wδ,ξ where ξ ∈ τ , W is the Weyl group, and (w) is the signature of w ∈ W .Any function f ∈ L 1 (G) has the Fourier series The oscillating multiplier where K z,β is a central kernel defined by and exp ξ ∈ T is conjugate to y.Let Q be a fixed fundamental domain of T and Up to sets of measure zero, {Q ν } is a sequence of mutually disjoint subsets in the Lie algebra τ .
Any element y ∈ G is conjugate to exactly one element in exp(Q).We denote y ∼ exp ξ if y is conjugate to exp ξ.

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]) An exceptional atom is an L ∞ function bounded by 1.In order to define a regular atom, one considers a faithful unitary representation Π of G such that Π(G) ≈ U (L, C).Then G can be identified as a submanifold in a real vector space V underlying End(C L ).A regular p-atom for 0 < p ≤ 1 is a function a(x) supported in a ball B(x 0 , ρ) such that where ℘ is any polynomial on V of degree less than or equal to for any integer where each a(x) is either a regular p-atom, or an exceptional atom.The "norm" f H p a is the infimum of all expressions ( |c k | p ) 1/p for which we have a representation f = c k a k .As we discussed in [5] (see also [3], [4]), to show that the operator T z,β is bounded on H p (G), it suffices to prove 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.
In this section, we introduce some notation in [8].Let B s be the set of all simple roots in A and let B L be the set of all largest roots.For θ ∈ Q, introduce the following sets When R is a large number, elements in I θ and J θ are independent.We define two facets and let F I,J be the affine subspace generated by F I,J so that A positive root γ is R−singular of type I at θ, 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 A positive root γ is R-singular of type II at θ, 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 some Both R−singular roots of types I and II are called singular roots.By this definition, it is easy to see that if γ is a positive non-singular root, then For a singular root α of type I, let S α be the orthogonal symmetry with respect to the hyperplane α = 0.For a singular root β of type II, let S β be the orthogonal symmetry with respect to the hyperplane α = 2π.Let W I,J be the group generated by This group is a finite subgroup of the affine Weyl group.Now we define Also, we write the root system where A s is the set of all singular (R−singular) roots and A ns is the set of all non-singular roots.Denote by µ R the number of singular roots and denote Thus The above definition of D(θ) and Γ (R) θ can be defined on the torus T itself.In fact, if x ∈ T , then x = exp θ for some θ in Q.We define 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 for all ξ in Γ (R) θ .

Derivatives on central functions.
Fixing a vector basis of g C , say Y 1 , Y 2 , . . ., Y n , we denote the element , where J = (j 1 , . . ., j n ).As J varies over all possible n−tuples, the {Y J } forms a basis of the complex universal enveloping algebra U (g) of g.Similarly, we fix a basis Θ 1 , . . ., Θ m of τ C and use the notation We can find the following two theorems in [8].
Theorem 2.1.Let p, q be positive integers.Assume that f is a C ∞ central function.Then for each I with |I| ≤ p and J with |J| ≤ q, there exists a constant C such that Theorem 2.2.Let p be a positive integer.Assume f (y) = d(y) −1 g(y) and that g is a C ∞ central function which is skew-invariant by the Weyl group.
For each I with |I| ≤ p, there exists a constant C such that

Estimates on the kernel K z,β , (z) > n
In this section, we denote γ = (z) and assume γ > n.First, we recall the following lemma, which is an easy modification of results in [12] or [17].
for all x ∈ supp(Φ), then for large |λ|, By the definition, it is easy to see where where c 1 and c 2 are two fixed positive numbers such that Since (z) = γ > n, the above summation is absolutely convergent.By the Poisson summation formula (see [11]) we obtain where For simplicity, we write Also, without loss of generality, we may use the Euclidean norm | .| instead of . .Thus, for ν ∈ N, Fix a small ρ > 0. Let ϕ(t) be a C ∞ function on (0, ∞) that satisfies and let ϕ ∞ (t) = ϕ(ρt), ϕ 0 (t) = 1 − ϕ(ρt).For each ν ∈ N, we write where Furthermore, for y ∼ exp ξ, we define four central kernels: Then, we decompose the kernel K z,β by ξ).We will give different estimates on these kernels. and Proof.For |ξ| ≥ 1000nρ 1−β , there is at least one coordinate ξ j satisfying On the other hand, if H lies on the support of ϕ ∞ we have By this observation, we perform integration by parts on H j −variable to obtain This shows (3.1).As a consequence, we obtain Thus, by the Weyl integral formula we have .We have, for any multi-index J and any positive integer L, Proof.It is easy to see that where P k is a homogeneous polynomial of degree k.Without loss of generality, we may write, for each k, The support condition of ψ implies that the phase function satisfies Thus the lemma follows by integration by parts N times for a suitably large N .
then we have Proof.As in the previous lemma, we need to show for each k, and Therefore, By polar coordinates, where S m−1 is the unit sphere in R m with the induced Lebesgue measure dσ(H ), and the phase function . By this observation, it is easy to see that after using integration by parts N times for a sufficiently large N , we have For the term L k,2 (ξ), it is easy to check that, in some direction H j , Again, performing integration by parts in the H j direction for sufficiently many times, we obtain For the term L k,3 (ξ), changing variables we have Recalling that the support of Γ 3 lies in the set without loss of generality, we may also assume that |ξ| is small such that We now have is a C ∞ function supported in the set S. By Lemma 3.1 we have (3.4) We now obtain the lemma by combining (3.2),(3.3)(3.4).
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 where exp ξ ∼ y and that there is a σ > 0 such that for all ξ ∈ Q, 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 then we have where y ∼ exp ξ.
Proof.By Theorem 2.1, without loss of generality, we may write By [22,Ch.4](or see [8]), we have , and J m−2 2 (t) is the Bessel function of order m−2 2 .An easy computation shows (see [5]), Thus, we obtain 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 is an analytic function, we may assume Using a derivative formula for the Bessel function (see [23]) we have , where ∞ 0 e it β e ±it|ξ| ϕ 0 (t)ψ(t)t −z+ n−1 2 −µ+k dt.

Lemma 3 . 5 .
Assume d(u, I) ≤ η.For any non-negative integer L and any multi-index M with |M | = q, we have