Necessary and sufficient conditions for boundedness of commutators of maximal function on the $p$-adic vector spaces

1.   Introduction and statement of main results

It is well known that the commutators of a great variety of operators appearing in Harmonic Analysis are intimately related to the regularity properties of the solutions of certain partial differential equations, see for example ^{[5,6,9,10,13,32]}. A first result in this direction was established by Coifman, Rochberg and Weiss in ^[11], where the authors studied the commutator $[b, T]$ generated by the classical singular integral operator $T$ and a suitable function $b$ is given by

They gave a characterization of ${\rm BMO}(\mathbb{R}^{d})$ in virtue of the $L^{q}$-boundedness of the above commutator. In ^[22] the author extended the results in ^[11] to functions belonging to a Lipschitz functional space and gave a characterization in terms of the boundedness of the commutators of singular integral operators with symbols in this class. Milman and Schonbek ^[27] established a commutator result by real interpolation techniques. As an application, they obtained the $L^{q}$-boundedness of the commutators of maximal function $[b, M]$ when $b\in {\rm BMO}(\mathbb{R}^{d})$ and $b\geq 0$. This operator can be used in studying the product of a function in $H^{1}$ and a function in BMO (see ^[7] for instance). Bastero, Milman and Ruiz ^[4] studied the necessary and sufficient conditions for the boundedness of the commutators of maximal function $[b, M]$ and sharp maximal function $[b, M^{\sharp}]$ on $L^{q}(\mathbb{R}^{d})$ spaces when $1 < q < \infty$. Recently, Guliyev et al. ^[17] gave the characterization of fractional maximal operator and its commutators on Orlicz spaces in the Dunkl setting. For more information about the characterization of the commutator of maximal operator, see also ^{[2,3,12,43,44,45]} and the references therein.

Motivated by ^[3,4] and ^[19], we will study the characterization of BMO functions in the context of $p$-adic field spaces. For a prime number $p$, let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers. It is defined as the completion of the field of rational numbers $\mathbb{Q}$ with respect to the non-Archimedean $p$-adic norm $|\cdot|_{p}$. This norm is defined as follows: $|0|_{p} = 0$. If any non-zero rational number $x$ is represented as $x = p^{\gamma}\frac{m}{n}$, where $m$ and $n$ are integers which are not divisible by $p$, and $\gamma$ is an integer, then $|x|_{p} = p^{-\gamma}$. It is not difficult to show that the norm satisfies the following properties:

It follows from the second property that when $|x|_{p}\neq|y|_{p}$, then $|x+y|_{p} = \max\{|x|_{p}, |y|_{p}\}$. From the standard $p$-adic analysis ^[40], we see that any non-zero $p$-adic number $x\in\mathbb{Q}_{p}$ can be uniquely represented in the canonical series

where $a_{j}$ are integers, $0\leq a_{j}\leq p-1$, $a_{0}\neq 0$. The series $(1.2)$ converges in the $p$-adic norm because $|a_{j}p^{j}|_{p} = p^{-\gamma}$.

The space $\mathbb{Q}_{p}^{d}$ consists of points ${\bf x} = (x_{1}, x_{2}, \ldots, x_{d})$, where $x_{j}\in\mathbb{Q}_{p}$, $j = 1, 2, \ldots, d$. The $p$-adic norm on $\mathbb{Q}_{p}^{d}$ is $|{\bf x}|_{p}: = \max_{1\leq j\leq d}|x_{j}|_{p}$ for ${\bf x}\in\mathbb{Q}_{p}^{d}$. Denote by $B_{\gamma}({\bf a}) = \{{\bf x}\in\mathbb{Q}_{p}^{d}:|{\bf x}-{\bf a}|_{p}\leq p^{\gamma}\}$, the ball with center at ${\bf a}\in\mathbb{Q}_{p}^{d}$ and radius $p^{\gamma}$, and by $S_{\gamma}({\bf a}): = \{{\bf x}\in\mathbb{Q}_{p}^{d}:|{\bf x}-{\bf a}|_{p} = p^{\gamma}\}$ the sphere with center at ${\bf a}\in\mathbb{Q}_{p}^{d}$ and radius $p^{\gamma}$, $\gamma\in\mathbb{Z}$. It clear that $S_{\gamma}({\bf a}) = B_{\gamma}({\bf a})\backslash B_{\gamma-1}({\bf a})$, and $B_{\gamma}({\bf a}) = \bigcup_{k\leq\gamma}S_{k}({\bf a})$.

Since $\mathbb{Q}_{p}^{d}$ is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Harr measure $d{\bf x}$ on $\mathbb{Q}_{p}^{d}$ (up to positive constant multiple) which is translation invariant. We normalize the measure $d{\bf x}$ so that

where $|E|_{H}$ denotes the Harr measure of a measurable subset $E$ of $\mathbb{Q}_{p}^{d}$. From this integral theory, it is easy to obtain that $|B_{\gamma}({\bf a})|_{H} = p^{\gamma d}$ and $|S_{\gamma}({\bf a})|_{H} = p^{\gamma d}(1-p^{-d})$ for any ${\bf a}\in\mathbb{Q}_{p}^{d}$.

In what follows, we say that a (real-valued) measurable function $f$ defined on $\mathbb{Q}_{p}^{d}$ is in $L^{q}(\mathbb{Q}_{p}^{d})$, $1\leq q\leq\infty$, if it satisfies

Here the integral in $(1.3)$ is defined as

if the limit exists. We now mention some of the previous works on harmonic analysis on the $p$-adic field, see ^{[18,26,33,34,35]} and the references therein.

For a function $f\in L_{\text{loc}}^{1}(\mathbb{Q}_{p}^{d})$, we defined the Hardy-Littlewood maximal function of $f$ on $\mathbb{Q}_{p}^{d}$ by

In ^[24,25], Kim proved $L^{q}$ boundedness of the version of maximal function $\mathcal{M}^{p}$ and gave some properties similar to the Euclidean setting.

The maximal commutator of $\mathcal{M}^{p}$ with a locally integrable function $b$ is defined by

The first part of this paper is to study the boundedness of $\mathcal{M}_{b}^{p}$ when the symbol belongs to a BMO space (see in Section $2$). Some characterizations of the BMO space via such commutator are given. Our first result can be stated as follows.

Theorem 1.1. Let $b$ be a locally integrable function on $\mathbb{Q}_{p}^{d}$. The following statements are equivalent:

$\rm (1)$ $b\in{{\rm BMO}}(\mathbb{Q}_{p}^{d})$;

$\rm (2)$ $\mathcal{M}_{b}^{p}$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for all $q$ with $1 < q\leq\infty$;

$\rm (3)$ $\mathcal{M}_{b}^{p}$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for some $q$ with $1 < q\leq\infty$.

We remark that the boundedenss of the commutators of maximal function unknown until the author made some progress in ^[19] for a partial case on the $p$-adic vector space. In an attempt to close the gap in his work, we came up with some new results.

On the other hand, similar to $(1.1)$, we can define the commutator of the $p$-adic version of maximal function $\mathcal{M}^{p}$ with a locally integrable function $b$ by

In this paper we show that a slightly extended form of positivity is a necessary and suffcient condition to characterize the boundedness of $[b, \mathcal{M}^{p}]$. To see what this condition should be we observe that if $\mathcal{M}^{p}$ were a linear operator, given that everything we do is modulo bounded operators, the correct requirement would appear to be that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ with its negative part $b^{-}$ bounded. Indeed, the suffciency of the condition $b\in {\rm BMO}(\mathbb{Q}_{p}^{d})$ with $b^{-}$ bounded formally follows from Theorem $1.1$, the fact that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ and the estimate

where $b^{-} = -\min\{0, b\}$. We summarize the previous discussion with the following

Theorem 1.2. If $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ and $b^{-}$ is bounded, then the commutator $[b, \mathcal{M}^{p}]$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for all $q\in(1, \infty].$

The purpose of this paper is to prove the converse of Theorem $1.2$ and to show that a similar characterization also holds for the sharp maximal operator.

Our main result for $[b, \mathcal{M}^{p}]$ can be now stated as follows.

Theorem 1.3. Let $b$ be a locally integrable function on $\mathbb{Q}_{p}^{d}$. The following statements are equivalent:

$\rm (1)$ $b\in{{\rm BMO}}(\mathbb{Q}_{p}^{d})$ and $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$;

$\rm (2)$ $[b, \mathcal{M}^{p}]$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for all $q$ with $1 < q\leq\infty$;

$\rm (3)$ $[b, \mathcal{M}^{p}]$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for some $q$ with $1 < q\leq\infty$;

$\rm (4)$ There exists $q\in[1, \infty)$ such that

where $\mathcal{M}_{B_{\gamma}({\bf x})}^{p}$ denote the maximal operator with respect to a $p$-adic ball which is defined by

Here, the supremum is take over all the $p$-adic $B_{\gamma_{0}}({\bf y})$ with $B_{\gamma_{0}}({\bf y})\subseteq B_{\gamma}({\bf x})$ for a fixed $p$-adic ball $B_{\gamma}({\bf x})$.

$\rm (5)$ For all $q\in[1, \infty)$ such that

Recall that the $p$-adic version of sharp function is given by

where $f_{B_{\gamma}({\bf x})}$ denotes the average of $f$ over $B_{\gamma}({\bf x})$, i.e., $f_{B_{\gamma}({\bf x})} = \frac{1}{|B_{\gamma}({\bf x})|_{H}}\int_{B_{\gamma}({\bf x})}f({\bf y})d{\bf y}$.

Next we consider commutators with the $p$-adic version of sharp function. The results are similar to those in Theorem $1.3$.

Theorem 1.4. Let $b$ be a locally integrable function on $\mathbb{Q}_{p}^{d}$ and $1\leq\delta < \infty$. The following statements are equivalent:

$\rm (1)$ $b\in{{\rm BMO}}(\mathbb{Q}_{p}^{d})$ and $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$;

$\rm (2)$ $[b, \mathcal{M}_{p}^{\sharp}]$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for all $q$ with $1 < q\leq\infty$;

$\rm (3)$ $[b, \mathcal{M}_{p}^{\sharp}]$ is bounded on $L^{q}(\mathbb{Q}_{p}^{d})$ for some $q$ with $1 < q\leq\infty$;

$\rm (4)$ There exists $q\in[1, \infty)$ such that

$\rm (5)$ For all $q\in[1, \infty)$ such that

It is well-known that the Morrey space introduced by Morrey in ^[28] in order to study regularity questions which appear in the Calculus of Variations, and the $p$-adic version of Morrey spaces defined as follows: for $1\leq q\leq\infty$ and $0\leq\lambda\leq d$,

where

Note that $L^{q, 0}(\mathbb{Q}_{p}^{d}) = L^{q}(\mathbb{Q}_{p}^{d})$ and $L^{q, d}(\mathbb{Q}_{p}^{d}) = L^{\infty}(\mathbb{Q}_{p}^{d})$.

These spaces These spaces describe local regularity more precisely than Lebesgue spaces and appeared to be quite useful in the study of the local behavior of solutions to partial differential equations, a priori estimates and other topics in PDE, such as applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients and the potential analysis, see ^{[1,8,14,23,38]}.

The following theorems we investigate boundedness of maximal commutator and commutator of maximal function on the $p$-adic version of Morrey spaces.

Theorem 1.5. Let $1 < q < \infty$ and $0\leq\lambda \leq d$. The following statements are equivalent:

$\rm(1)$ $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$;

$\rm(2)$ $\mathcal{M}_{b}^{p}$ is bounded on $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$.

Theorem 1.6. Let $1 < q < \infty$ and $0\leq\lambda \leq d$. The following statements are equivalent:

$\rm(1)$ $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ and $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$;

$\rm(2)$ $[b, \mathcal{M}^{p}]$ is bounded on $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$.

The rest of the present paper is organized as follows: In Section $2$, we will give some definitions and lemmas. The proof of Theorems $1.1$–$1.4$ are presented in Section 3. In Secction 4, we will give the proof of Theorems $1.5$ and $1.6$. By $A\lesssim B$ we mean that $A\leq CB$ with some positive constant $C$ independent of appropriate quantities. The positive constants $C$ varies from one occurrence to another. For a real number $q$, $1 < q < \infty$, $q^{\prime}$ is the conjugate number of $q$, that is, $1/q+1/q^{\prime} = 1$.

2.   Preliminary definitions and lemmas

To prove our main results, we need the following definitions and lemmas.

Definition 2.1. Let $f\in L_{{\rm loc}}^{1}(\mathbb{Q}_{p}^{d})$ be given. If $\|f^{\sharp}\|_{L^{\infty}(\mathbb{Q}_{p}^{d})} < \infty$, then we say that $f$ is a function of bounded mean oscillation on $\mathbb{Q}_{p}^{d}$. We denote the space of such function by ${\rm BMO}(\mathbb{Q}_{p}^{d})$; that is say,

For $f\in {\rm BMO}(\mathbb{Q}_{p}^{d})$, we write

In ^[24], Kim gave the following property of BMO functions whcih is similar to Euclidean setting.

Lemma 2.1. If $1 < q < \infty$ and $f\in {\rm BMO}(\mathbb{Q}_{p}^{d})$ is given, then we have the following properties;

$\rm (a)$ The norm $\|f\|_{{\rm BMO}(\mathbb{Q}_{p}^{d})}$ is equivalent to the norm $\|f\|_{{\rm BMO}^{q}}(\mathbb{Q}_{p}^{d})$, where the norm $\|f\|_{{\rm BMO}^{q}}(\mathbb{Q}_{p}^{d})$ defined by

$\rm (b)$ For any $\lambda$ with $0 <\lambda <c_{2}/\|f\|_{{\rm BMO}(\mathbb{Q}_{p}^{d})}$, where $c_{2}$ is the constant given by Theorem 5.16 in ^[24],

Let

denote the family of all the $p$-adic balls, which differ from those of the Euclidean case, see ^[25].

Lemma 2.2. The family $\mathcal{F}_{p}$ has the following properties:

$\rm (a)$ If $\gamma\leq \gamma^{\prime}$, then either $B_{\gamma}({\bf x})\cap B_{\gamma^{\prime}}({\bf y}) = \emptyset$ or $B_{\gamma}({\bf x})\subset B_{\gamma^{\prime}}({\bf y})$.

$\rm (b)$ $B_{\gamma}({\bf x}) = B_{\gamma}({\bf y})$ if and only if ${\bf y}\in B_{\gamma}({\bf x})$.

A continuously increasing function on $[0, \infty]$, say $\Psi : [0, \infty]\rightarrow [0, \infty]$ such that $\Psi(0) = 0$, $\Psi(1) = 1$ and $\Psi(\infty) = \infty$, will be referred to as an Orlicz function. If $\Psi$ is a Orlicz function, then

is the complementary Orlicz function to $\Psi$.

The Orlicz space denoted by $L^{\Psi}(\mathbb{Q}_{p}^{d})$ consists of all measurable function $g : \mathbb{Q}_{p}^{d}\rightarrow \mathbb{R}$ such that

for some $\alpha > 0$.

Let us define the $\Psi$-average of $g$ over a $p$-adic ball $B_{\gamma}({\bf x})$ of $\mathbb{Q}_{p}^{d}$ by

When $\Psi$ is a Young function, that is, a convex Orlicz function, the quantity

is well known Luxemburg norm is the space $L^{\Psi}(\mathbb{Q}_{p}^{d})$ which can be found in ^[31].

A Young function $\Psi$ is said to satisfy the $\nabla_{2}$-condition, denoted $\Psi\in\nabla_{2}$, if for some $K > 1$

It should be noted that $\Psi(t)\equiv t$ fails the $\nabla_{2}$-condition.

If $f\in L^{\Psi}(\mathbb{Q}_{p}^{d})$, the maximal function of $f$ with respect to $\Psi$ is defined by setting

The following generalized Hölder's inequality (see ^[31])

holds for any the complementary Young function $\Psi$ associated to $\Phi$.

The main example that we will consider to use the Young function $\Phi(t) = t(1+\log^{+}t)$ with maximal function defined by $\mathcal{M}_{L(\log L)}^{p}$. The complementary Young function is given by $\Psi(t)\approx e^{t}$ with the corresponding maximal function denoted by $\mathcal{M}_{\exp L}^{p}$.

Let $\mathfrak{M}(\mathbb{Q}_{p}^{d})$ denote the set of all measurable functions on $\mathbb{Q}_{p}^{d}$. The Zygmund class $L(\log^{+}L)(\mathbb{Q}_{p}^{d})$ is the set of all $f\in \mathfrak{M}(\mathbb{Q}_{p}^{d})$ such that

where $\log^{+}t = \max\{\log t, 0\}$ and $t > 0$. Generally, this is not a linear set. Nevertheless, considering the class

we obtain a linear set, the Zygmund space.

The size of $\mathcal{M}^{p}(\mathcal{M}^{p})$ is given by the following.

Lemma 2.3. Let $f\in \mathfrak{M}(\mathbb{Q}_{p}^{d})$. Then there exist two constants $c$ and $c^{\prime}$ such that for any ${\bf x}\in\mathbb{Q}_{p}^{d}$

This lemma, in the same form but in the context of $\mathbb{R}^{d}$ and spaces of homogeneous type which can be found in ^[29,30]. A similar estimate is also given in both ^{[15,16,20,42]}. The idea of deducing $L\log L$ behavior of a function from integrability of its maximal function goes back to E. Stein in ^[39].

In order to prove Lemma $2.3$, we need the following lemma.

Lemma 2.4. Let $f\in\mathfrak{M}(\mathbb{Q}_{p}^{d})$ and $\alpha > 0$. Then we have the following estimatetes for $\omega_{H}(\alpha) = |\{{\bf x}\in \mathbb{Q}_{p}^{d}:\, \mathcal{M}^{p}(f)({\bf x}) > \alpha\}|_{H}$:

with constants $c$ and $c^{\prime}$ which do not depend on $f$ or $\alpha$.

Proof. Firstly, we give the proof of the right hand side inequality in $(2.3)$. Write $f = f_{1}+f_{2}$, where

Since $|f_{2}({\bf x})|\leq \alpha/2$ implies that $\mathcal{M}^{p}(f_{2})({\bf x})\leq \alpha/2$. Then we have $\mathcal{M}^{p}(f)({\bf x})\leq \mathcal{M}^{p}(f_{1})({\bf x})+\mathcal{M}^{p}(f_{2})({\bf x})\leq \mathcal{M}^{p}(f_{1})({\bf x})+\alpha/2$. Thus, by the weak $(1, 1)$ boundedness of maximal function $\mathcal{M}^{p}$, we have

which gives the right hand of inequality $(2.3)$.

On the other hand, we may assume that $f\in L^{1}(\mathbb{Q}_{p}^{d})$ (otherwise we truncate and apply a limiting process). Then we use the $p$-adic version of Calderón-Zygmund decomposition (see [24,Corollary 3.4]) for $f$ and $\alpha$. we have non-overlapping $p$-adic balls $B_{j}\in\mathcal{F}_{p}$, such that

for any $j$, and $|f({\bf x})|\leq \alpha$ for a.e. ${\bf x}\notin \bigcup_{j}B_{j}$. Now, since $x\in B_{j}$ implies that $\mathcal{M}^{p}(f)({\bf x}) > \alpha$, we can write

and $(2.3)$ is proved with $c^{\prime} = p^{-d}$.

Proof of Lemma $2.3$. Firstly, we give the proof of the left hand side inequality in $(2.2)$. By the definition of the Luxemburg norm, the left hand side of inequality $(2.2)$ will follow from showing that for some constant $c_{0} > 1$, $c_{0}$ independent of $f$,

where we denote $\lambda_{B_{\gamma}({\bf x})} = (c_{0}/|B_{\gamma+1}({\bf x})|_{H})\int_{B_{\gamma+1}({\bf x})}\mathcal{M}^{p}(f)({\bf y})d{\bf y}$.

Let $h = |f|/\lambda_{B_{\gamma}({\bf x})}$. Recall that $h_{B_{\gamma}({\bf x})} = \frac{1}{|B_{\gamma}({\bf x})|_{H}}\int_{B_{\gamma}({\bf x})}h({\bf y})d{\bf y}$ so that $0\leq h_{B_{\gamma+1}({\bf x})}\leq1/c_{0}$ by the $p$-adic version of Lebesgue differentiation theorem (see [24,Corollary 2.11]) and the definition of $\lambda_{B_{\gamma}({\bf x})}$. Using the formula

which holds for any Young function $\Phi$ and Harr mearsure $\nu$ (see [31,p. 406]), we have

where we use the nonation $h(E) = \int_{E}h({\bf x})d{\bf x}$ for any measurable set $E$ on $\mathbb{Q}_{p}^{d}$. Recalling that $h_{B_{\gamma+1}({\bf x})}\leq$ $1/c_{0}$, we have

For the second term $II$, by using Lemma $2.4$, we have

by using the definition of $\lambda_{B_{\gamma}({\bf x})}$. Therefore, we conclude that

if $c_{0}$ is large enough.

On the other hand, let ${\bf x}\in\mathbb{Q}_{p}^{d}$ and fix a $p$-adic ball $B_{\gamma}({\bf x})\subset\mathbb{Q}_{p}^{d}$. Let $f = f_{1}+f_{2}$, where $f_{1} = f\chi_{B_{\gamma+1}({\bf x})}$. Then

Now, $D_{2}({\bf x})$ is comparable to $\inf_{{\bf z}\in B_{\gamma}({\bf x})}\mathcal{M}^{p}(f)({\bf z})$ (see [24,p. 1298] for instance) and hence $D_{2}({\bf x})\leq C \mathcal{M}^{p}(f)({\bf x})$. To estimate $D_{1}({\bf x})$ we claim that

for all $f$ such that supp $f\subset B_{\gamma}({\bf x})$. By homogeneity we can take $f$ with $\|f\|_{L\log L, B_{\gamma}({\bf x})} = 1$ which implies

Hence, it is enough to prove

for all $f$ with supp $f\subset B_{\gamma}({\bf x})$. Indeed, by using Lemma $2.4$, we have

This imlies that $(2.6)$ holds. Hence, by using generalized Hölder's inequality and using $(2.5)$ with $B_{\gamma}({\bf x})$ replaced by $B_{\gamma+1}({\bf x})$, we have

This completes the proof of Lemma $2.3$.

The following $p$-adic version of Kolmogorov's inequality will be used in the proof Lemma $2.6$.

Lemma 2.5. Let $B_{\gamma}({\bf x})$ be any $p$-adic ball and $0 < q_{0} < q < \infty$. Then we have

Proof. Let $t$ be some positive real number which will be determined later. Then, by using Lemma 2.4 in ^[25], we have

Taking $t = \|f\|_{L^{q, \infty}(B_{\gamma}({\bf x}), d{\bf y}/|B_{\gamma}({\bf x})|_{H})}$, we conclude that

This completes the proof of the lemma.

For $\delta > 0$ and $f\in L_{\rm{loc}}^{1}(\mathbb{Q}_{p}^{d})$, the $p$-adic version of maximal function is defined by

The following lemma is true which play key role in the proof of our results.

Lemma 2.6. Let $0 < \delta < 1$ and $b\in {\rm BMO}(\mathbb{Q}_{p}^{d})$. then there exists a constant $C > 0$ such that

for all $f\in L_{\rm{loc}}^{1}(\mathbb{Q}_{p}^{d})$.

This lemma has been studied in ^[2,3] for the Euclidean setting which improves the known inequality

where $M_{\delta}^{\sharp}$, $C_{b}$ and $M$ denote the sharp maximal function, commutator of maximal function and maximal function in Euclidean case, respectively. Inequality $(2.7)$ is key tool to prove the boundedness of commutator of maximal function and it has attracted much more attention, see ^{[21,36,37,41]}.

Proof. Actually, the method stem from Agcayazi et al. ^[2], they have investigated the corresponding theorem in Euclidean case. Following their method, it is easy to give this lemma on $p$-adic vector spaces as well. For completeness, we give the deails.

Let ${\bf x}\in\mathbb{Q}_{p}^{d}$ and fix $p$-adic ball $B_{\gamma}({\bf x})$, it is enough to show that,

Now, we split $f = f_{1}+f_{2}$, where $f_{1} = f\chi_{B_{\gamma+1}({\bf x})}$. Since for any ${\bf y}\in\mathbb{Q}_{p}^{d}$

it follows that

For the first term $A_{1}({\bf x})$, by using Hölder's inequality and Lemma $2.1$, we obtain

For the second term $A_{2}({\bf x})$. Combining Lemma $2.5$ and the weak-$(1, 1)$ boundedness of $\mathcal{M}^{p}$ gives that

By using generalized Hölder's inequality $(2.1)$, we obtain

Since by (b) of Lemma $2.1$, there is a constant $C > 0$ such that for any $p$-adic ball $B_{\gamma}({\bf x})$,

we arrive at

For the third term $A_{3}({\bf x})$. This case is easy, since $A_{3}({\bf x})$ is comparable to $\inf_{{\bf y}\in B_{\gamma}({\bf x})}\, \mathcal{M}^{p}((b-b_{B_{\gamma+1}({\bf x})})f)({\bf y})$ (see [24,p. 1298] for instance), then

Again by using generalized Hölder's inequality $(2.1)$ and (b) of Lemma $2.1$, we conclude that

Finally, combining $(2.8)$–$(2.10)$ together with Lemma $2.3$, we conclude that

This finishes the proof of the lemma.

Lemma 2.7. Let $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$. Then there exists a positive constant $C$ such that

for all $f\in L_{{\rm loc}}^{1}(\mathbb{Q}_{p}^{d})$.

Proof. By using the $p$-adic version of Lebesgue differentiation theorem (see [24,Corollary 2.11])

the statement follows from Lemma $2.6$.

Considering the characteristic function $\chi_{B_{\gamma}({\bf x})}$, we have the following property (see ^[19]).

Lemma 2.8. Let $1\leq q < \infty$ and $0 < \lambda < d$, then there exist a constant $C > 0$ such that

3.   Proof of Theorems $1.1$–$1.4$

Proof of Theeorem $1.1$. Combining Lemma $2.7$ and $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ together with the stong $(q, q)$-type boundedness of $\mathcal{M}^{p}$ $(1 < q\leq\infty)$ (see [25,Theorem 1.1]) gives that (2) and (3) hold.

(3) $\Longrightarrow$ (1): Assume that $\mathcal{M}_{b}^{p}$ is bounded from $L^{q}(\mathbb{Q}_{p}^{d})$ to $L^{q}(\mathbb{Q}_{p}^{d})$ for some $1 < q\leq\infty$. For any $p$-adic ball $B_{\gamma}({\bf x})\subset\mathbb{Q}_{p}^{d}$, by using Hölder's inequality implies that

This together with Lemma $2.1$ implies that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$.

The proof of Theorem $1.1$ is completed since (2) $\Longrightarrow$ (1) follows from (3) $\Longrightarrow$ (1).

Proof of Theorem $1.2$. Combining $(1.4)$ and $f\leq \mathcal{M}^{p}(f)$ together with Lemma $2.7$ follows sthat

Thus, by using stong $(q, q)$-type boumdedness of $\mathcal{M}^{p}$ (see [25,Theorem 1.1]) implies that

We conclude that Theorem $1.2$ is proven.

Proof of Theorem $1.3$. Since the implications (2) $\Longrightarrow$ (3) and (5) $\Longrightarrow$ (4) follow readily, we only need to prove (1) $\Longrightarrow$ (2), (3) $\Longrightarrow$ (4), (4) $\Longrightarrow$ (1) and (2) $\Longrightarrow$ (5).

(1) $\Longrightarrow$ (2): The conclusion follows from Theorem $1.2$.

(3) $\Longrightarrow$ (4): By using Lemma $2.2$, it is easy to obtain that

for any fixed $p$-adic ball $B_{\gamma}({\bf x})\subset \mathcal{F}_{p}$ and all $y\in B_{\gamma}({\bf x})$. Thus, we have

which gives that (4) since the $p$-adic ball $B_{\gamma}({\bf x})\subset\mathbb{Q}_{p}^{d}$ is arbitrary.

(4) $\Longrightarrow$ (1): To prove $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$, it suffices to verify that there is a constant $C > 0$ such that for any $p$-adic ball $B_{\gamma}({\bf x})\subset\mathbb{Q}_{p}^{d}$,

For any fixed $p$-adic ball $B_{\gamma}({\bf x})$, let $E = \{{\bf y}\in B_{\gamma}({\bf x}):\, b({\bf y})\leq b_{B_{\gamma}({\bf x})}\}$ and $B_{\gamma}({\bf x})$, let $F = \{{\bf y}\in B_{\gamma}({\bf x}):\, b({\bf y}) > b_{B_{\gamma}({\bf x})}\}$. The following equality is trivially true:

Since for any ${\bf y}\in E$ we have $b({\bf y})\leq b_{B_{\gamma}({\bf x})}\leq \mathcal{M}_{B_{\gamma}({\bf x})}^{p}(b)({\bf y})$, then for any ${\bf y}\in E$,

Thus, we can conclude that

On the other hand, it follows from Hölder's inequality and $(1.5)$ that

Combining the above ineuality with $(3.3)$ it follows that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$.

In order to prove $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$, it suffices to show $b^{-} = 0$, where $b^{-} = -\min\{b, 0\}$. Let $b^{+} = |b|-b^{-}$, then $b = b^{+}-b^{-}$. For any fixed $p$-adic ball $B_{\gamma}({\bf x})$, observe that

for ${\bf y}\in B_{\gamma}({\bf x})$ and therefore we have that for ${\bf y}\in B_{\gamma}({\bf x})$,

Then, it follows from $(1.5)$ that for any $p$-adic ball ${\bf y}\in B_{\gamma}({\bf x})$,

Thus, $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$ follows from the $p$-adic version of Lebesgue differentiation theorem (see [24,Corollary 2.11]).

(2) $\Longrightarrow$ (5): This proof is similar to (3) $\Longrightarrow$ (4), we omit the details. Hence, the proof of Theorem $1.4$ is completed.

Proof of Theorem $1.4$. Similar to prove Theorem $1.3$, we only need to give the proof of (1) $\Longrightarrow$ (2), (3) $\Longrightarrow$ (4) and (4) $\Longrightarrow$ (1).

(1) $\Longrightarrow$ (2): Note that for any ${\bf x}\in\mathbb{Q}_{p}^{d}$, we have

For any $p$-adic ball $B_{\gamma}({\bf x})\subset \mathbb{Q}_{p}^{d}$, by using triangle inequality, we conclude that

Comining $(3.4)$ and $(3.5)$ together with $\mathcal{M}_{p}^{\sharp}(f)\leq 2\mathcal{M}^{p}(f)$ gives that

Since $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})\Longrightarrow |b|\in{\rm BMO}(\mathbb{Q}_{p}^{d})$, then by using Theorem $1.1$, $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$ and the stong $(q, q)$-type boumdedness of $\mathcal{M}^{p}$ (see [25,Theorem 1.1]), we have

(3) $\Longrightarrow$ (4): The proof of this case follows the procedure in ^[4]. Let $B_{\gamma}({\bf x})$ be a fixed $p$-adic ball as before. For another $p$-adic ball $B_{\gamma^{\prime}}({\bf y})$, this gives that

Without loss of generality, we may assume that $\gamma\leq \gamma^{\prime}$. Then by using Lemma $2.2$, we have that $B_{\gamma}({\bf x})\cap B_{\gamma^{\prime}}({\bf y}) = \emptyset$ or $B_{\gamma}({\bf x})\subset B_{\gamma^{\prime}}({\bf y})$. If $B_{\gamma}({\bf x})\cap B_{\gamma^{\prime}}({\bf y}) = \emptyset$, then we obtain

If $B_{\gamma}({\bf x})\subset B_{\gamma^{\prime}}({\bf y})$, then we have

where the last inequality is due to $\gamma^{\prime}, \gamma\in\mathbb{Z}$ and $1\leq\gamma^{\prime}-\gamma\in\mathbb{Z}$. On the other hand, for ${\bf y}\in B_{\gamma}({\bf x})$, we consider a $p$-adic ball $B_{\gamma^{\prime}}({\bf y})$ always containing $B_{\gamma}({\bf x})$ such that $|B_{\gamma^{\prime}}({\bf y})|_{H} = p|B_{\gamma}({\bf x})|_{H}$. This implies that $(\chi_{B_{\gamma}({\bf x})})_{p}^{\sharp}({\bf y}) = 2(p-1)/p^{2}$ for any ${\bf y}\in B_{\gamma}({\bf x})$. Hence, we have

(4) $\Longrightarrow$ (1): We proceed as in the corresponding portion of the proof of Theorem $1.3$, but some extra diffculties appear.

First, our claim is to prove that

Picking a $p$-adic ball $B_{\gamma^{\prime}}({\bf y})$ containing $B_{\gamma}({\bf x})$ such that $|B_{\gamma^{\prime}}({\bf y})|_{H} = p|B_{\gamma}({\bf x})|_{H}$. Then, we have

On the other hand

and so

Therefore, $(3.7)$ and $(3.8)$ lead us to $(3.6)$.

We can now achieve that $b\in {\rm BMO}(\mathbb{Q}_{p}^{d})$. In fact, let $E = \{y\in B_{\gamma}({\bf x}):\, b({\bf y})\leq b_{B_{\gamma}({\bf x})}\}$. Then, by using $(3.6)$ and $(1.6)$ gives that

In order to prove that $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$ we also use $(3.6)$. We start from the following fact

Averaging on $B_{\gamma}({\bf x})$, we have

Letting $\gamma\rightarrow -\infty$ with ${\bf y}\in B_{\gamma}(\bf x)$, the $p$-adic version of Lebesgue differentiation theorem assures that

and the desired result follows. This finishes the proof of Theorem $1.4$.

4.   Proof of Theorems $1.5$–$1.6$

Proof of Theorem $1.5$. Applying the similar agrument as in the proof of Theorem 1.1 in ^[25], we have that for any $p$-adic ball $B_{\gamma}({\bf x})$

Assume that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$. By using $(4.1)$ and Lemma $2.7$, we have

Thus, we conclude that

Conversely, if $\mathcal{M}_{b}^{p}$ is bounded from $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$ to $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$, then for any $p$-adic ball $B_{\gamma}({\bf x})\subset \mathbb{Q}_{p}^{d}$

where in the last step we have used Lemma $2.8$.

It follows from Lemma $2.1$ that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$. This finishes the proof of Theorem $1.5$.

Proof of Theorem $1.6$. (1) $\Longrightarrow$ (2): Assume that $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$ and $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$, then by using $(1.4)$ and Theorem $1.5$, we show that $[b, \mathcal{M}^{p}]$ is bounded from $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$ to $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$.

(2) $\Longrightarrow$ (1): Assume that $[b, \mathcal{M}^{p}]$ is bounded from $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$ to $L^{q, \lambda}(\mathbb{Q}_{p}^{d})$. Similar to estimate for $(3.3)$, we have that for any $p$-adic ball $B_{\gamma}({\bf x})\subset\mathbb{Q}_{p}^{d}$,

where inn the last step we have used Lemma $2.8$. Thus, by using Theorem $1.3$, we give that $b\in{\rm BMO}(\mathbb{Q}_{p}^{d})$ and $b^{-}\in L^{\infty}(\mathbb{Q}_{p}^{d})$. This finish the proof of Theorem $1.6$.

Acknowledgments

This work was in part supported by National Natural Science Foundation of China (Grant No. 12071473) and Shandong Jianzhu University Foundation (Grant No. X20075Z0101).

Conflict of interest

The authors declare that they have no conflict of interest and competing interests. All procedures were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. All authors contributed equally to this work. The manuscript is approved by all authors for publication. Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.