The critical delay of the consensus for a class of multi-agent system involving task strategies

Yipeng Chen; Yicheng Liu; Xiao Wang; Yipeng Chen; Yicheng Liu; Xiao Wang

doi:10.3934/nhm.2023021

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 513-531. doi: 10.3934/nhm.2023021

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Research article

The critical delay of the consensus for a class of multi-agent system involving task strategies

Department of Mathematics, National University of Defense Technology, Changsha, P.R. China

Received: 26 August 2022 Revised: 25 November 2022 Accepted: 10 January 2023 Published: 18 January 2023

The time delay may induce oscillatory behaviour in multi-agent systems, which may destroy the consensus of the system. Therefore, the critical delay that is the maximum value of the delay to guarantee the consensus of the system, is an important performance index of multi-agent systems. This paper studies the influence of the processing delay on the consensus for a class of multi-agent system involving task strategies. The first-order system with a single delay and the second-order system with two different delays are investigated respectively. A critical delay independent of strategies and a critical region of the 2-D plane that depends on strategies is obtained for the first-order and the second-order system respectively. Specifically, a geometric method was used for the case of two different delays. Several numerical simulations are presented to explain the results.

Keywords:

Citation: Yipeng Chen, Yicheng Liu, Xiao Wang. The critical delay of the consensus for a class of multi-agent system involving task strategies[J]. Networks and Heterogeneous Media, 2023, 18(2): 513-531. doi: 10.3934/nhm.2023021

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Abstract

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

$\begin{equation} Hf(x) = \frac{1}{x}\int_{0}^{x}f(t)dt,\quad x > 0, \end{equation}$

(1.1)

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

$\begin{equation} \|H f\|_{L^{q}(\mathbb{R}^{+})}\leq\frac{q}{q-1}\|f\|_{L^{q}(\mathbb{R}^{+})}, \quad 1 < q < \infty, \end{equation}$

(1.2)

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

$\begin{equation} Hf({\bf{x}}) = \frac{1}{|{\bf{x}}|^{n}} \int_{|{\bf{t}}| \leq |{\bf{x}}|} f({\bf{t}}) d{\bf{t}}, \end{equation}$

(1.3)

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

$x = p^{\gamma}s/t,$

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:

$\begin{equation} |x+y|_p\le\max\{|x|_p,|y|_p\}. \end{equation}$

(1.4)

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:

$\begin{equation} x = p^\gamma\sum\limits_{j = 0}^{\infty}\alpha_jp^j, \end{equation}$

(1.5)

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

$\begin{equation} |{\bf{x}}|_{p} = \max\limits_{1\leq i \leq n}|x_{i}|_{p}. \end{equation}$

(1.6)

Let

$B_{\gamma}({\bf{a}}) = \{{\bf{x}} \in \mathbb{Q}_p^n:|{\bf{x}}-{\bf{a}}|_{p}\leq p^{\gamma}\}, \ S_{\gamma}({\bf{a}}) = \{{\bf{x}} \in \mathbb{Q}_p^n:|{\bf{x}}-{\bf{a}}|_{p} = p^{\gamma}\}$

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

$\int_{B_{0}}d{\bf{x}} = |B_{0}|_{H} = 1,$

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

$H^{p} f({\bf{x}}) = \frac{1}{|{\bf{x}}|_{p}^{n}} \int_{ |{\bf{t}}|_{p}\leq |{\bf{x}}|_{p}}f({\bf{t}})d{\bf{t}}$

and its commutator

$H^{p}_{b} f({\bf{x}}) = bH^{p}(f)-H^{p}(bf)$

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:

$\| f\|_{L^q(w,\mathbb{Q}_p^n)} = \bigg(\int_{\mathbb{Q}_p^n}|f({\bf{x}})|^q w({\bf{x}})d{\bf{x}}\bigg)^\frac{1}{q} < \infty.$

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

$\bigg(\frac{1}{|B|}\int_{B}w({\bf{x}})d{\bf{x}}\bigg)\bigg(\frac{1}{|B|}\int_{B}w({\bf{x}})^{-\frac{1}{q-1}}d{\bf{x}}\bigg)^{1/q}\leq C.$

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

$\frac{1}{|B|}\int_{B}w({\bf{x}})d{\bf{x}}\leq Cess\inf\limits_{{\bf{x}}\in B}w({\bf{x}}),$

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

$\begin{equation*} \|f\|_{CMO^{q}(w,\mathbb{Q}_p^n)} = \sup\limits_{ \gamma \in Z}\bigg(\frac{1}{w(B_{\gamma})}\int_{B_{\gamma}}|f(\bf x)-f_{B_{\gamma}}|^q w({\bf{x}})^{1-q} d{\bf{x}}\bigg)^{1/q}, \end{equation*}$

where

$\begin{equation} f_{B_{\gamma}} = \frac{1}{|B_{\gamma}|}\int_{B_{\gamma}}f({\bf{x}})d{\bf{x}}. \end{equation}$

(1.7)

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

$K^{\alpha,r}_{q}(w_{1},w_{2}) = \{f \in L^q_{\rm{loc}} (w_{2}, \mathbb{Q}_p^n\backslash\{0\}): \|f\|_{K^{\alpha,r}_{q}(w_{1},w_{2})} < \infty\},$

where

$\begin{equation} \| f\|_{K^{\alpha,r}_{q}(w_{1},w_{2})} = \bigg(\sum\limits_{k = -\infty}^{\infty}w_{1}(B_k)^{\alpha r/n}\|f\chi_{k}\|^{r}_{L^q(w_{2},\mathbb{Q}_p^n)}\bigg)^{1/r} \end{equation}$

(1.8)

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

$MK^{\alpha,\lambda}_{r,q}(w_{1},w_{2}) = \{f \in L^q_{loc} (w_{2}, \mathbb{Q}_p^n\backslash\{0\}): \|f\|_{MK^{\alpha,\lambda}_{r,q}(w_{1},w_{2})} < \infty\},$

where

$\begin{equation} \| f\|_{MK^{\alpha,\lambda}_{r,q}(w_{1},w_{2})} = \sup\limits_{k_{0}\in\mathbb{Z}}w_{1}(B_{k_0})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{k_{0}}w_{1}(B_k)^{\alpha r/n}\|f\chi_{k}\|^{r}_{L^{q}(w_{2},\mathbb{Q}_p^n)}\bigg)^{1/r}. \end{equation}$

(1.9)

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

$\begin{equation*} \|f\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)} = \sup\limits_{ B \subset \mathbb{Q}_p^n}\frac{1}{w(B)^{\beta/n}}\bigg(\frac{1}{w(B)}\int_{B}|f(\bf x)-f_{B}|^q w({\bf{x}})^{1-q} d{\bf{x}}\bigg)^{1/q}, \end{equation*}$

where

$\begin{equation} f_{B} = \frac{1}{|B|}\int_{B}f({\bf{x}})d{\bf{x}}. \end{equation}$

(1.10)

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

$C_{1}\frac{|A|}{|B|}\leq\frac{w(A)}{w(B)}\leq C_{2}\bigg(\frac{|A|}{|B|}\bigg)^{\mu},$

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 $i$ , $k\in\mathbb{Z},$

$\begin{equation*} |b_{B_{i}}-b_{B_{k}}|\leq C(i-k)\|b\|_{CMO^{q}(w,\mathbb{Q}_p^n)}\frac{w(B_{k})}{|B_{k}|}. \end{equation*}$

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

$\int_{B}w(x)^{1-q'}dx\leq C|B|^{q'}w(B)^{1-q'},$

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

$\|H^{p}_{b}f\|_{\dot{K}^{\alpha,r_{2}}_{q}(w,w^{1-q})}\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_{p}^{n})}\|f\|_{\dot{K}^{\alpha,r_{1}}_{q}(w,w)}$

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

$\|H^{p}_{b}f\|_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_{p}^{n})}\|f\|_{L^{q}(w,\mathbb{Q}_p^n)}$

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

$\begin{eqnarray*} \begin{aligned}\|H^{p}_{b}f\|_{M\dot{K}^{\alpha,\lambda}_{r_{2},q}(w,w^{1-q})}\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_{p}^{n})}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)} \end{aligned} \end{eqnarray*}$

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

$\begin{equation} \begin{aligned}[b] &\|(H^{p}_{b}f)\chi_{k}\|^{q}_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\\ & = \int_{S_{k}}|{\bf{x}}|_{p}^{-qn} \bigg|\int_{|{\bf{t}}|_{p} \leq|{\bf{x}}|_{p}}f({\bf{t}})(b({\bf{x}}) -b({\bf{t}})) d{\bf{t}}\bigg|^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ &\leq Cp^{-kqn}\int_{S_{k}}\bigg(\int_{|{\bf{t}}|_{p}\leq p^{k}}|f({\bf{t}})(b({\bf{x}}) -b({\bf{t}}))| d{\bf{t}}\bigg)^{q} w({\bf{x}})^{1-q}d{\bf{x}} \\ & = Cp^{-kqn}\int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}}|f({\bf{t}})(b({\bf{x}})-b({\bf{t}}))| d{\bf{t}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ &\leq Cp^{-kqn} \int_{S_{k}} \bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}} |f({\bf{t}})(b({\bf{x}})-b_{B_{k}})| d{\bf{t}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ &\quad+Cp^{-kqn} \int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k} \int_{S_{i}} |f({\bf{t}}) (b({\bf{t}})-b_{B_{k}})|d{\bf{t}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ & = I+II. \end{aligned} \end{equation}$

(2.1)

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

$\begin{equation} \begin{aligned}[b] \int_{S_{i}}f({\bf{t}})d{\bf{t}}&\leq\bigg(\int_{S_{i}}|f({\bf{t}})|^{q}w({\bf{t}})d{\bf{t}}\bigg)^{1/q}\bigg(\int_{S_{i}}w({\bf{t}})^{-q'/q}d{\bf{t}}\bigg)^{1/q'}\\ &\leq C\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}|B_{i}|w(B_{i})^{-1/q}. \end{aligned} \end{equation}$

(2.2)

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

$\begin{equation} \begin{aligned}[b] I &\leq Cp^{-kqn}\|b\|^{q}_{CMO^{q}(w,\mathbb{Q}_p^n)}w(B_{k})\bigg(\sum\limits_{i = -\infty}^{k}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}|B_{i}|w(B_{i})^{-1/q}\bigg)^{q}\\ &\leq Cp^{-knq}\|b\|^{q}_{CMO^{q}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}|B_{i}|\bigg(\frac{w(B_{k})}{w(B_{i})}\bigg)^{1/q}\bigg)^{q}\\ &\leq C\|b\|^{q}_{CMO^{q}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}. \end{aligned} \end{equation}$

(2.3)

Now, we estimate $II$ as follows

$\begin{equation} \begin{aligned}[b] II&\leq Cp^{-kqn} \int_{S_{k}} \bigg(\sum\limits_{i = -\infty}^{k} \int_{S_{i}} |f({\bf{t}}) (b({\bf{t}})-b_{B_{i}})|d{\bf{t}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ &\quad+Cp^{-kqn}\int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}}|f({\bf{t}})(b_{B_{k}}-b_{B_{i}})|d{\bf{t}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}} \\ & = II_{1}+II_{2}. \end{aligned} \end{equation}$

(2.4)

Next, applying Hölder's inequality to deduce

$\begin{equation} \begin{aligned}[b] \int_{S_{i}}|f({\bf{t}})(b({\bf{t}})-b_{B_{i}})|d{\bf{t}}&\\\leq& \bigg(\int_{S_{i}}|f({\bf{t}})|^{q}w({\bf{t}})d{\bf{t}}\bigg)^{1/q}\bigg(\int_{S_{i}}|b({\bf{t}})-b_{B_{i}}|^{q'}w({\bf{t}})^{-q'/q}d{\bf{t}}\bigg)^{1/q'}\\ &\leq w(B_{i})^{-1/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\|b\|_{CMO^{q'}(w,\mathbb{Q}_p^n)}. \end{aligned} \end{equation}$

(2.5)

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

$\begin{equation} \begin{aligned}[b] II_{1} &\leq Cp^{-kqn}\int_{S_{k}}w({\bf{x}})^{1-q}d{\bf{x}}\bigg(\sum\limits_{i = -\infty}^{k} \|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}w(B_{i})^{1/q'}\bigg)^{q} \\ &\leq Cp^{-kqn}|B_{k}|^{q}w(B_{k})^{1-q}\|b\|^{q}_{CMO^{q'}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k} \|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}w(B_{i})^{1/q'}\bigg)^{q}\\ &\leq C\|b\|^{q}_{CMO^{q'}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}\bigg(\frac{w(B_{i})}{w{(B_{k})}} \bigg)^{1-1/q}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}\\ & \leq C\|b\|^{q}_{CMO^{q'}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n\mu/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}. \end{aligned} \end{equation}$

(2.6)

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)

$\begin{equation} \begin{aligned}[b] II_{2}&\leq Cp^{-kqn}\\ &\quad\times\int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}}|f({\bf{y}})(i-k)\|b\|_{CMO^{r}(w,\mathbb{Q}_p^n)}\frac{w(B_{i})}{|B_{i}|}|d{\bf{y}}\bigg)^{q}w({\bf{x}})^{1-q}d{\bf{x}}\\ &\leq Cp^{-kqn}\|b\|^{q}_{CMO^{r}(w,\mathbb{Q}_p^n)}|B_{k}|^{q}w(B_{k})^{1-q}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)\frac{w(B_{i})^{1-1/q}}{|B_{i}|}|B_{i}|\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}\\ &\leq C\|b\|^{q}_{CMO^{r}(w,\mathbb{Q}_p^n)}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)\bigg(\frac{w(B_{i})}{w(B_{k})}\bigg)^{1-1/q}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}\\ &\leq C\|b\|^{q}_{CMO^{r}(w,\mathbb{Q}_p^n)}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)n\mu/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{q}. \end{aligned} \end{equation}$

(2.7)

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

$\begin{align*} &\|H^{p}_{b}f\|_{\dot{K}^{\alpha,r_{2}}_{q}(w,w^{1-q})}\\& = \bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{2}/n}\|(H^{p}_{b}f)\chi_{k}\|^{r_{2}}_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\bigg)^{1/r_{2}}\\ &\leq\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{1}/n}\|(H^{p}_{b}f)\chi_{k}\|^{r_{1}}_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\bigg)^{1/r_{1}}\\ &\leq C\|b\|_{CMO^{q}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{r_{1}}\bigg)^{1/r_{1}}\\&\quad+C\|b\|_{CMO^{q'}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n\mu/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{r_{1}}\bigg)^{1/r_{1}}\\&\quad+C\|b\|_{CMO^{r}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)n\mu/q'}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{r_{1}}\bigg)^{1/r_{1}}\\ & = J. \end{align*}$

Therefore,

$\begin{align*} J^{r_{1}}&\leq C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\\&\quad\times\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\alpha r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k} (k-i)p^{(i-k)n\mu/q'} \|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{r_{1}}\\ &\leq C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\\&\quad\times\sum\limits_{k = -\infty}^{\infty}\bigg(\sum\limits_{i = -\infty}^{k} (k-i)p^{(i-k)n\mu/q'-\alpha} \|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{r_{1}}. \end{align*}$

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

$\begin{align*} J^{r_{1}}&\leq C\|b\|^{r_{1}}_{CMO^{r \max\{q,q'\}}(w,\mathbb{Q}_p^n)}\\&\quad\times\sum\limits_{k = -\infty}^{\infty}\sum\limits_{i = -\infty}^{k}(k-i)^{r_{1}}w(B_{i})^{\alpha r_{1}/n}p^{(i-k)(n\mu/q'-\alpha)r_{1}}\|f\chi_{i}\|^{r_{1}}_{L^{q}(w,\mathbb{Q}_p^n)}\\ & = C\|b\|^{r_{1}}_{CMO^{r \max\{q,q'\}}(w,\mathbb{Q}_p^n)}\\&\quad\times\sum\limits_{k = -\infty}^{\infty}w(B_{i})^{\alpha r_{1}/n}\|f\chi_{i}\|^{r_{1}}_{L^{q}(w,\mathbb{Q}_p^n)}\sum\limits_{k = i}^{\infty}(k-i)^{r_{1}}p^{(i-k)(n\mu/q'-\alpha)r_{1}}\\ & = C\|b\|^{r_{1}}_{CMO^{r \max\{q,q'\}}(w,\mathbb{Q}_p^n)}\|f\|^{r_{1}}_{\dot{K}^{\alpha,r_{1}}_{q}(w,w)}. \end{align*}$

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

$\begin{align*} J^{r_{1}} &\leq C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sum\limits_{k = -\infty}^{\infty}\sum\limits_{i = -\infty}^{k}w(B_{i})^{\alpha r_{1}/n}\|f\chi_{i}\|^{r_{1}}_{L^{q}(w,\mathbb{Q}_p^n)}p^{(i-k)(n\mu/q' -\alpha)r_{1}/2}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)^{r_{1}'}p^{(i-k)(n\mu/q' -\alpha)r_{1}'/2}\bigg)^{r_{1}/r_{1}'}\\ & = C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sum\limits_{k = -\infty}^{\infty}w(B_{i})^{\alpha r_{1}/n}\|f\chi_{i}\|^{r_{1}}_{L^{q}(w,\mathbb{Q}_p^n)}\sum\limits_{k = i}^{\infty}p^{(i-k)(n\mu/q' -\alpha)r_{1}/2}\\ & = C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\|f\|^{r_{1}}_{\dot{K}^{\alpha,r_{1}}_{q_{1}}(w,w)}. \end{align*}$

Hence, the proof of theorem is completed.

Proof of Theorem 2.7: From theorem 2.5, we have

$\|(H^{p}_{b})f\chi_{k}\|_{L^{q}(w^{1-q} ,\mathbb{Q}_p^n)}\leq C\|b\|^{r_{1}}_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)(n\mu/q')}\|f\chi_{i}\|_{L^{q}(w,\mathbb{Q}_p^n)}.$

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

$\begin{eqnarray*} \begin{aligned}&\|H^{p}_{b}f\|_{M\dot{K}^{\alpha,\lambda}_{r_{2},q}(w,w^{1-q})}\\& = \sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{k_{0}}w(B_{k})^{\alpha r_{2}/n}\|(H^{p}_{b}f)\chi_{k}\|^{r_{2}}_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\bigg)^{1/r_{2}}\\ &\leq\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{k_{0}}w(B_{k})^{\alpha r_{1}/n}\|(H^{p}_{b}f)\chi_{k}\|^{r_{1}}_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\bigg)^{1/r_{1}}\\ &\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\\&\quad\times\bigg(\sum\limits_{k = -\infty}^{k_{0}}w(B_{k})^{\lambda r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)n\mu/q'}\bigg(\frac{w(B_{k})}{w(B_{i})}\bigg)^{\lambda r_{1}/n}\\&\quad\times w(B_{i})^{-\lambda/n}\bigg(\sum\limits_{l = -\infty}^{i}w(B_{l})^{\alpha r_{1}/n}\|f\chi_{i}\|^{r_{1}}_{L^{q}(w,\mathbb{Q}_p^n)}\bigg)^{1/r_{1}}\bigg)^{r_{1}}\bigg)^{1/r_{1}}\\ &\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\\&\quad\times\bigg(\sum\limits_{k = -\infty}^{k_{0}}w(B_{k})^{\lambda r_{1}/n}\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)(n\mu/q'-\alpha+\lambda)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)}\bigg)^{r_{1}}\bigg)^{1/r_{1}}\\ &\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\\&\quad\times\bigg(\sum\limits_{k = -\infty}^{k_{0}}w(B_{k})^{\lambda r_{1}/n}\bigg)^{1/r_{1}}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)}\\ &\leq C\|b\|_{CMO^{r\max\{q,q'\}}(w,\mathbb{Q}_p^n)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)}. \end{aligned} \end{eqnarray*}$

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 $i$ , $k\in\mathbb{Z},$

$\begin{equation*} |b_{B_{i}}-b_{B_{k}}|\leq C(i-k)\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}w(B_{i})^{\beta/n}\frac{w(B_{k})}{|B_{k}|}. \end{equation*}$

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

$\|H^{p}_{b}f\|_{\dot{K}^{\alpha,r_{2}}_{q_{2}}(w,w^{1-q_{2}})}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{\dot{K}^{\alpha,r_{1}}_{q_{1}}(w,w)}$

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

$\|H^{p}_{b}f\|_{L^{q}(w^{1-q},\mathbb{Q}_p^n)}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{L^{q}(w,\mathbb{Q}_p^n)}$

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

$\begin{eqnarray*} \begin{aligned}\|H^{p}_{b}f\|_{M\dot{K}^{\alpha,\lambda}_{r_{2},q_{2}}(w,w^{1-q_{2}})}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q_{1}}(w,w)} \end{aligned} \end{eqnarray*}$

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

$\begin{equation} \begin{aligned}[b] &\|(H^{p}_{b}f)\chi_{k}\|^{q_{2}}_{L^{q_{2}}(w^{1-q_{2}},\mathbb{Q}_p^n)}\\ &\leq Cp^{-kq_{2}n} \int_{S_{k}} \bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}} |f({\bf{t}})(b({\bf{x}})-b_{B_{k}})| d{\bf{t}}\bigg)^{q_{2}}w({\bf{x}})^{1-q_{2}}d{\bf{x}} \\ &\quad+Cp^{-kq_{2}n} \int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k} \int_{S_{i}} |f({\bf{t}}) (b({\bf{t}})-b_{B_{k}})|d{\bf{t}}\bigg)^{q_{2}}w({\bf{x}})^{1-q_{2}}d{\bf{x}} \\ & = L+LL. \end{aligned} \end{equation}$

(3.1)

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

$\begin{equation} \begin{aligned}[b] L &\leq Cp^{-kq_{2}n}\|b\|^{q_{2}}_{{Lip_{\beta}}(w,\mathbb{Q}_p^n)}w(B_{k})^{1+\beta q_{2}/n}\bigg\{\sum\limits_{i = -\infty}^{k}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}|B_{i}|w(B_{i})^{-1/q_{1}}\bigg\}^{q_{2}}\\ &\leq Cp^{-knq_{2}}\|b\|^{q_{2}}_{{Lip_{\beta}}(w,\mathbb{Q}_p^n)}\bigg\{\sum\limits_{i = -\infty}^{k}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}|B_{i}|\bigg(\frac{w(B_{k})}{w(B_{i})}\bigg)^{1/q_{1}}\bigg\}^{q_{2}}\\ &\leq C\|b\|^{q_{2}}_{{Lip_{\beta}}(w,\mathbb{Q}_p^n)}\bigg(p^{(k-i)n/q_{1}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}. \end{aligned} \end{equation}$

(3.2)

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

$\begin{equation} \begin{aligned}[b] LL&\leq Cp^{-kq_{2}n} \int_{S_{k}} \bigg(\sum\limits_{i = -\infty}^{k} \int_{S_{i}} |f({\bf{t}}) (b({\bf{t}})-b_{B_{i}})|d{\bf{t}}\bigg)^{q_{2}}w({\bf{x}})^{1-q_{2}}d{\bf{x}} \\ &\quad+Cp^{-kq_{2}n}\int_{S_{k}}\bigg(\sum\limits_{i = -\infty}^{k}\int_{S_{i}}|f({\bf{t}})(b_{B_{k}}-b_{B_{i}})|d{\bf{t}}\bigg)^{q_{2}}w({\bf{x}})^{1-q_{2}}d{\bf{x}} \\ & = LL_{1}+LL_{2}. \end{aligned} \end{equation}$

(3.3)

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

$\begin{equation} \begin{aligned}[b] \int_{S_{i}}|f({\bf{t}})(b({\bf{t}})-b_{B_{i}})|d{\bf{t}}&\\\leq&\bigg(\int_{S_{i}}|f({\bf{t}})|^{q_{1}}w({\bf{t}})d{\bf{t}}\bigg)^{1/q_{1}}\bigg(\int_{S_{i}}|b({\bf{t}})-b_{B_{i}}|^{q_{1}'}w({\bf{t}})^{-q_{1}'/q_{1}}d{\bf{t}}\bigg)^{1/q_{1}'}\\ &\leq w(B_{i})^{-1/q_{1}'+\beta/n}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}. \end{aligned} \end{equation}$

(3.4)

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

$\begin{equation} \begin{aligned}[b] LL_{1} &\leq Cp^{-kq_{2}n}\int_{S_{k}}w({\bf{x}})^{1-q_{2}}d{\bf{x}}\bigg(\sum\limits_{i = -\infty}^{k} \|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}w(B_{i})^{1/q_{1}'+\beta/n}\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}\\ &\leq Cp^{-kq_{2}n}|B_{k}|^{q_{2}}w(B_{k})^{1-q_{2}}\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k} \|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}w(B_{i})^{1/q_{1}'+\beta/n}\bigg)^{q_{2}}\\ &\leq C\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}\bigg(\frac{w(B_{i})}{w{(B_{k})}} \bigg)^{1-1/q_{2}}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}\\ & \leq C\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n\mu/q_{2}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}. \end{aligned} \end{equation}$

(3.5)

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

$\begin{equation} \begin{aligned}[b] LL_{2} &\leq Cp^{-kq_{2}n}\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}|B_{k}|^{q_{2}}w(B_{k})^{1-q_{2}}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)w(B_{k})^{\beta/n}\frac{w(B_{i})}{|B_{i}|}|B_{i}|w(B_{i})^{-1/q_{1}}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}\\ & = C\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)\bigg(\frac{w(B_{i})}{w(B_{k})}\bigg)^{1-1/q_{1}}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}\\ &\leq C\|b\|^{q_{2}}_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\\&\quad\times\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)n\mu/q_{1}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{q_{2}}. \end{aligned} \end{equation}$

(3.6)

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

$\begin{align*} &\|H^{p}_{b}f\|_{M\dot{K}^{\alpha,\lambda}_{r_{2},q_{2}}(w,w^{1-q_{2}})}\\ &\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\beta r_{2}/n}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n/q_{1}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{r_{2}}\bigg)^{1/r_{2}}\\&\quad+C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\beta r_{2}/n}\bigg(\sum\limits_{i = -\infty}^{k}p^{(i-k)n\mu/q_{2}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{r_{2}}\bigg)^{1/r_{2}}\\&\quad+C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\sup\limits_{k_{0}\in\mathbb{Z}}w(B_{k_{0}})^{-\lambda/n}\bigg(\sum\limits_{k = -\infty}^{\infty}w(B_{k})^{\beta r_{2}/n}\bigg(\sum\limits_{i = -\infty}^{k}(k-i)p^{(i-k)n\mu/q_{1}'}\|f\chi_{i}\|_{L^{q_{1}}(w,\mathbb{Q}_p^n)}\bigg)^{r_{2}}\bigg)^{1/r_{2}}\\ & = S_{1}+S_{2}+S_{3}. \end{align*}$

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

$S_{1}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)},\quad \alpha < n/q_{1}'+\lambda,$

$S_{2}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)},\quad \alpha < n\mu/q_{2}'+\lambda,$

$S_{3}\leq C\|b\|_{Lip_{\beta}(w,\mathbb{Q}_p^n)}\|f\|_{M\dot{K}^{\alpha,\lambda}_{r_{1},q}(w,w)},\quad \alpha < n\mu/q_{1}'+\lambda.$

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.

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