Show Summary Details
More options …

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …

# Parabolic Marcinkiewicz integrals on product spaces and extrapolation

Mohammed Ali
• Corresponding author
• Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
• Email
• Other articles by this author:
/ Mohammed Al-Dolat
Published Online: 2016-09-26 | DOI: https://doi.org/10.1515/math-2016-0061

## Abstract

In this paper, we study the the parabolic Marcinkiewicz integral ${\mathcal{M}}_{\mathrm{\Omega },h}^{{\rho }_{1,}{\rho }_{2}}$ on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.

MSC 2010: 40B20; 40B15; 40B25

## 1 Introduction and the main result

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

Let αi(i = 1,2,…, N) be fixed real numbers such that αi ≥ 1. Define the function F: RN × R+R by $F\left(z,\rho \right)=\sum _{i=1}^{N}\frac{{z}_{i}^{2}}{{\rho }^{2{\alpha }_{i}}}$. It is clear that for each fixed zRN, 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 RN, and (RN ρ) is called the mixed homogeneity space related to $\left\{{\alpha }_{i}{\right\}}_{i=0}^{N}$.

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

$Aλ=diag{λα1,λα2,⋯,λαN}=λαl0⋱0λαN.$

The change of variables related to the space (RN, ρ) is given by the transformation ${x}_{1}={\rho }^{{\alpha }_{1}}\mathrm{cos}{\vartheta }_{1}\cdots \mathrm{cos}{\vartheta }_{N-2}\mathrm{cos}{\vartheta }_{N-1}$

${x}_{2}={\rho }^{{\alpha }_{2}}\mathrm{cos}{\vartheta }_{1}\cdots \mathrm{cos}{\vartheta }_{N-2}\mathrm{sin}{\vartheta }_{N-1}$

${x}_{N-1}={\rho }^{{\alpha }_{N-1}}\mathrm{cos}{\vartheta }_{1}\mathrm{sin}{\vartheta }_{2}$,

${x}_{N}={\rho }^{{\alpha }_{N}}\mathrm{sin}{\vartheta }_{1}$

where x =(x1 x2 xN) ∈ RN Thus, $dx={\rho }^{\alpha -1}{J}_{N}\left({x}^{\mathrm{\prime }}\right)d\rho d\sigma \left({x}^{\mathrm{\prime }}\right)$, where ρα−1JN(xʹ) is the Jacobian of the above transforms,

$α=∑k=1NαkandJN=∑k=1Nαk(xk′)2.$

It was shown in [1] that JN(xʹ) is a C(SN−1 function in the variable xʹ ∈ SN−1, and that a real constant LN ≥ 1 exists so that 1 ≤ JN (xʹ) ≤ LN.

Let KΩ,ρ(u) = Ω(u)ρ(u)1−α where #x03A9; is a real valued and measurable function on RN with Ω ∈ L1 (SN−1) that satisfies the conditions

$Ω(Aλz)=Ω(z),∀λ>0and∫sN−1Ω(z′)JN(z′)dσ(z′)=0.$

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

$μΩf(z)=∫0∞|FΩ,t(z)|2dtt31/2,$

where

$FΩ,t(z)=∫ρ(u)<_tKΩ,ρ(u)f(z−u)du.$

In particular, the authors of [2] proved that the parabolic Littlewood-Paley operator μΩ is bounded for p ∈ (1, ∞) provided that Ω ∈ Lq (SN−1) for q > 1. Subsequently, the study of the Lp boundedness of μΩ under various conditions on the function Ω has been studied by many authors (see for example [37]).

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

$TΩf(z)=p.v.∫RNΩ(u)ρ(u)αf(z−u)du.$

The class of the operators TΩ 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 Lp estimates of the Marcinkiewicz integral on product spaces has been started (see for example [9, 10].)

Let αi, βj be fixed real numbers with αi, βj ≥ 1 (i = 1,2,…, n and j = 1, 2, …, m). For τ1 = a1 + ib1, τ2 = a2 + ib2 (a1, b1, a2, b2R with a1, a2 > 0), let ${K}_{\mathrm{\Omega },h}^{{\rho }_{1},{\rho }_{2}}\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)=\mathrm{\Omega }\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)h\left({\rho }_{1}\left(x\right),\phantom{\rule{thickmathspace}{0ex}}{\rho }_{2}\left(y\right)\right){\rho }_{1}\left(x{\right)}^{{\tau }_{1}-\alpha }{\rho }_{2}\left(y{\right)}^{{\tau }_{2}-\beta }$, where $\alpha =\sum _{i=1}^{n}{\alpha }_{i},\beta =\sum _{j=1}^{m}{\beta }_{j},h$ is a measurable function on R+ × R+, and Ω is a real valued and measurable function on Rn × Rm with Ω ∈ L1(SN−1 × Sm−1) satisfying the conditions

$Ω(Aλ1x,Aλ2y)=Ω(x,y),∀λ1,λ2>0,$(1)$∫sn−1Ω(x′,.)Jn(x′)dσ(x′)=∫sm−1Ω(.,y′)Jm(y′)dσ(y′)=0.$(2)

We define the parabolic Marcinkiewicz integral operator ${\mathcal{M}}_{\mathrm{\Omega },h}^{{\rho }_{1,}{\rho }_{{}_{2}}}$ for fS(Rn × Rm) by

$MΩ,hρ1,ρ2f(x,y)=∫0∞∫0∞|Fts(x,y)|2dtdsts1/2,$(3)

where

$Ft,s(x,y)=1tτ1sτ2∫ρ1(u)<_t∫ρ2(v)sKΩhρ1,ρ2,(u,v)f(x−u,y−v)dudv.$

By specializing to the case h = 1 and τ1 = τ2 = 1, plus considering the cases α1 = … = αn = 1 and β1 = … = βm = 1, we obtain that ρ1(u) = |u|, ρ2(v) = |v|, α = n, β = m, and (Rn × Rm, ρ1, ρ2 = (Rn × Rm, |·|, |·|). In this case ${\mathcal{M}}_{\mathrm{\Omega },h}^{{\rho }_{1},{\rho }_{2}}$ (denoted by ${\mathcal{M}}_{\mathrm{\Omega }}$ 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 ${\mathcal{M}}_{\mathrm{\Omega }}$ if $\mathrm{\Omega }\in L\left(\mathrm{log}L{\right)}^{2}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$. Later, it was verified in [12] that ${\mathcal{M}}_{\mathrm{\Omega }}$ is bounded for all 1 < p < ∞ provided that $\mathrm{\Omega }\in L\left(\mathrm{log}L{\right)}^{2}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$. This result was improved (for p = 2) in [13] in which the author established that ${\mathcal{M}}_{\mathrm{\Omega }}$ is bounded on L2(Rn × Rm) for all $\mathrm{\Omega }\in L\left(\mathrm{log}L\right)\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$. Recently, Al-Qaseem et al found in [14] that the boundedness of ${\mathcal{M}}_{\mathrm{\Omega }}$ is obtained under the condition $\mathrm{\Omega }\in L\left(\mathrm{log}L\right)\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ for 1 < p < ∞. Furthermore, they proved that the exponent 1 is the best possible.

Al-Qassem in [15], found that ${{\mathcal{M}}_{\mathrm{\Omega }}}_{,h}$ is bounded on Rn × Rm(1 < p < ∞) provided that h is a bounded radial function and Ω is a function in certain block space ${B}_{q}^{\left(0,0\right)}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ for q > 1. He also established the optimality of the condition in the sense that the space ${B}_{q}^{\left(0,0\right)}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ cannot be replaced by ${B}_{q}^{\left(0,\epsilon \right)}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ for any −1 < ε < 0.

On the other hand, Al-Salman in [9] extended the result in [14]. In fact, he proved that ${\mathcal{M}}_{\mathrm{\Omega },1}^{{\rho }_{1,}{\rho }_{{}_{2}}}$ is bounded in Lp(Rn × Rm)(1 < p < ∞) provided that $\mathrm{\Omega }\in L\left(\mathrm{log}L\right)\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$.

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

$TΩ,hf(x,y)=p.v.∫Rn×RmΩ(u,v)h(ρ1(u),ρ2(v))ρ1(u)αρ2(v)βf(x−u,y−v)dudv$

is being under investigation by one of our graduate students. In fact, he shall prove the Lp boundedness of TΩ, h when $\mathrm{\Omega }\in {L}^{q}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ for some q > 1 and h ∈ Δγ(R+ × R+) for some γ > 1, where Δγ(R+ × R+) (for γ > 1) denotes the collection of all measurable functions h: R+ × R+C satisfying

$supR1,R2>01R1R2∫0R1∫0R2|h(t,s)|γdtds1/γ<∞.$

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 Lp(Rn) and the results concerning the classical Marcinkiewicz in the two-parameter setting, a question arises naturally Does the Lp boundedness of the operators ${\mathcal{M}}_{\mathrm{\Omega },h}^{{\rho }_{1},{\rho }_{2}}$ hold under the conditions when Ω belongs to the space L(log L)(Sn−1} × Sm−1) or whether it belongs to the block space ${B}_{q}^{\left(0,0\right)}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ and h ∈ Δγ(R+ × R+) for some γ, q > 1?

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

Let 003E ∈ Lq(Sn−1 × Sm−1) for some 1 < q ≤ 2 and h ∈ Δγ(R+ × R+) for some γ > 1. Then there exists a constant Cp (independent of Ω, h, γ, and q) such that

$MΩ,hρ1,ρ2(f)Lp(Rn×Rm)≤CpA(γ)q−1hΔγ(R+×R+)ΩLq(Sn−1×Sm−1)fLp(Rn×Rm)$

for |1/p−1/2| < min{1/2, 1γʹ}, where $A\left(\gamma \right)=\left\{\begin{array}{ll}\gamma & if\phantom{\rule{thickmathspace}{0ex}}\gamma >2,\\ {\left(\gamma -1\right)}^{-1}& if\phantom{\rule{thickmathspace}{0ex}}1<\gamma \le 2\end{array}\right\$.

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

Suppose that Ω satisfies (1)–(2) and h ∈ Δγ(R+ × R+) for some γ > 1.

1. If $\mathrm{\Omega }\in {B}_{q}^{\left(0,0\right)}\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$ for some q > 1, then

$MΩ,hρ1,ρ2(f)Lp(Rn×Rm)≤CpA(γ)hΔγ(R+×R+)fLp(Rn×Rm)1+ΩBq(0.0)(sn−1×sm−1)$

for |1/p−1/2| < min{1/2, 1/γʹ};

2. if $\mathrm{\Omega }\in L\left(\mathrm{log}L\right)\left({\mathrm{S}}^{n-1}×{\mathrm{S}}^{m-1}\right)$, then

$MΩ,hρ1,ρ2(f)Lp(Rn×Rm)≤CpA(γ)hΔγ(R+×R+)fLp(Rn×Rm)(1+ΩL(logL)(Sn−1×Sm−1))$

for |1/p−1/2| < min{1/2, 1/γʹ}.

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.

([18]). Suppose that ${\lambda }_{i}^{\mathrm{\prime }}s$ and ${\alpha }_{i}^{\mathrm{\prime }}s$ are fixed real numbers, and $\mathrm{\Gamma }\left(t\right)=\left({\lambda }_{1}{t}^{{\alpha }_{1}},\phantom{\rule{thickmathspace}{0ex}}\cdots ,\phantom{\rule{thickmathspace}{0ex}}{\lambda }_{N}{t}^{{\alpha }_{N}}\right)$ is a function from R+ to RN For suitable f, let ${\mathcal{M}}_{\tau }$ be the maximal operator defined on RN by

$MΓf(x)=suph>0⁡1h|∫0hf(x−Γ(t)dt|$

for xRN Then for 1 < p ≤ ∞, there exists a constant Cp > 0 such that

$MΓfLp(RN)≤CpfLp(RN).$

The constant Cp is independent of ${\lambda }_{i}^{\mathrm{\prime }}s$ and f.

Suppose that ${a}_{i}^{\mathrm{\prime }}s,\phantom{\rule{thickmathspace}{0ex}}{b}_{i}^{\mathrm{\prime }}s,\phantom{\rule{thickmathspace}{0ex}}{\alpha }_{i}^{\mathrm{\prime }}s$, and ${\beta }_{i}^{\mathrm{\prime }}s$ are fixed real numbers. Let $\mathrm{\Gamma }\left(t\right)=\left({a}_{1}{t}^{{\alpha }_{1}},\phantom{\rule{thickmathspace}{0ex}}\cdots ,\phantom{\rule{thickmathspace}{0ex}}{a}_{n}{t}^{{\alpha }_{n}}\right)$ and $\mathrm{\Lambda }\left(t\right)=\left({b}_{1}{t}^{{\beta }_{1}},\phantom{\rule{thickmathspace}{0ex}}\cdots ,\phantom{\rule{thickmathspace}{0ex}}{b}_{m}{t}^{{\beta }_{m}}\right)$, and let ${\mathcal{M}}_{\mathrm{\Gamma },\mathrm{\Lambda }}$ be the maximal operator defined on Rn × Rm by

$MΓ,Λf(x,y)=suph1,h2>0⁡1h1h2|∫0h1∫0h2f(x−Γ(t),y−Λ(r))dtdr|$

for (x, y) ∈ Rn × Rm Then for 1 < p ≤ ∞, there exists a constant Cp > 0 (independent of ${a}_{i}^{\mathrm{\prime }}s,\phantom{\rule{thickmathspace}{0ex}}{b}_{i}^{\mathrm{\prime }}s$ and f) such that

$MΓ,ΛfLp(Rn×Rm)≤CpfLp(Rn×Rm)$

for all fLp Rn × Rm).

It is easy to prove this lemma by using Lemma 2.1 and the inequality ${\mathcal{M}}_{\mathrm{\Gamma },\mathrm{\Lambda }}f\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)\le {\mathcal{M}}_{\mathrm{\Lambda }}\circ {\mathcal{M}}_{\mathrm{\Gamma }}f\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)$, where ${\mathcal{M}}_{\mathrm{\Gamma }}f\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)={\mathcal{M}}_{\mathrm{\Gamma }}f\left(.,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}y\right)\left(x\right),\phantom{\rule{thickmathspace}{0ex}}{\mathcal{M}}_{\mathrm{\Lambda }}f\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)={\mathcal{M}}_{\mathrm{\Lambda }}f\left(x,\phantom{\rule{thickmathspace}{0ex}}\cdot \right)\left(y\right)$, and ∘ denotes the composition of operators.

([5]). Suppose that 0 ≤ v ≤ 1. Let m denote the distinct numbers ofi}. Then for u, ξ ∈ RN, we have

$∫12e−iAλu.ξdλλ≤C|u⋅ξ|−vm,$

where C is independent of u and ξ.

Let θ ≥ 2. For a measurable function h: R+ × R+C and Ω: Sn−1 × Sm−1R, we define the family of measures σΩ, h, t, s: t, sR+} and its corresponding maximal operators ${\sigma }_{\mathrm{\Omega },h,t}^{\ast }$ and MΩ, h θ on Rn × Rm by

$∫Rn×RmfdσΩ,h,t,s=1tτ1sτ2∫1/2t≤ρ1(u)≤t∫1/2s≤ρ2(v)≤sf(u,v)h(ρ1(u),ρ2(v))Ω(u,v)ρ1(u)α−τ1ρ2(v)β−τ2dudv,$$σΩ,h∗f(x,y)=supt,s∈R+σΩ,h,t,s∗f(x,y),$

and

$MΩ,h,θf(x,y)=supi,j∈Z⁡∫θiθi+1∫θθj+1σΩ,h,t,s∗f(x,y)dtdsts,$

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

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

Suppose that Ω ∈ Lq(Sn−1 × Sm−1) for some q > 1 and satisfies the cancellation conditions (1)-(2). For t, s > 0, let

$Gt,s(ρ1,ρ2)=∫sn−1×sm−1e−i{Atρ1x⋅ξ+Asρ2y⋅η}Ω(x,y)Jn(x)Jm(y)dσ(x)dσ(y).$

Then there are constants C and w with $0 such that

$∫1/21∫1/21|Gt,s(ρ1,ρ2)|2dρ1dρ2ρ1ρ2≤C|Atξ|±wm1q′|Asη|±wm2q′ΩLq(Sn−1×Sm−1)2,$(4)

where m1, m2 denote the distinct numbers ofi}, {βj}, respectively

Let Ω∈ Lq(Sn-1× Sm-1) for some q > 1, h ∈ Δγ(R+× R+) for some γ > 1 and θ = 2q′ γ′ Then there are constants C and w (as in Lemma 2.4) such that

$σΩ,h,t,s≤ChΔγR+×R+ΩLq(sn−1×sm−1);$(9)

$∫θiθi+1∫θθj+1σ^Ω,h,t,s(ξ,η)2dtdsts≤Cln2(θ)hΔγ(R+×R+)2ΩLq(Sn−1×Sm−1)2×|Aθiξ|±2wm1q′κ|Aθjη|±2wm2q′κ$(10)

hold for all i, jZ, where $\kappa =max\left\{2,\gamma \prime \right\}$ The constant C is independen of i, j ξ, η, q, and θ.

Assume that Ω∈L1(Sn−1×Sm−1) and h∈Δγ(R+×R+) for some γ′ > 1. Then for any f∈Lp(Rn×Rm) with γ′<p≤∞ there exists a cons tant C such that

$∥σΩ∗,h(f)∥Lp(Rn×Rm)≤Cp∥h∥Δγ(R+×R+)∥Ω∥L1(Sn−1×Sm−1)∥f∥Lp(Rn×Rm).$

The following lemma can be obtained by applying the arguments (with only minor modifications) used in [4, 19].

Let Ω∈Lq(Sn−1 × Sm−1) for some 1 > q ≤ 2 and θ = 2γʹ. Assume that h ∈ Δγ (R+ × R+) for some γ > 1. Then for any functions {gi, j(.,.), i, jZ} on Rn × Rm, there exists a constant Cr such that

$(i∑j∈Z∫θjθj+1∫θiθi+1σΩ,h,t,s∗gi,j2dsdtst)12Lr(Rn×Rm)≤crA(γ)q−1hΔγ(R+×R+)ΩLq(Sn−1×Sm−1)i∑j∈Z|gij|21/2Lr(Rn×Rm)$

for any r satisfiying |1/r−1/2| < min{1/2, 1/γʹ}.

## 3 Proof of the main result

We prove Theorem 1.1 by applying the same approaches as in [5, 14], which have their roots in [20]. Let us assume that h ∈ Δγ(R+ × R+) for some γ > 1. Then by Minkowskl’s inequality, we get that

$MΩ,hρ1,ρ2f(x,y)=∫R+×R+∑i,j=0∞⁡1tρ1sρ2∫2−i−1t<ρ1(u)≤2−it∫2−j−1s<ρ2(v)≤2−jS×f(x−u,y−v)KΩ,hρ1,ρ2(u,v)dudv2dtdsts1/2≤∑i,j=0∞∫R+×R+1tρ1s02∫2−i−1t<ρ1(u)≤2−it∫2−j−1s<ρ2(v)≤2−jS×f(x−u,y−v)KΩ,hρ1,ρ2(u,v)dudv2dtdsts1/2≤2a1+a2(2a1−1)(2a2−1)∫R+×R+σΩ,h,ts∗f(x,y)|2dtdsts1/2.$(14)

Take θ = 2qʹγʹ For iZ, let $\left\{{\mathrm{\Gamma }}_{i}{\right\}}_{-\mathrm{\infty }}^{\mathrm{\infty }}$ be a smooth partition of unity in (0, ∞) adapted to the interval ${\mathcal{I}}_{i}=\left[\left({\theta }^{-i-1}\right),\phantom{\rule{thickmathspace}{0ex}}\left({\theta }^{-i+1}\right)\right]$. Specifically, we require the following:

$Γi∈C∞,0≤Γi≤1,∑i∈ZΓi2(t)=1,suppΓi⊆Ii,anddκΓi(t)dtk≤Cktk,$

where Ck is independent of the lacunary sequence θi; iZ}. Define the multiplier operators Mi, j on Rn × Rm by $\left(\stackrel{^}{{M}_{i,j}f}\right)\left(\xi ,\phantom{\rule{thickmathspace}{0ex}}\eta \right)={\mathrm{\Gamma }}_{i}\left({\rho }_{1}\left(\xi \right)\right){\mathrm{\Gamma }}_{j}\left({\rho }_{2}\left(\eta \right)\right)\stackrel{^}{f}\left(\xi ,\phantom{\rule{thickmathspace}{0ex}}\eta \right)$. Then for any fS(Rn × Rm) and i, jZ, we have $f\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)=\sum _{k,l\in \mathrm{Z}}\left({M}_{i+k,j+l}f\right)\left(x,\phantom{\rule{thickmathspace}{0ex}}y\right)$. Therefore, by Minkowski’s inequality we obtain

$∫R+×R+|σΩh,t,s∗f(x,y)|2dtdsts1/2≤C∑k,l∈ZSk,lf(x,y),$(15)

where

$Sk,lf(x,y)=∫0∞∫0∞|Yk,l(x,y,t,s)|2dtdsts1/2,Yk,l(x,y,t,s)=∑i,j∈ZσΩh,t,s∗Mi+k,j+l∗f(x,y)χ[θi,θi+1)×[θj,θj+1)(t,s).$

Thus, it suffices to show for v > 0, that

$∥Sk,l(f)∥Lp(Rn×Rm)≤CpA(γ)q−12−v2(|k|+|l|)∥h∥Δγ(R+×R+)∥Ω∥Lq(Sn−1×Sm−1)∥f∥Lp(Rn×Rm)$

holds for any p with |1/p−1/2| < min{1/γʹ, 1/2}.

Let us first consider the L2 boundedness of Sk, l(f). By using Plancherels theorem, Fubinis theorem, Lemma 2.5, plus the approaches used in [10], we obtain that

$∥Sk,l(f)∥L2(Rn×Rm)2≤∑i,j∈z∫Δi+k,j+l∫θiθi+1∫θjθj+1|σ^Ω,h,t,s(ξ,η)|2dtdstsf^(ξ,η)2dξdη≤Cpln2⁡(θ)∥h∥Δγ(R+×R+)2∥Ω∥Lq(Sn−1×Sm−1)2×∑ij∈z∫Δi+kj+l|Aθjξ|±2wκm1q′|Aθiη|±2wκm2q′|f^(ξ,η)|−dξdη≤Cpln2⁡(θ)2−δ(|k|+|l|)∥h∥Δγ(R+×R+)2∥Ω∥Lq(Sn−1×Sm−1)2∑i,j∈z∫Δi+k,j+lf^(ξ,η)2dξdη≤CpA(γ)q−122−δ(|k|+|l|)∥h∥Δγ(R+×R+)2∥Ω∥Lq(Sn−1×Sm−1)2∥f∥L2(Rn×Rm)2,$(16)

where ${\mathrm{\Delta }}_{i,j}=\left\{\left(\xi ,\phantom{\rule{thickmathspace}{0ex}}\eta \right)\in {\mathbf{R}}^{n}×{\mathbf{R}}^{m}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left({\rho }_{1}\left(\xi \right),\phantom{\rule{thickmathspace}{0ex}}{\rho }_{2}\left(\eta \right)\right)\in {\mathcal{I}}_{i}×{\mathcal{I}}_{j}\right\}$ and 0 < δ < 1.

Now, let us compute the Lp-norm of Sk, l(f) for p = r. By using Lemma 2.7, and applying the Littlewood-Paley theory plus [[10], (3.20) pp. 1242] see also [[9], Proposition 2.1]), we obtain

$∥Sk,l(f)∥Lr(Rn×Rm)≤Ci∑j∈Z∫θiθi+1∫θjθj+1(|σΩh,t,s∗Mi+k,j+l∗f|)2dtdsts1/2Lr(Rn×Rm)≤CrA(γ)q−1∥h∥Δγ(R+×R+)∥Ω∥Lq(Sn−1×Sm−1)∑i,j∈Z|Mi+k,j+l∗f|21/2Lr(Rn×Rm)≤CrA(γ)q−1∥h∥Δγ(R+×R+)∥Ω∥Lq(Sn−1×Sm−1)∥f∥Lr(Rn×Rm).$

By Interpolation between the last inequality and (16), we reach (16). Consequently, the proof of Theorem 1.1 is complete.

## Acknowledgement

The authors would like to thank the referees for their careful reading and valuable comments. Also, the authors would like to acknowledge Dr Hussain Al-Qassem for his suggestions on this note.

## References

• [1]

Fabes E., Riviére N., Singular integrals with mixed homogeneity, Studia Math., 1966, 27(1), 19-38 Google Scholar

• [2]

Ding Y., Xue Q., Yabuta K., Parabolic Littlewood-Paley g-function with rough kernels, Acta Math. Sin. (Engl. Ser.), 2008, 24(12), 2049-2060 Google Scholar

• [3]

Al-Salman A., A note on parabolic Marcinkiewicz integrals along surfaces, Proce. of A. Razmadze Math. Inst., 2010, 154, 21-36 Google Scholar

• [4]

Ali M., Abo-Shgair E., On certain estimates for parabolic Marcinkiewicz integral and extrapolation, Inter. J. Pure App. Math., 2014, 96(3), 391-405 Google Scholar

• [5]

Chen Y., Ding Y., Lp Bounds for the parabolic Marcinkiewicz integral with rough kernels, J. Korean Math. Soc., 2007, 44(3), 733745 Google Scholar

• [6]

Chen Y., Ding Y., The parabolic Littlewood-Paley operator with Hardy space kernels, Canad. Math. Bull., 2009, 52(4), 521-534Google Scholar

• [7]

Wang F., Chen Y., Yu W., Lp Bounds for the parabolic Littlewood-Paley operator associated to surfaces of revolution, Bull. Korean Math. Soc., 2012, 29(4), 787-797 Google Scholar

• [8]

Nagel A., Riviére N., Wainger S., On Hilbert transforms along curves, II, Amer. J. Math., 1976, 98(2), 395-403 Google Scholar

• [9]

Al-Salman A., Parabolic Marcinkiewicz Integrals along Surfaces on Product Domains, Acta Math. Sinica, English Series, 2011, 27(1), 1-18 Google Scholar

• [10]

Liu F., Wu H., Rough Marcinkiewicz Integrals with Mixed Homogeneity on Product Spaces, Acta Math. Sin. (Engl. Ser.), 2013, 29(7), 1231-1244 Google Scholar

• [11]

Ding Y., L2-boundedness of Marcinkiewicz integral with rough kernel, Hokk. Math. J., 1998, 27(1), 105-115 Google Scholar

• [12]

Chen J., Fan D., Ying Y., Rough Marcinkiewicz integrals with L(log L)2 kernels, Adv. Math., (China), 2001, 30(2), 179-181 Google Scholar

• [13]

Choi Y., Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Anal. Appl., 2001, 261(1), 53-60 Google Scholar

• [14]

Al-Qassem H., Al-Salman A., Cheng L., Pan Y., Marcinkiewicz integrals on product spaces, Studia Math., 2005, 167(3), 227-234 Google Scholar

• [15]

Al-Qassem H., Rough Marcinkiewicz integral operators on product spaces, Collec. Math., 2005, 36(3), 275-297 Google Scholar

• [16]

Al-Qassem H., Pan Y., On certain estimates for Marcinkiewicz integrals and extrapolation, Collec. Math., 2009, 60(2), 123-145 Google Scholar

• [17]

Sato S., Estimates for singular integrals and extrapolation, arXiv:0704.1537v1 Google Scholar

• [18]

Ricci F., Stein E., Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier, 1992, 42(3), 637-670 Google Scholar

• [19]

Ali M., Janaedeh E., Marcinkiewicz integrals on product spaces and extrapolation, Glob. J. Pure App. Math., 2016, 12(2), 451-1463 Google Scholar

• [20]

Duoandikoetxea D., Multiple singular integrals and maximal functions along hypersurfaces, Ann. Inst. Fourier, 1986, 36(4), 185-206 Google Scholar

Accepted: 2016-08-25

Published Online: 2016-09-26

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

Export Citation