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15. The vee-layered permutation 9 2 1 5 4 3 6 8 7 10 11 and its image under
the bijection in Theorem 5.6, the $\bf{S}$-Motzkin path
$EUEUDEDUEUEDDUD$.

######  Theorem 5.7.

We have $|\mathcal{U}_{2k+1}(132,3412)|=\frac{1}{2k+1}{3k\choose k}$.

###### Proof.

The result follows from Theorem 5.6 and some previous work. Theorem 6.1 in [8]
states that there exists a bijection
$\text{swu}:\mathcal{U}_{2k+1}(231)\to\mathcal{U}_{2k+1}(132)$. In the proof
of Theorem 5.1 in [12], Defant shows that
$\text{swu}(\text{Av}(231,1423))=\text{Av}(132,3412)$. Consequently, swu
yields a bijection between $\mathcal{U}_{2k+1}(231,1423)$ and
$\mathcal{U}_{2k+1}(132,3412)$. Thus
$|\mathcal{U}_{2k+1}(132,3412)|=|\mathcal{U}_{2k+1}(231,1423)|=\frac{1}{2k+1}{3k\choose
k}$. ∎

##  6\. A Bijection with a Subclass of Schröder Paths

We now introduce another well-studied type of lattice path. A Schröder path of
semilength $k$ is a path from $(0,0)$ to $(2k,0)$ that consists of an equal
number of up steps and down steps, as well as some number of (2,0) steps
(called horizontal steps and denoted $H$ in the up-down sequence) and at no
point crosses below the horizontal axis. The $k^{\text{th}}$ Schröder number,
denoted $\mathscr{S}_{k}$, is defined to be the number of Schröder paths of
semilength $k$.

Figure 16. The Schröder path $UHUUUDDUHDDD$ of semilength 7.

It is well-known that the Schröder paths can be divided into two equinumerous
subclasses: paths that have a horizontal step on the horizontal axis and paths
that do not have a horizontal step on the horizontal axis. Let
$\mathcal{S}_{k}$ be the set of paths in the latter subclass of length $k$. It
follows that $|\mathcal{S}_{k}|=\mathscr{S}_{k}/2$ for $k\geq 1$ (and
$|\mathcal{S}_{0}|=\mathscr{S}_{0}=1$); these numbers are (somewhat aptly)
known as the _little Schröder numbers_ 333Sometimes they are called (also
somewhat aptly) the small Schröder numbers..

######  Theorem 6.1.

We have $|\mathcal{U}_{2k+1}(231,1432)|=|\mathcal{S}_{k}|$.

###### Proof.

Consider $\pi\in\mathcal{U}_{2k+1}(231,1432)$ and some consecutive subsequence
of $\pi$, called $\rho$. Since $\rho$ avoids 231, we can write
$\rho=\lambda\rho_{m}\mu$, where $\rho_{m}$ is the largest entry in $\rho$ and
$\lambda$ is below $\mu$. We now consider two cases.

Case 1: The subsequence $\lambda$ is nonempty. Then, since a point in
$\lambda$ lies below and to the left of $\mu$ and $\rho_{m}$ lies above and
horizontally between $\lambda$ and $\mu$, we have that $\mu$ must not contain
21 in order for $\rho$ to avoid 1432. Thus, $\mu$ is increasing. If $\mu$ has
size $n\geq 2$, then $\rho_{m}$ and $\mu_{n}$ are both ascent tops and thus NE
endpoints of hooks $H_{i}$ and $H_{j}$, respectively, in $\pi$’s CHC. If
$\pi_{i},\pi_{j}$ are the SW endpoints of $H_{i},H_{j}$, respectively, then
$i<j$ in order for the hooks to not intersect. But then $\rho_{m}$ would be
the NE endpoint of $H_{j}$, and not $\mu_{n}$, because $\rho_{m}$ is to the
left of and immediately above $\mu_{n}$. Thus, $\mu$ must have size at most 1,
giving us that $\rho=\lambda\oplus\tau$, where $\tau$ is $1$ or $21$.

Case 2: The subsequence $\lambda$ is empty. Then there are no clear
restrictions on $\mu$.

We first perform this decomposition process on $\pi$. In case 1, we can repeat
this decomposition process on $\lambda$; in case 2, we can repeat it on $\mu$.
We repeat until the unknown subsequences are empty. The result is the
decomposition $\pi=L1R$ where $L$ is decreasing and $R$’s normalization is
$\tau_{1}\oplus\cdots\oplus\tau_{\ell}$ where each $\tau_{i}$ is $1$ or $21$.
Thus $\pi$ is a permutation that is vee, except points to the right of the
vertical can be replaced with the block 21. We call this type of permutation
vee-step. Note that, by definition, the set of vee-step permutations is the
set of vee-layered permutations (described in the proof of Theorem 5.6) in
which each layer is at most size 2.

Now we define a map from $\mathcal{U}_{2k+1}(231,1432)$ to $\mathcal{S}_{k}$
as follows: Given $\pi\in\mathcal{U}_{2k+1}(231,1432)$, we write $\pi=L1R$ as
before and label every point in $L$ with the symbol $U$. Recall $R$ has
normalization $\tau_{1}\oplus\cdots\oplus\tau_{\ell}$ where $\tau_{i}=1$ or
$21$. Treating each $\tau_{i}$ as a single unit, if $\tau_{i}=1$ we label the
corresponding point in $R$ with a $D$ and if $\tau_{i}=21$ we label the
corresponding pair points in $R$ with a single $H$. Then, starting at the
lowest point 1 in $\pi$, we read off the labels from bottom to top to obtain a
path $\Lambda$. Since $\pi$ is uniquely sorted, it has $k$ ascents and $k$
descents. The steps labeled with $H$ contain both an ascent top and a descent
bottom, so $\Lambda$ has an equal number of $U$’s and $D$’s. Moreover, $\pi$
being sorted implies not only that every prefix of $\Lambda$ contains as many
$U$’s as $D$’s, but also that there are no horizontal steps in $\Lambda$ along
the horizontal axis. This is because $H$ corresponds to a 21 block, which
contains an ascent top followed by a descent bottom. If this occurred along
the horizontal axis, there would be a prefix subsequence
$\pi_{1}\cdots\pi_{m}$ with more ascent tops than descent bottoms, meaning
that there must be an ascent top that is not a NE endpoint, which contradicts
Lemma 2.2. Thus, $\Lambda\in\mathcal{S}_{k}$. The map is easily reversible,
making it a bijection. Therefore
$|\mathcal{U}_{2k+1}(231,1432)|=|\mathcal{S}_{k}|$.∎

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17. The vee-step permutation 732154689 and its image under the bijection in
Theorem 6.1, the Schröder path UUHDUDD.

##  7\. Counting $\mathcal{U}_{2k+1}(231,4312)$ with a Generating Function

In [8], Defant showed how to decompose permutations in
$\mathcal{U}_{2k+1}(231,4132)$ to obtain an identity proving these
permutations are in bijection with Pallo comb intervals. In this section, we
modify Defant’s method to instead decompose the permutations in
$\mathcal{U}_{2k+1}(231,4312)$, which are counted by the same sequence.

######  Theorem 7.1.

We have

$\sum_{k\geq 0}|\mathcal{U}_{2k+1}(231,4312)|x^{k}=C(xC(x))$,

where $C(x)=\frac{1-\sqrt{1-4x}}{2x}$ is the generating function of the
sequence of Catalan numbers.

###### Proof.

Recall that a uniquely sorted permutation $\pi=\pi_{1}\cdots\pi_{2k+1}$ is
called _nice_ if the point $(1,\pi_{1})$ is the southwest endpoint of a hook
whose northeast endpoint is $(2k+1,2k+1)$. Consider
$\pi\in\mathcal{U}_{2k+1}(231,4312)$ and, for now, assume that $\pi$ is nice.
Since $\pi$ avoids 231, we are able to write $\pi=\pi_{1}\lambda\mu(2k+1)$,
where $\lambda\in S_{\pi_{1}-1}$(which is nonempty because, by assumption, $1$
is a descent) and $\mu$ is a permutation of $\pi_{1}+1,\ldots,2k$. Since
$\lambda$ is a subsequence of $\pi$, it avoids 231. Since $\pi$ avoids 4312,
$\lambda$ must avoid 312, so $\lambda$ is layered.

Note that the first layer in $\lambda$ cannot be a single point. If it were,
$\lambda_{2}$ would be an ascent top and thus a NE endpoint of a hook. But the
only descent top before $\lambda_{2}$ is $\pi_{1}$, which lies above
$\lambda_{2}$, a contradiction.

Let $m$ be the largest integer such that the normalization of
$\lambda_{1}\cdots\lambda_{2m+1}$ is in $\mathcal{U}_{2m+1}(231,312)$ and such
that $\lambda_{2m+1}$ is the first entry in a layer of size at least two. We
let $\tau=\lambda_{1}\cdots\lambda_{2m+1}$. Since the first layer of $\lambda$
has size at least two, and the single entry $\lambda_{1}$ is in
$\mathcal{U}_{1}(231,312)$, such a $\tau$ always exists. Let $\sigma$ be the
remaining entries of $\lambda$, so that $\tau\sigma=\lambda$.

Now, let $\pi^{\prime}=\pi_{1}\sigma\mu$, that is, the permutation obtained by
removing $\tau$ and $(2k+1)$ from $\pi$. We claim that
$\pi^{\prime}\in\mathcal{U}_{2k-2m-1}(231,4312)$. Because $\pi^{\prime}$ is a
subsequence of $\pi$, it avoids 231 and 4312. Since to obtain $\pi^{\prime}$
we remove one permutation of length $2m+1$ and one of length 1, we have that
$\pi^{\prime}$ has length $2k-2m-1$.

Since $\pi\in\mathcal{U}_{2k+1}$ and $\tau\in\mathcal{U}_{2m+1}$, the
permutations $\pi$ and $\tau$ have $k$ descents and $m$ descents,
respectively. Thus when we take out $\tau$, $\pi^{\prime}$ loses $m$ descents.
Because all of $\sigma$ lies below $\pi_{1}$, we have that $\pi_{1}$ remains a
descent top in $\pi^{\prime}$. Since $(2k+1,2k+1)$ has the maximal height in
the plot of $\pi$, the point $(2k,\pi_{2k})$ was not a descent top of $\pi$,
so when we remove $2k+1$, $\pi^{\prime}$ does not lose any descents.
Originally $\lambda_{2m+1}$ was not the last entry in a decreasing block, so
it was a descent top in $\pi$ but now is not in $\tau$. Thus $\pi^{\prime}$
has $k-m-1=\frac{2k-2m-1-1}{2}$ descents.

All that is left to check is that $\pi^{\prime}$ has a CHC. Note that $\tau$
is, when considered its own permutation, uniquely sorted, and appears all the
way on the left of $\pi$, save for $\pi_{1}$. Thus in $\pi$ all hooks with an
endpoint in $\tau$ have the other endpoint in $\tau$, except for the last
point in $\tau$, which is a descent top in $\pi$ and thus is the SW endpoint
of some hook $H_{i}$. Then when we remove $\tau$ and $(2k+1,2k+1)$ to obtain
$\pi^{\prime}$, all hooks in $\pi^{\prime}$ are preserved except $H_{1}$,
which is missing a NE endpoint, and $H_{i}$, which is missing a SW endpoint.
We create the hook $H_{1}^{\prime}$ connecting $\pi^{\prime}_{1}$ to the NE
endpoint of $H_{i}$, which resolves the issue and results in a CHC of
$\pi^{\prime}$. Thus, $\pi^{\prime}\in\mathcal{U}_{2k-2m-1}(231,4312)$. Now,
let $\pi^{\prime\prime}$ be the normalization of $\pi^{\prime}$ and
$\tau^{\prime}$ be the normalization of $\tau$. We have obtained
$(\pi^{\prime\prime},\tau^{\prime})\in\mathcal{U}_{2k-2m-1}(231,4312)\times\mathcal{U}_{2m+1}(231,312)$
from the original $\pi$.

We now show that this process is reversible. Given some
$(\pi^{\prime\prime},\tau^{\prime})\in\mathcal{U}_{2k-2m-1}(231,4312)\times\mathcal{U}_{2m+1}(231,312)$,
we can insert the plot of $\tau^{\prime}$ under $\pi^{\prime\prime}_{1}$ and
merge its last block with the first block in
$\pi^{\prime\prime}_{2}\cdots\pi^{\prime\prime}_{2k-2m-1}$. Lastly, we append
$(2k+1)$ at the end to recover the original $\pi$. Thus, this decomposition is
bijective, so nice permutations in $\mathcal{U}_{2k+1}(231,4312)$ correspond
to pairs of permutations in
$\mathcal{U}_{2k-2m-1}(231,4312)\times\mathcal{U}_{2m+1}(231,312)$.

From Lemma 3.3, we have that $|\mathcal{U}_{2m+1}(231,312)|=C_{m}$, and thus
$\sum_{m\geq 0}|\mathcal{U}_{2m+1}(231,312)|x^{m}\\\ =C(x)$. Reindexing gives
$\sum_{n\geq 1}|\mathcal{U}_{n}(231,312)|=xC(x^{2})$. Let $B(x)=\sum_{k\geq
0}|\mathcal{U}_{2k+1}(231,4312)|x^{k}$ and $\widetilde{B}(x)=\sum_{n\geq
1}|\mathcal{U}_{n}(231,4312)|x^{n}$. Then nice permutations in
$\mathcal{U}_{2k+1}(231,4312)$ are counted by the generating function
$[\widetilde{B}(x)][xC(x^{2})][x]=x^{2}C(x^{2})\tilde{B}(x)$. The rest of the
proof is identical to that of Theorem 8.1 in Defant’s article [8]: using the
generating function for nice permutations in $\mathcal{U}_{2k+1}(231,4312)$ to
count all such permutations, we can prove that
$\tilde{B}(x)=x+xC(x^{2})\tilde{B}(x)^{2}$, from which it follows that
$B(x)=C(xC(x))$, as desired. Refer to Defant’s proof for details. ∎

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18. Decomposing the nice permutation $\pi$ into $\tau$ and $\pi^{\prime}$.

##  8\. Conclusion

The theorems in this paper prove nine out of the eighteen conjectures in [8],
which enumerate classes of the form
$\mathcal{U}_{2k+1}(\tau^{(1)},\tau^{(2)})$, where $\tau^{(1)}$ has length 3
and $\tau^{(2)}$ has length 4. The nine remaining conjectures are given again
in Table 2.

Patterns |  OEIS Sequence
---|---
$312,1243$ |  A122368
$231,1243$ |  A001700
$132,2341$ $132,4123$ |  A109081
$312,2341$ |  A006605

Patterns |  OEIS Sequence
---|---
$312,3241$ |  A279569
$312,4321$ |  A063020
$132,4231$ |  A071725
$231,4321$ |  A056010
|

Table 2. The remaining conjectural OEIS sequences enumerating sets of the form
$\mathcal{U}_{2k+1}(\tau^{(1)},\tau^{(2)})$.

It should be noted that the remaining conjectures include
$\mathcal{U}_{2k+1}(231,1243)$ being counted by ${2k-1\choose k}$, which is
the same sequence counting the two classes in Section 4. Decomposition gives
that for a permutation $\pi$ in this class, we can write $\pi=DD^{\prime}\tau
I$ where $D,D^{\prime}$ are decreasing, $I$ is increasing, and $\tau$ is vee
and is below $D$ and $I$ but is above $D^{\prime}$. However, the author was
unable to find a way to count these permutations and encourages the reader to
try.

The other eight remaining classes are more difficult because, while some are
counted by special lattice paths according to the OEIS, these paths seem to
either not have the correct length or not have the correct properties to
create bijections like the ones in this paper. For example, the elements of
$\mathcal{U}_{2k+1}(132,2341)$ and $\mathcal{U}_{2k+1}(132,4123)$ are counted
by Motzkin paths of length $2k-3$ with no downsteps in even positions. But
general Motzkin paths do not require a specific number of down and up steps
and $2k-3$ is less than $2k+1$, so it would be difficult to find a bijection
that associates descents and ascents in these permutations with certain
patterns of steps. That said, these remaining classes could be enumerated
through other methods such as the direct counting that we do in Theorem 4.1,
or, more likely, generating functions. Another possible route would be
defining new types of lattice paths that are in bijection with these classes
and then enumerating the new paths, which should be easier since there is a
greater body of work on the properties of and counting of lattice paths.

Also, according to the data generated by the author of [8], the 24 sequences
counting classes of the form $\mathcal{U}_{2k+1}(\tau)$ where $\tau$ has
length four appear to be new and thus studying these classes is likely to be
more challenging than the classes studied in this paper. According to the same
author, the sequence counting $\mathcal{U}_{2k+1}(231,4123)$ is also not in
the OEIS.

Beyond this, the natural next step is to enumerate classes of the form
$\mathcal{U}(\tau^{(1)},\tau^{(2)})$, where $\tau^{(1)}$ and $\tau^{(2)}$ both
have length 4, for there has been a large deal of work devoted to counting
general permutations avoiding two patterns of length four that could prove to
be very useful [2, 17, 21]. Moreover, some of these classes have nice
descriptions, such as the class of skew-merged permutations (which avoid 3412
and 2143) and the class of separable permutations (which avoid 2413 and 3142).

##  9\. Acknowledgments

The author would like to thank Professor Joe Gallian and advisors Colin Defant
and Aaron Berger for organizing the Duluth Research Experience for
Undergraduates, the lovely and supportive program where this research took
place. The author would also like to thank Defant for introducing this problem
and being a constant resource throughout the process of solving it. Lastly, a
special thanks to Defant, Gallian, Berger, and former Duluthian Marisa Gaetz
for providing invaluable feedback on earlier iterations of the present paper.

This research was funded by NSF/DMS grant 1659047 and NSA grant
H98230-18-1-0010.

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# On weighted Compactness of Commutator of semi-group maximal function
associated to Schrödinger operators

Shifen Wang  Shifen Wang: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>and  Qingying Xue∗
Qingying Xue: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>
(Date: December 3, 2020)

###### Abstract.

Let $\mathcal{T}^{*}$ be the semi-group maximal function associated to the
Schrödinger operator $-\Delta+V(x)$ with $V$ satisfying an appropriate reverse
Hölder inequality. In this paper, we show that the commutator of
$\mathcal{T}^{*}$ is a compact operator on $L^{p}(w)$ for $1<p<\infty$ if
$b\in\text{CMO}_{\theta}(\rho)(\mathbb{R}^{n})$ and $w\in
A_{p}^{\rho,\theta}(\mathbb{R}^{n})$. Here $\text{
CMO}_{\theta}(\rho)(\mathbb{R}^{n})$ denotes the closure of
$\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$ in the
$\text{BMO}_{\theta}(\rho)(\mathbb{R}^{n})$ (which is larger than the
classical $\text{BMO}(\mathbb{R}^{n})$ space) topology. The space where $b$
belongs and the weighs class $w$ belongs are more larger than the usual
$\text{CMO}(\mathbb{R}^{n})$ space and the Muckenhoupt $A_{p}$ weights class,
respectively.

###### Key words and phrases:

Schrödinger operator, semi-group maximal operator, commutator, compactness.
2010 Mathematics Subject Classification. Primary 42B25, Secondary 35J10.

The second author was supported partly by NNSF of China (Nos. 11671039,
11871101) and NSFC-DFG (No. 11761131002).

∗ Corresponding author, e-mail address<EMAIL_ADDRESS>
##  1\. Introduction

Consider the Schrödinger operator

$L=-\Delta+V(x)$

in $\mathbb{R}^{n},\ n\geq 3$. Where $\Delta$ is the Laplacian operator on
$\mathbb{R}^{n}$ and the function $V$ is a nonnegative potential belonging to
certain reverse Hölder class $RH_{q}$ with an exponent $q>n/2$, that is, there
exists a constant $C>0$ such that

$\Big{(}\frac{1}{|B(x,r)|}\int_{B(x,r)}V^{q}(y)dy\Big{)}^{\frac{1}{q}}\leq
C\Big{(}\frac{1}{|B(x,r)|}\int_{B(x,r)}V(y)dy\Big{)},$

for every ball $B(x,r)\subset\mathbb{R}^{n}$. It is worth pointing out that if
$V\in RH_{q}$ for some $q>1$, then there exists $\epsilon>0$, such that $V\in
RH_{q+\epsilon}$ (see [G]). On the other hand, the Hölder inequality gives
that $RH_{q_{1}}\subset RH_{q_{2}}$ if $q_{1}\geq q_{2}>1$. Therefore, the
assumption $q>n/2$ is equivalent to the case $q\geq n/2$. Thoughout this
paper, we always assume that $V\not\equiv 0$ and $V\in RH_{q}$ with $q\geq
n/2$.

The Schrödinger operator $L$ with nonnegative potentials is very useful in the
study of certain subelliptic operators. For instance, by taking the partial
Fourier transform in the $t$ variable, the operator
$-\Delta_{x}-V(x)\partial^{2}_{t}$ is reduced to $-\Delta_{x}+V(x)\xi^{2}$
(See [S]). Some basic results on $L$, including certain estimates of the
fundamental solutions of $L$ and the boundedness on $L^{p}$ of Riesz
transforms $\Delta L^{-1/2}$ were obtained by Fefferman [F], Shen [S1] and
Zhong [Z].

Attentions have also been paid to the study of function spaces associated to
$L$. It was Dziubański and Zienkiewicz [DZ2] who characterized the Hardy space
$H^{1}_{L}$ related to the Schrödinger operator. Later on, for $0<p\leq 1$,
the $H^{p}_{L}$ space with potentials from reverse Hölder classes were studied
in [DZ] and [DZ1]. Subsequently, Yang et al. [YYZ] characterize the localized
Hardy spaces by establishing the boundedness of Riesz transforms, maximal
operators and endpoint estimates of fractional integrals associated with $L$.

For the classical Schrödinger operators $L$ , there are many interesting
results of its associated Riesz transforms, which essentially and heavily
depend on the properties of $e^{-tL}$. The properties of semi-group $e^{-tL}$
such as the positivity, Gaussian estimates and off-diagonal estimates play a
fundamental role in the study of Riesz transform. The maximal function defined
by the semigroup $e^{-tL}\ (t>0)$ or the Riesz transforms $\Delta L^{-1/2}$
were further generalized by Lin, et. al. [LLLY] to the setting of Heisenberg
groups.

In order to introduce more results, we need to give some definitions. The
semi-group maximal function associated to the Schrödinger operator $L$ is
defined by

(1.1)
$\mathcal{T}^{*}(f)(x)=\sup_{t>0}|e^{-tL}f(x)|=\sup_{t>0}\Big{|}\int_{\mathbb{R}^{n}}k_{t}(x,y)f(y)dy\Big{|},$

where $k_{t}$ is the kernel of the operator $e^{-tL}$. As in [S1], we will use
the auxiliary function $\rho$ defined for $\mathbb{R}^{n}$ as

(1.2) $\rho(x)=\sup_{r>0}\Big{\\{}r:\frac{1}{r^{n-2}}\int_{B(x,r)}V(y)dy\leq
1\Big{\\}}.$

######  Remark 1.1.

Under the above assumptions on $V$, it is easy to get that $0<\rho(x)<\infty$.
In particular, $\rho(x)=1$ with $V=1$. For more details concerning the
function $\rho(x)$ and its applications in studying the Schrödinger operator
$L$, we refer the reader to [F, T, S1].

For $1<p<\infty$, the $A_{p}^{\rho,\theta}$ weights class is defined as
follows.

######  Definition 1.2.

($A_{p}^{\rho,\theta}$ weights class, [BHS1]). Let $w$ be a nonnegative,
locally integrable function on $\mathbb{R}^{n}$. For $1<p<\infty$, we say that
a weight $w$ belongs to the class $A_{p}^{\rho,\theta}$ if there exists a
positive constant $C$ such that for all balls $B=B(x,r)$, it holds that

(1.3)
$\Big{(}\frac{1}{|B|}\int_{B}w(y)dy\Big{)}\Big{(}\frac{1}{|B|}\int_{B}w(y)^{-1/(p-1)}dy\Big{)}^{p-1}\leq
C\Big{(}1+\frac{r}{\rho(x)}\Big{)}^{\theta p}.$

$w$ is said to satisfy the $A_{1}^{\rho,\theta}$ condition if there exists a
constant $C$ such that for all balls $B$

$M^{\theta}_{V}(w)(x)\leq Cw(x),\ a.e.\ x\in\mathbb{R}^{n},$

where

$M^{\theta}_{V}(f)(x)=\sup_{x\in
B}\frac{1}{\Big{(}1+\frac{r}{\rho(x)}\Big{)}^{\theta}|B|}\int_{B}|f(y)|dy.$