Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

1 Introduction

Throughout this article, let Rⁿ (n ≥ 2) be the n-dimensional Euclidean space and Sⁿ⁻¹ be the unit sphere in Rⁿ equipped with the normalized Lebesgue surface measure dσ = dσ(·). Let α₁, α₂,…,α_n be fixed real numbers in the interval [1, ∞). Define the function H: Rⁿ × R⁺ → R by H(x,ρ)=∑i=1nxi2ρ2αi with x = (x₁, x₂,…,x_n) ∈ Rⁿ. Then, for each fixed x ∈ Rⁿ, the function H(x, ρ) is a strictly decreasing function in ρ > 0. We denote the unique solution of the equation H(x, ρ) = 1 by ρ = ρ(x). Fabes and Riviére showed in ref. [1] that (Rⁿ, ρ) is a metric space, which is known by the mixed homogeneity space related to {αi}i=1n.

For ρ > 0, let A_ρ be the diagonal n × n matrix:

The change of variables related to the space (Rⁿ, ρ) is given by the transformation

Hence, dx = ρ^α−1J(x′)dρdσ(x′), where ρ^α−1J(x′) is the Jacobian of the above transforms,

The authors of ref. [1] showed that J(x′) is a C∞(Sn−1) function, and there exists a constant L, such that 1 ≤ J(x′) ≤ L.

For a suitable mapping Φ: Rⁿ → Rⁿ, we define the generalized parabolic parametric Marcinkiewicz integral operators ℳΩ,h,Φ,λ(r), initially for C0∞ functions on Rⁿ, by

where r > 1; λ = τ + σi (τ, σ ∈ R with τ > 0); h: R⁺ → C is a measurable function; and Ω is a real valued function on Rⁿ, integrable on Sⁿ⁻¹ and satisfies the conditions.

We point out if α₁ = ⋯ = α_n = 1, then we have α = n, ρ(x) = |x| and (Rⁿ,ρ) = (Rⁿ,|·|). In this case, ℳΩ,h,Φ,λ(r) is denoted by ℳΩ,h,Φ,λ(r). Also, when Φ(u) = u, h = 1, and r = 2, then the operator ℳΩ,h,Φ,λ(r,c), denote by ℳΩ,λ, reduces to the classical parametric Marcinkiewicz integral operator. Historically, the operator ℳΩ,λ was introduced by Stein [2] and proved the L^p (1 < p ≤ 2) boundedness of ℳΩ,1 provided that Ω ∈ Lip_α(Sⁿ⁻¹) with 0 < α ≤ 1. Subsequently, this result was investigated and improved by many researchers (see, for example, refs. [3,4,5,6]). The study of the boundedness of the operator ℳΩ,λ was performed by Hörmander [7]. As a matter of fact, he showed that if λ > 0 and Ω ∈ Lip_α(Sⁿ⁻¹) with α > 0, then L^p(Rⁿ) (1 < p < ∞) boundedness of ℳΩ,λ is satisfied. Later on, the study of the operator ℳΩ,h,Φ,λ(2,c) under very various conditions on the kernels has been considered by many authors. For more information about the importance and the recent advances on the study of such operators, we refer the readers to refs. [8,9,10,11,12,13,14,15], as well as ref. [16], and the references therein.

Conversely, there has been a considerable amount of mathematicians with respect to the study of the boundedness of the generalized parametric Marcinkiewicz integrals ℳΩ,h,Φ,λ(r,c). This operator was first introduced by Chen et al. [17] and showed that whenever Φ(u) = u, h ≡ 1, and Ω ∈ L^q(Sⁿ⁻¹) for some q > 1, then a positive constant C exists such that

holds for all 1 < p, r < ∞, where f belongs to the homogeneous Triebel-Lizorkin space Ḟp,rα(Rn). Afterward, Le [18] improved the aforementioned result. Precisely, he established the inequality (1.3) for all p, r ∈ (1,∞) under the conditions that Φ(u) = u, Ω ∈ L(log L)(Sⁿ⁻¹) and h∈Δmax{r′,2}(R+). For the significance and recent advances on the study of such operators, readers may refer to [16,19,20,21,22,23].

Although many problems concerning the boundedness of the operator ℳΩ,h,Φ,λ(r,c) remain open, the investigation to verify the boundedness of the parametric Marcinkiewicz operators with mixed homogeneity has been started.

Again when Φ(u) = u, λ = h = 1, and r = 2, then the operator ℳΩ,h,Φ,λ(r) recovers the classical parabolic Marcinkiewicz integral operator, denoted by μ_Ω, which was introduced by Ding et al. [24]. In particular, Ding et al. [24] proved that the parabolic Littlewood-Paley operator μ_Ω is of type (p,p) for all p ∈ (1,∞) provided that Ω ∈ L^q(Sⁿ⁻¹) for q > 1. Subsequently, the study of the L^p boundedness of ℳΩ,h,Φ,λ(2) under various conditions on the kernel functions has been carried out by many researchers (see, for example, refs. [25,26,27,28,29,30]).

Let us recall the definition of the Triebel-Lizorkin spaces. For 1 < p, r < ∞ and α ∈ R, the homogeneous Triebel-Lizorkin space Ḟp,rα(Rn) is defined by

where S′ denotes the tempered distributions class on Rⁿ, Λk^(ξ)=Γ(2−kξ) for k ∈ Z, and Γ∈C0∞(Rn) is a radial function satisfying the following conditions:

(i) 0 ≤ Γ ≤ 1;

(ii) suppΓ⊂{ξ:12≤|ξ|≤2};

(iii) Γ(ξ)≥c>0if35≤|ξ|≤53;

(iv) ∑j∈ZΓ(2−jξ)=1(ξ≠0).

The following properties of the Triebel-Lizorkin space are well known (for more details, see ref. [31]).

(a) S(Rn) is dense in Ḟp,rα(Rn);

(b) Ḟp,20(Rn)=Lp(Rn)for1<p<∞;

(c) Ḟp,r1α(Rn)⊂Ḟp,r2α(Rn)ifr1<r2;

(d) (Ḟp,rα(Rn))⁎=Ḟp′,r′−α(Rn).

Let Δ_γ(R⁺) (for γ ≥ 1) denote the collection of all measurable functions h:[0,∞) → C, satisfying

Also, let Nγ(R+) denote the set of all measurable functions h: R⁺ → C that satisfy the condition

where dk(h)=supj∈Z2−j|E(j,k)| with E(j,1) = {ρ ∈ (2^j,2^j+1]:|h(ρ)| ≤ 2} and E(j,k) = {ρ ∈ (2^j,2^j+1]:2^k−1 < |h(ρ)| ≤ 2^k} for k ≥ 2.

It is clear that Δγ(R+)=L∞(R+)⊂Δγ1(R+)⊂Δγ2(R+)for1<γ2<γ1<∞ and Δγ(R+)⊂Nβ(R+)foranyβ>0and1<γ<∞.

In this study, the class F denoted the set of all positive, increasing C1 functions ϕ:(0,∞) → R⁺ satisfying the following conditions:

(i) tϕ′(t) ≥ C_ϕϕ(t) for all t > 0; and

(ii) ϕ(2t) ≤ c_ϕϕ(t) for all t > 0,where C_ϕ, c_ϕ are independent of t. There are many model examples for the class F such as t^d with d > 0, t^ι(ln(1 + t)^κ) with ι, κ > 0, real-valued polynomials P on R with positive coefficients and P(0) = 0, and so on.

Let us recall some useful spaces related to our work. For κ > 0, the space L(log L)^κ(Sⁿ⁻¹) is denoted to the set of all measurable functions Ω that satisfies

The block space that was introduced in ref. [32] is denoted by Bq(0,ν)(Sn−1) (for ν > −1 and q > 1).

The main results of this paper are formulated as follows:

By the conclusions from Theorems 1.1 and 1.2 and following the same extrapolation arguments used in refs. [9,20,29,33,34], we have the following:

(i) IfΩ∈Bq(0,1r−1)(Sn−1)forsomeq>1, then

(ii) If Ω ∈ L(log L)^1/r(Sⁿ⁻¹), then

(i) IfΩ∈Bq(0,0)(Sn−1)forsomeq>1, then

(ii) If Ω ∈ L(log L)(Sⁿ⁻¹), then

It is worth mentioning to the following remark related to our results and their optimality.

(2) Walsh [4] proved that ℳΩ,1 is bounded on L²(Rⁿ) whenever Ω ∈ L(log L)^1/2(Sⁿ⁻¹). Furthermore, he showed that the condition Ω ∈ L(log L)^1/2(Sⁿ⁻¹) is optimal in the sense that the operator ℳΩ,1 may lose the L² boundedness if Ω is assumed to be in the space Ω ∈ L(log L)^ε(Sⁿ⁻¹) for some 0 < ε < 1/2.

(3) If Φ(u) = u, then, Al-Qassem et al. [20] established the boundedness of the parametric Marcinkiewicz integral operator ℳΩ,h,Φ,λ(r,c) under the same our conditions on Ω, h, and r.

(4) The L^p boundedness of the parametric Marcinkiewicz operators with mixed homogeneity ℳΩ,h,Φ,λ(2) was satisfied [29] only when h∈N1/2(R+), Ω ∈ L(log L)^1/2(Sⁿ⁻¹), and Φ is given as in Theorem 1.1.

Here and henceforth, the letter C denotes a positive constant that may be different at different occurrences and independent of the essential variables.

2 Some notations and lemmas

In this section, we give some lemmas, which we shall need in the proof of the main results. Let N=max1≤j≤ndeg(Pj). For 1 ≤ s ≤ N and 1 ≤ l ≤ n, let Pl(s)(t)=∑i=1sci,lti and P(s)(t)=(P1(s)(t),…,Pn(s)(t)). Set P⁽⁰⁾(t) = 0, Pl(t)=∑i=1Nci,lti with 1 ≤ l ≤ n and Φs(u)=(P1(s)(ϕ(ρ(u)))u1′,P2(s)(ϕ(ρ(u)))u2′,…,Pn(s)(ϕ(ρ(u)))un′).

Let θ ≥ 2. For a suitable measurable function h: R⁺ → C, a suitable function ϕ: R⁺ → R, and Ω: Sⁿ⁻¹ → R, we define the family of measures {σΩ,ϕ,h,ts:=σh,ts:t∈R+,1≤s≤N} and the corresponding maximal operators σh,s⁎ and M_h,θ,s on Rⁿ by

and

where

We shall need the following lemma from ref. [29].

By using Lemma 2.2 from [29], we directly obtain the following lemma.

To prove Theorem 1.1, we employ the next lemmas with arguments similar to those in refs. [20] and [29].