Weighted $L^{p}$ norms of Marcinkiewicz functions on product domains along surfaces

1.   Introduction and statement of results

Let $\mathbb{R}^{n}(n\geq2$) be an $n$-dimensional Euclidean space, $\mathbb{S}^{n-1}$ the unit sphere in $\mathbb{R}^{n}$ equipped with normalized Lebesgue measure $d\sigma$, and set $\mathbb{R}_{+} = (0, \infty)$. Furthermore, we let $y^{\prime} = \dfrac{y}{|y|}\in\mathbb{S}^{n-1}$ for $y\neq0$ and let $\Omega\in L^{1}(\mathbb{S}^{n-1})$ be a homogeneous function of degree zero on $\mathbb{R}^{n}$ that satisfies

The classical Marcinkiewicz integral operator introduced by E. M. Stein in ^[1] is given by

When $\Omega\in Lip_{\alpha}(\mathbb{S}^{n-1})(0 < \alpha\leq1)$, Stein ^[1] proved that $\mu_{\Omega}$ maps $L^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$ for all $1 < p\leq2$. In ^[2], A. Benedek, A. Calderón, and R. Panzone proved that $\mu_{\Omega}$ is bounded on $L^{p}$ for all $1 < p < \infty$ provided that $\Omega\in C^{1}\left(\mathbb{S}^{n-1}\right)$. In ^[3], Walsh proved that $\mu_{\Omega}$ is bounded on $L^{2}(\mathbb{R}^{n})$ under the weak condition $\Omega\in L(\log^{+}L)^{\frac{1}{2}}(\mathbb{S}^{n-1})$. Moreover, he showed that the $L^{2}$ boundedness of $\mu_{\Omega}$ may fail if the condition $\Omega\in L(\log^{+}L)^{\frac{1}{2}}(\mathbb{S}^{n-1})$ is replaced by $\Omega\in L(\log L)^{\frac{1}{2}-\varepsilon}(\mathbb{S}^{n-1})$ for some $\varepsilon > 0$. In 2002, Al-Salman et al. ^[4] improved Walsh's result by showing that the condition $\Omega\in L(\log^{+}L)^{\frac {1}{2}}(\mathbb{S}^{n-1})$ is also sufficient for the $L^{p}$ boundedness of $\mu_{\Omega}$ for all $p\in(1, \infty)$. For further results and background information about the operator $\mu_{\Omega}$, we refer readers to ^{[4,5,6,7,8,9]} and references therein, among others.

In 1990, Torchinsky and Wang studied the $L^{p}$ boundedness of the operator $\mu_{\Omega}$ on weighted spaces. In fact, they showed in ^[10] that $\mu_{\Omega}$ is bounded on $L^{p}(\omega)\, \, (1 < p < \infty)$ if $\Omega\in Lip_{\alpha}(\mathbb{S}^{n-1})\, (0 < \alpha\leq1)$ and $\omega\in A_{p}$ (the Muckenhoupt weight class, see^[11]). Subsequently, Ding et al. ^[12] proved that $\mu_{\Omega}$ is bounded on $L^{p}(\omega)$ for $p\in(1, \infty)$ provided that $\Omega\in L^{q}(\mathbb{S}^{n-1}), q > 1$, and $\omega^{q^{\prime}}\in A_{p}(\mathbb{R}^{n})$. In ^[13], Lee et al. proved a weighted norm inequality for $\mu_{\Omega}$ under the assumption that $\Omega$ is in the Hardy space $H^{1}(\mathbb{S}^{n-1})$ and the weight $\omega$ is in the class $\tilde{A}_{p}^{I}(\mathbb{R}^{n})$ of radial weights introduced by Duoandikoetxea in ^[14]. In ^[15], Al-Salman studied weighted inequalities of the generalized operator

where $\Psi:(0, \infty)\rightarrow \mathbb{R}$ is a smooth function satisfying the following growth conditions

for some $d\neq0$ and $t\in(0, \infty)$ where $C_{1}, C_{2}, C_{3}$, and $C_{4}$ are positive constants independent of $t$. We shall let $\mathcal{G}$ be the class of all smooth mappings $\Psi:(0, \infty)\rightarrow \mathbb{R}$ that satisfy the growth conditions (1.4)–(1.5). It is clear that $\mathcal{G}$ contains all power functions $t^{\alpha}(\alpha\neq0)$. It is shown in ^[15] that $\mu_{\Omega, \Psi}$ is bounded on $L^{p}(\omega)$ for $p\in(1, \infty)$ provided that $\omega\in\tilde{A}_{p}^{I}$ and that $\Omega$ is in the optimal space $L(\log^{+}L)^{\frac{1}{2}}(\mathbb{S} ^{n-1})$. Here, we remark that for any $q > 1$ and $0 < \alpha\leq1$, the following inclusions hold and that they are proper

and

In ^[8], Ding considered the analogy of the operator $\mu_{\Omega}$ on the product domain setting. For $\Omega\in L^{1} (\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying

consider the Marcinkiewicz integral operator on the product domains $\mathcal{U} _{\Omega}$ defined by

where

and

Ding proved that $\mathcal{U}_{\Omega}$ is bounded on $L^{2}(\mathbb{R} ^{n}\mathbb{\times R}^{m})$ when $\Omega$ satisfies the additional assumption of $\Omega\in L(\log^{+}L)^{2}(\mathbb{S}^{n-1}\mathbb{\times S}^{m-1})$, i.e.,

In 2002, Chen et al. ^[7] improved the result of Ding and showed that $\mathcal{U}_{\Omega}$ is bounded on $L^{p}(\mathbb{R}^{n}\mathbb{\times R}^{m})\, \, (1\leq p < \infty)$ under the same condition on $\Omega$. Later, Choi ^[16] proved that the $L^{2}$ boundedness of $\mathcal{U}_{\Omega}$ still holds under the very weak condition $\Omega\in L(\log L)(\mathbb{S} ^{n-1}\times\mathbb{S}^{m-1})$. Subsequently, Al-Qassem et al. ^[17] substantially improved Choi's result by showing that $\mathcal{U}_{\Omega}$ is bounded on $L^{p}(\mathbb{R}^{n}\mathbb{\times R}^{m})$ for all $1 < p < \infty$ under the same condition $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$. Moreover, they proved that the condition $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ is nearly optimal in the sense that the $L^{2}$ boundedness may fail if the function is assumed to be in $L(\log L)^{\alpha}(\mathbb{S}^{n-1} \times\mathbb{S}^{m-1})\backslash L(\log L)(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$ for any $\alpha < 1$. For further results for Marcinkiewicz integral operators on product domains, we cite ^{[16,17,18,19,20,21,22]}, among others.

Motivated by the work in ^[15] and ^[18], we consider the weighted $L^{p}$ boundedness of the Marcinkiewicz integral operator on product domains along surfaces. For suitable mappings $\Phi, \Psi:[0, \infty)\rightarrow \mathbb{R}$, consider the $\mathcal{U}_{\Omega, \Phi, \Psi}$ given by

where

and $\Lambda(t^{\prime}, s^{\prime}) = \{(u, v)\in\mathbb{R}^{n}\times \mathbb{R}^{m}:|u|\leq2^{t^{\prime}}\, \, \text{and}\, \, |v|\leq2^{s^{\prime}} \}$. By specializing to the case $\Phi(t) = \Psi(t) = t$, the operator reduces to the classical Marcinkiewicz integral operator $\mathcal{U}_{\Omega}$ on product domains. Integral operators on product domains along surfaces have been considered by several authors. For background information, we advise the readers to consult ^{[17,18,19,20,21,22,23]} and references therein.

In order to state our results in this paper, we recall the definition of radial weights $\tilde{A}_{p}^{I}(\mathbb{R}^{n})$ introduced in ^[14]:

Definition 1.1. Let $\omega(t)\geq0$; and $\omega\in L_{loc}^{1}(\mathbb{R}_{+})$. For $1 < p < \infty$, we say that $\omega\in A_{p}(\mathbb{R}_{+})$ if there is a positive constant $C$ such that, for any interval $I\subseteq\mathbb{R}_{+}$,

We say that $\omega\in A_{1}(\mathbb{R}_{+})$ if there is a positive constant $C$ such that

where $\omega^{\ast}$ is the Hardy-Littlewood maximal function of $\omega$ on $\mathbb{R}_{+}$.

Definition 1.2. Let $1\leq p\leq\infty$. We say that $\omega\in\tilde{A}_{p}(\mathbb{R}_{+})$ if

where either $\nu_{i}\in A_{1}(\mathbb{R}_{+})$ is decreasing or $\nu_{i}^{2}\in A_{1}(\mathbb{R}_{+}), \, \, \, i = 1, 2.$

Definition 1.3. For $1 < p < \infty$, we let

Let $A_{p}^{I}(\mathbb{R}^{n})$ be the weight class defined by exchanging the cubes in the definition of $A_{p}$ for all $n$-dimensional intervals with sides parallel to coordinate axes. It is well known that $\bar{A} _{p}(\mathbb{R_{+}})\subseteq\tilde{A}_{p}(\mathbb{R_{+}})$ (see ^[24]). Moreover, if $\omega(t)\in\bar{A}_{p}(\mathbb{R_{+}})$, then $\omega(|x|)$ is the Mukenhoupt weighted class $A_{p}(\mathbb{R}^{n})$ whose definition can be found in ^[14]. We let $\tilde{A}_{p}^{I} = \tilde {A}_{p}\cap{A}_{p}^{I}$.

We shall need the following lemma:

Lemma 1.4. If $1 < p < \infty$, then the weight class $\tilde{A}_{p}^{I}(\mathbb{R}_{+})$ has the following properties:

(ⅰ) $\tilde{A}_{p_{1}}^{I}\subset\tilde{A}_{p_{2}}^{I}$, if $1\leq p_{1} < p_{2} < \infty;$

(ⅱ) For any $\omega\in\tilde{A}_{p}^{I}$, there exists an $\varepsilon > 0$ such that $\omega^{1+\varepsilon}\in\tilde{A} _{p}^{I};$

(ⅲ) For any $\omega\in\tilde {A}_{p}^{I}$ and $p > 1$, there exists an $\varepsilon > 0$ such that $p-\varepsilon > 1$ and $\omega\in\tilde {A}_{p-\varepsilon}^{I}$;

(ⅳ) $\omega \in\tilde{A}_{p}^{I}$ if and only if $\omega^{1-p^{\prime}}\in \tilde{A}_{p^{\prime}}^{I}$.

For any weights $\omega_{1}$ and $\omega_{2}$, we let $L^{p}(\mathbb{R} ^{n}\times\mathbb{R}^{m}, \, \omega_{1}(x)dx, \, \omega_{2}(y)dy)$ $(1 < p < \infty)$ be the weighted $L^{p}$ space associated with the weight $\omega_{1}$ and $\omega_{2}$, i.e., $L^{p}(\mathbb{R}^{n}\times\mathbb{R} ^{m}, \omega_{1}(x)dx, \omega_{2}(y)dy) = L^{p}(\omega_{1}, \omega_{2})$ consists of all measurable functions $f$ with $\Vert f\Vert_{L^{p}(\omega_{1}, \omega_{2})} < \infty$, where

In light of the above discussion, the following natural question arises:

Question: Let $\mathcal{U}_{\Omega, \Phi, \Psi}$ be given by (1.8) and assume that $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7). Assume that $\Phi\mathit{, }\Psi\in\mathcal{G}$, $\omega_{1}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n})$ and $\omega_{2} \in\tilde{A}_{p}^{I}(\mathbb{R}^{m})$ for some $1 < p < \infty.$ Is $\mathcal{U}_{\Omega, \Phi, \Psi}$ bounded on $L^{p}(\omega _{1}, \omega_{2})$?

In the following we shall answer the above question in the affirmative. In fact, we shall prove that the weighted $L^{p}$ boundedness holds for various classes of mappings $\Phi$ and $\Psi$.

Theorem 1.5. Suppose that $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7), $\omega_{1}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n})$, and $\omega_{2}\in\tilde{A}_{p}^{I}(\mathbb{R}^{m})$. If $\Phi \mathit{, }\Psi\in\mathcal{G}$, then $\mathcal{U}_{\Omega, \Phi, \Psi}$ is bounded on $L^{p}(\omega_{1}, \omega_{2})$ for $1 < p < \infty$.

We remark here that, by specializing to the case $\Phi(t) = \Psi(t) = t$, we obtain that the classical operator $\mathcal{U}_{\Omega}$ is bounded on $L^{p} (\omega_{1}, \omega_{2})$ for $1 < p < \infty$. This result, as far as we know, is not known previously. We shall prove in this paper that the weighted boundedness in Theorem 1.5 holds for a more mappings $\Phi$ and $\Psi$. In order to state our second result, we recall the following class of mappings introduced in ^[5]:

Definition 1.6. A function $\psi:[0, \infty)\rightarrow \mathbb{R}$ is said to belong to the class $\mathcal{PC} _{\lambda}(d)$ ($d > 0$) if there exist $\lambda\in\mathbb{R}$, a polynomial $P$, and $\varphi\in\mathcal{C}^{d+1} [0, \infty)$ such that

say that In fact, we prove the following:

The class $\mathcal{PC}_{\lambda}(d)$ was introduced in ^[5]. It is shown in ^[5] that the class $\cup_{d\geq0}\mathcal{PC}_{\lambda}(d)$ properly contains the class of polynomials $\mathcal{P}_{d}$ of degree less than or equal $d$ as well as the class of convex increasing functions. Examples of functions in $\cup_{d\geq0}\mathcal{PC}_{\lambda}(d)$ that are neither convex nor polynomial are widely available. A particular example is the function $\theta(t) = -t^{2}+t^{2}\ln(1+t)$. Our second result in this paper is the following:

Theorem 1.7. Suppose that $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7), $\omega_{1}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n})$, and $\omega_{2}\in\tilde{A}_{p}^{I}(\mathbb{R}^{m})$. If $\Phi \in\mathcal{PC}_{\lambda}(d), \, \Psi\in\mathcal{PC}_{\alpha}(b)$ for $d, \, b > 0$ and $\lambda, \, \alpha\in\mathbb{R}$, then $\mathcal{U}_{\Omega, \Phi, \Psi}$ is bounded on $L^{p}(\omega _{1}, \omega_{2})$ for $1 < p < \infty$ with $L^{p}$ bounds independent of $\lambda, \alpha\in\mathbb{R}$ and the coefficients of the particular polynomials involved in the standard representations of $\Phi$ and $\Psi$.

We remark here that; Theorem 1.7 is the analogy of Theorem 1.3 ^[15] in the product domain setting. On the other hand, Theorem 1.7 is a generalization of the corresponding result in ^[18]. More specifically, if $\omega_{1}(x) = \omega_{2}(x) = 1$, then Theorem 1.7 reduces to Theorem 1.3 in ^[18].

We point out here that the method employed in this paper is based on interpolation between good $L^{2}$ estimates and crude $L^{p}$ estimates. The $L^{2}$ estimates depend heavily on the nature of the involved surface. This is clearly expressed interns of the obtained oscillatory estimates. On the other hand, the $L^{p}$ estimates depend on proving the boundedness of the corresponding maximal functions. The the method employed can be used to study the weighted $L^{p}$ boundedness of more general classes of Marcinkiewicz integral operators along surfaces.

Throughout this paper, the letter $C$ will stand for a constant that may vary at each occurrence, but it is independent of the essential variables.

2.   Weighted estimates for certain square and maximal functions

This section is devoted to obtaining weighted estimates of certain square functions and maximal functions. For positive real numbers $a$ and $\, b$ and a Schwartz function $\Phi\in$ $\mathcal{S}(\mathbb{R}^{n}\times\mathbb{R}^{m})$, we let

where

It can be observed here that if $\Phi(x, y) = \Phi^{(1)}(x)\Phi^{(2)}(y)$ and $f(x, y) = f_{1}(x)f_{2}(y)$, then

where $S_{\Phi^{(1)}, a}$ and $S_{\Phi^{(2)}, b}$ are the square functions in the one parameter setting defined in ^[15]. Thus, by Lemma 2.1 in ^[15], it follows that for two Muckenhoupt weights $\omega_{1}, \omega_{2}\in A_{p}$, we have

Therefore, it is natural to question if (2.2) holds for general $\Phi\in$ $\mathcal{S}(\mathbb{R}^{n}\times\mathbb{R}^{m})$ and $f\in L^{p}(\omega_{1}, \omega_{2})$. In the following lemma, which is analogues to Lemma 2.1 in ^[15], we answer this question in the affirmative:

Lemma 2.1. Given $a, \, b > 2$ and let $\psi, \, \theta$ be $\mathcal{C}^{\infty}$ functions on $\mathbb{R}$ that satisfy the following conditions:

(ⅰ) $supp(\psi)\subseteq\left[\dfrac{4}{5a}, \dfrac{5a} {4}\right] \text{ and}\, \, \, supp(\theta)\subseteq\left[\dfrac{4}{5b}, \dfrac{5b}{4}\right].$

(ⅱ) $\left\vert \dfrac{d^{l}\psi}{du^{l}}(u)\right\vert, \left\vert \dfrac{d^{l}\theta}{du^{l}}(u)\right\vert \leq\dfrac{C_{l}}{u^{l} }$ for all $u$ and $l\geq0$ where $C_{l}$ is independent of $a$ and $b$.

Let $\Upsilon\in\mathcal{S}(\mathbb{R}^{n}\times\mathbb{R}^{m})$ be given by $\widehat{\Upsilon}(\xi, \eta) = \psi(|\xi|^{2}).\, \theta(|\eta|^{2})$ and let $S_{\Upsilon, a, b}$ be the square function $S_{\Upsilon, a, b}$ given by (2.1) with $\Phi$ is replaced by $\Upsilon$. Then, for $1 < p < \infty$, $\omega_{1}\in A_{p}(\mathbb{R}^{n})$, and $\omega_{2}\in A_{p}(\mathbb{R}^{m})$, there exists a constant $C_{p}$ independent of $a, \, b$ such that

Proof: For $(\xi, \eta)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$, let

By the assumption $(ii)$, we have

and

for every multi-index $\alpha, \beta$ with $|\alpha|, |\beta|\geq0$, where $C_{\alpha}, \, C_{\beta}$ are constants independent of $a$ and $b$. We set

Then, by (2.4)–(2.5), and a vector-valued analogy of the argument in [25, p. 245–246], we obtain

for $|\alpha|\leq1$ and $|\beta|\leq1$ where $C$ is a constant independent of $a$ and $b$.

Now, let

and

Then,

By Plancherel's theorem, we obtain

and

Hence, by the Corollary on page 205 in ^[25], and (2.4), (2.6), and (2.11), we have

for $w_{1}{(x)}\in A_{p}(\mathbb{R}^{n}).$

Thus, by (2.13) and following similar arguments as in ^[26], we get

for each $y\in\mathbb{R}^{m}$ with $C$ independent of $y$. Then, by integration over $\mathbb{R}^{m}$, we get

By repeating the argument between (2.13) and (2.15) for $g_{\Upsilon, b}(f)(., y)$, and replacing $x$ by $y$, we get

Finally, by (2.10), inequality (2.3) follows vector-valued analogues of the argument in the proof of the Theorem $3$ in [26, p. 128], and (2.15)–(2.16). This ends the proof of Lemma 2.1.

Now, for $\Omega\in L^{1}(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ and suitable mappings $\Phi, \Psi:(0, \infty)\longrightarrow\mathbb{R}$, we define the family of measures $\{\sigma_{\Omega, \, \Phi, \Psi, a^{t^{\prime} }, b^{s^{\prime}}}:{t^{\prime}}, \, {s^{\prime}}\in\mathbb{R}\}$ by

We let $\mathcal{M}_{\Omega, \, \Phi, \, \Psi, a, b}$ be the maximal function corresponding to the family $\{\sigma_{\Omega, \, \Phi, \, \Psi, a^{t^{\prime} }, b^{s^{\prime}}}:{t^{\prime}}, \, {s^{\prime}}\in\mathbb{R}\}$, i.e.,

Then, we have the following lemma:

Lemma 2.2. Suppose that $\Omega\in L^{1} (\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.7). For $a, b > 2$ and suitable $\Phi, \Psi:(0, \infty)\longrightarrow\mathbb{R}$, let $\mathcal{M}_{\Omega, \, \Phi, \, \Psi, a, b}$ be the maximal function defined by (2.18). Suppose that (i) $\Phi\mathit{, } \Psi\in\mathcal{G}$; or (ii) $\Phi\in\mathcal{PC}_{\lambda} (d_{1}), \, \Psi\in\mathcal{PC}_{\alpha}(d_{2})$ for $d_{1}, \, d_{2} > 0$ and $\lambda, \, \alpha\in\mathbb{R}$. Then, for $1 < p < \infty$ and $\omega_{1}\in A_{p})(\mathbb{R}^{n}), \, \omega _{2}\in A_{p})(\mathbb{R}^{m})$, there exists a constant $C_{p}$ independent of $\Omega, \, a,$ and $b$ such that

Proof: We shall start by verifying (2.19) under assumption (ii) on the functions $\Phi$ and$\Psi$. Notice that

Thus, it is enough to show that

We define the one parameter maximal functions

and

Then,

By polar coordinates, we have

where

Now, we have

where (2.24) follows by change of variables and (1.4). By (8) in ^[14] and since $\omega_{2}\in\tilde{A_{p}}(\mathbb{R}^{m})$, we get

where $C_{p}$ is a constant independent of $v^{\prime}$. By a similar argument, for $\omega_{1}\in\tilde{\mathit{A}_{p}}(\mathbb{R}^{n})$, we get

where

Thus, by (2.23), (2.26), and Minkowski's inequality, we get

Similarly, for $\omega_{1}\in\tilde{\mathit{A}_{p}}(\mathbb{R}^{n})$, we get

Now, by (2.28) and following a similar argument as in ^[26], we have

for $x\in\mathbb{R}^{n}$ where $C$ is a constant independent of $x$. Then, by integration with respect to $x$, we get

Thus, by following a similar argument as in (2.30)–(2.31) on $\mathcal{M}_{\Omega, \Phi}$, replacing $x$ by $y$, we get

Thus, by (2.22), we have

Hence, (2.19) follows by (2.20) and (2.33). This ends the proof of (2.19) under assumption (ⅱ) on the functions $\Phi$ and$\Psi$. To prove (2.19) under assumption (ⅰ) on the functions $\Phi$ and$\Psi$, we follow a similar argument as above and make use of estimates developed in the proof of Lemma 2.2 in ^[15]. We omit the details. This ends the proof of the lemma.

Next, we prove the following weighted inequalities for square functions:

Lemma 2.3. Suppose that $\Vert\Omega\Vert_{L^{1} }\leq1$. Suppose also that $a$, $b$, $\Psi$, $\Phi$, and $\Upsilon$ are as in Lemma 2.2. Let For $t, s\in\mathbb{R}$, let $\sigma_{\Omega, \, \Phi, \, \Psi, a, b}$ be given by (2.17) where $t^{\prime}$ and $s^{\prime}$ are replaced by $t$ and $s$, respectively. Assume that (i) $\Phi\mathit{, }\Psi \in\mathcal{G}$; or (ii) $\Phi\in\mathcal{PC}_{\lambda}(d_{1}), \, \Psi\in\mathcal{PC}_{\alpha}(d_{2})$ for $d_{1}, \, d_{2} > 0$ and $\lambda, \, \alpha\in\mathbb{R}$. Then, for $1 < p < \infty,$ $j, \, k\in\mathbb{Z}$, $\omega_{1}\in\tilde{A}_{p} ^{I}(\mathbb{R}^{n}),$ and $\, \omega_{2}\in\tilde{A}_{p} ^{I}(\mathbb{R}^{m})$, and there exists a constant $C_{p}$ independent of $a, \, b, \, j, \, k$, and $\Omega$ such that

Proof: Notice that

Next, by Lemma 2.2, we have

Now, by duality, choose a non-negative function $g(x, y)$ with $\Vert g\Vert_{{L^{p^{\prime}}}(\omega_{1}^{1-p^{\prime}}, \omega_{2}^{1-p^{\prime}})}\leq1$ such that

where $\tilde{g}(x, y) = g(-x, -y)$. Thus, by Lemma 2.1, (2.36), and Hölder's inequality, we get

By an application of Lemma 2.2, we get

Hence, by interpolation between (2.35) and (2.38) in a vector-valued setting, we get

where the last inequality is obtained by Lemma 2.1. This completes the proof of Lemma 2.3.

3.   Preliminary estimates

This section is establish some preliminary estimates that are needed to prove our results.

Lemma 3.1. Let $\Omega\in L^{2}(\mathbb{S} ^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7) with $\left\Vert \Omega\right\Vert _{1}\leq1$ and $\left\Vert \Omega\right\Vert _{2}\leq A$ for some $A > 2$. Suppose that $\Phi\mathit{, }\Psi\in\mathcal{G}$ with powers $d_{1}, d_{1}$ in (1.4)–(1.5). For $t, \, s\in\mathbb{R}$, let $\sigma_{A, {t}, {s}}^{(\Phi, \Psi)}$ be the measure defined via the Fourier transform by

where

Then, there exists $\varepsilon\in(0, \frac{1}{2})$ such that

where the constant $C$ is independent of $A, s,$ and $t$.

Proof: We shall assume that $d_{1}, d_{2} > 0$. The other cases follows by similar argument. The estimate (3.3) is clear. To see the estimate (3.4), notice that

where

and $g(A, \Psi, \eta)$ has similar definition as $g(A, \Phi, \xi)$. By integration by parts along with the assumptions (1.4)–(1.5), and the observations $g(A, \Phi, \xi)\leq1$ and $g(A, \Psi, \eta)\leq1$, there exists $\varepsilon\in(0, \frac{1}{2})$ such that

By (3.8), (3.9)–(3.10), Hölder's inequality, and assumption on $\Omega$, we have

where

Since $\varepsilon < 1/2$, we have $\tilde{C}_{\varepsilon} < \infty$. Thus, by (3.9)–(3.10), (3.8), and an interpolation, we get

which implies (3.4) since $A^{\frac{1}{2\log_{2}A}} < 2$. To verify (3.5), we first notice that

which by interpolation implies

By the cancellation property (1.6), Hölder's inequality, the assumption on $\Omega$, (3.9), and (3.13), we have

where the last inequality follows by the same reasoning for $\tilde {C}_{\varepsilon}$ above. Thus, (3.5) follows by (3.14), (3.3), and an interpolation. The verifications of other estimates follows by a similar argument with minor modifications. We omit the details. This completes the proof of the lemma.

Now, by the same argument as in ^[18], we have the following lemma:

Lemma 3.2. Let $\Omega\in L^{2}(\mathbb{S} ^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7) with $\left\Vert \Omega\right\Vert _{1}\leq1$ and $\left\Vert \Omega\right\Vert _{2}\leq A$ for some $A > 2$. Let $\Phi\in\mathcal{PC}_{\lambda}(d_{1}), \, \Psi\in\mathcal{PC} _{\alpha}(d_{2})$ for $d_{1}, \, d_{2} > 0$ and $\lambda, \, \alpha\in\mathbb{R}$. Suppose that

where $P, Q, \varphi_{1}$, and $\varphi_{2}$ are as in the definition of the spaces $\mathcal{PC}_{\lambda}(d_{1})$ and $\Psi\in\mathcal{PC}_{\alpha}(d_{2})$. For $t, \, s\in\mathbb{R}$, let $\sigma_{A, {t}, {s}}^{(\Phi, \Psi)}$, $\sigma_{A, {t}, {s}} ^{(\Phi, Q)}, \sigma_{A, {t}, {s}}^{(P, \Psi)},$ and $\sigma_{A, {t}, {s} }^{(P, Q)}$ be the measures defined by (3.1) with proper modifications. Then,

(ⅰ) $\Vert\sigma_{A, {t}, {s}}^{(\Phi, \Psi)}\Vert\leq C;$

(ⅱ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t, s}}^{(\Phi, \Psi)}(\xi, \eta)\right\vert \leq C\, \, |\lambda\, \varphi_{1}\, (A^{t-1})\, \xi\, |^{-\frac{1}{2\, (d_{1}+1)\log _{2}A}}\, \, \, |\alpha\, \varphi_{2}\, (A^{s-1})\, \eta\, |^{-\frac{1} {2\, (d_{2}+1)\log_{2}A}}; & \end{array}$

(ⅲ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t, s}}^{(\Phi, \Psi)}(\xi, \eta)-\widehat{\sigma }_{A, {t, s}}^{(P, \Psi)}(\xi, \eta)\right\vert \leq C\, \, \left\vert \lambda\varphi_{1}(A^{t})\, \xi\right\vert ^{\frac{1}{2\log_{2}A} }\, \, \left\vert \alpha\varphi_{2}(A^{s-1})\eta\right\vert ^{-\frac{1} {2(d_{2}+1)\, \log_{2}A}}; & \end{array}$

(ⅳ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t, s}}^{(\Phi, \Psi)}(\xi, \eta)-\widehat{\sigma }_{A, {t, s}}^{(\Phi, Q)}(\xi, \eta)\right\vert \leq C\, \left\vert \, \lambda \, \varphi_{1}(A^{t-1})\, \xi\right\vert ^{-\frac{1}{2\, (d_{1}+1)\, \log_{2}A} }\, \, \, \left\vert \, \alpha\, \varphi_{2}(A^{s})\, \eta\right\vert ^{\frac {1}{2\log_{2}A}}; & \end{array}$

(ⅴ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t, s}}^{(\Phi, \Psi)}(\xi, \eta)-\widehat{\sigma }_{A, {t, s}}^{(P, \Psi)}(\xi, \eta)-\widehat{\sigma}_{A, {t, s}}^{(\Phi, Q)} (\xi, \eta)+\widehat{\sigma}_{A, {t, s}}^{(P, Q)}(\xi, \eta)\right\vert & \\ \leq C|\lambda\varphi_{1}(A^{t})\, \xi|^{\frac{1}{2\log_{2}A}}\, |\alpha \varphi_{2}(A^{s})\, \eta|^{\frac{1}{2\log_{2}A}}; & \end{array}$

(ⅵ) $\left\vert \widehat{\sigma}_{A, {t, s}}^{(\Phi, Q)} (\xi, \eta)-\widehat{\sigma}_{A, {t, s}}^{(P, Q)}(\xi, \eta)\right\vert \leq C\, |\lambda\varphi_{1}(A^{t})\, \xi|^{\frac{1}{2\log_{2}A}};$

(ⅶ) $\left\vert \widehat{\sigma}_{A, {t}, {s}}^{(P, \Psi)} (\xi, \eta)-\widehat{\sigma}_{A, {t}, {s}}^{(P, Q)}(\xi, \eta)\right\vert \leq C\, \left\vert \, \alpha\, \varphi_{2}(A^{s})\, \eta\right\vert ^{\frac{1} {2\log_{2}A}},$

where $C$ is independent of $\kappa$ and $(\xi, \eta)\in(\mathbb{R}^{n}, \mathbb{R}^{m})$.

We end this section by the following estimates contained in the argument in ^[18].

Lemma 3.3. Let $\Omega\in L^{2}(\mathbb{S} ^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7) with $\left\Vert \Omega\right\Vert _{1}\leq1$ and $\left\Vert \Omega\right\Vert _{2}\leq A$ for some $A > 2.$ Suppose that $P(w) = \sum_{k = 0}^{d_{1}}c_{k, 1}\, w^{k}$ and $Q(z) = \sum_{k = 0}^{d_{2}}c_{k, 2}\, z^{k}$ are polynomials of degrees $d_{1}$ and $d_{2}$, respectively. For $0\leq l\leq d_{1}$ and $0\leq s\leq d_{2}$, let

with the convention that $\sum\limits_{j\in\emptyset} = 0$. For $t, \, s\in\mathbb{R}$, $0\leq l\leq d_{1}$, and $0\leq o\leq d_{2}$, let ${\sigma}\, _{A, {t}, {s}}^{(l, o)}$ be defined by (3.1) where $\Phi$ and $\Psi$ are replaced by $P_{l}$ and $Q_{o}$, respectively. For $0\leq l\leq d_{1}, \, 0\leq o\leq d_{2}$, let ${\sigma}\, _{A, {t}, {s}}^{(l, o)} = {\sigma} \, _{A, {t}, {s}}^{(P_{l}, Q_{o})}$. Then, for $1\leq l\leq d_{1}$ and $\, 1\leq o\leq d_{2}$, we have

(ⅰ) $\Vert\sigma_{A, {t}, {s}}^{(l, o)}\Vert\leq C;$

(ⅱ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l, o)}(\xi, \eta)\right\vert \leq C\, \, |c_{l, 1}A^{l(t-1)}\, \, {l}!\, \xi|^{-\frac{1}{2\, l\, \log_{2}A}} |\, c_{o, 2}\, (A^{o(s-1)}\, {o}!\eta|^{-\frac{1}{2\, o\log_{2}A}}; & \end{array}$

(ⅲ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l, o)}(\xi, \eta)-\widehat{\sigma }_{A, {t}, {s}}^{(l-1, o)}(\xi, \eta)\right\vert \leq C\, \, |c_{l, 1}\, A^{lt} \xi|^{\frac{1}{2\log_{2}A}}\, \, |c_{o, 2}\, A^{o(s-1)}\, {o}!\, \, \eta|^{-\frac {1}{2\, o\log_{2}A}}; & \end{array}$

(ⅳ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l, o)}(\xi, \eta)-\widehat{\sigma }_{A, {t}, {s}}^{(l, o-1)}(\xi, \eta)\right\vert \leq C\, \left\vert c_{l, 1} \, A^{l(t-1)}\, l!\, \xi\right\vert ^{-\frac{1}{2\, l\, \log_{2}A}}\, \, \left\vert c_{o, 2}A^{os}\, \eta\right\vert ^{\frac{1}{2\log_{2}A}}; & \end{array}$

(ⅴ) $\begin{array} [c]{ll} \left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l, o)}(\xi, \eta)-\widehat{\sigma }_{A, {t}, {s}}^{(l-1, o)}(\xi, \eta)-\widehat{\sigma}_{A, {t}, {s}}^{(l, o-1)} (\xi, \eta)+\widehat{\sigma}_{A, {t}, {s}}^{(l-1, o-1)}(\xi, \eta)\right\vert & \\ \leq C\, |c_{l, 1}A^{lt}\, \xi|^{\frac{1}{\kappa+1}}\, |c_{o, 2}\, A^{os} \, \eta|^{\frac{1}{2\log_{2}A}}; & \end{array}$

(ⅵ) $\left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l, o-1)} (\xi, \eta)-\widehat{\sigma}_{A, {t}, {s}}^{(l-1, o-1)}(\xi, \eta)\right\vert \leq C\, \, |c_{l, 1}\, A^{lt}\, \xi|^{\frac{1}{2\log_{2}A}};$

(ⅶ) $\left\vert \widehat{\sigma}_{A, {t}, {s}}^{(l-1, o)} (\xi, \eta)-\widehat{\sigma}_{A, {t}, {s}}^{(l-1, o-1)}(\xi, \eta)\right\vert \leq C\, \left\vert c_{o, 2}A^{os}\, \eta\right\vert ^{\frac{1}{2\log_{2}A}},$

where $C$ is independent of $A$ and $(\xi, \eta)\in(\mathbb{R}^{n}, \mathbb{R}^{m}).$

4.   Proof of results

This section is devoted for the proofs of Theorems 1.5 and 1.7. To this end, we prove the following proposition:

Proposition 4.1. Suppose that $\Omega\in L^{2}(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$ satisfying (1.6)–(1.7) with $\left\Vert \Omega\right\Vert _{1}\leq1$ and that $\left\Vert \Omega\right\Vert _{2}\leq A$ for some $A > 2.$ Suppose also that $\omega_{1}\in\tilde{A}_{p}^{I} (\mathbb{R}^{n})$ and $\omega_{2}\in\tilde{A}_{p}^{I}(\mathbb{R} ^{m}),$ $1 < p < \infty$. Assume that the mappings $\Phi, \Psi$ satisfies (i) $\Phi\mathit{, }\Psi\in\mathcal{G}$; or (ii) $\Phi\in\mathcal{PC}_{\lambda}(d_{1}), \, \Psi\in\mathcal{PC}_{\alpha}(d_{2})$ for $d_{1}, \, d_{2} > 0$ and $\lambda, \, \alpha\in\mathbb{R}$. Then, for $1 < p < \infty$, we have

with constants $C_{p}$ independent of $A$.

Proof: We shall prove (4.1) under the assumption (ⅱ) on the mappings $\Phi$ and $\Psi$. The proof under the assumption (ⅰ) follows by similar argument with minor modifications. We write $\Phi$ and $\Psi$ as

where $P$ and $Q$ are polynomials of degrees $d_{1}$ and $d_{2}$ as in the statement of Lemma 3.3. We let $\{c_{k, 1}\}, \{c_{k, 2}\}, P_{l}, Q_{o},$ and ${\sigma}\, _{A, {t}, {s}}^{(l, o)}$ be as in Lemma 3.3. Let ${\sigma} \, _{A, {t}, {s}}^{(d_{1}+1, d_{2}+1)}$ be the measure $\sigma_{A, {t}, {s}} ^{(\Phi, \Psi)}$ in Lemma 3.1. By simple change of variables, we have

where

and

Thus, to prove (4.1), it suffices to show that

with constant $C_{p}$ independent of $A$. Let $\{\sigma_{A, {t}, {s}} ^{(l, o)}:0\leq l\leq d_{1}, \, 0\leq o\leq d_{2}\}$ be as in Lemma 3.3. Notice that

Following the same arguments in ^[18], for $1\leq l\leq d_{1}, \, 1\leq o\leq d_{2}$, $1 < p < \infty,$ $j, \, k\in\mathbb{Z}$, $\omega_{1}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n}),$ and $\, \omega_{2}\in \tilde{A}_{p}^{I}(\mathbb{R}^{m})$, we can find linear transformations $L_{l}:\mathbb{R}^{n}\mathbb{\rightarrow R}^{n}$ and $Q_{s}:\mathbb{R} ^{m}\mathbb{\rightarrow R}^{m}$ and measures $\{\tau_{A, t, s}^{(l, o)} :t, \, s\in\mathbb{R}, \}$ such that

and

where

and

Thus, by (4.14) and Minkowski's inequality, we obtain that

where

Now, by a similar argument as in ^[27], choose two collections of $\mathcal{C^{\infty}}$ functions $\{\varpi _{i}^{(l)}\}_{i\in\mathbb{Z}}$ and $\{\varpi_{i}^{(o)}\}_{i\in\mathbb{Z}}$ on $(0, \infty)$ satisfying the following properties:

where $C_{r}$ is independent of $A$. Define the measures $\{\upsilon_{i} ^{(l)}:i\in\mathbb{Z}\}$ on $\mathbb{R}^{n}$ and $\{\upsilon_{i}^{(o)} :i\in\mathbb{Z}\}$ on $\mathbb{R}^{m}$ by

By (4.18), we immediately obtain

where $\lfloor t\rfloor$ is the greatest integer function such that $t-1 < \lfloor t\rfloor < t$, and similarly for $\lfloor s\rfloor$ (see ^[6,20]). Hence, by taking the inverse Fourier transform for (4.20), we get

Thus, by (4.21), we obtain

where

By (4.7)–(4.12) and the Plancherel theorem, we get

where

Next, by (4.13), for $1 < p < \infty$ and $\omega_{1}\in\tilde{A} _{p}^{I}(\mathbb{R}^{n}), \, \omega_{2}\in\tilde{A}_{p}^{I}(\mathbb{R}^{m})$, there exists a positive constant $C_{p}$ independent of $i, j$, and $A$ such that

Now, we have three cases:

Case 1. $p > 2$. Choose a $q > p$ and $\varepsilon > 0$, such that $\omega_{1}^{1+\varepsilon}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n})\subset \tilde{A}_{q}^{I}(\mathbb{R}^{n})$ and $\omega_{2}^{1+\varepsilon}\in\tilde {A}_{p}^{I}(\mathbb{R}^{m})\subset\tilde{A}_{q}^{I}(\mathbb{R}^{m})$. Thus, by (4.25) we get

which when combined with (4.24) and the interpolation theorem with change of measures, we have

for $0 < \gamma < 1$ and $p > 2$.

Case 2. $1 < p < 2$. Choose a $1 < q < p$ and $\varepsilon > 0$ such that $\omega_{1}\in\tilde{A}_{p}^{I}(\mathbb{R}^{n}), \, \omega_{2}\in\tilde{A} _{p}^{I}(\mathbb{R}^{m})$ and $\omega_{1}^{1+\varepsilon}\in\tilde{A}_{q} ^{I}(\mathbb{R}^{n}), \, \omega_{2}^{1+\varepsilon}\in\tilde{A}_{q} ^{I}(\mathbb{R}^{m})$. Thus, by (4.25) we get

for some positive constant $C_{p}$ independent of $A$. Then, by the same argument as in Case $1$, we obtain (4.27) for $0 < \gamma < 1$ and $1 < p < 2$.

Case 3. $p = 2$. We choose $\varepsilon > 0$ such that $\omega _{1}^{1+\varepsilon}\in\tilde{A}_{2}^{I}(\mathbb{R}^{n}), \omega_{2} ^{1+\varepsilon}\in\tilde{A}_{2}^{I}(\mathbb{R}^{n})$. Then, we follow a similar argument as in the previous two cases and get

for $0 < \gamma < 1$ and $p = 2$.

Finally, by (4.15), (4.22), and (4.27)–(4.29), we get (4.5). This completes the proof of Proposition 4.1.

Proof (of Theorem 1.5): Assume that $\Omega\in L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})$. We write $\Omega$ as

where $\Omega_{k}$ satisfies (1.6)–(1.7), $\Vert\Omega_{\kappa }\Vert_{1}\leq4$, $\, \Vert\Omega_{k}\Vert_{2}\leq2^{2(k+1)}$, and the estimate

By (4.30) and Minkowski's inequality, we have

Thus, by Proposition 4.1 with $A = 2^{2(k+1)}$, we have

This completes the proof.

Proof (of Theorem 1.7): The proof follows a similar argument as in the proof of Theorem 1.5. We omit the details.

5.   Conclusions

In this paper, we proved the weighted $L^{p}$ boundedness of Marcinkiewicz integral operators along surfaces. We considered surfaces that are determined by functions satisfying some growth conditions or mappings that are more general than polynomials and convex functions. We proved the weighted $L^{p}$ boundedness of related square functions and maximal functions. The argument in this paper can be used to treat more general integral operators. This shall be the topic of future research.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Authors cordially thank the reviewers for their useful comments on the manuscript. The authors would like to thank Sultan Qaboos University for paying the APC.

Conflict of interest

The authors declare no conflicts of interest.