Weighted L^p boundedness of maximal operators with rough kernels

1.   Introduction and statement of results

Let $\mathbb{S}^{d-1}$ be the unit sphere in the $d$-dimensional Euclidean space $\mathbb{R}^{d}(d \geq 2)$ which is equipped with the normalized Lebesgue surface measure $d\sigma = d\sigma (\cdot)$.

Let $h:\mathbb{R}^{+}\rightarrow \mathbb{C}$ be a radial function satisfying

and let $\mho$ be a homogeneous function of degree zero on $\mathbb{R}^{d}$ with $\mho \in L^{1}(\mathbb{S}^{d-1})$ and

where $x^{\prime } = x/|x|$ for $x\in \mathbb{R}^{d}\setminus \{0\}$.

For a Schwartz function $f\in (\mathbb{R}^{d})$, we consider the maximal operator $\mathfrak{M}_{\mho, P}$ given by

where $P:\mathbb{R}^{d}\rightarrow \mathbb{R}$ is a real-valued polynomial.

We notice that if $P(y)\equiv 0$, then the operator $\mathfrak{M}_{\mho, P}$ is reduced to be the classical maximal operator denoted by $\mathfrak{M} _{\mho }$, which was introduced by Chen and Lin in ^[1]. The authors of ^[1] proved the boundedness of $\mathfrak{M}_{\mho }$ on $L^{p}(\mathbf{\mathbb{R}}^{d})$ for $2d/(2d-1) < p < \infty$ if $\mho \in \mathcal{C(\mathbb{S}}^{d-1})$, and they showed that the range of $p$ is the best possible. This result was extended in ^[2] in which the author confirmed the $L^{p}$ boundedness of $\mathfrak{M}_{\mho }$ for all $p\in \lbrack 2, \infty)$ whenever $\mho \in L(\log L)^{1/2}(\mathfrak{ \mathbb{S}}^{d-1})$, and that the condition $\mho \in L(\log L)^{1/2}(\mathfrak{\mathbb{S}}^{d-1})$ is optimal in the sense that the operator $\mathfrak{M}_{\mho }$ may fail to be bounded on $L^{2}(\mathbf{\mathbb{R}} ^{d})$ when $\mho \in L(\log L)^{r}(\mathfrak{\mathbb{S}}^{d-1})$ for any $r\in (0, 1/2)$. On the other hand, the author of ^[3] proved that $\mathfrak{M}_{\mho }$ is bounded on $L^{p}(\mathbf{\mathbb{R}}^{d})$ for $p\geq 2$ if $\mho$ lies in the block spaces $B_{q}^{(0, -1/2)}(\mathbb{S} ^{d-1})$ with $q > 1$, and they also proved that if the kernel $\mho$ belongs to $B_{q}^{(0, r)}(\mathbb{S}^{d-1})$ for some $r\in (-1, -1/2)$, then $\mathfrak{M }_{\mho }$ may not be bounded in $L^{2}(\mathbf{\mathbb{R}}^{d})$. In ^[4], the author generalized the above results. In fact, he proved that $\mathfrak{M}_{\mho, P}$ is bounded on $L^{p}(\mathbf{\mathbb{R}} ^{d})$ for all $p\geq 2$ provided that $\mho \in B_{q}^{(0, -1/2)}(\mathbb{ S}^{d-1})\cup L(\log L)^{1/2}(\mathfrak{\mathbb{S}}^{d-1})$. Subsequently, the investigation of the boundedness of $\mathfrak{M}_{\mho, P}$ on $L^{p}(\mathbf{\mathbb{R}}^{d})$ under various conditions has attracted the attention of many authors: For background information ^[5,6,7,8], importance and the development ^[9,10,11], and recent advances and studies ^[12,13].

On the other hand, in ^[14] Y. Ding and H. Qingzheng proved the weighted $L^{p}$ boundedness of $\mathfrak{M}_{\mho }$ as described in the following theorem.

Theorem A. Let $d\geq 2$. Assume $\mho \in L^{2}(\mathbb{S}^{d-1})$ satisfies (1.1). Then,

if $p$ and $\omega$ satisfy one of the following conditions:

(a) $2\leq p < \infty$ and $\omega \in A_{p/2}(\mathbb{R}^{d})$;

(b) $2d/(2d-1) < p < 2$, $\omega (x) = \left\vert x\right\vert ^{{\alpha }},$ and $\frac{1}{2}(1-d)(2-p) < \alpha <$ $\frac{1}{2}(2dp-2d-p),$ where $A_{p}$ is the Muckenhoupt's weight class, and the weighted $L^{p}(\mathbf{\ \omega, \mathbb{R}}^{d})$ with $\omega \geq 0$ is defined by

Subsequently, Al-Qassem in ^[15] generalized the above result as in the following theorem:

Theorem B. Suppose that $\mho \in L^{q}(\mathbb{S}^{d-1})$ for some $q > 1$ and it satisfies (1.1). Then,

if $p$ and $\omega$ satisfy one of the following conditions:

$(a)$ $\delta \leq p < \infty$ and $\omega \in A_{p/\delta }$;

$(b)$ $2d\delta /(2d+d\delta -2) < p < 2$, $\omega (x) = \left\vert x\right\vert ^{{\alpha }},$ $\frac{1}{2}(1-d)(2-p) < \alpha <$ $\frac{1}{2} (2dp-2d-p)$, where $\delta = \max \{2, q^{\prime }\}$ and $q^{\prime }$ is the dual exponent of $q.$

In view of the results in ^[4] concerning the $L^{p}$ boundedness of $\mathfrak{M}_{\mho, P}$ and of the results in ^[15] concerning the weighted $L^{p}$ boundedness of $\mathfrak{M}_{\mho }$, it is natural to ask wether the weighted $L^{p}$ boundedness of $\mathfrak{M}_{\mho, P}$ holds under the same conditions as assumed in Theorem B. We shall obtain an answer to this question in the affirmative as described in the following theorem.

Theorem 1.1. Let $\mho \in L^{q}(\mathbb{S}^{d-1})$ with $q > 1$. Suppose that $P:\mathbb{R}^{d}\rightarrow \mathbb{R}$ is a polynomial of degree $k$, then the estimate

holds for $\delta \leq p < \infty$ and $\omega \in A_{p/\delta }$, where $\delta = \max \{2, q^{\prime }\}$.

Now let us give some results which follow as a consequence of Theorem 1.1. For $\gamma \in (1, \infty)$, we let ${L^{\gamma }(\mathbb{R}^{+}, \frac{ds}{s})}$ be the set of all measurable functions $h:\mathbb{R} ^{+}\rightarrow \mathbb{R}$ such that

Consider the maximal operator $\mathfrak{M}_{\mho, P}^{(\gamma)}$ given by

where $P:\mathbb{R}^{d}\rightarrow \mathbb{R}$ is a real-valued polynomial, $f\in \mathcal{S}({\mathbb{R}}^{d})$ and $1\leq \gamma \leq 2$.

The study of the boundedness of the operator $\mathfrak{M}_{\mho, P}^{(\gamma)}$ started in ^[1] in which the authors proved that if $\mho \in \mathcal{C(\mathbb{S}}^{d-1})$ and $h\in {L^{\gamma }(\mathbb{R} ^{+}, \frac{ds}{s})}$ for some $1\leq \gamma \leq 2$, then the $L^{p}(\mathbb{ R}^{d})$ boundedness of the operator $\mathfrak{M}_{\mho, 0}^{(\gamma)}$ is satisfied for $(\gamma d)^{\prime } < p < \infty$. For more information about the investigation of $\mathfrak{M}_{\mho, P}^{(\gamma)}$, under various conditions and some past studies, readers are referred to see ^[16,17,18] and the references therein. In this work, an extension and improvement over the result in ^[1] shall be obtained by proving the weighted $L^{p}$ of $\mathfrak{M}_{\mho, P}^{(\gamma)}$ when the condition $\mho \in \mathcal{C(\mathbb{S}}^{d-1})$ is replaced by the weaker condition $\mho \in L^{q}(\mathbb{S}^{d-1})$ with $q > 1$. Precisely, we have the following:

Theorem 1.2. Let $\mho \in L^{q}(\mathbb{S}^{d-1})$ with $q > 1$. Let $\omega \in A_{p/\delta }$ and $h\in {L^{\gamma }(\mathbb{R}^{+}, \frac{ds}{s})}$ with $1\leq \gamma \leq 2$. Then, we have

for $(\delta \gamma ^{\prime })/2\leq p < \infty$.

Concerning the boundedness of a certain class of oscillatory singular integrals, we have the following:

Theorem 1.3. Assume that $\mho \in L^{q}(\mathbb{S}^{d-1})$ with $q > 1$. Let $\omega \in A_{p/\delta }$ and $h\in {L^{\gamma }(\mathbb{R}^{+}, \frac{ds }{s})}$ for some $1 < \gamma \leq 2$. Then, the oscillatory singular integral operator $T_{\mho, P}^{(\gamma)}$ given by

is bounded on $L^{p}(\omega, \mathbb{R}^{d})$ for $(\delta \gamma ^{\prime })/2\leq p < \infty$, and it is bounded on $L^{p}(\omega, \mathbb{R}^{d})$ for $1 < p\leq (\frac{\delta \gamma ^{\prime }}{2})^{\prime }$ and $\omega ^{1-p^{\prime }}\in A_{p^{\prime }/\delta }$.

For background information and related work about the operator, see ^{[19,20,21,22,23,24]}.

We point out that the generalized Marcinkiewicz operator concerning the operator $\mathfrak{M}_{\mho, P}^{(\gamma)}$ is given by

As an immediate consequence of the fact

for $1\leq \gamma \leq 2$, we obtain the following result:

Theorem 1.4. Let $\mho$, $\omega$, $P$, and $\gamma$ be given as in Theorem 1.2. Then, the generalized Marcinkiewicz integral $\mathcal{M }_{\mho, P}^{(\gamma)}$ is bounded on $L^{p}(\omega, \mathbb{R}^{d})$ for $(\delta \gamma ^{\prime })/2\leq p < \infty$ with $1 < \gamma \leq 2$.

It is clear that for the special case $P = 0$ and $\gamma = 2$, the operator $\mathcal{M}_{\mho, 0}^{(2)}$ reduces to the classical Marcinkiewicz integral operator, which was introduced in ^[25], in which the author proved that the operator is bounded on $L^{p}(\mathbb{R}^{d})$ only for $1 < p\leq 2$ whenever $\mho \in Lip_{\eta }(\mathbb{S}^{d-1})$ for some $0 < \eta \leq 2$. Thereafter, the study of the operator $\mathcal{M}_{\mho, P}^{(\gamma)}$ under several conditions has been discussed by many mathematicians (see, for instance ^{[4,26,27,28,29,30]}).

Throughout the rest of the paper, the letter $C$ stands for a positive constant which is independent of the essential variables and its value is not necessary the same at each occurrence.

2.   Preliminary lemmas

In this section, we give some preliminary lemmas to prove our main results. Let us start with the following lemma, which is found in ^[4].

Lemma 2.1. Let $\mho \in L^{q}(\mathbb{S}^{d-1})$, $q > 1$ be a homogeneous function of degree zero. Suppose that

is a polynomial of degree $k > 1$ such that $\left\vert u\right\vert ^{k}$ is not one of its terms. For $j\in \mathbb{Z}$, define $I_{j, \mho }:\mathbb{R} ^{d}\rightarrow \mathbb{R}$ by

Then, there exist constants $C > 0$ and $0 < \epsilon < 1$ such that

We need the following lemma from ^[15].

Lemma 2.2. Let $\mho \in L^{q}(\mathbb{S}^{d-1})$ for some $q > 1$ and $\omega \in {A}_{p/q^{\prime }}(\mathbb{R}^{+})$ with $1 < p < \infty$. Assume that the maximal function ${M}_{\mho }$ is given by

Then there exists a positive constant $C_{p}$ such that

for any $f\in L^{p}(\omega, \mathbb{R}^{d})$ with $q^{\prime }\leq p < \infty$.

The next lemma can be proved by employing the same argument as in the proof of Theorem 1.1 in ^[15].

Lemma 2.3. Let $\omega \in A_{p/\delta }$ and $\mho \in L^{q}(\mathbb{S}^{d-1})$ with $q > 1$. Then, there is a constant $C_{p} > 0$ such that

for all $\delta \leq p < \infty$.

Proof. Let $\left\{ \psi _{j}\right\} _{j\in \mathbb{Z}}$ be a smooth partition of unity in $(0, \infty)$ with the following properties:

For $j\in \mathbb{Z}$, define the operator $\Upsilon _{j}$ in $\mathbb{R}^{d}$ by

Then, for $f\in \mathcal{S}(\mathbb{R}^{d})$, we have that

where

By following the same argument utilized in the proof of Theorem 1.1 in ^[15], along with invoking Lemma 2.1, we obtain that

for some constant $\tau \in (0, 1)$ and for all $\delta \leq p < \infty$. Consequently, by (2.4) and (2.5), we get (2.2) for all $\delta \leq p < \infty$. □

3.   Proof of the main results

Proof of Theorem 1.1. We shall use some of the ideas from ^[4]. Precisely, we use the induction on the degree of the polynomial $P$. It is clear that if the degree of $P$ is $0$, then by Lemma 2.3 we get

for all $\delta \leq p < \infty$. Now, if the degree of $P$ is $1$, then we deduce that there is $\overrightarrow{c}\in \mathbb{R}^{d}$ so that $P(u) = \overrightarrow{c}\cdot u$. Hence, if we set $g(u) = e^{-iP(u)}f(u)$, then by (3.1) we get that

Next, suppose that (1.4) holds for any polynomial $P$ whose degree is less than or equal to $k\geq 1$. We need to prove that the inequality (1.4) is also satisfied for any polynomial of degree $k+1$. Let

be a polynomial of degree $k+1$. Without loss of generality, we may assume that $P$ does not contain $\left\vert u\right\vert ^{k+1}$ as one of its terms, and $\sum\limits_{\left\vert \eta \right\vert = k+1}\left\vert \lambda _{\eta }\right\vert = 1.$

For ${j\in \mathbb{Z}}$, let $\left\{ \psi _{j}\right\}$ and $\Upsilon _{j}$ be chosen as those in (2.3). Set

Then, $\Gamma _{\infty }(s)+\Gamma _{0}(s) = 1$, supp$\left(\Gamma _{\infty }(s)\right) \subseteq \lbrack 2^{-1}, \infty)$, and supp$\left(\Gamma _{0}(s)\right) \subseteq (0, 1].$ Hence, we get by Minkowski's inequality that

where

and

Let us estimate $\left\Vert \mathfrak{M}_{\mho, P, \infty }(f)\right\Vert _{L^{p}(\omega, \mathbb{R}^{d})}$. Define

Then, by the generalized Minkowski's inequality, we have

Case 1. When $q\geq 2$. In this case, we have $2\leq p < \infty$ and $\omega \in A_{p/2}$. Let us consider first the case $p > 2$. By duality, there is $g\in L^{(p/2)^{\prime }}(\omega ^{1-{(p/2)^{\prime }}}, \mathbb{R} ^{d})$ such that $\left\Vert g\right\Vert _{L^{(p/2)^{\prime }}(\omega ^{1-{ (p/2)^{\prime }}}, \mathbb{R}^{d})}\leq 1$ and

Hence, by Hölder's inequality, we get

where $\widetilde{g}(y) = g(-y)$ and $M^{\ast }(f)$ is the Hardy-Littlewood maximal function. Thus,

for $2 < p < \infty$ and $\omega \in A_{p/2}$.

Now, for the case $p = 2$ and $\omega \in A_{1}$, we have

where the last inequality is obtained by the fact that $M ^{\ast }\omega (x)\leq C\omega (x)$ for a.e.$x\in\mathbb{R}^{{d}}$.

Since for any $\omega \in A_{p/2}$ there exists $\alpha > 0$ such that $\omega ^{1+\alpha }\in A_{p/2}$, by (3.4) and (3.5), we get that

for $2 < p < \infty$ and $\omega \in A_{p/2}$.

Nowwe will obtain a sharp unweighted $L^{2}$ estimate of $\mathfrak{M}_{\mho, P, \infty, j}(f)$. By Fubini's theorem, Plancherel's theorem and Lemma 2.1 we get

Thus, using the Stein-Weiss interpolation theorem with change of measure ^[31], we may interpolate between (3.6) and (3.7) to obtain

for $2\leq p < \infty$, $\omega \in A_{p/2}$, and for some $\varepsilon \in (0, 1)$. Consequently, by (3.3) and (3.8), we conclude that

for $2\leq p < \infty$ and $\omega \in A_{p/2}$.

Case 2. When $1 < q < 2$. In this case, we have $q^{\prime }\leq p < \infty$ and $\omega \in A_{p/q^{\prime }}$. Since $p > 2$, by duality, there exists $F\in L^{(p/2)^{\prime }}(\omega ^{1-{(p/2)^{\prime }}}, \mathbb{ R}^{d})$ such that $\left\Vert F\right\Vert _{L^{(p/2)^{\prime }}(\omega ^{1- {(p/2)^{\prime }}}, \mathbb{R}^{d})}\leq 1$ and

Hence, by Hölder's inequality, we get

where $\widetilde{F}(y) = F(-y)$. The last inequality holds since $(p/2)^{\prime } > q/(2-q)$ and by invoking Lemma 2.2. Therefore, we have

for $q^{\prime }\leq p < \infty$ and $\omega \in A_{p/q^{\prime }}$. By the last inequality and (3.3), we have that

for $\delta \leq p < \infty$ and $\omega \in A_{p/\delta }$.

Now, let us estimate the $\left\Vert \mathfrak{M}_{\mho, P, 0}(f)\right\Vert _{L^{p}(\omega, \mathbb{R}^{d})}$. Take $Q(x) = \sum\limits_{\left\vert \eta \right\vert \leq k}\lambda _{\eta }x^{\eta }$, and let $\mathfrak{M}_{\mho, Q, 0}(f)$ and $\mathfrak{M}_{\mho, P, Q, 0}(f)$ be given by

and

By Minkowski's inequality, we deduce that

Since the degree of the polynomial $Q$ is less than or equal to $k$, we have that

for all $\delta \leq p < \infty$ and $\omega \in A_{p/\delta }$. By noticing that

and using the Cauchy-Schwartz inequality, we obtain

Therefore, by following the same arguments as above, we obtain that

for all $\delta \leq p < \infty$ and $\omega \in A_{p/\delta }$. Hence, by (3.13) and (3.14), we deduce that

Consequently, by (3.2), (3.9), (3.11) and (3.15), the proof of Theorem 1.1 is complete.

Proof of Theorem 1.2. By duality, it is easy to get that

for all $1 < \gamma \leq 2$. Hence,

where $S:L^{p}(\omega, \mathbb{R}^{d})\rightarrow L^{p}(L^{\gamma ^{\prime }}(\mathbb{R}^{+}, \frac{ds}{s}), \omega, \mathbb{R}^{d})$ is a linear operator given by

Now, if $\gamma = 1$, $f\in L^{\infty }(\mathbb{R}^{d})$ and $h\in L^{1}(\mathbb{R}^{+}, \frac{ds}{s}),$ then we have that

and, hence,

which, in turn, implies

Since $L^{\infty }(\mathbb{R}^{d}, \omega) = L^{\infty }(\mathbb{R}^{d})$, we have

On the other hand, by Theorem 1.1 we get

for $\delta \leq p < \infty$. Therefore, by applying the interpolation theorem for the Lebesgue mixed normed spaces to (3.16) and (3.17), we deduce that

for all $(\delta \gamma ^{\prime })/2\leq p < \infty$ with $1 < \gamma \leq 2$.

Proof of Theorem 1.3. To begin, we notice that $\left(T_{\mho, P}^{(\gamma)}f\right) (x) = \lim_{\varepsilon \rightarrow 0}T_{\mho, P, \varepsilon }^{(\gamma)}f(x),$ where $T_{\mho, P, \varepsilon }^{(\gamma)}$ is the truncated singular integral operator given by

By Hölder's inequality, we deduce

Hence,

Therefore, by Theorem 1.2, we get that $T_{\mho, P}^{(\gamma)}$ is bounded on $L^{p}(\omega, \mathbb{R}^{d})$ for $(\delta \gamma ^{\prime })/2\leq p < \infty$ and $\omega \in A_{p/\delta }$. On the other hand, by a standard duality argument, we get that $T_{\mho, P}^{(\gamma)}$ is bounded on $L^{p}(\omega, \mathbb{R}^{d})$ for $1 < p\leq (\frac{\delta \gamma ^{\prime }}{2})^{\prime }$ and $\omega ^{1-p^{\prime }}\in A_{p^{\prime }/\delta }$. The proof is complete.

4.   Conclusions

In this work, we studied the mapping properties of the maximal integral operators $\mathfrak{M}_{\mho, P}^{(\gamma)}$. In fact, we proved the weighted space $L^p(\omega, \mathbb{R}^d)$ boundedness of $\mathfrak{M}_{\mho, P}^{(\gamma)}$ for all $(\delta \gamma ^{\prime })/2\leq p < \infty$ whenever $\omega \in A_{p/\delta }$, $\mho\in L^{q}(\mathbb{S} ^{d-1})$, and $1\leq \gamma \leq 2$. Then, as consequence of the this result, we confirmed the weighted $L^p(\omega, \mathbb{R}^d)$ boundedness of the operators $T_{\mho, P}^{(\gamma)}$ and $\mathcal{M}_{\mho, P}^{(\gamma)}$. The results of this paper are substantial extensions and improvements of the main results in ^[4] and ^[15].

Author contributions

Mohammed Ali: Writing-original draft, Formal Analysis, Commenting; Hussain Al-Qassem: Writing-original draft, Commenting. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors would like to express their gratitude to the editor for handling the full submission of the manuscript.

Conflict of interest

The authors declare that they have no conflicts of interest in this paper.