111Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000,
Pakistan
∗Corresponding Author<EMAIL_ADDRESS>
2Hakkari University, Faculty of Education, Department of Mathematics
Education, Hakkari 30000, Turkey

# Sharp Weak Bounds for $p$-adic Hardy operators on $p$-adic Linear Spaces

Amjad Hussain1,∗ ,  Naqash Sarfraz1 and  Ferit Gürbüz2

###### Abstract.

In this article, we establish sharp weak endpoint estimates for $p$-adic
fractional Hardy operator. In addition, sharp weak bounds for $p$-adic Hardy
operator on $p$-adic central Morrey space are also obtained.

###### Key words and phrases:

$p$-adic fractional Hardy operator; Boundedness; $p$-adic Weak type spaces;
Sharp bounds

###### 2010 Mathematics Subject Classification:

42B35, 26D15, 46B25, 47G10

##  1\. Introduction

Let $p$ be a prime number, if a non-zero rational number $x$ can be written in
the form

$x=\frac{s}{t}p^{k},$

where the integer $k=k(x)\in\mathbb{Z}$ and $s,t\in\mathbb{Z}$ are not
multiples of $p,$ then the function:

$|\cdot|_{p}:\mathbb{Q}\setminus\\{0\\}\rightarrow\mathbb{R},$

defined as:

$|0|_{p}=0,\quad|x|_{p}=p^{-k},$

satisfies all the axioms of a field norm with an additional property that:

$|x+y|_{p}\leq\max\\{|x|_{p},|y|_{p}\\},$ (1.1)

and is commonly known as $p$-adic norm.

Likewise, the completion of field of rational numbers $\mathbb{Q}$ with
respect to ultrametric $p$-adic norm $|\cdot|_{p}$ is denoted by
$\mathbb{Q}_{p}$ and is known as the field of $p$-adic numbers. In expanded
form, a $p$-adic number $x\neq 0$ can be written as (see [24]):

$x=p^{k}(\alpha_{0}+\alpha_{1}p+\alpha_{1}p^{2}+...)$ (1.2)

where $\alpha_{i},k\in\mathbb{Z},\alpha_{0}\neq
0,\alpha_{i}\in\\{0,1,2,...,p-1\\},i=1,2,\cdot\cdot\cdot.$ The series (1.2) is
convergent with respect to $p$-adic norm, since
$|p^{k}\alpha_{i}p^{i}|_{p}\leq p^{-k-i}.$

The higher dimensional space $\mathbb{Q}_{p}^{n}$ represents the vector space
over $\mathbb{Q}_{p}$ which consists of all points
$\mathbf{x}=(x_{1},x_{2},...,x_{n}),$ where
$x_{i}\in\mathbb{Q}_{p},i=1,2,...,n.$ The $p$-adic norm on the very space
$\mathbb{Q}_{p}^{n}$ is:

$|\mathbf{x}|_{p}=\max_{1\leq j\leq n}|x_{j}|_{p}.$

In non-Archimedean geometry, the ball and and its boundary are defined,
respectively, as:

$B_{k}(\mathbf{a})=\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{x}-\mathbf{a}|_{p}\leq
p^{k}\\},\
S_{k}(\mathbf{a})=\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{x}-\mathbf{a}|_{p}=p^{k}\\}.$

We denote $B_{k}(\mathbf{0})=B_{k}$ and $\ S_{k}(\mathbf{0})=S_{k}$ for
convenience. Additionally, for every $\mathbf{a}_{0}\in\mathbb{Q}_{p}^{n},\
\mathbf{a}_{0}+B_{k}=B_{k}(\mathbf{a}_{0})$ and
$\mathbf{a}_{0}+S_{k}=S_{k}(\mathbf{a}_{0})$.

The local compactness and commutativity of the group $\mathbb{Q}_{p}^{n}$
under addition implies the existence of Haar measure $d\mathbf{x}$ on
$\mathbb{Q}_{p}^{n}$, such that

$\int_{B_{0}}d\mathbf{x}=|B_{0}|=1,$

where the notation $|B|$ refers to the Haar measure of a measurable subset $B$
of $\mathbb{Q}_{p}^{n}.$ Also, it is not difficult to show that
$|B_{k}(\mathbf{a})|=p^{nk}$, $|S_{k}(\mathbf{a})|=p^{nk}(1-p^{-n}),$ for any
$\mathbf{a}\in\mathbb{Q}_{p}^{n}.$

Let $w(\mathbf{x})$ be a nonnegative locally integrable function on
$\mathbb{Q}_{p}^{n}$ and and $w(E)$ the weighted measure of measurable subset
$E\subset\mathbb{Q}_{p}^{n},$ that is $w(E)=\int_{E}w(x)dx$ respectively. The
space of all complex-valued functions $f$ with norm conditions:

$\|f\|_{L^{q}(w;\mathbb{Q}_{p}^{n})}=\bigg{(}\int_{\mathbb{Q}_{p}^{n}}|f(\mathbf{x})|^{p}w(\mathbf{x})d\mathbf{x}\bigg{)}^{1/q}<\infty,$

is denoted by $L^{q}(w,\mathbb{Q}_{p}^{n}),(0<q<\infty),$ and is known as
weighted Lebesgue space. Note that
$L^{q}(1,\mathbb{Q}_{p}^{n})=L^{q}(\mathbb{Q}_{p}^{n}).$

We define the weighted $p$-adic weak Lebesgue space
$L^{q,\infty}(w;\mathbb{Q}_{p}^{n})$ as the set of all measurable functions
$f$ satisfying:

$\|f\|_{L^{q,\infty}(w,\mathbb{Q}_{p}^{n})}=\sup_{\lambda>0}\lambda
w\bigg{(}\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|f(\mathbf{x})|>\lambda\\}\bigg{)}^{1/q}<\infty.$

When $w=1,$ we get the weak Lebesgue space $L^{q,\infty}(\mathbb{Q}_{p}^{n})$
defined in [26]. Next, we give the definition of strong and weak $p$-adic
central Morrey spaces, respectively.

######  Definition 1.1.

[28] Let $1\leq q<\infty$ and $-1/q\leq\lambda<0.$ A function $f\in
L^{p}_{loc}(\mathbb{Q}_{p}^{n})$ is said to belong to central Morrey space if:

$\displaystyle\begin{aligned}
\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=\sup_{\gamma\in\mathbb{Z}}\bigg{(}\frac{1}{|B_{\gamma}|_{H}^{{1+\lambda
q}}}\int_{B_{\gamma}}|f(\mathbf{x})|^{q}d\mathbf{x}\bigg{)}^{1/q}<\infty.\end{aligned}$

When $\lambda=-1/q,$ then
$\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})=L^{q}(\mathbb{Q}_{p}^{n}).$ It is not
hard to see that $\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})$ is reduced to {0}
whenever $\lambda<-1/q.$

######  Definition 1.2.

[29] Let $1\leq q<\infty$ and $-1/q\leq\lambda<0.$ The $p$-adic weak central
Morrey space $W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})$ is defined as

$\displaystyle\begin{aligned}
W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})=\\{f:\|f\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}<\infty\\},\end{aligned}$

where

$\displaystyle\begin{aligned}
\|f\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=\sup_{\gamma\in\mathbb{Z}}|B_{\gamma}|_{H}^{-\lambda-1/q}\|f\|_{WL^{q}(B_{\gamma})},\end{aligned}$

and $\|f\|_{WL^{q}(B_{\gamma})}$ is the local $p$-adic $L^{q}$-norm of $f(x)$
restricted to the ball $B_{\gamma}$, that is

$\displaystyle\begin{aligned}
\|f\|_{WL^{q}(B_{\gamma})}=\sup_{\lambda>0}|\\{\mathbf{x}\in
B_{\gamma}:|f(\mathbf{x})|>\lambda\\}|^{1/q}.\end{aligned}$

It is clear that if $\lambda=-1/q,$ then
$W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})=L^{q,\infty}(\mathbb{Q}_{p}^{n}).$
Also, $\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})\subseteq
W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})$ for $1\leq q<\infty$ and
$-1/q<\lambda<0.$

The study of $p$-adic analysis is considered important in modern ages,
because, $p$-adic analysis is a natural base for development of various models
of ultrametric diffusion energy landscape [3]. It also attracts great deal of
interest towards quantum mechanics [24], theoretical biology [10], string
theory [25], quantum gravity [1, 6], spin glass theory [2, 22]. In [3], it was
shown that the $p$-adic analysis can be efficiently applied both to relaxation
in complex speed systems and processes combined with the relaxation of a
complex environment. Besides, $p$-adic analysis has an imperative role in
pseudo-differential equations and harmonic analysis, see for example [4, 8, 9,
19].

For a non-negative integrable function $f$ on $\mathbb{R}^{+},$ the one
dimensional Hardy operator is defined as:

$hf(x)=\frac{1}{x}\int_{0}^{x}f(y)dy,\quad x>0,$ (1.3)

Hardy, in [17], established the following integral inequality for the Hardy
operator:

$\|hf\|_{L^{q}(\mathbb{R}^{+})}\leq\frac{q}{q-1}\|f\|_{L^{q}(\mathbb{R}^{+})},\quad
1<q<\infty.$ (1.4)

Later, Faris in [11] gave the following extension of one dimensional Hardy
operator:

$\mathcal{H}f(\mathbf{x})=\frac{1}{|\mathbf{x}|^{n}}\int_{|\mathbf{y}|\leq|\mathbf{x}|}f(\mathbf{y})d\mathbf{y}.$
(1.5)

In [7], the authors computed the norm of Hardy operator on the Lebesgue space,
whereas Fu et al., in [12], obtained the boundedness of $\mathcal{H}$ on power
weighted Lebesgue space $L^{q}(|\mathbf{x}|^{\alpha})$ for $1<q<\infty.$ For
more details about the boundedness of Hardy operator we included some
references [5, 18, 23, 33].

On the other hand, the fractional Hardy operator is obtained by merely
interchanging $|\cdot|^{n}$ with $|\cdot|^{n+\alpha}$ ($0\leq\alpha<n$) in
(1.5). The weak and strong type optimal bounds for the fractional Hardy and
adjoint Hardy operator has also spotlighted many researchers in the past, see
for example [12, 14, 15, 31, 32].

For $f\in L_{\rm{loc}}(\mathbb{Q}_{p}^{n})$ and $0\leq\alpha<n,$ Wu, in
([27]), defined the $p$-adic fractional Hardy operator as:

$H^{p}_{\alpha}f(\mathbf{x})=\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}f(\mathbf{y})d\mathbf{y},\qquad\mathbf{x}\in\mathbb{Q}_{p}^{n}\setminus\\{\mathbf{0}\\}.$

If $\alpha=0,$ the fractional $p$-adic Hardy operator is reduced to $p$-adic
Hardy operators, see [13]. Some other papers showing the boundedness of
$p$-adic Hardy type operator include [16, 20, 21, 28].

In this article, we give the sharp weak endpoint estimates for $p$-adic
fractional Hardy operator on $p$-adic Lebesgue space. Moreover, Sharp weak
bounds for $p$-adic Hardy operator on $p$-adic central morrey space are also
obtained.

##  2\. Sharp weak endpoint estimates for $p$-adic fractional Hardy Operator

Our main result for this section read as:

######  Theorem 2.1.

Let $0<\alpha<n$ and $n+\gamma>0.$ If $f\in L^{1}(\mathbb{Q}_{p}^{n}),$ then

$\displaystyle\begin{aligned}
\|H^{p}_{\alpha}f\|_{L^{(n+\gamma)/(n-\alpha),\infty}(|\mathbf{x}|_{p}^{\gamma};\mathbb{Q}_{p}^{n})}\leq&C\|f\|_{L^{1}(\mathbb{Q}_{p}^{n})},\end{aligned}$

where the constant

$C=\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}$

is optimal.

###### Proof.

Since

$\displaystyle\begin{aligned}
|H^{p}_{\alpha}f(\mathbf{x})|=&\bigg{|}\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}f(\mathbf{y})d\mathbf{y}\bigg{|}\\\
=&\bigg{|}\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}f(\mathbf{y})d\mathbf{y}\bigg{|}\\\
\leq&|\mathbf{x}|_{p}^{-(n-\alpha)}\|f\|_{L^{1}(\mathbb{Q}_{p}^{n})}.\end{aligned}$
(2.1)

Let $C_{1}=\|f\|_{L^{1}(\mathbb{Q}_{p}^{n})},$ then

$\displaystyle\begin{aligned}
\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|H^{p}_{\alpha}f(\mathbf{x})|>\lambda\\}\subset\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{x}|_{p}<(C_{1}/\lambda)^{1/(n-\alpha)}\\}.\end{aligned}$

Thus,

$\displaystyle\begin{aligned}
&\|H^{p}_{\alpha}f\|_{L^{(n+\gamma)/(n-\alpha),\infty}(|x|_{p}^{\gamma};\mathbb{Q}_{p}^{n})}\\\
\leq&\sup_{\lambda>0}\lambda\bigg{(}\int_{\mathbb{Q}_{p}^{n}}\chi_{\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|H^{p}_{\alpha}f(\mathbf{x})|>\lambda\\}}(\mathbf{x})|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
\leq&\sup_{\lambda>0}\lambda\bigg{(}\int_{\mathbb{Q}_{p}^{n}}\chi_{\big{\\{}\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{x}|_{p}<\big{(}C_{1}/\lambda\big{)}^{1/(n-\alpha)}\big{\\}}}(\mathbf{x})|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\sup_{\lambda>0}\lambda\bigg{(}\int_{|\mathbf{x}|_{p}<\big{(}C_{1}/\lambda\big{)}^{1/(n-\alpha)}}|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\sup_{\lambda>0}\lambda\bigg{(}\sum_{j=-\infty}^{\log_{p}\big{(}C_{1}/\lambda\big{)}^{1/(n-\alpha)}}\int_{S_{j}}|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&(1-p^{-n})^{(n-\alpha)/(n+\gamma)}\sup_{\lambda>0}\lambda\bigg{(}\sum_{j=-\infty}^{\log_{p}\big{(}C_{1}/\lambda\big{)}^{1/(n-\alpha)}}p^{j(n+\gamma)}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\sup_{\lambda>0}\lambda\bigg{(}\frac{C_{1}}{\lambda}\bigg{)}\\\
=&\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\|f\|_{L^{1}(|\mathbf{x}|_{p}^{\beta})}.\end{aligned}$
(2.2)

To show that the constant

$\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)},$

appeared in (LABEL:NT1) is optimal, we employ the idea of use of power
function given in [30], hence, we let

$\displaystyle\begin{aligned}
f_{0}(\mathbf{x})=\chi_{\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{x}|_{p}\leq
1\\}}(\mathbf{x}),\end{aligned}$

then

$\displaystyle\begin{aligned}
\|f_{0}\|_{L^{1}(\mathbb{Q}_{p}^{n})}=1.\end{aligned}$

Also,

$\displaystyle\begin{aligned}
H_{\alpha}^{p}f_{0}(\mathbf{x})=&\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}f_{0}(\mathbf{y})d\mathbf{y}\\\
=&\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}\chi_{\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|\mathbf{y}|_{p}\leq
1\\}}(\mathbf{y})d\mathbf{y}\\\
=&\frac{1}{|\mathbf{x}|_{p}^{n-\alpha}}\begin{cases}\int_{|\mathbf{y}|_{p}\leq|\mathbf{x}|_{p}}d\mathbf{y},&|\mathbf{x}|_{p}\leq
1;\\\ \int_{|\mathbf{y}|_{p}\leq
1}d\mathbf{y},&|\mathbf{x}|_{p}>1.\end{cases}\end{aligned}$

Since $|B_{\log_{p}|\mathbf{x}|_{p}}|_{H}=|\mathbf{x}|_{p}^{n}|B_{0}|_{H},$
therefore,

$\displaystyle\begin{aligned}
H^{p}_{\alpha}f_{0}(\mathbf{x})=\begin{cases}|\mathbf{x}|_{p}^{\alpha},&|\mathbf{x}|_{p}\leq
1;\\\
|\mathbf{x}|_{p}^{\alpha-n},&|\mathbf{x}|_{p}>1.\end{cases}\end{aligned}$

Now,

$\displaystyle\begin{aligned}
\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|H^{p}_{\alpha}f_{0}(\mathbf{x})|>\lambda\\}=&\\{|\mathbf{x}|_{p}\leq
1:|\mathbf{x}|_{p}^{\alpha}>\lambda\\}\cup\\{|\mathbf{x}|_{p}>1:|\mathbf{x}|_{p}^{\alpha-n}>\lambda\\}.\end{aligned}$

Since $0<\alpha<n,$ therefore, when $\lambda\geq 1,$ then

$\displaystyle\begin{aligned}
\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|H^{p}_{\alpha}f_{0}(\mathbf{x})|>\lambda\\}=\emptyset,\end{aligned}$

and when $0<\lambda<1,$ then

$\displaystyle\begin{aligned}
\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:|H^{p}_{\alpha}f_{0}(\mathbf{x})|>\lambda\\}=\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:(\lambda)^{1/n}<|\mathbf{x}|_{p}<(1/\lambda)^{1/n-\alpha}\\}.\end{aligned}$

Ultimately we are down to:

$\displaystyle\begin{aligned}
&\|\mathcal{H}^{p}_{\alpha}f_{0}\|_{L^{(n+\gamma)/(n-\alpha))),\infty}(|\mathbf{x}|_{p}^{\gamma};\mathbb{Q}_{p}^{n})}\\\
=&\sup_{0<\lambda<1}\lambda\bigg{(}\int_{\mathbb{Q}_{p}^{n}}\chi_{\\{\mathbf{x}\in\mathbb{Q}_{p}^{n}:(\lambda)^{1/\alpha}<|\mathbf{x}|_{p}<(1/\lambda)^{1/(n-\alpha)}\\}}(\mathbf{x})|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\sup_{0<\lambda<1}\lambda\bigg{(}\int_{(\lambda)^{1/\alpha}<|\mathbf{x}|_{p}<(1/\lambda)^{1/(n-\alpha)}}|\mathbf{x}|_{p}^{\gamma}d\mathbf{x}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&(1-p^{-n})^{(n-\alpha)/(n+\gamma)}\sup_{0<\lambda<1}\lambda\bigg{(}\sum_{j=\log_{p}\lambda^{1/\alpha}+1}^{\log_{p}\lambda^{1/(\alpha-n)}}p^{j(n+\gamma)}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&(1-p^{-n})^{(n-\alpha)/(n+\gamma)}\sup_{0<\lambda<1}\lambda\bigg{(}\frac{p^{(\log_{p}\lambda^{1/\alpha}+1)(n+\gamma)}-p^{(\log_{p}\lambda^{1/(\alpha-n)}+1)(n+\gamma)}}{1-p^{(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&(1-p^{-n})^{(n-\alpha)/(n+\gamma)}\sup_{0<\lambda<1}\lambda\bigg{(}\frac{{\lambda^{(n+\gamma)/\alpha}}-{\lambda^{(n+\gamma)/(\alpha-n)}}}{p^{-(n+\gamma)}-1}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&(1-p^{-n})^{(n-\alpha)/(n+\gamma)}\sup_{0<\lambda<1}\bigg{(}\frac{1-{\lambda^{(n+\gamma)/\alpha}}{\lambda^{(n+\gamma)/(n-\alpha)}}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\sup_{0<\lambda<1}\bigg{(}1-{\lambda^{(n+\gamma)/\alpha}}{\lambda^{(n+\gamma)/(n-\alpha)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\\\
=&\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{(n-\alpha)/(n+\gamma)}\|f_{0}\|_{L^{1}(\mathbb{Q}_{p}^{n})}.\end{aligned}$
(2.3)

We thus conclude from (LABEL:NT1) and (LABEL:n1) that

$\displaystyle\begin{aligned}
\|{H}^{p}_{\alpha}\|_{{L^{1}(\mathbb{Q}_{p}^{n})\rightarrow
L^{(n+\gamma)/(n-\alpha),\infty}(|\mathbf{x}|_{p}^{\gamma};\mathbb{Q}_{p}^{n})}}=\bigg{(}\frac{1-p^{-n}}{1-p^{-(n+\gamma)}}\bigg{)}^{1/q}.\end{aligned}$

∎

##  3\. Sharp bound for $p$-adic Hardy operator

The present section investigates the boundedness of $p$-adic Hardy operator on
$p$-adic weak central Morrey spaces. It is shown the constant obtained in this
case is also optimal.

######  Theorem 3.1.

Let $1\leq q<\infty,$ $-1/q\leq\lambda<0$ and if
$f\in\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n}),$ then

$\displaystyle\begin{aligned}
\|Hf\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}\leq\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})},\end{aligned}$

and the constant $1$ is optimal.

###### Proof.

Applying Hölder’s inequality:

$\displaystyle\begin{aligned}
|H^{p}f(\mathbf{x})|\leq&\frac{1}{|\mathbf{x}|_{p}^{n}}\bigg{(}\int_{B(0,|\mathbf{x}|_{p})}|f(\mathbf{y})|^{q}d\mathbf{y}\bigg{)}^{1/q}\bigg{(}\int_{B(0,|\mathbf{x}|_{p})}d\mathbf{y}\bigg{)}^{1/q^{\prime}}\\\
=&|\mathbf{x}|_{p}^{n\lambda}\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$

Let $C_{2}=\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.$ Since
$\lambda<0,$ then

$\displaystyle\begin{aligned}
\|Hf\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}\leq&\sup_{\gamma\in\mathbb{Z}}\sup_{t>0}t|B_{\gamma}|_{H}^{-\lambda-1/q}\big{|}\\{\mathbf{x}\in
B_{\gamma}:C_{2}|\mathbf{x}|_{p}^{n\lambda}>t\\}\big{|}^{1/q}\\\
=&\sup_{\gamma\in\mathbb{Z}}\sup_{t>0}t|B_{\gamma}|_{H}^{-\lambda-1/q}\big{|}\\{|\mathbf{x}|_{p}\leq
p^{\gamma}:|\mathbf{x}|_{p}<(t/C_{2})^{1/n\lambda}\\}\big{|}^{1/q}.\end{aligned}$

If $\gamma\leq\log_{p}(t/C_{2})^{1/n\lambda},$ then for $\lambda<0,$ we obtain

$\displaystyle\begin{aligned}
&\sup_{t>0}\sup_{\gamma\leq\log_{p}(t/C_{2})^{1/n\lambda}}t|B_{\gamma}|_{H}^{-\lambda-1/q}\big{|}\\{|\mathbf{x}|_{p}\leq
p^{\gamma}:|\mathbf{x}|_{p}<(t/C_{2})^{1/n\lambda}\\}\big{|}^{1/q}\\\
&\leq\sup_{t>0}\sup_{\gamma\leq\log_{p}(t/C_{2})^{1/n\lambda}}tp^{-\gamma
n\lambda}\\\ &=C_{2}\\\
&=\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$

If $\gamma>\log_{p}(t/C_{2})^{1/n\lambda},$ then for $\lambda+1/q>0,$ we get

$\displaystyle\begin{aligned}
&\sup_{t>0}\sup_{\gamma>\log_{p}(t/C_{2})^{1/n\lambda}}t|B_{\gamma}|_{H}^{-\lambda-1/q}|\\{|\mathbf{x}|_{p}\leq
p^{\gamma}:|\mathbf{x}|_{p}<(t/C_{2})^{1/n\lambda}\\}|^{1/q}\\\
&\leq\sup_{t>0}\sup_{\gamma>\log_{p}(t/C_{2})^{1/n\lambda}}tp^{-\gamma
n(\lambda+1/q)}(t/C_{2})^{1/q\lambda}\\\ &=C_{2}\\\
&=\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$

Therefore,

$\displaystyle\begin{aligned}
\|Hf\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}\leq\|f\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$
(3.1)

Conversely, to prove that the constant $1$ is optimal, we let

$f_{0}(\mathbf{x})=\chi_{\\{|\mathbf{x}|_{p}\leq 1\\}}(\mathbf{x}),$

then,

$\|f_{0}\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=\sup_{\gamma\in\mathbb{Z}}\bigg{(}\frac{1}{|B_{\gamma}|_{H}^{{1+\lambda
q}}}\int_{B_{\gamma}}\chi_{\\{|\mathbf{x}|_{p}\leq
1\\}}(\mathbf{x})d\mathbf{x}\bigg{)}^{1/q}.$

If $\gamma<0,$ then

$\displaystyle\begin{aligned} \sup_{\begin{subarray}{c}\gamma\in\mathbb{Z}\\\
\gamma<0\end{subarray}}\bigg{(}\frac{1}{|B_{\gamma}|_{H}^{{1+\lambda
q}}}\int_{B_{\gamma}}d\mathbf{x}\bigg{)}^{1/q}=\sup_{\begin{subarray}{c}\gamma\in\mathbb{Z}\\\
\gamma<0\end{subarray}}p^{-n\gamma\lambda}=1,\end{aligned}$

since $\lambda<0.$ If $\gamma\geq 0,$ then using the condition that
$\lambda+1/q>0,$ we have

$\displaystyle\begin{aligned} \sup_{\begin{subarray}{c}\gamma\in\mathbb{Z}\\\
\gamma\geq 0\end{subarray}}\bigg{(}\frac{1}{|B_{\gamma}|_{H}^{{1+\lambda
q}}}\int_{B_{0}}d\mathbf{x}\bigg{)}^{1/q}=\sup_{\begin{subarray}{c}\gamma\in\mathbb{Z}\\\
\gamma\geq 0\end{subarray}}p^{-n\gamma(\lambda+1/q)}=1.\end{aligned}$

Therefore,

$\|f_{0}\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=1.$

Moreover,

$\displaystyle\begin{aligned}
H^{p}f_{0}(\mathbf{x})=\begin{cases}1,&|\mathbf{x}|_{p}\leq 1;\\\
|\mathbf{x}|_{p}^{-n},&|\mathbf{x}|_{p}>1,\end{cases}\end{aligned}$

which implies that $|H^{p}f_{0}(\mathbf{x})|\leq 1.$

Next, in order to construct weak central Morrey norm we divide our analysis
into following two cases:
Case 1. When $\gamma\leq 0,$ then

$\displaystyle\begin{aligned} \|Hf_{0}\|_{WL^{q}(B_{\gamma})}=\sup_{0<t\leq
1}t|\\{\mathbf{x}\in B_{\gamma}:1>t\\}|^{1/q}=p^{n\gamma/q},\end{aligned}$

and

$\displaystyle\begin{aligned}
\|Hf_{0}\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=\sup_{\gamma<0}|B_{\gamma}|_{H}^{-\lambda-1/q}\|f\|_{WL^{q}(B_{\gamma})}=\sup_{\gamma<0}p^{-n\gamma\lambda}=1=\|f_{0}\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$

Case 2. When $\gamma>0,$ we have

$\displaystyle\begin{aligned} \|Hf_{0}\|_{WL^{q}(B_{\gamma})}=&\sup_{0<t\leq
1}t|\\{\mathbf{x}\in
B_{\mathbf{0}}:1>t\\}\cup\\{1\leq|\mathbf{x}|_{p}<p^{\gamma}:|\mathbf{x}|_{p}^{-n}>t\\}|^{1/q}.\end{aligned}$

For further analysis, this case is further divided into the following
subcases:
Case 2(a). If $1<\gamma<\log_{p}t^{-1/n},$ then

$\displaystyle\begin{aligned} \|Hf_{0}\|_{WL^{q}(B_{\gamma})}=\sup_{0<t\leq
1}t\\{1+p^{n\gamma}-1\\}^{1/q}=\sup_{0<t\leq 1}tp^{n\gamma/q}.\end{aligned}$

Case 2(b). If $1<\log_{p}t^{-1/n}<\gamma,$ then:

$\displaystyle\begin{aligned} \|Hf_{0}\|_{WL^{q}(B_{\gamma})}=&\sup_{0<t\leq
1}t(1+t^{-1}-1)^{1/q}=&\sup_{0<t\leq 1}t^{1-1/q}.\end{aligned}$

Now, for $1\leq q<\infty$ and $-1/q\leq\lambda<0,$ from case 2(a) and 2(b), we
obtain

$\displaystyle\begin{aligned}
&\|Hf_{0}\|_{W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}\\\
&=\max\bigg{\\{}\sup_{0<t\leq
1}\sup_{1<\gamma\leq\log_{p}(1/t)^{1/n}}tp^{-n\gamma\lambda},\sup_{0<t\leq
1}\sup_{1<\log_{p}(1/t)^{1/n}<\gamma}t^{1-1/q}p^{-n\gamma(\lambda+1/q)}\bigg{\\}}\\\
&=\max\bigg{\\{}\sup_{0<t\leq 1}t^{1+\lambda},\sup_{0<t\leq
1}t^{1+\lambda}\bigg{\\}}\\\
&=1=\|f_{0}\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}.\end{aligned}$ (3.2)

Finally, using (3.1) and (LABEL:2), we arrive at:

$\displaystyle\begin{aligned}
\|H\|_{\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})\rightarrow
W\dot{B}^{q,\lambda}(\mathbb{Q}_{p}^{n})}=1.\end{aligned}$

∎

Conflict of Interest: The authors declare that they have no conflict of
interest.

Data Availability Statement: No data were used to support this study.

Funding Statement: This research is partially supported by Higher Education
Commission (HEC) NRPU Programme 2017-18 [7098/Federal/ NRPU/R&D/HEC/ 2017] and
the Quaid-I-Azam University Research Fund [URF-2019].

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