Some weighted estimates for the commutators of $p$-adic Hardy operator on two weighted $p$-adic Herz-type spaces

1.   Introduction

In mathematical analysis Hardy operator is considered an important averaging operator as it plays a vital role in many branches of mathematics, such as complex analysis, partial differential equations and harmonic analysis (for example, see ^{[2,7,8,10,29]}). In ^[6], Hardy introduced the one-dimensional Hardy operator

for a measurable function $f \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$. The operator in (1.1) satisfies the below inequality

where the constant $q/(q-1)$ is sharp. An extension of the operator $H$ on higher dimensional space $\mathbb R^n$ was defined in ^[3] by Faris as

where $|{\bf{x}}| = \left(\sum_{i = 1}^{n} x_i^{2}\right)^{1/2}$ for ${\bf{x}} = (x_1, \ldots, x_n)$. Furthermore, Christ and Grafakos ^[1] acquired the exact value of the norm of operator $H$ defined by (1.3). Recently, Hardy operator has gained a tremendous amount of consideration, see for example ^{[16,20,24,30,33,34]} and the references therein.

In the past few decades there has been a relentless attention in $p$-adic models appearing in various branches of science. The applications of $p$-adic analysis are found mainly in the field of mathematical physics (see, for example, ^[15,26,27]). Importantly, many current researchers are paying a valiant effort to harmonic analysis on $p$-adic field ^{[9,10,11,13,17,21,22,25]}.

Let $\mathbb{Q}$ be a field of rational numbers and $p$ a prime number. We introduce a so called p-adic norm $|x|_p$ on $\mathbb{Q}$ by a rule $|x|_p = \{0\}\cup\{p^{-\gamma}:\gamma\in\mathbb{Z}\},$ where $\gamma = \gamma(x)$ is defined from the following representation

integers $s$ and $t$ are coprime to $p$. $|\cdot|_p$ fulfills all the axioms of a real norm along with the following non-Archimedean property:

The field of $p$-adic numbers $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ with respect to $|\cdot|_{p}.$ Any nonzero $p$-adic number can be written in canonical form (see ^[28]) as:

where $\alpha_j, \gamma\in\mathbb Z, \alpha_j\in\frac{\mathbb{Z}}{p\mathbb{Z}_{p}}, \alpha_{0}\neq0.$ Interestingly, the series in (1.5) is convergent with respect to $|\cdot|_{p}$ because $|p^\gamma\alpha_{k}p^{j}|_{p} = p^{-\gamma-j}.$

The higher dimensional $p$-adic vector space $\mathbb Q_p^n$ consists of points ${\bf{x}} = (x_{1}, x_{2}, ..., x_{n}),$ where $x_i\in\mathbb Q_p, i = 1, 2, ..., n,$ with the following norm

Let

be the ball and sphere respectively with center at ${\bf{a}} \in \mathbb{Q}_p^n$ and radius $p^{\gamma}.$ If ${\bf{a}} = {\bf{0}},$ we may write $B_\gamma({\bf{0}}) = B_\gamma, \ S_\gamma({\bf{0}}) = S_\gamma.$

It is well known that the space $\mathbb{Q}_p^n$ is locally compact commutative group under addition, then there exists a translation invariant Haar measure $d{\bf{x}}$ which is normalized such that

where $|A|_{H}$ represents the Haar measure of a measurable subset $A$ of $\mathbb{Q}_p^n.$ Moreover, one can easily show that $|B_{\gamma}({\bf{a}})|_{H} = p^{n\gamma}$ and $|S_{\gamma}({\bf{a}})|_{H} = p^{n\gamma}(1-p^{-n})$, for any ${\bf{a}}\in \mathbb{Q}_p^n$.

In what follows the $p$-adic Hardy operator

and its commutator

were defined and studied for $f, b\in L_1^{loc}(\mathbb{Q}_{p}^{n})$ in ^[4]. In the same paper, Fu et al. acquired the boundedness of $p$-adic Hardy operator and its commutator on Lebesgue spaces and Herz spaces. On the Morrey-Herz spaces, the $p$-adic Hardy type operators and their commutators are reported in ^[5]. For complete comprehension of $p$-Hardy operator and its commutator, we refer the publications ^{[12,18,31,32]}.

The purpose of the current article is to discuss the weighted central bounded mean oscillations and weighted $p$-adic Lipschitz estimates of $H^{p}_{b}$ on two weighted $p$-adic Herz spaces and $p$-adic Morrey-Herz spaces. Throughout this article a letter $C$ denotes a constant whose value may change at its different places. It is mandatory to recall the definitions of relevant $p$-adic function spaces before moving to our results.

Suppose $w({\bf{x}})$ is a nonnegative function on $\mathbb{Q}_p^n.$ The weighted measure of $A$ is denoted and defined as $w(A) = \int_{A}w({\bf{x}})d{\bf{x}}$. The weighted $p$-adic Lebesgue space $L^q(w, \mathbb{Q}_p^n), (0 < q < \infty)$ is defined to be the space of all measurable functions f on $\mathbb{Q}_p^n$ such that:

The theory of $A_{q}$ weights on $\mathbb{R}^{n}$ was introduced by Benjamin Muckenhoupt in ^[19]. Let us recall the definition of $A_{q}$ weights in $p$-adic setting.

Definition 1.1. ^[23] A weight function $w\in A_{q}(1\leq q < \infty)$ if there exists a constant $C$ free from choice of $B\subset\mathbb{Q}_p^n$ such that

For the case $q = 1, w\in A_{1}$, we have

for every $B\subset\mathbb{Q}_p^n.$

Remark 1.2. A weight function $w\in A_{\infty}$ if it undergoes the stipulation of $A_{q}(1\leq q < \infty)$ weights.

Definition 1.3. Suppose $w$ is a weight function and $1\leq q < \infty.$ The $p$-adic space $CMO^{q}(w, \mathbb{Q}_p^n)$ is defined by

where

Definition 1.4. ^[22] Suppose $w_{1}$ and $w_{2}$ are weight functions, $0 < r, q < \infty$ and $\alpha \in \mathbb{R}$. Then the two weighted $p$-adic Herz space $K^{\alpha, r}_{q}(w_{1}, w_{2})$ is defined as

where

and $\chi_{k}$ is the characteristic function of the sphere $S_{k} = B_{k}\setminus B_{k-1}$.

Remark 1.5. Obviously $K^{0, q}_{q}(w_{1}, w_{2}) = L^q(w_{2}, \mathbb{Q}_p^n).$

Definition 1.6. ^[22] Suppose $w_{1}$ and $w_{2}$ are weight functions, $0 < r, q < \infty,$ $\alpha \in \mathbb{R}$ and $\lambda\geq0.$ Then the two weighted $p$-adic Morrey-Herz space $MK^{\alpha, \lambda}_{r, q}(w_{1}, w_{2})$ is defined as follows

where

Remark 1.7. It is evident that $MK^{\alpha, 0}_{r, q}(w_{1}, w_{2}) = K^{\alpha, r}_{q}(w_{1}, w_{2}).$

Definition 1.8. ^[23] Suppose $1\leq q < \infty,$ $0 < \beta < 1$ and $w$ is a weight function. The $p$-adic space $Lip_{\beta}(w, \mathbb{Q}_p^n)$ is defined as

where

2.   Weighted $CMO$ estimates of $H^{p}_{b}$ on two weighted $p$-adic Herz-type spaces

The following section discusses the weighted $CMO$ estimates of $H^{p}_{b}$ on two weighted $p$-adic Herz-type spaces. We open up the section with few lemmas which are useful in proving key results.

Lemma 2.1. ^[14] Suppose $w\in A_{1},$ then there exist constants $C_{1}, C_{2}$ and $0 < \mu < 1$ such that

for any measurable subset $A$ of a ball $B$.

Remark 2.2. If $w\in A_{1}$, then it follows from lemma (2.1) that there exist constants $C$ and $\mu\quad(0 < \mu < 1)$ such that $\frac{w(B_{k})}{w(B_{i})}\leq Cp^{(k-i)n}$ as $i < k$ and $\frac{w(B_{k})}{w(B_{i})}\leq Cp^{(k-i)n\mu}$ as $i\geq k$.

Lemma 2.3. ^[23] Suppose $w\in A_{1}$ and $b\in CMO^{q}(w, \mathbb{Q}_p^n),$ then there is a constant $C$ such that for $i$, $k\in\mathbb{Z},$

Lemma 2.4. ^[23] Suppose $w\in A_{1},$ then for $1 < q < \infty,$

where $1/q+1/q' = 1$.

Now we state the result about the boundedness of $H^{p}_{b}$ on two weighted $p$-adic Herz-type spaces.

Theorem 2.5. Let $0 < r_{1}\leq r_{2} < \infty$, $1\leq r, q < \infty$ and let $w\in A_{1}.$ If $\alpha < \frac{n\mu}{q'}$, then the inequality

holds for all $b\in CMO^{r \max\{q, q'\}}(w, \mathbb{Q}_p^n)$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

If $\alpha = 0, r_{1} = r_{2} = q$, then we have the following result.

Corollary 2.6. Let $1\leq r, q < \infty$ and $w\in A_{1}$, then

holds for all $b\in CMO^{r \max\{q, q'\}}(w, \mathbb{Q}_p^n)$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

Theorem 2.7. Let $0 < r_{1}\leq r_{2} < \infty, \; 1\leq r, q < \infty$ and let also $w\in A_{1}$ and $\lambda > 0.$ If $\alpha < \frac{n\mu}{q'}+\lambda,$ then

holds for all $b\in CMO^{r\max\{q, q'\}}(w, \mathbb{Q}_{p}^{n})$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

Proof of Theorem 2.5: First, by the definition we have

Since $w\in A_{1}\subset A_{q},$ making use of Hölder's inequality along with lemma 2.4, we have

To estimate $I,$ by the application of Hölder's inequality, Remark 2.2 along with inequality (2.2), we are down to

Now, we estimate $II$ as follows

Next, applying Hölder's inequality to deduce

By the application of Hölder's inequality, inequality (2.5), lemma 2.4 and Remark 2.2, we are in a position to estimate $II_{1}.$

Next task is to estimate $II_{2}.$ For this, we use Hölder's inequality, lemmas 2.3 and 2.4, Remark 2.2 and inequality (2.2)

From (2.3), (2.6) and (2.7) together with Jensen's Inequality, we have

Therefore,

In what follows we consider two cases, $0 < r_{1}\leq 1$ and $r_{1} > 1.$

Case 1: When $0 < r_{1}\leq 1$ and $\alpha < n\mu/q'$, we have

Case 2: Whenever $r_{1} > 1$, an application of Hölder's inequality with $\alpha < n\mu/q'$, we get

Hence, the proof of theorem is completed.

Proof of Theorem 2.7: From theorem 2.5, we have

By definition of weighted $p$-adic Morrey-Herz spaces and Jensen's Inequality along with $\alpha < n\mu/q'+\lambda$, $\lambda > 0$ and $1 < r_{1} < \infty,$ we reach at

3.   Weighted $p$-adic Lipschitz estimates of $H^{p}_{b}$ on two weighted $p$-adic Herz-type Spaces

The current section deals the weighted $p$-adic Lipschitz estimates of $H^{p}_{b}$ on two weighted $p$-adic Herz-type spaces. The outset of a section is with a lemma which is helpful in proving main results.

Lemma 3.1. ^[23] Suppose $w\in A_{1}$ and $b\in Lip_{\beta}(w, \mathbb{Q}_p^n),$ then there is a constant $C$ such that for $i$, $k\in\mathbb{Z},$

Now, we state the result about the boundedness of commutator of $p$-adic Hardy operator on two weighted $p$-adic Herz-type spaces.

Theorem 3.2. Let $0 < r_{1}\leq r_{2} < \infty$, $1\leq q_{1}, q_{2} < \infty, \;1/q_{1}-1/q_{2} = \beta/n$ and let $w\in A_{1}.$ If $\alpha < \frac{n\mu}{q_{1}'}$, then the inequality

holds for all $b\in Lip_{\beta}(w, \mathbb{Q}_p^n)$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

If $\alpha = 0, r_{1} = q_{1} = p$ and $r_{2} = q_{2} = q,$ then we have the following corollary.

Corollary 3.3. Let $1\leq q < \infty$, $1/q_{1}-1/q_{2} = \beta/n$ and $w\in A_{1}$, then

holds for all $b\in Lip_{\beta}(w, \mathbb{Q}_p^n)$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

Theorem 3.4. Let $0 < r_{1}\leq r_{2} < \infty$, $1\leq q_{1}, q_{2} < \infty, \;1/q_{1}-1/q_{2} = \beta/n$ and let $w\in A_{1}.$ If $\alpha < \frac{n\mu}{q_{1}'}+\lambda$, then

holds for all $b\in Lip_{\beta}(w, \mathbb{Q}_p^n)$ and $f\in L_{\rm loc}(\mathbb{Q}_p^n).$

Proof of Theorem 3.2: In a similar fashion as that of theorem 2.5, we get

For the evaluation of $L,$ we apply Hölder's inequality, Remark 2.2, $\beta/n = 1/q_{1}-1/q_{2}, \;w\in A_{1}\subset A_{q_{1}},$ and inequality (2.2) to get

In order to evaluate $LL$, we proceed as follows

The following preparation will do world of good to estimate $LL_{1}.$ Using Hölder's inequality, we have

To evaluate $LL_{1},$ we apply Hölder's inequality, inequality (3.4), lemma 2.4 and Remark 2.2.

Next step is to evaluate $LL_{2}.$ For this we use Hölder's inequality, lemmas 3.1 and 2.4, inequality (2.2), and Remark 2.2 to get

Remaining proof is more or less same to the proof of theorem 2.5. Thus, we conclude the theorem.

Proof of Theorem 3.4: Let $\alpha < n\mu/q_{1}'+\lambda$. By the definition of weighted $p$-adic Morrey-Herz spaces together with inequalities (3.2), (3.5), and (3.6), we have

Next by applying the similar arguments as in theorem 2.7, we get

So, the proof of the theorem is finished.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudia Arabia for funding this work through research groups program under grant number R.G. P-2/29/42.

Conflict of interest

The authors declare that they have no conflict of interest.