Parabolic Marcinkiewicz integrals on product spaces and extrapolation

1 Introduction and the main result

Let R^N (N = n or m), N ≥ 2 be the N-dimensional Euclidean space, and let S^N−1 be the unit sphere in R^N equipped with the normalized Lebesgue surface measure dσ = dσ Also, let pʹ denote the exponent conjugate to p; that is 1/p+1/pʹ = 1.

Let α_i(i = 1,2,…, N) be fixed real numbers such that α_i ≥ 1. Define the function F: R^N × R⁺ → R by F(z,ρ)=∑i=1Nzi2ρ2αi. It is clear that for each fixed z ∈ R^N, the function F(z, ρ) is a decreasing functions in ρ > 0.

The unique solution of the equation F(z, ρ) = 1 is denoted by ρ(z). In [1], Fabes and Riviére showed that ρ(z) is a metric space in R^N, and (R^N ρ) is called the mixed homogeneity space related to {αi}i=0N.

For λ > 0, let A_λ be the diagonal N × N matrix

The change of variables related to the space (R^N, ρ) is given by the transformation x1=ρα1cosϑ1⋯cosϑN−2cosϑN−1

x2=ρα2cosϑ1⋯cosϑN−2sinϑN−1

⋮

xN−1=ραN−1cosϑ1sinϑ2,

xN=ραNsinϑ1

where x =(x₁x₂ x_N) ∈ R^N Thus, dx=ρα−1JN(x′)dρdσ(x′), where ρ^α−1J_N(xʹ) is the Jacobian of the above transforms,

It was shown in [1] that J_N(xʹ) is a C^∞(S^N−1 function in the variable xʹ ∈ S^N−1, and that a real constant L^N ≥ 1 exists so that 1 ≤ J_N (xʹ) ≤ L_N.

Let K_Ω,ρ(u) = Ω(u)ρ(u)^1−α where #x03A9; is a real valued and measurable function on R^N with Ω ∈ L¹ (S^N−1) that satisfies the conditions

The parabolic Marcinkiewicz integral μ_Ω, which was introduced by Ding, Xue and Yabuta in [2], is defined by

where

In particular, the authors of [2] proved that the parabolic Littlewood-Paley operator μ_Ω is bounded for p ∈ (1, ∞) provided that Ω ∈ L^q (S^N−1) for q > 1. Subsequently, the study of the L^p boundedness of μ_Ω under various conditions on the function Ω has been studied by many authors (see for example [3–7]).

We point out that the class of the operators μ_Ω is related to the class of the parabolic singular integral operators

The class of the operators T_Ω belongs to the class of singular Radon transforms, which was studied by by many mathematicians (we refer the readers, in particular, to [1, 8]).

Although some open problems related to the boundedness of parabolic Marcinklewicz integral in the one- parameter setting remain open, the investigation of L^p estimates of the Marcinkiewicz integral on product spaces has been started (see for example [9, 10].)

Let α_i, β_j be fixed real numbers with α_i, β_j ≥ 1 (i = 1,2,…, n and j = 1, 2, …, m). For τ₁ = a₁ + ib₁, τ₂ = a₂ + ib₂ (a₁, b₁, a₂, b₂ ∈ Rwitha₁, a₂ > 0), let KΩ,hρ1,ρ2(x,y)=Ω(x,y)h(ρ1(x),ρ2(y))ρ1(x)τ1−αρ2(y)τ2−β, where α=∑i=1nαi,β=∑j=1mβj,h is a measurable function on R⁺ × R⁺, and Ω is a real valued and measurable function on Rⁿ × R^m with Ω ∈ L¹(S^N−1 × S^m−1) satisfying the conditions

We define the parabolic Marcinkiewicz integral operator MΩ,hρ1,ρ2 for f ∈ S(Rⁿ × R^m) by

where

By specializing to the case h = 1 and τ₁ = τ₂ = 1, plus considering the cases α₁ = … = α_n = 1 and β₁ = … = β_m = 1, we obtain that ρ₁(u) = |u|, ρ₂(v) = |v|, α = n, β = m, and (Rⁿ × R^m, ρ₁, ρ₂ = (Rⁿ × R^m, |·|, |·|). In this case MΩ,hρ1,ρ2 (denoted by MΩ is just the classical Marcinkiewicz integral on product domains, which was studied by many mathematicians. For instance, the author of [11] gave the L² boundedness of MΩ if Ω∈L(log⁡L)2(Sn−1×Sm−1). Later, it was verified in [12] that MΩ is bounded for all 1 < p < ∞ provided that Ω∈L(log⁡L)2(Sn−1×Sm−1). This result was improved (for p = 2) in [13] in which the author established that MΩ is bounded on L²(Rⁿ × R^m) for all Ω∈L(log⁡L)(Sn−1×Sm−1). Recently, Al-Qaseem et al found in [14] that the boundedness of MΩ is obtained under the condition Ω∈L(log⁡L)(Sn−1×Sm−1) for 1 < p < ∞. Furthermore, they proved that the exponent 1 is the best possible.

Al-Qassem in [15], found that MΩ,h is bounded on Rⁿ × R^m(1 < p < ∞) provided that h is a bounded radial function and Ω is a function in certain block space Bq(0,0)(Sn−1×Sm−1) for q > 1. He also established the optimality of the condition in the sense that the space Bq(0,0)(Sn−1×Sm−1) cannot be replaced by Bq(0,ε)(Sn−1×Sm−1) for any −1 < ε < 0.

On the other hand, Al-Salman in [9] extended the result in [14]. In fact, he proved that MΩ,1ρ1,ρ2 is bounded in L^p(Rⁿ × R^m)(1 < p < ∞) provided that Ω∈L(log⁡L)(Sn−1×Sm−1).

We point out that the parabolic singular integral operator on product domains of the form

is being under investigation by one of our graduate students. In fact, he shall prove the L^p boundedness of T_{Ω, h} when Ω∈Lq(Sn−1×Sm−1) for some q > 1 and h ∈ Δ_γ(R⁺ × R⁺) for some γ > 1, where Δ_γ(R⁺ × R⁺) (for γ > 1) denotes the collection of all measurable functions h: R⁺ × R⁺ → C satisfying

In view of the result in [4]; that is the parabolic Marcinkiewicz integral in the one-parameter setting, defined as in (3), is bounded on L^p(Rⁿ) and the results concerning the classical Marcinkiewicz in the two-parameter setting, a question arises naturally Does the L^p boundedness of the operators MΩ,hρ1,ρ2 hold under the conditions when Ω belongs to the space L(log L)(Sⁿ⁻¹} × S^m−1) or whether it belongs to the block space Bq(0,0)(Sn−1×Sm−1) and h ∈ Δ_γ(R⁺ × R⁺) for some γ, q > 1?

We shall obtain an affirmative answer to this question, as described in the following theorems.

The conclusion from Theorem 1.1 and the application of an extrapolation method as in [16, 17] lead to the following theorem.

Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but is independent of the essential variables.

2 Preparation

In this section, we give some auxiliary lemmas used in the sequel. We shall recall the following lemma due to Ricci and Stem.

Let θ ≥ 2. For a measurable function h: R⁺ × R⁺ → C and Ω: Sⁿ⁻¹ × S^m−1 → R, we define the family of measures σ_{Ω, h, t, s}: t, s ∈ R⁺} and its corresponding maximal operators σΩ,h,t∗ and M_{Ω, h θ} on Rⁿ × R^m by

and

where |σ_{Ω, h, t, s}| is defined in the same way as σ_{Ω, h, t, s}, but with replacing h by |h| and Ω by |Ω|. We write ‖σ_{t, s}‖ for the total variation of σ_{t, s} and a^±r = min{a^r, a^−r}.

In order to obtain Theorem 1.1, we need to prove the following lemmas.