A Multiplier Theorem for Herz-Type Hardy Spaces Associated with the Dunkl Transform

Theorem 1. Let ℓ be the least integer greater than α + 1 and let m be a bounded C^ℓ-function on ℝ∖{0} which satisfies the Hörmander condition M_α(2, ℓ) as follows:

Theorem 2. Let 0 < p ≤ 1, β = (1/p) − (1/2), and ℓ be an integer greater than 2(α + 1)β. If m satisfies the Hörmander condition M_α(2, ℓ), then the operator $$ is bounded on $$.

Lemma 3. Let ℓ be the least integer greater than α + 1. If m satisfies the Hörmander condition M_α(2, ℓ), then there is a constant C independent of m, such that if q = 1 or s − ℓ + α + 1 < (α + 1)/q ≤ α + 1, the following inequality holds:

Notation. For all x, y, z ∈ ℝ, we put

• (i)

  For x, y ∈ ℝ and a continuous function f on ℝ, we have

  $$ ()

• (ii)

  For all x ∈ ℝ, the operator τ_x can be extended to L^p(μ_α)  (p ≥ 1), and for f ∈ L^p(μ_α), we have

  $$ ()

• (iii)

  For all x, λ ∈ ℝ and f ∈ L¹(μ_α), we have

  $$ ()

Let p, q, r ∈ [1, ∞] such that 1/p + 1/q = 1/r + 1. The convolution product of f ∈ L^p(μ_α) and g ∈ L^q(μ_α) is defined by

Definition 4. Let β ∈ ℝ, p ∈ ]0, ∞[, and q ∈ [1, ∞].

• (i)

  The homogeneous weighted Herz space $$ is the space constituted by all functions $$, such that

  $$ ()
  where χ_k is the characteristic function of A_k = {x ∈ ℝ/2^k−1 ≤ |x| ≤ 2^k}.

• (ii)

  The nonhomogeneous weighted Herz space $$ is defined, as usual, by $$. Moreover, $$.

Note that $$.

Definition 5. Let β ∈ ℝ, p ∈ ]0, ∞], and q ∈ ]1, ∞]. The Herz-type Hardy space $$ is the space of distributions f ∈ S′(ℝ) such that $$. Moreover, we define

Definition 6. Let q ∈ ]1, ∞] and β ≥ 1 − 1/q. A measurable function a on ℝ is called a (central) (β, q, s)-atom if it satisfies the following:

• (i)

  supp  (a)⊂[−r, r], for some r > 0,

• (ii)

  ∥a∥_q,α ≤ r^−2(α+1)β,

• (iii)

  ∫_ℝ a(x)x^kdμ_α(x) = 0, k = 0,1, …, s, where s = [2(α + 1)(β − 1 + 1/q)] and [·] denotes the integer part function.

Theorem 7. Let 0 < p ≤ 1 < q ≤ ∞ and β ≥ 1 − 1/q  . Then, $$ if and only if, for all j ∈ ℕ∖{0}, there exist a (β, q, s)-atom a_j and λ_j ∈ ℂ, such that $$ and $$. Moreover,

Definition 8. For 0 < p ≤ 1. Set s ≥ [2(α + 1)(1/p − 1)], ε > s/2(α + 1), $$, and b = 1/2 + ε. A central (p, s, ε)-molecule is a function M ∈ L²(μ_α) satisfying the following:

• (i)

  M(x)|x|^2(α+1)b ∈ L²(μ_α),

• (ii)

  $$,

• (iii)

  ∫_ℝ M(x)x^kdμ_α(x) = 0, k = 0,1, …, s.

Proposition 9. Let (p, s, ε) be the triple cited in the previous definition. Every central (p, s, ε)-molecule M belongs to $$ and $$, where the constant C is independent of M.

Proof. Let M be a central (p, s, ε)-molecule and suppose that $$. In the general case, letting $$, we have $$.

Let E₀ = {|x| ≤ 1}, E_k = {2^k−1 < |x| ≤ 2^k}, and M_k = Mχ_k, k = 1, 2, 3, …, where χ_k is the characteristic function of E_k. For each k, there exists a unique polynomial Q_k, of degree at most s, such that if P_k = Q_kχ_k; then

Using some ideas in [2], we can show that each (M_k − P_k) is a multiple of a central (β, 2, s)-atom with a sequence of coefficients in l^p. We also show that the sum $$ can be written as an infinite linear combination of central (β, ∞, s)-atom with a sequence of coefficients in l^p. Since a (β, ∞, s)-atom is also (β, 2, s)-atom, hence,

Lemma 10. Let a be a (β, 2, s)-atom. For all integer 0 ≤ k ≤ s and every 1 ≤ u ≤ ∞, there exists a constant C independent of a, such that

Proof. (i) Let a be a (β, 2, s)-atom. Consider that r > 0 such that   supp (a)⊂[−r, r] and that ∥a∥_2,α ≤ r^−2(α+1)β. From (9), (iii) of Definition (19), and the estimate for the remainder in Taylors’ formula, it follows that

Corollary 11. Let 0 < p ≤ 1. Then, the generalized Hilbert transform H_α defined by

Proof. From Proposition 3.6 in [3], the generalized Hilbert transform H_α is a multiplier operator $$ with m(ξ) = −sign (ξ); then the proof of the corollary follows from Theorem 2.