Some new characterizations of boundedness of commutators of $p$-adic maximal-type functions on $p$-adic Morrey spaces in terms of Lipschitz spaces

1.   Introduction

In the contemporary era, $p$-adic analysis is so significant that there is a lot of research being done on theories that are only concerned with $p$-adic objects, a case in point is the $p$-adic Hodge theory ^[1], Coleman's theory of $p$-adic integration ^[2], $p$-adic geometry ^[3], the theory of $p$-adic differential equations ^[4], the $p$-adic Langlands correspondence ^[5], study of $p$-adic cohomologies ^[6], and the study of $p$-adic modular forms ^[7]. In this connection, numerous of these concepts and advancements are present in the proof of Fermet's last theorem ^[8]. Recently, they have been applied in mathematical physics ^[9] and harmonic analysis ^{[10,11,12,13,14,15,16]}.

Ostrowski's theorem states that any nontrivial norm on the field of rational numbers $\mathbb{Q}$ is either the $p$-adic norm $|\cdot|_{p},$ or the real norm $|\cdot|,$ where $p$ is a prime number. The former norm is stated as follows, if any rational number $r$ is represented as $r = p^{\theta}\frac{m}{n},$ where $\theta = \theta\mathbf{(x)}\in\mathbb{Z},$ $(p, m, n) = 1$ and $m, n\in\mathbb{Z},$ then

This norm exhibits an ultrametric property

A symbol $\mathbb{Q}_p$ is the field of $p$-adic numbers, and is the completion of field of rational numbers $\mathbb{Q}$ with respect to the norm $|\cdot|_{p}.$ Any $\mathbb{Q}_p\ni r\neq0$ is uniquely represented as, see ^[9]

where $\gamma_{k}, \theta\in\mathbb{Z},$ $\gamma_{k}\in\frac{\mathbb{Z}}{p\mathbb{Z}_{p}}, \gamma_{0}\neq0.$ It is eminent that the series in (1.1) is convergent as $|\gamma_{k}p^{k}| = p^{-k}.$

The $n$-dimensional field $\mathbb{Q}_p^n$ is defined as $n$-tuples of $p$-adic numbers, $(r_{1}, r_{2}, \cdot\cdot\cdot, r_{n}),$ where $\mathbb{Q}_p\ni r_{k},$ $k = 1, 2, \cdot\cdot\cdot, n.$ The $n$-dimensional $p$-adic numbers inherit many properties from the $p$-adic numbers. They form a complete metric space with respect to the $n$-dimensional $p$-adic metric $d(\mathbf{r}, \mathbf{s}) = |\mathbf{r}-\mathbf{s}|_{p}$, which measures the divisibility of $n$-tuples by powers of $p$. The $n$-dimensional $p$-adic metric induces a topology on $\mathbb{Q}_p^n$, allowing for the study of continuity, convergence, and limit concepts in this space. The norm on $\mathbb{Q}_p^n$ is

The $p$-adic ball $B_{\theta}(\mathbf{x})$ and $p$-adic sphere $S_{\theta}(\mathbf{x})$ with radius $p^{\theta}$ and center $\mathbf{x}$ are defined by

Since $\mathbb{Q}_p^n$ is a locally compact commutative group, then there exists a Haar measure $d\mathbf{y}$ on the additive group $\mathbb{Q}_p^n,$ which is normalized by

From standard analysis, we get $|B_{\theta}(\mathbf{x})|_{h} = p^{n\theta}$ and $|S_{\theta}(\mathbf{x})|_{h} = p^{n\theta}(1-p^{-n})$, for any $\mathbf{x}\in \mathbb{Q}_p^n.$

A measurable function $b$ defined on $\mathbb{Q}_{p}^{n}$ is in $L^{p}(\mathbb{Q}_{p}^{n})$ $(1\leq p\leq\infty),$ if it satisfies

The commutators of harmonic analysis are vital integral operators and play a crucial role in examining the regularity characteristics of solutions to various partial differential equations, for instance ^{[17,18,19,20]}. Suppose $T$ is a classical singular integral operator with an another function $b$, then the commutator $[b, T]$ generated by $T$ is defined as follows:

In ^[21], authors have studied the $L^{p}$ boundedness of (1.2) with $b\in B\dot{M}O(\mathbb{R}^n)$. These results were extended with $b\in\Lambda_{\delta}(\mathbb{R}^n)$ in ^[22]. Since then, a great attention has been paid with studying the commutators of operators; see for instance, ^{[17,23,24,25]}.

In what follows, for $f\in L^{1}_{loc}(\mathbb{Q}_p^n)$, we define the $p$-adic sharp maximal function $M^{p, \sharp}$ and $p$-adic fractional maximal function $M^{p}_{\alpha}$ as

and

where $f_{B_{\theta}(\mathbf{x})} = \frac{1}{|B_{\theta}(\mathbf{x})|_{h}}\int_{B_{\theta}(\mathbf{x})}f(\mathbf{t})d\mathbf{t}.$ When $\alpha = 0,$ we get the Hardy Littlewood maximal function $M^{p},$ which is defined as:

Significant work has been done intensively in the past on $M^{p}$ by many researchers; see for example, ^[26,27,28] and the references therein.

The $p$-adic fractional commutator of $M^{p}_{\alpha}$ with $b\in L^{1}_{loc}(\mathbb{Q}_p^n)$ is defined by

On the other hand, nonlinear commutators of respectively $M^{p, \sharp}$ and $M^{p}_{\alpha}$ with a locally integrable function $b$ are defined by

and

When $\alpha = 0,$ $[b, M^{p}_{\alpha}]$ reduces to $[b, M^{p}]$, see ^[29]. In $p$-adic setting, boundedness of commutators of $p$-adic maximal function is a new area, and we only found some work in ^[29]. In that paper, the authors acquired the boundedness of commutators of $M^{p}$ on $p$-adic function spaces with $b\in B\dot{M}O(\mathbb{Q}_p^n)$. However, in the case of Euclidean, commutators of maximal-type functions have spotlighted many researchers. For example, in ^[17], Bastero et al. obtained the boundedness of commutators of maximal and sharp functions on Lebesgue spaces with $b\in B\dot{M}O(\mathbb{R}^n)$. Furthermore, the results of ^[17] are extended in ^[30]. Zhang ^[31] further obtained the characterizations of nonlinear commutators of the Hardy Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces with $b\in\Lambda_{\delta}(\mathbb{R}^n).$ Recently, Xuechun et al. ^[32] established new characterizations of Lipschitz space in terms of the boundedness of $[b, M^{\sharp}]$ and $[b, M^{\alpha}]$ in the context of variable Lipschitz space.

As we observed in the above work, the characterization of nonlinear commutators of $M^{p, \sharp}$ and $M^{p}_{\alpha}$ remains widely open. Therefore, we obtain some characterizations of $p$-adic versions of Lipschitz spaces via the boundedness of $M^{p, \sharp}$ and $M^{p}_{\alpha}$ on $p$-adic Morrey spaces, by considering $b$ from Lipschitz spaces under certain assumptions. Throughtout this article, a letter $C$ represents a constant with different or the same values at different places, and $\chi_{B_{\theta}}$ is the characteristic function of $B_{\theta}(\mathbf{x})$.

Definition 1.1. Let $1\leq p <\infty$ and $0\leq\lambda\leq n$. The $p$-adic Morrey space $L^{p, \lambda}(\mathbb{Q}_{p}^{n})$ is defined as follows:

where

Remark 1.1. It is evident that $L^{p, -1/p}(\mathbb{Q}_{p}^{n}) = L^{p}(\mathbb{Q}_{p}^{n})$ and $L^{p, 0}(\mathbb{Q}_{p}^{n}) = L^{\infty}(\mathbb{Q}_{p}^{n}).$

Definition 1.2. The Lipschitz space $\Lambda_{\delta}(\mathbb{Q}_{p}^{n}),$ $(\delta\in\mathbb{R}^{+})$ is the space of all measurable functions $b$ on $\mathbb{Q}_{p}^{n}$ such that

Next, we have the $p$-adic version of the Lipschitz space $\tilde{\Lambda}_{\delta}(\mathbb{Q}_{p}^{n}),$ which is the space of all measurable functions $b$ on $\mathbb{Q}_{p}^{n}$ with the following norm:

where $b_{B_{\theta}(\mathbf{x})} = \frac{1}{|B_{\theta}(\mathbf{x})|_{h}}\int_{B_{\theta}(\mathbf{x})}b(\mathbf{t})d\mathbf{t}.$

In the following section, we state some characterizations of the $p$-adic version of Lipschitz spaces via the boundedness of the commutators of $[b, M^{p, \sharp}],$ $M^{\alpha}_{\alpha, b},$ and $[b, M^{p}_{\alpha}].$

2.   Some characterizations of $p$-adic version of Lipschitz spaces

Theorem 2.1. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $0 <\lambda <n-q\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta}{n-\lambda} = \frac{1}{q}$. Then, $[b, M^{p, \sharp}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n)$ with $b\geq0.$

Theorem 2.2. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $0 <\lambda <n-q\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta+\alpha}{n-\lambda} = \frac{1}{q}$. Then, $M^{p}_{\alpha, b}:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n).$

Theorem 2.3. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $0 <\lambda <n-q\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta+\alpha}{n-\lambda} = \frac{1}{q}$. Then, $[b, M^{p}_{\alpha}]:$$L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n)$ with $b\geq0.$

Since $L^{p, -1/p}(\mathbb{Q}_{p}^{n}) = L^{p}(\mathbb{Q}_{p}^{n}).$ So, we have the characterizations in terms of the boundedness of operators $[b, M^{p, \sharp}],$ $M^{p}_{b},$ and $[b, M^{p}]$ on Lebesgue spaces.

Corollary 2.1. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta}{n} = \frac{1}{q}$. Then, $[b, M^{p, \sharp}]:$ $L^{q}(\mathbb{Q}_p^n)\rightarrow L^{p}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n)$ with $b\geq0.$

Corollary 2.2. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $0 <\lambda <n-q\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta+\alpha}{n} = \frac{1}{q}$. Then, $M^{p}_{\alpha, b}:$ $L^{q}(\mathbb{Q}_p^n)\rightarrow L^{p}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n).$

Corollary 2.3. Suppose $b$ is a locally integrable function, $1 <q <n/\delta,$ $0 <\lambda <n-q\delta,$ $\delta\in(0, 1),$ and $\frac{1}{p}+\frac{\delta+\alpha}{n} = \frac{1}{q}$. Then, $[b, M^{p}_{\alpha}]:$$L^{q}(\mathbb{Q}_p^n)\rightarrow L^{p}(\mathbb{Q}_p^n)$ if and only if $b\in\Lambda_{\delta}(\mathbb{Q}_p^n)$ with $b\geq0.$

In order to prove the above results, we need some lemmas and remarks. We begin with a very useful result.

Lemma 2.1. The $p$-adic space $\Lambda_{\delta}(\mathbb{Q}_{p}^{n})$ coincides with $\tilde{\Lambda}_{\delta}(\mathbb{Q}_{p}^{n}),$ for $0 <\delta<1.$

Proof. Consider a ball $B_{\theta}(\mathbf{x})$ and $\mathbf{t}\in B_{\theta}(\mathbf{x})$, then from the definition (1.2), we have

We further proceed as

which implies that

Therefore,

On the other hand, let $b\in\tilde{\Lambda}_\delta(\mathbb{Q}_p^n).$ For any $\mathbf{t}, \mathbf{z}\in\mathbb{Q}_p^n$ with $\mathbf{t}\neq \mathbf{z}.$ We set $B = B(\mathbf{t}, |\mathbf{t}-\mathbf{z}|_{p})$ and $B' = B'(\mathbf{z}, |\mathbf{t}-\mathbf{z}|_{p}).$ Then we have

Estimates of all terms on the right-hand side of (2.2) are more or less the same. So, we will estimate the first term. Let $B_{j} = B(t, p^{-j}|t-\mathbf{x}|_{p})$ for $j\geq1$ and $B_{0} = B.$ We proceed as

Consequently,

From (2.1) and (2.3), we have completed the proof.□

In what follows, taking into account the characteristic function $\chi_{B_{\theta}(\mathbf{x})},$ we have the following property:

Lemma 2.2. Suppose $1\leq q <\infty$ and $0 <\lambda <n,$ then

Next, the fractional integral operator on $\mathbb{Q}_p^n$ is introduced by Taibleson ^[33] and is defined by

The following lemma shows the boundedness of $T^{p}_{\alpha}$ on $p$-adic Morrey spaces, which is proved in a book ^[33].

Lemma 2.3. Suppose $1 <q <n/\alpha,$ $0 <\alpha < n,$ $0 <\lambda <n-q,$ and $\frac{1}{p}+\frac{\alpha}{n-\lambda} = \frac{1}{q},$ then $T^{p}_{\alpha}$ is bounded from $L^{q, \lambda}(\mathbb{Q}_p^n)$ to $L^{p, \lambda}(\mathbb{Q}_p^n).$

Remark 2.1. From the condition of Lemma 2.3, we get

Therefore,

From here, we deduce that $M^{p}_{\alpha}$ is bounded from $L^{q, \lambda}(\mathbb{Q}_p^n)$ to $L^{p, \lambda}(\mathbb{Q}_p^n).$

Proof of Theorem 2.1. Consider any $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}$ with $b \geq 0,$ we prove that $[b, M^{p, \sharp}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n)$. Let $f\in L^{q, \lambda}(\mathbb{Q}_p^n).$ From definition (1.2), we deduce

From Remark 2.1 and equation (2.4), we obtain

Hence, $[b, M^{p, \sharp}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n).$

Conversely, suppose that $[b, M^{p, \sharp}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n).$ We need to show $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}$ and $b \geq 0.$ Consider any fixed $p$-adic ball $B_{\theta}\mathbf{\mathbf{(x)}}$, and $\mathbf{t}\in {B_\theta}\mathbf{\mathbf{(x)}}.$ We see in ^[29] that

By above expression, Eq (1.5) and the boundedness of $[b, M^{p, \sharp}]$, we reach at

which implies that

Now, consider a $p$-adic ball $B_{\theta}\mathbf{\mathbf{(x)}}\subset \mathbb{Q}_p^n$. From ^[29], we see that for any $\mathbf{t}\in {B_\theta}\mathbf{\mathbf{(x)}},$

Now to achieve $b\in \Lambda_{\delta}(\mathbb{Q}_p^n)$, we let $A = \{\mathbf{t}\in {B_\theta}\mathbf{\mathbf{(x)}}: b\mathbf{\mathbf{(t)}}\leq b_{{B_\theta}\mathbf{\mathbf{(x)}}}\}.$ Moreover, consider any $\mathbf{t}\in A$ and we get $b\mathbf{\mathbf{(t)}}\leq b_{{B_\theta}\mathbf{\mathbf{(x)}}}\leq |b_{{B_\theta}\mathbf{\mathbf{(x)}}}| \leq 2M^{p, \sharp}{(b\chi_{B_{\theta}\mathbf{\mathbf{(x)}}}})\mathbf{\mathbf{(t)}},$ then

Since $\frac{1}{p} = \frac{1}{q}-\frac{\delta}{n-\lambda}$, then using (2.7) along with Hölder's inequality, Lemma 2.2, and (2.5), we ultimately have

This shows that $b \in {{\widetilde{\Lambda}_\delta}(\mathbb{Q}_p^n)}.$ This, along with Lemma 2.1, shows $b\in\Lambda_{\delta}(\mathbb{Q}_p^n).$

The final task is to show that $b\geq 0.$ For this its suffices to show $b^{-} = 0,$ where $b^{-} = \min\{b, 0\}$ and $b^{+} = |b|-b^{-}.$ Consider a $p$-adic ball $B_{\theta}\mathbf{\mathbf{(x)}}$. Using (2.6), we observe that

for any $\mathbf{t}\in {{B_\theta}\mathbf{\mathbf{(x)}}}$.

Now averaging on a ball $B_{\theta}\mathbf{\mathbf{(x)}}$, we deduce that

On the other hand, from (2.8), we have

From this and (2.9), we get

By letting $\theta\rightarrow \infty$ with $t\in B_{\theta}\mathbf{\mathbf{(x)}},$ the Lebesgue differentiation theorem in the $p$-adic field ensures that

Consequently, $b^{-} = 0$, and hence $b\geq 0$ holds true, which complete the proof of theorem.

Proof of Theorem 2.2. Suppose $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}.$ We show that $M_{\alpha, b}:$ $L^{q}(\mathbb{Q}_p^n)\rightarrow L^{p}(\mathbb{Q}_p^n).$ From the definition of (1.2) and Eq (1.4), we deduce

By this and boundedness of $M_{\alpha+\delta}$ from $L^{q, \lambda}(\mathbb{Q}_p^n)$ to $L^{p, \lambda}(\mathbb{Q}_p^n)$ (see Remark 2.1), we eventually have

Conversely, suppose that $[M_{\alpha, b}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n),$ we show that $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}.$ For this, consider a $p$-adic ball $B_{\theta}\mathbf{(x)}$, we are down to

From (2.12) and that $[M_{\alpha, b}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n),$ we obtain

which implies that

Since $\frac{1}{p} = \frac{1}{q}-\frac{\delta+\alpha}{n},$ making use of Hölder's inequality, Lemma 2.2, and (2.13), we have

This shows that $b\in\tilde{\Lambda_{\delta}}(\mathbb{Q}_p^n).$ From this and Lemma 2.1, we have $b\in\Lambda_{\delta}(\mathbb{Q}_p^n)$, which finishes the proof.

Before proving Theorem 2.3, we define the $p$-adic fractional maximal operator $M^{p}_{B_{\theta}(\mathbf{x})}$ with respect to a $p$-adic ball as follows:

where supremum is taken over all balls $B_{\theta_{0}}(\mathbf{t})$ such that $B_{\theta_{0}}(\mathbf{t})\subseteq B_{\theta}(\mathbf{x}).$

Proof of Theorem 2.3. Assume that $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}$ and $b\geq 0.$ We show that $[b, M_{\alpha}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n)$. Let $f\in L^{q, \lambda}(\mathbb{Q}_p^n).$ From definitions of (1.2), we reach at

From (2.15) and Theorem 2.2, we acquire

Consequently, $[b, M^p_{\alpha}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n).$

Conversely, suppose $[b, M^p_{\alpha}]:$ $L^{q, \lambda}(\mathbb{Q}_p^n)\rightarrow L^{p, \lambda}(\mathbb{Q}_p^n).$ We need to show that

First, we opt for the former one, and in order to do so, we need the following preparation:

Consider a $p$-adic ball $B_{\theta}\mathbf{(x)}.$ For all $\mathbf{t}\in B_{\theta}\mathbf{(x)}$, we have

and

Then, from this and (1.6), we have

which implies that

From (2.17) and the boundedness of $[b, M^p_{\alpha}],$ we obtain

which implies that

Furthermore, consider a $p$-adic ball $B_{\theta}\mathbf{(x)}$, suppose $A = \{\mathbf{t}\in {B_\theta}\mathbf{(x)}: b\mathbf{(t)}\leq {B_\theta}\mathbf{(x)}\}.$ Now, for any $\mathbf{t}\in A$, we have

Thus,

Since $\frac{1}{p} = \frac{1}{q}-\frac{\delta+\alpha}{n},$ from (2.19), Hölder's inequality, Lemma 2.2, and (2.18), we sum up that

which implies that $b \in {{\widetilde{\Lambda}_\delta}(\mathbb{Q}_p^n)}$, so, it follows from this and Lemma 2.1 that $b \in {{\Lambda_\delta}(\mathbb{Q}_p^n)}$.

Next, we show the latter one in equation (2.16). For this, its suffices to show $b^{-} = 0,$ where $b^{-} = \min\{b, 0\}$ and $b^{+} = |b|-b^{-}.$ Consider any fixed $p$-adic ball $B_{\theta}\mathbf{(x)}$ and for any $\mathbf{t}\in {B_{\theta}\mathbf{(x)}},$ we have

Therefore, for $\mathbf{t}\in {B_{\theta}\mathbf{(x)}}$, we obtain

Then, by this and (2.20), we have

Therefore,

By letting $\theta\rightarrow \infty$ together with the Lebesgue differentiation theorem in $p$-adic field, we have $b^{-} = 0.$ Hence $b\geq0,$ which finishes the proof of theorem.

3.   Conclusions

Necessary and sufficient conditions for the boundedness of commutators of $p$-adic sharp maximal functions, $p$-adic fractional maximal functions, and $p$-adic fractional maximal commutators on $p$-adic Morrey spaces are studied by considering the symbol function as a Lipschitz spaces. Wavelet characterization of $p$-adic Lebesgue spaces can be obtained as a future prospect.

Author contributions

Naqash Sarfraz: Conceptualization, data curation, investigation, methodology, writing-original draft; Muhammad Bilal Riaz: Formal analysis, methodology, writing-original draft; project management, funding acquisition, supervision; Qasim Ali Malik: Validation, visualization, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

The Author Muhammad Bilal Riaz thankful to Ministry of Education, Youth and Sports of the Czech Republic for their support through the e-INFRA CZ (ID:90254).

Conflict of interest

The authors declare that they have no conflict of interest.