11institutetext: National Research University Higher School of Economics,
Moscow, Russia

# Decision Concept Lattice vs. Decision Trees and Random Forests

Egor Dudyrev
Sergei O. Kuznetsov 11 0000-0002-2144-3308 11 0000-0003-3284-9001

###### Abstract

Decision trees and their ensembles are very popular models of supervised
machine learning. In this paper we merge the ideas underlying decision trees,
their ensembles and FCA by proposing a new supervised machine learning model
which can be constructed in polynomial time and is applicable for both
classification and regression problems. Specifically, we first propose a
polynomial-time algorithm for constructing a part of the concept lattice that
is based on a decision tree. Second, we describe a prediction scheme based on
a concept lattice for solving both classification and regression tasks with
prediction quality comparable to that of state-of-the-art models.

###### Keywords:

Concept Lattice Decision Trees Random Forest.

##  1 Introduction

In this work we propose an approach to combining the ideas based on concept
lattices and decision trees, which are extensively used in practical machine
learning (ML), in order to create a new ML model which generates good
classifiers and regressors in polynomial time.

Formal Concept Analysis (FCA) is a mathematically-founded theory well suited
for developing models of knowledge discovery and data mining [18], [11], [8].
One of the serious obstacles to the broad use of FCA for knowledge discovery
is that the number of formal concepts (i.e. patterns found in a data) can grow
exponentially in the size of the data [15]. Sofia algorithm [7] offers a
solution to this problem by constructing only a limited amount of most stable
concepts.

Learning decision trees (DT) [5] is one of the most popular supervised machine
learning approaches. Most famous methods based on ensembles of decision trees
– aimed at increasing the accuracy of a single tree – are random forest (RF)
[6] and gradient boosting over decision trees [9]. Both algorithms are
considered among the best in terms of accuracy [17].

There are a number of papers which highlight the connection between the
concept lattice and the decision tree. The work [4] states that a decision
tree can be induced from a concept lattice. In [16] the author compares the
ways the concept lattice and the decision tree can be used for supervised
learning. Finally, in [12] the authors provide a deep mathematical explanation
on the connection between the concept lattice and the decision tree.

In this paper we develop the previous work in a more practical way. We show
that the decision tree (and its ensembles) can induce a subset of concepts of
the concept lattice. We propose a polynomial-time algorithm to construct a
supervised machine learning model based on a concept lattice with prediction
quality comparable to that of the state-of-the-art models.

##  2 Basic Definitions

For standard definitions of FCA and decision trees we refer the reader to [10]
and [5], respectively.

In what follows we describe algorithms for binary attributes, numerical data
can be processed by means of interval pattern structures or can be scaled to
binary contexts [13].

##  3 Construct a Concept Lattice via a set of Decision Trees

######  Definition 1 (Classification rule)

Let $M$ be a set of attributes of a context $\mathbb{K}$ and $Y$ be a set of
“target” values. A pair $(\rho,\hat{y}_{\rho}),\rho\subseteq
M,\hat{y}_{\rho}\in Y$ is a classification rule where $\rho$ is called a
premise and $\hat{y}_{\rho}$ is a target prediction.

Applied to object $g\subseteq G$ it can be interpreted as “if the description
of $g$ falls under the premise $\rho$, then object $g$ should have the target
value $\hat{y}_{\rho}$” or “$\text{if }\rho\subseteq
g^{\prime}\Rightarrow\hat{y}_{\rho}$”.

In the case of classification task $Y$ can be represented either as a set
$\\{0,1\\}$: $Y=\\{y\in\\{0,1\\}\\}_{i=1}^{|G|}$ or a set of probabilities of
a positive class: $Y=\\{y\in[0,1]\\}_{i=1}^{|G|}$. In the case of regression
task target value $Y$ is a set of real valued numbers:
$Y=\\{y\in\mathbb{R}\\}_{i=1}^{|G|}$.

We can define a decision tree $DT$ as a partially ordered set (poset) of
classification rules:

$DT\subseteq\\{(\rho,\hat{y}_{\rho})\mid\rho\subseteq M,\hat{y}_{\rho}\in
Y\\}$ (1)

where by _the order of classification rules_ we mean the inclusion order on
their premises:

$(\rho_{1},\hat{y}_{\rho_{1}})\leq(\rho_{2},\hat{y}_{\rho_{2}})\Leftrightarrow\rho_{1}\subseteq\rho_{2}$
(2)

Here we assume that a decision tree is a binary tree, i.e. its node is either
a leaf (has no children) or has exactly 2 children nodes.

The other property of a decision tree is that each premise of its
classification rules describes its own unique subset of objects:

$\forall(\rho_{1},\hat{y}_{\rho_{1}})\in
DT,\nexists(\rho_{2},\hat{y}_{\rho_{2}})\in
DT:\rho_{1}^{\prime}=\rho_{2}^{\prime}$ (3)

These simple properties result in an idea that 1) we can construct a concept
lattice by closing premises of a decision tree, 2) join semilattice of such
concept lattice is isomorphic to a decision tree.

###### Proposition 1

Let $\mathbb{K}=(G,M,I)$ be a formal context, $L(\mathbb{K})$ be a lattice of
the context $\mathbb{K}$. A subset of formal concepts $L_{DT}(\mathbb{K})$
forming a lattice can be derived from the decision tree $DT(\mathbb{K})$
constructed from the same context as:

$L_{DT}=\\{(\rho^{\prime},\rho^{\prime\prime})\mid\forall(\rho^{\prime},\hat{y}_{\rho})\in
DT(\mathbb{K})\\}\cup\\{(M^{\prime},M)\\}$ (4)

###### Proposition 2

Join-semilattice of a concept lattice $L_{DT}$ is isomorphic to the decision
tree $DT$.

###### Proof

Given two classification rules
$(\rho_{1},\hat{y}_{\rho_{1}}),(\rho_{1},\hat{y}_{\rho_{1}})\in DT$ let us
consider two cases:

  1. 1.

$\rho_{1}\subseteq\rho_{2}\Rightarrow(\rho_{1}^{\prime},\rho_{1}^{\prime\prime})\leq(\rho_{2}^{\prime},\rho_{2}^{\prime\prime})$

  2. 2.

$\rho_{1}\not\subseteq\rho_{2},\rho_{2}\not\subseteq\rho_{1}\Rightarrow\exists
m\in M:m\in\rho_{1},\neg m\in\rho_{2}\\\
\Rightarrow(\rho_{1}^{\prime},\rho_{1}^{\prime\prime})\not\leq(\rho_{2}^{\prime},\rho_{2}^{\prime\prime}),(\rho_{2}^{\prime},\rho_{2}^{\prime\prime})\not\leq(\rho_{1}^{\prime},\rho_{1}^{\prime\prime})$

Thus the formal concepts from the join-semilattice of $L_{DT}$ possess the
same partial order as the classification rules from $DT$.

Since we can construct a concept lattice from a decision tree and there is a
union operation for concept lattices then we can construct a concept lattice
which will correspond to a “union” of a number of independent decision trees
(i.e. a random forest).

###### Proposition 3

Let $\mathbb{K}=(G,M,I)$ be a formal context, $L(\mathbb{K})$ be a lattice of
the context $\mathbb{K}$. A subset of formal concepts $L_{RF}(\mathbb{K})$ of
the concept lattice $L(\mathbb{K})$ forming a lattice can be obtained from a
random forest, i.e. from a set of $m$ decision trees constructed on subsets of
a formal context $DT_{i}(K_{i}),i=1,...,m,K_{i}\subseteq\mathbb{K}$:

$L_{RF}(\mathbb{K})=\bigcup_{i=1}^{m}L_{DT_{i}}(K_{i})$ (5)

The size of the lattice $L_{RF}$ is close to the size of the underlying random
forest $RF$: $|L_{RF}|\sim|RF|\sim O(mG\log(G))$, where $m$ is the number of
trees in $RF$ [3]. According to [2] the time complexity of constructing a
decision tree is $O(MG^{2}\log(G))$. Several algorithms for constructing
decision trees and random forests are implemented in various libraries and
frameworks like Sci-kit learn111https://scikit-
learn.org/stable/modules/ensemble.html#random-forests ,
H2O222http://h2o-release.s3.amazonaws.com/h2o/master/1752/docs-
website/datascience/rf.html ,
Rapids333https://docs.rapids.ai/api/cuml/stable/api.html#random-forest . The
latter is even adapted to be run on GPU.

Thus, our lattice construction algorithm has two steps:

  1. 1.

Construct a random forest $RF$

  2. 2.

Use random forest $RF$ to construct a concept lattice $L_{RF}$ (by eq. 3)

Both strong and weak side of this algorithm is that it relies on a supervised
machine learning model, so it can be applied only if target labels $Y$ are
given. In addition, the result set of concepts may not be optimal w.r.t. any
concept interestingness measure [14]. Though it is natural to suppose that
such set of concepts should be reasonable for supervised machine learning
tasks.

##  4 Decision Lattice

Given a formal concept $(A,B)$ we can use its intent $B$ as a premise of a
classification rule $(B,\hat{y}_{B})$.

The target prediction $\hat{y}_{B}$ of such classification rule
$(B,\hat{y}_{B})$ can be estimated via an aggregation function over the set
$\\{y_{g}\mid\forall g\in A\\}$. In what follows we use the average
aggregation function:

$\hat{y}_{B}=\frac{1}{|A|}\sum_{\forall g\in A}y_{g}$ (6)

Let us define a decision lattice (DL) as a poset of classification rules.

###### Definition 2

Let $M$ be a set of attributes of a formal context $\mathbb{K}$ and $Y$ be a
set of target values. Then a poset of classification rules is called a
decision lattice $DL$ if a premise of each classification rule of $DL$
describes its own unique subset of objects (similar to $DT$ in equation 3)

Decision lattice $DL$ can be constructed from a concept lattice $L$ as
follows:

$DL=\\{(B,\hat{y}_{B})\mid(A,B)\in L\\}$ (7)

where $\hat{y}_{B}$ can be computed in various ways (we use the equation 6).

To get a final prediction $\hat{y}_{g}$ for an object $g$ a decision tree $DT$
firstly selects all the classification rules $DT^{g}$ describing the object
$g$. Then it uses the target prediction of the maximal classification rule
from $DT^{g}$

$\displaystyle DT^{g}$ $\displaystyle=\\{(\rho,\hat{y}_{\rho})\in
DT\mid\rho\subseteq g^{\prime}\\}$ (8) $\displaystyle DT^{g}_{max}$
$\displaystyle=\\{(\rho,\hat{y}_{\rho})\in
DT^{g}\mid\nexists(\rho_{1},\hat{y}_{\rho_{1}})\in
DT^{g}:\rho\subset\rho_{1}\\}$ (9) $\displaystyle\hat{y}_{g}$
$\displaystyle=\hat{y}_{\rho},\quad(\rho,\hat{y}_{\rho})\in DT^{g}_{max}$ (10)

We use the same algorithm to get a final prediction $\hat{y}_{g}$ for an
object $g$ by a decision lattice $DL$. The only difference is that when the
subset $DT^{g}_{max}$ always contains only one classification rule a subset
$DL^{g}_{max}$ may contain many. In this case we average the predictions of
maximal classification rules $DL^{g}_{max}$:

$\displaystyle\hat{y}_{g}$
$\displaystyle=\frac{1}{|DL^{g}_{max}|}\sum_{(\rho,\hat{y}_{\rho})\in
DL^{g}_{max}}\hat{y}_{\rho}$ (11)

| attributes M | label Y
---|---|---
| firm | smooth | color | form | fruit
objects G |  |  | yellow | green | blue | white | round | oval | cubic |
apple |  | X | X |  |  |  | X |  |  | 1
grapefruit |  |  | X |  |  |  | X |  |  | 1
kiwi |  |  |  | X |  |  |  | X |  | 1
plum |  | X |  |  | X |  |  | X |  | 1
toy cube | X | X |  | X |  |  |  |  | X | 0
egg | X | X |  |  |  | X |  | X |  | 0
tennis ball |  |  |  |  |  | X | X |  |  | 0
mango |  | X |  | X |  |  |  | X |  | 1
Table 1: _Fruit context_ and _fruit_ labels

Let us consider the _fruit context_ $\mathbb{K}=(G,M,I)$ and _fruit_ label $Y$
presented in Table 1. We want to compare the way decision lattice makes an
estimation of the label $y_{mango}$ of object _mango_ when this object is
included in the train or the test context.

Figure 1 represents decision lattices constructed upon _fruit context_ with
(on the left) and without (in the center) _mango_ object. In both cases we
show only the classification rules which cover (describe) _mango_ object.

The left picture represents a decision lattice with 8 classification rules and
1 single maximal classification rule: (“color_is_green & form_is_oval &
smooth”, 1). Therefore we use this classification rule to predict the target
label of _mango_.

The picture in the center shows a decision lattice with 6 classification rules
and 2 maximal classification rules: (“color_is_green & form_is_oval”, 1),
(“form_is_oval & smooth”, 1/2). We average the target predictions of these
classification rules to get a final prediction of $3/4$ as shown in the
picture on the right.

Figure 1: Example of prediction of mango object

##  5 Experiments

We compare our decision lattice (DL) approach with the most popular machine
learning models on real world datasets. One can reproduce the results by
running Jupyter notebooks stored on GitHub [1]. Decision lattice models are
implemented in open-source Python library for FCA which is called FCApy and
located in the same GitHub repository.

We use ensembles of 5 and 10 decision trees to construct decision lattice
models DL_RF_5 and DL_RF_10, respectively. The same ensemble of 5 decision
trees is used by random forest models RF_5. Thus, we can compare prediction
qualities of DL_RF_5 and RF_5 based on the same set of decision trees (and,
consequently, classification rules).

The non-FCA model we use for the comparison are decision tree (DT), random
forest (RF) and gradient boosting (GB) from sci-kit learn library, gradient
boostings from LightGBM (LGBM), XGBoost (XGB), CatBoost (CB) libraries.

We also test Sofia algorithm [7] as a polynomial-time approach to construct a
decision lattice DL_Sofia. We compute only 100 of most stable concepts by
Sofia algorithm because of its time inefficiency.

Metadata of the datasets is given in Table 3.

Dataset name | Task type | # Instances | # Attrsibutes
---|---|---|---
adult 444https://archive.ics.uci.edu/ml/datasets/Adult | Bin. class. | 48842 | 14
amazon 555https://www.kaggle.com/c/amazon-employee-access-challenge/data | Bin. class. | 32770 | 10
bank 666https://archive.ics.uci.edu/ml/datasets/bank+marketing | Bin. class. | 45211 | 17
breast 777https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+(Diagnostic) | Bin. class. | 569 | 32
heart 888https://archive.ics.uci.edu/ml/datasets/heart+Disease | Bin. class. | 303 | 75
kick 999https://www.kaggle.com/c/DontGetKicked/data?select=training.csv | Bin. class. | 72984 | 34
mammographic 101010http://archive.ics.uci.edu/ml/datasets/mammographic+mass | Bin. class. | 961 | 6
seismic 111111https://archive.ics.uci.edu/ml/datasets/seismic-bumps | Bin. class. | 2584 | 19
boston 121212https://archive.ics.uci.edu/ml/machine-learning-databases/housing | Regression | 506 | 14
calhouse 131313https://scikit-learn.org/stable/datasets/real_world.html#california-housing-dataset | Regression | 20640 | 8
diabetes 141414https://scikit-learn.org/stable/datasets/toy_dataset.html#diabetes-dataset | Regression | 442 | 10
Table 2: Description of the datasets

| boston | calhouse | diabetes | mean delta
---|---|---|---|---
| train | test | train | test | train | test | train | test
model |  |  |  |  |  |  |  |
DL_RF_5 | 0.02 | 0.06 | 0.14 | 0.05 | 0.05 | 0.00 | 0.07 | 0.04
DL_RF_10 | 0.01 | 0.07 | 0.13 | 0.04 | 0.01 | 0.01 | 0.05 | 0.04
DL_Sofia | 0.29 | 0.20 |  |  | 0.40 | 0.11 | 0.35 | 0.16
DT | 0.00 | 0.05 | 0.00 | 0.09 | 0.00 | 0.12 | 0.00 | 0.09
RF_5 | 0.05 | 0.01 | 0.14 | 0.04 | 0.15 | 0.02 | 0.12 | 0.03
RF | 0.04 | 0.00 | 0.06 | 0.02 | 0.12 | 0.00 | 0.07 | 0.01
GB | 0.05 | 0.00 | 0.17 | 0.02 | 0.16 | 0.00 | 0.13 | 0.01
LGBM | 0.04 | 0.01 | 0.13 | 0.00 | 0.11 | 0.00 | 0.09 | 0.01
CB | 0.02 | 0.00 | 0.13 | 0.00 | 0.06 | 0.00 | 0.07 | 0.00
best result | 0.00 | 0.14 | 0.00 | 0.21 | 0.00 | 0.31 | 0.00 | 0.22
Table 3: Weighted Average Percentage Error (best model delta)

For each dataset we use 5-fold cross-validation. We compute F1-score to
measure the predictive quality of classification and weighted average
percentage error (WAPE) to that of regression. In Tables 4–3 we show the
difference between the metric value of the model and the best obtained metric
value among all methods.

As can be seen from Tables 4–3 DL_RF model does not always show the best
result among all the tested models, though its prediction quality is
comparable to the state-of-the-art.

DL_Sofia model shows the worst results. There may be 2 reasons for this.
First, it uses only a hundred of concepts. Second, we use Sofia algorithm to
find one of the most stable concepts, but not the ones which minimize the
loss.

Figure 2 shows the time needed to construct a lattice by the sets of 5
($DL\\_RF\\_5$) and 10 ($DL\\_RF\\_10$) decision trees and by Sofia algorithm
($DL\\_Sofia$). The lattice can be constructed in a time linear in the number
of objects in the given data.

| adult | amazon | bank | breast | heart | kick | mamm. | seismic | mean delta
---|---|---|---|---|---|---|---|---|---
| train | test | train | test | train | test | train | test | train | test | train | test | train | test | train | test | train | test
model |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
DL_RF_5 | -0.35 | -0.06 | -0.01 | -0.00 | -0.41 | -0.16 | -0.01 | -0.01 | -0.03 | -0.02 | -0.59 | -0.03 | -0.03 | -0.02 | -0.24 | -0.15 | -0.21 | -0.05
DL_RF_10 | -0.33 | -0.05 | -0.01 | -0.00 | -0.37 | -0.14 | -0.00 | -0.00 | -0.01 | 0.00 | -0.58 | -0.03 | -0.01 | -0.02 | -0.13 | -0.15 | -0.18 | -0.05
DL_Sofia |  |  |  |  |  |  | -1.00 | -0.95 | -0.33 | -0.27 |  |  | -0.87 | -0.72 | -1.00 | -0.15 | -0.80 | -0.52
DT | 0.00 | -0.10 | -0.00 | -0.01 | 0.00 | -0.24 | 0.00 | -0.06 | 0.00 | -0.15 | 0.00 | -0.05 | -0.00 | -0.08 | 0.00 | 0.00 | -0.00 | -0.09
RF_5 | -0.35 | -0.05 | -0.01 | -0.00 | -0.41 | -0.12 | -0.01 | -0.01 | -0.05 | -0.07 | -0.60 | -0.02 | -0.04 | -0.02 | -0.36 | -0.07 | -0.23 | -0.05
RF | -0.00 | -0.04 | 0.00 | 0.00 | 0.00 | -0.11 | 0.00 | 0.00 | 0.00 | -0.00 | -0.00 | -0.01 | 0.00 | -0.03 | -0.00 | -0.12 | -0.00 | -0.04
GB | -0.36 | -0.02 | -0.01 | -0.00 | -0.47 | 0.00 | 0.00 | -0.01 | -0.07 | -0.01 | -0.62 | -0.00 | -0.07 | 0.00 | -0.42 | -0.09 | -0.25 | -0.02
LGBM | -0.31 | -0.00 | -0.01 | -0.00 | -0.32 | -0.04 | 0.00 | -0.02 | 0.00 | -0.03 | -0.60 | -0.00 | -0.03 | -0.02 | -0.00 | -0.11 | -0.16 | -0.03
CB | -0.31 | 0.00 | -0.01 | -0.00 | -0.31 | -0.05 | 0.00 | -0.01 | -0.01 | -0.01 | -0.59 | 0.00 | -0.04 | -0.01 | -0.33 | -0.13 | -0.20 | -0.02
best result | 1.00 | 0.65 | 0.98 | 0.97 | 1.00 | 0.48 | 1.00 | 0.95 | 1.00 | 0.76 | 1.00 | 0.35 | 0.95 | 0.81 | 1.00 | 0.15 | 0.99 | 0.64
Table 4: F1 score (best model delta) Figure 2: Time needed to construct a
lattice

##  6 Conclusions

In this paper we have introduced a new concept-based method to classification
and regression. The proposed method constructs concept-based classifiers
obtained with decision trees and random forests. This method is quite
efficient and can be used for big datasets. We have shown that our approach is
non-inferior to the predictive quality of the state-of-the-art competitors.

In the future work we plan to extend the algorithm for constructing decision
trees in the case of data given by pattern structures.

## Acknowledgments

The work of Sergei O. Kuznetsov on the paper was carried out at St. Petersburg
Department of Steklov Mathematical Institute of Russian Academy of Science and
supported by the Russian Science Foundation grant no. 17-11-01276

## References

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# Magnetic field measurement in TMC-1C using 22.3 GHz CCS Zeeman splitting

Atanu Koley1,2 , Nirupam Roy2, Emmanuel Momjian3, Anuj P. Sarma4, Abhirup
Datta5
1Joint Astronomy Programme, Indian Institute of Science, Bangalore 560012,
India
2Department of Physics, Indian Institute of Science, Bangalore 560012, India
3National Radio Astronomy Observatory, P. O. Box O, 1003 Lopezville Road,
Socorro, NM 87801, USA
4Physics and Astrophysics Department, DePaul University, 2219 N. Kenmore Ave.,
Chicago, IL 60614, USA
5Discipline of Astronomy, Astrophysics and Space Engineering, Indian Institute
of Technology Indore, Indore 453552, India E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Measurement of magnetic fields in dense molecular clouds is essential for
understanding the fragmentation process prior to star formation. Radio
interferometric observations of CCS 22.3 GHz emission, from the starless core
TMC-1C, have been carried out with the Karl G. Jansky Very Large Array to
search for Zeeman splitting of the line in order to constrain the magnetic
field strength. Toward a region offset from the dust peak, we report a
detection of the Zeeman splitting of the CCS 21 \- 10 transition, with an
inferred magnetic field of $\sim$2 mG. If we interpret the dust peak to be the
core of TMC-1C, and the region where we have made a detection of the magnetic
field to be the envelope, then our observed value for the magnetic field is
consistent with a subcritical mass-to-flux ratio envelope around a core with
supercritical mass-to-flux ratio. The ambipolar diffusion timescale for the
formation of the core is consistent with the relevant timescale based on
chemical modeling of the TMC-1C core. This work demonstrates the potential of
deep CCS observation to carry out future measurements of magnetic field
strengths in dense molecular clouds and, in turn, understand the role of the
magnetic field in star formation.

###### keywords:

ISM: general – ISM: individual objects: TMC-1C – ISM: magnetic fields – ISM:
molecules – Radio lines: ISM

††pubyear: 2022††pagerange: Magnetic field measurement in TMC-1C using 22.3
GHz CCS Zeeman splitting–3

##  1 Introduction

Starless cores are an ideal place to study star formation in molecular clouds.
Here, due to the absence of existing stars, no feedback effect influences the
star formation process. Starless cores are broadly divided into two
categories: evolved and young starless cores (Tafalla et al., 2004; Roy et
al., 2011; Navarro-Almaida et al., 2021). Young starless cores are in the
preliminary stages of their evolution and do not have a sufficiently dense
central nucleus, whereas evolved cores are believed to be out of the
equilibrium stage and in the process of forming stars.

It is well known that stars are formed from dense molecular clouds through
gravitational collapse, although the details of the complex processes leading
to star formation remain unclear. Although there are several theories on cloud
fragmentation and star formation, the debate is still ongoing for whether
magnetic field or turbulence is the most dominant factor in the star formation
process (Mestel & Spitzer, 1956; Padoan & Nordlund, 1999; Uchida et al., 2001;
Crutcher & Kemball, 2019; Das et al., 2021; Li, 2021; Mignon-Risse et al.,
2021). To get a better understanding of the process of star formation, it is
crucial to measure the magnetic field strength and its morphology in different
density regimes in star-forming prestellar cores. In molecular clouds, Zeeman
splitting and dust polarization are the two main methods for studying the
magnetic field, with the former providing the line of sight magnetic field
strength using spectral lines. One of the spectral lines suitable for magnetic
field measurements via Zeeman splitting is the 21 \- 10 22.3 GHz transition of
the CCS molecule that has a relatively high Zeeman splitting factor (0.767 Hz
$\mu$G-1). Furthermore, the critical density of CCS, $\sim$ 104 cm-3, is well
matched to the typical densities of prestellar dense cores ($\sim$ 104 $-$ 105
cm-3; Nakamura et al. 2019). This means that CCS should be observable in these
prestellar cores if it is abundant and therefore it is an obvious choice to
measure the magnetic field in such prestellar dense cores. Apart from Zeeman
splitting measurements, CCS is used to probe density as well as the
evolutionary age of dense cores.

The TMC-1C core has been studied extensively using various molecular
transitions (Potapov et al. 2016; Spezzano et al. 2016). There are existing
single dish Zeeman splitting measurements of similar cores TMC-1 and L1521E at
45.4 GHz (yielding magnetic field strength of $\sim$117 $\pm$ 21 $\mu$G in
TMC-1 and $\sim$160 $\pm$ 42 $\mu$G in L1521E; Nakamura et al. 2019; Shinnaga
et al. 1999). In this Letter, we present interferometric observations of CCS
toward the TMC-1C core with the aim of measuring the Zeeman effect. The
kinematical properties of TMC-1C, along with that of TMC-1 and L1544, based on
the observations of CCS and NH3, will be reported in a forthcoming paper
(Koley, A., 2022, in prep.). We decided to target TMC-1C based on earlier
interferometric observations (Roy et al., 2011), indicating that the CCS 22.3
GHz line intensity and narrow line profile are suitable for Zeeman splitting
measurements.

Figure 1: Integrated CCS 22.3 GHz line intensity image (primary beam
corrected) of TMC-1C. The black square (from $\alpha=04^{\rm h}41^{\rm
m}36\aas@@fstack{s}11$,
$\delta=25{\degr}59{\arcmin}22\aas@@fstack{\prime\prime}58$ to $\alpha=04^{\rm
h}41^{\rm m}34\aas@@fstack{s}44$,
$\delta=25{\degr}59{\arcmin}43\aas@@fstack{\prime\prime}74$) denotes the
region in which we report the detection of the magnetic field due to the
Zeeman effect.

This Letter is organized in the following way. Summary of the observations and
data reduction are provided in section §2. The results on the Zeeman splitting
measurement and magnetic field estimation are presented in §3, and discussed
in §4. Finally, we present the conclusions in section §5.

Figure 2: Left: Average Stokes $I$ (upper panel—black histogram-like line) and
Stokes $V$ (lower panel—black histogram-like line) profiles of CCS 22.3 GHz
line toward the black square region in Figure 1. The magenta curve in the
upper panel shows the Gaussian component that we fitted to the Stokes $I$
profile. In the lower panel, the magenta solid line is the fit to the Stokes
$V$ profile for the estimated $B_{\text{los}}$ = $-1.80\pm 0.38$ mG. Right:
Average Stokes $I$ (upper panel—black histogram-like line) and Stokes $V$
(lower panel—black histogram-like line) profiles toward the same region marked
by the black square. The solid magenta line in the upper panel is the observed
data (that is, the same as the black histogram-like line, but plotted as a
continuous curve). The magenta solid line in the lower panel is the fit to the
Stokes $V$ profile for the estimated $B_{\text{los}}$ = $-$2.40 $\pm$ 0.44 mG
from the one-sided derivative of Stokes $I$.

##  2 Observations and Data Analysis

The data for this work were taken using the Karl G. Jansky Very Large Array
(VLA) as part of the proposal VLA/13A-112. The observations were carried out
in 2013 February and March (over 13 observing sessions). More details of the
observations are provided in Appendix A. All the initial flagging and
calibration of the data were performed both in the scripted CASA (Common
Astronomy Software Applications package) pipeline of the National Radio
Astronomy Observatory (NRAO)111The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. and manually in AIPS (Astronomical
Image Processing System; version 31DEC19). This was done as a consistency
check of the analysis. There was no noticeable difference found between the
output of the two methods. For imaging, we used the CASA task tclean with the
multiscale deconvolver to pick up all the different structures in the core in
total intensity. The scales that we used correspond to
$3\aas@@fstack{\prime\prime}2$, $7\aas@@fstack{\prime\prime}2$,
$16\aas@@fstack{\prime\prime}0$, $24\aas@@fstack{\prime\prime}0$, and
$32\aas@@fstack{\prime\prime}0$. We note that multiscale CLEANing produces a
single image that is restored with a synthesised beam that corresponds to the
full angular resolution of the data. Due to the low signal to noise, we did
not CLEAN the Stokes $V$ image cube.

##  3 Results

Magnetic field values in the interstellar medium are relatively low ($\sim$
$\mu$G - mG). Thus, the expected Stokes $V$ signal for typical magnetic field
values in a dense core is very weak, requiring deep observations with high
spectral resolution to detect the Zeeman splitting signal. In astrophysical
environments, spectral lines are significantly broadened by thermal and non-
thermal motions. Therefore, the Zeeman splitting is much smaller than the line
width, which in turn results in the measurement of only the line-of-sight
component ($B_{\rm los}=B{\rm cos}\theta$) of the magnetic field (Crutcher et
al., 1993; Sault et al., 1990). In this small splitting approximation, the
fitting formulae for Stokes $V$ with Stokes $I$ is:

$\displaystyle V=aI+\frac{b}{2}\frac{dI}{d\nu}$ (1)

where $a$ is the gain difference between the left circular polarization (LL)
and the right circular polarization (RR), $b=zB\mathrm{cos}\theta$, and $z$ is
the Zeeman splitting factor (Greisen, 2017). We used XGAUS and ZEMAN in AIPS
to fit the observed Stokes $I$ and $V$ profiles.

The integrated intensity map of the CCS 22.3 GHz emission in TMC-1C is shown
in Figure 1. This figure reveals a very patchy appearance spread over the
field of view. Figure 2 shows the Stokes $I$ and $V$ profiles toward the
region highlighted by the black box in Figure 1. The Stokes $V$ profile
(bottom panels of Figure 2) has an S-shape that usually suggests a Zeeman
splitting detection. Typically, at least 6 channels across the spectral line
is considered to be necessary in order to claim a Zeeman splitting detection.
Since the profile in Figure 2 marginally spans 6 channels “null-to-null”, we
claim that the observations reported in this Letter represent a detection. To
eliminate the possibility of any systematic effects that may lead to a false
S-curve, we checked each individual session’s data, and found that the Stokes
$V$ profiles were indeed noise-like with no spurious features that may be
attributed to instrumental effects. We obtained the magnetic field value in
two different ways: (1) We fitted the observed Stokes $I$ profile with a
single Gaussian component, then found the derivative of that component and
fitted it to Stokes $V$ using equation (1) to derive $B_{\text{los}}=-1.80\pm
0.38$ mG, and (2) We calculated the one-sided derivative of the observed
Stokes $I$ profile and fitted it to Stokes $V$, again using equation (1) to
find $B_{\text{los}}=-2.44\pm 0.44$ mG. Within the errors, both methods agree
on the magnetic field value, which we adopt to be 2 mG for further analysis in
section §4.

We note that, for single dish observations, the beam squint can produce false
Zeeman signal in presence of velocity gradient of the line emission. For
interferometric observations, the prominent effect is leakage of Stokes I into
Stokes V (Sault et al., 1990). In our modeling, leakage is included in Eqn. 1
to account for this. Further, our detected Stokes V signal indicates a
magnetic field strength of $\sim 2$ mG. As the observed line profile is fairly
narrow, we can clearly rule out, from our observations, any large velocity
gradient that will be required, along with beam squint, to produce a Zeeman-
like signal.

##  4 Discussion

This work presents the interferometric study of CCS 22.3 GHz emission from the
TMC-1C region. The emission is found to be widespread and patchy, while the
line profile is very narrow and consistent with earlier shallow observations
(Roy et al., 2011). For one region within the primary beam of the VLA, we also
have a detection of the magnetic field, based on the Zeeman splitting of the
transition. If the signal is real, the determined magnetic field is about 2
mG.

###  4.1 CCS Emission

Suzuki et al. (1992) measured the column density of CCS in TMC-1C with the
Nobeyama 45 meter telescope and found N(CCS) $\sim$ 3.1 $\times$ 1013 cm-2. As
the diffuse emission is resolved out, only 33% of this column density is
detected in our VLA observations. In Figure 3, the integrated intensity of CCS
22.3 GHz line is superimposed on the 1200 $\mu$m dust emission from a larger
region (Kauffmann et al., 2008). The white square denotes the area in which we
have detected the magnetic field. A clear indication of a spatial offset
between the CCS and dust emission is observed as seen in Figure 3. The CCS
emission peaks are shifted from the dust emission peaks, and there is very
little CCS emission from the region where the dust emission peaks within the
VLA’s primary beam. CCS is a carbon-bearing species, and it is depleted during
the evolution of the core. Thus, the depletion of CCS is observed toward the
extremely dense nucleus where dust or nitrogen-bearing species are present in
copious amounts. Tatematsu et al. (2014) and Seo et al. (2019) have observed a
similar offset in the CCS emission from Orion A and L1495-B218, respectively.

We note that there is some variation of the spectral shape of the CCS emission
across the region, with a hint of two components in the Stokes $I$ profile in
some regions (including the region where Zeeman splitting is detected). The
broader component is relatively weak, similar to what is observed for NH3 in
L1688 (Choudhury et al., 2020). The reduced chi-square for a two-component
Gaussian fit to the Stokes $I$ spectra is slightly better than that of a
single component fit, and indicates that the weak, broad component is likely
to be real. With very similar central velocities, these are likely to be
associated components, similar to two NH3 components in L1688 (Choudhury et
al. 2020, also see Kama et al. 2015). However, the spectral resolution and the
sensitivity of the current observations are not adequate to reliably fit the
Stokes $V$ signal with two components; hence, we have used the one-sided
derivative and single component fit methods to determine an effective single
average value for the magnetic field.

###  4.2 Magnetic Field

In this section we discuss the role of the magnetic field during the evolution
of a prestellar core, which can be determined by the ratio of the observed to
the critical mass-to-flux ratios, given by the equation:

${\lambda_{c}\equiv\frac{(M/\Phi)_{\rm obs}}{(M/\Phi)_{\rm
critical}}=7.6\times 10^{-21}{\frac{N(\text{H}_{2})}{B_{\text{total}}}}}$ (2)

where N(H2) is in cm-2, $B_{\rm total}$ is in $\mu$G and $({M}/{\Phi})_{\rm
critical}={1}/{\sqrt{4\pi^{2}G}}$ (Nakano & Nakamura, 1978). Earlier, Caselli
et al. (2008) have reported N(H2) to be $8.5\times 10^{22}$ cm-2 towards the
dust emission peak in TMC-1C. With a factor of 2 lower dust emission (compared
to the dust emission peak) toward the region enclosed by the white box in
Figure 3, these observations, therefore, imply N(H2) $\sim 4.25\times 10^{22}$
cm-2 for the region in which we have reported a detection of the magnetic
field. This is consistent with a number density of (1.4 $\pm$ 0.8) $\times$
105 cm-3 (Navarro-Almaida et al., 2021) and linear size of $\sim$ 0.1 pc
(Barranco & Goodman, 1998) for this region. Based on their chemical modeling,
Suzuki et al. (1992) predicted a fractional abundance ${n_{\rm CCS}}/{n_{\rm
H_{2}}}=(2-6)\times 10^{-10}$ i.e., N(H2) $\sim(5-15)\times 10^{22}$ cm-2 from
the Nobeyama observations. This was for the position coinciding with the
pointing centre of our VLA observations, and with a factor of 1.5 lower dust
emission (compared to the VLA pointing centre) toward the region of our
interest, N(H2) is expected to be $\sim(3.3-10)\times 10^{22}$ cm-2. We note
that the value of N(H2) can also be estimated from the average 1200$\mu$m dust
emission intensity, which is $\sim$ 18 mJy beam-1 in the region marked by the
white box in Figure 3, and the conversion factor reported in Kauffmann et al.
(2008). This gives an N(H2) $\sim 1.2\times 10^{22}$ cm-2, which is
significantly lower than the above values and results in an unusually small
value of $\lambda_{c}$. Hence, for our analysis below, we adopt N(H2)
$\sim(3.3-10)\times 10^{22}$ cm-2.

Our observations of TMC-1C resulted in a detection of the Zeeman effect with a
line of sight magnetic field value of $\sim 2$ mG. As the CCS emission is weak
at the position of the dust peak, and the Stokes $V$ signal is detected only
in a region offset from that peak, the inferred magnetic field is possibly
associated with the surrounding envelope of the dense core, where the magnetic
field is expected to be dynamically more important. We note that magnetic
field of the order of a few mG for the density regime probed by CCS is broadly
consistent with earlier observations (e.g. Liu et al. 2022 and references
therein). To assess the implications of this measurement, we calculate the
mass-to-flux ratio by considering the line-of-sight magnetic field is
approximately equal to the total magnetic field ($B_{\rm total}=B_{\rm los}$).
For N(H2) $=(3.3-10)\times 10^{22}$ cm-2, this gives $\lambda_{c}=0.13-0.38$,
suggesting a subcritical envelope. This is consistent with the general picture
of supercritical starless accreting core, surrounded by a subcritical
envelope. Indeed, earlier large scale observations of the Taurus Molecular
Cloud have shown the cloud to be subcritical (Heyer et al., 2008; Heyer &
Brunt, 2012). A logical question to ask is whether the timescale of formation
and growth for the starless accreting core in TMC-1C is consistent with other
observations.

If during the stage of evolution and gravitational collapse, a supercritical
core is formed via the ambipolar diffusion process, we can make an order of
magnitude estimation of the ambipolar diffusion timescale $t_{\rm AD}$
(Stahler & Palla, 2004; Kudoh & Basu, 2009; Heitsch & Hartmann, 2014;
Hennebelle & Inutsuka, 2019a):

$t_{\rm AD}\approx 3\times 10^{6}\left(\frac{n_{{\rm H}_{2}}}{10^{4}~{}{\rm
cm}^{-3}}\right)^{3/2}\left(\frac{B}{30~{}\mu{\rm
G}}\right)^{-2}\left(\frac{L}{0.1~{}{\rm pc}}\right)^{2}\,{\rm yr}$ (3)

where $n_{{\rm H}_{2}}$ is the number density of the hydrogen molecule, $B$ is
the magnetic field and $L$ is the length scale over which the ambipolar
diffusion occurs in a system. For coherent dense cores with subsonic non-
thermal motions (Pineda et al., 2010), the relevant scale is the size of the
core (Hennebelle & Inutsuka, 2019b; Mouschovias & Morton, 1991). For TMC-1C,
the size of the core is $\approx$ 0.1 pc (Barranco & Goodman, 1998; Goodman et
al., 1998). The estimated number density of TMC-1C, from analysis based on
multiple observed transitions of CS, is as high as 1.4 $\pm$ 0.8 $\times$ 105
cm-3 (Navarro-Almaida et al., 2021). Hence, if we adopt $L$ = 0.1 pc, $n_{{\rm
H}_{2}}$ = 1.5 $\times$ 105 cm-3 and take $B$ $\approx$ 2 mG ($B_{\rm los}$),
the ambipolar diffusion will make the core significantly supercritical within
a timescale of $\sim$ 4 $\times$ $10^{4}$ yr. We note that the diffusion time
scale may further decrease by a factor of few if the ambient turbulent flow is
comparable to or greater than the Alfven velocity (Kudoh & Basu, 2014). This
may look like a rather short timescale compared to the age of TMC-1C,
estimated to be $>$ 3 $\times$ 105 yr (Schnee et al., 2007, based on N2H+
abundance) to $\approx$ 1 Myr (based on detailed astrochemical models;
Navarro-Almaida et al., 2021). However, Schnee et al. (2007) have also
concluded that the TMC-1C core is much younger or has undergone significant
mass infall from the surrounding envelope with a corresponding timescale of
$\sim$ 104 yr. Thus, the relevant timescale matches well with the ambipolar
diffusion timescale.

##  5 Conclusions

We have observed the 22.3 GHz CCS line toward the starless core TMC-1C with
the VLA. We report in this Letter the detection of CCS Zeeman splitting, the
first of its kind, with a radio interferometer. Our inferred line of sight
magnetic field toward TMC-1C is $\sim$2 mG. Future observations with higher
sensitivity and higher spectral resolution will be necessary to confirm these
results. The Zeeman detection is toward a region to the southwest of the 1200
$\mu$m dust peak. If we interpret the dust peak to be the core of TMC-1C, then
our observed magnetic field may be associated with the envelope region around
the core. This is consistent with the general expectation of a supercritical
core and a subcritical envelope in an accreting starless core like TMC-1C
where the magnetic field in the envelope may be dynamically important, and the
accreting core grows via the ambipolar diffusion process. The ambipolar
diffusion timescale of core formation is found to be in agreement with the
mass infall timescale for TMC-1C reported by previous authors. The detection
reported here is a crucial step highlighting the possibility of future
interferometric measurements of magnetic fields in dense molecular clouds
using CCS transitions.

## Acknowledgements

We thank the reviewer for many useful comments and valuable suggestions that
have helped in improving this paper significantly. We are grateful to Thushara
Pillai for useful discussions and valuable inputs. A.K. would like to thank
DST-INSPIRE (IF160553) for a fellowship.

## Data Availability

All data used in this study are available publicly from the VLA online
archive. All reduced data will be shared at reasonable request to the
corresponding author.

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##  Appendix A Details of observations

Details of the VLA observations are summarized in table 1.

Table 1: Summary of observational parameters. VLA proposal ID | 13A-112
---|---
Date of Observations | 2013 February 7–March 29
Configuration | D
Observing Band | K (18.0-26.5 GHz)
R.A. of field center (J2000) | $04{{}^{\rm h}}41{{}^{\rm m}}34\aas@@fstack{s}316$
Dec. of field center (J2000) | $+26{{}^{\circ}}00^{\prime}42\aas@@fstack{\prime\prime}592$
Calibrators | 3C 147
| J0440$+$2728
Spectral line: |
Rest Frequency | 22.344 GHz
Observing Bandwidth | 4 MHz
Number of channels | 512
Channel spacing | 0.1 km s-1
Primary beam (HPBW) | ${\sim 135^{\prime\prime}}$
Synthesised beam (FWHM) | $3\aas@@fstack{\prime\prime}57\times 3\aas@@fstack{\prime\prime}40$
On-source time |  ${\sim 13}$ hr $30$ min
RMS noise (per channel) |  ${\sim 0.8}$ mJy beam-1

##  Appendix B Comparison of CCS and dust emission

The archival $1200~{}\mu$m public data for TMC-1C region222Data available
through the Strasbourg astronomical Data Centre (CDS):
https://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=J/A%2bA/487/993/object is
used to compare the dust and CCS emission distribution in this region. The
dust map was acquired using the Max-Planck Millimeter Bolometer (MAMBO) array
at the IRAM 30-meter telescope (Kauffmann et al., 2008) . Figure 3 is an
overlay of the CCS 22.3 GHz integrated intensity with the MAMBO $1200~{}\mu$m
emission, that shows the offset of the dust peak with respect to the prominent
CCS emission. The region indicated by white square is inferred to be in the
envelope around the dense core where the Zeeman splitting signal is detected.

Figure 3: Overplot of the integrated intensities of CCS 22.3 GHz emission and
the 1200 $\mu$m dust emission (Kauffmann et al., 2008) at a resolution of
$10\aas@@fstack{\prime\prime}80\times 10\aas@@fstack{\prime\prime}50$. Dust
integrated intensity is shown in the color plot and CCS emission is in the
contours. The contour levels are at 1.76, 3.58, 5.40, and 7.22 Jy beam-1. The
white square denotes the region where we report the detection of the magnetic
field.


# Some non-standard biparametric Poincaré type inequalities through harmonic
analysis

María Eugenia Cejas  Departamento de Matemática, Facultad de Ciencias Exactas,
Universidad Nacional de La Plata, Calle 50 y 115, (1900) La Plata, Prov. de
Buenos Aires, Argentina<EMAIL_ADDRESS>,  Carolina Mosquera
Department of Mathematics, Facultad de Ciencias Exactas y Naturales,
University of Buenos Aires, Ciudad Universitaria Pabellón I, Buenos Aires 1428
Capital Federal Argentina<EMAIL_ADDRESS>,  Carlos Pérez  Department of
Mathematics, University of the Basque Country, IKERBASQUE (Basque Foundation
for Science) and BCAM –Basque Center for Applied Mathematics, Bilbao, Spain
<EMAIL_ADDRESS>and  Ezequiel Rela  Department of Mathematics, Facultad de
Ciencias Exactas y Naturales, University of Buenos Aires, Ciudad Universitaria
Pabellón I, Buenos Aires 1428 Capital Federal Argentina<EMAIL_ADDRESS>
###### Abstract.

We show some non-standard Poincaré type estimates in the biparametric setting
with appropriate weights. We will derive these results using variants from
classical estimates exploiting the interplay between maximal functions and
fractional integrals. We also provide a sharper result by using extrapolation
techniques.

###### Key words and phrases:

Poincaré - Sobolev inequalities. Muckenhoupt weights.

###### 1991 Mathematics Subject Classification:

Primary: 42B25. Secondary: 42B20.

M.E.C. is partially supported by grant PICT-2018-03017 (ANPCYT)

C.M. is partially supported by grants UBACyT 20020170100430BA, PICT 2018–03399
and PICT 2018–04027.

C. P. and E.R. have been supported by the Basque Government through the
IT1247-19 project, the BERC 2018- 2021 program and by the Spanish State
Research Agency through BCAM Severo Ochoa excellence accreditation
SEV-2017-0718. C.P is also supported by the project PID2020-113156GB-I00/AEI
/10.13039/501100011033 funded by Spanish State Research Agency and acronym
“HAPDE”.

E.R. is partially supported by grants PICT-2019–03968, PICT 2018-3399, UBACyT
20020190200230BA and PIP (CONICET) 11220110101018

##  1\. Introduction and Main Results

In the present article we will study non-standard variations of the very well
known Poincaré-Sobolev inequalities. More precisely, we are interested in
biparametric extensions of the classical Poincaré-Sobolev inequality of the
form

(1.1)
$\left(\frac{1}{|Q|}\int_{Q}\mbox{$|f-f_{Q}|$}^{p^{*}}\right)^{1/p^{*}}\leq
c\,\ell(Q)\left(\frac{1}{|Q|}\int_{Q}\mbox{$|\nabla f|$}^{p}\right)^{1/p},$

where $Q$ is any cube in $\mathbb{R}^{n}$ (from now on we will only consider
cubes with sides parallell to the coordinate axes) or euclidean ball, $p$ is a
parameter $p>1$ and $p^{*}$ is the so called Sobolev exponent given by the
condition $\frac{1}{p}-\frac{1}{n}=\frac{1}{p^{*}}$, defined for $p<n$.

The classical way of proving Poincaré-Sobolev inequalities is by exploiting
the interplay between averaged oscillations and fractional integral operators.
Recall that $\mathcal{I}_{1}$ is the standard notation for the fractional
integral or Riesz potential of order $1$ defined by

(1.2) $\mathcal{I}_{1}(f)(x)=\int_{\mathbb{R}^{n}}\frac{f(y)}{|x-y|^{n-1}}\
dy.$

The key estimate is the following pointwise estimate, allowing the control of
the oscillation by the fractional integral

(1.3) $|f(x)-f_{Q}|\leq c_{n}\,\mathcal{I}_{1}(|\nabla f|\chi_{Q})(x).$

It is an interesting fact that the previous estimate is equivalent to the
following averaged result,

(1.4) $\frac{1}{|Q|}\int_{Q}\mbox{$|f-f_{Q}|$}\leq
c_{n}\,\frac{\ell(Q)}{|Q|}\int_{Q}\mbox{$|\nabla f|$},$

as shown first in [FLW96] (see an extension in [PR19], Theorem 11.3). Then,
estimate (1.1) will follow from the appropriate $L^{p}\to L^{q}$ boundedness
properties of the classical fractional integral operators $I_{\alpha}$. We
remit to [AH95] for an extensive overview of this material. The key ideas go
back to Sobolev about 100 years ago.

In the late 90’s, in [FPW98] a new method to derive these estimates which
avoids the use of potential operators was introduced. This method was
considered very recently by the authors in [CMPR], using ideas from [PR19], in
the context of rectangles, or more generally in the context of muliparameter
analysis. We include below Theorem 1.1 as a sample of this kind of results
which do not use any pointwise estimate as (1.3) requiring the use of
fractional operators. This result provides an inequality holding for the
family $\mathfrak{R}$, defined by rectangles of the form $R=I_{1}\times I_{2}$
where $I_{1}\subset\mathbb{R}^{n_{1}}$ and $I_{2}\subset\mathbb{R}^{n_{2}}$
are cubes with sides parallel to the coordinate axes and $n:=n_{1}+n_{2}$.

######  Theorem 1.1.

Let $w\in A_{1,\mathfrak{R}}$ in $\mathbb{R}^{n}$ and let $p\geq 1.$ Let also

$\frac{1}{p}-\frac{1}{p^{*}}=\frac{1}{n}\frac{1}{(1+\log[w]_{A_{1,\mathfrak{R}}})}.$

Then, there exists a constant $c=c(n,p)>0$ such that for every Lipschitz
function $f$ and any $R=I_{1}\times I_{2}\in\mathfrak{R}$,

(1.5) $\left\|f-f_{R}\right\|_{L^{p^{*}}(R,\frac{wdx}{w(R)})}\leq
c\,[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\left(a_{1}(R)+a_{2}(R)\right),$

where

$a_{1}(R)=\ell(I_{1})\left\|\nabla_{1}f\right\|_{L^{p}(R,\frac{wdx}{w(R)})}\quad\text{
and }\quad
a_{2}(R)=\ell(I_{2})\left\|\nabla_{2}f\right\|_{L^{p}(R,\frac{wdx}{w(R)})}.$

We are using here the following notation: for a given function
$f:U\to\mathbb{R}$ defined on the open set
$U\subset\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$, we will write
$f(x)=f(x_{1},x_{2})$ where $x_{1}$ stands for the first $n_{1}$ variables and
$x_{2}$ stands for the remaining $n_{2}$ variables. $\nabla_{1}f$ will denote
the partial gradient of $f$ containing the $x_{1}$-derivatives and similarly
$\nabla_{2}f$ will denote the partial gradient of $f$ containing the
$x_{2}$-derivatives.

To the extent of our knowledge, these type of results were proved for the
first time in [ST93], but this theorem is a particular case of many others
results that can be found in [CMPR]. The latter approach provides much more
precise inequalities.

The present work is an outgrowth of [CMPR]. Indeed, it is quite surprising
that a result like (1.5) cannot be derived using a version of the pointwise
estimate (1.3) based on a multiparameter potential operator version of
$\mathcal{I}_{1}$ from (1.2). There is though a multiparametric counterpart of
the fractional integral operator introduced in [CUMP04] which leads to a
special pointwise inequality and hence to a non-standard Poincaré inequality
(1.9) and (1.10). The main point of this paper is to improve the $(1,1)$ non-
standard Poincaré inequality (1.10) to the $(p,p)$ case. We stress the fact
that unfortunately we cannot use the method in [CMPR] since it is not clear
how to handle the non-standard oscillation $\pi_{R}(f)$ appearing in (1.9).

###  1.1. Biparameter Poincaré inequalities

We include in this section the main contributions of this article, concerning
results on $(p,p)$-Poincaré type inequalities obtained by means of classical
arguments involving the interplay between fractional integrals and maximal
functions. This approach is known for the case of cubes, but requires some
extra work to adapt it to the bi-parametric geometry of rectangles given by
products of cubes. Recall that $I_{1}$ will always denote a cube in
$\mathbb{R}^{n_{1}}$ while $I_{2}$ will be a cube in $\mathbb{R}^{n_{2}}$. Let
$\delta_{1}=\ell(I_{1})$ and $\delta_{2}=\ell(I_{2}).$

Let us consider the multiparametric fractional operator

(1.6)
$Tf(x,y)=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\frac{f(\bar{x},\bar{y})}{|x-\bar{x}|^{n_{1}-1}|y-\bar{y}|^{n_{2}-1}}\,d\bar{x}\,d\bar{y}.$

Consider the following notations:

$f_{I_{1}}^{y}=\frac{1}{|I_{1}|}\int_{I_{1}}f(x,y)\,dx\qquad,\qquad
f_{I_{2}}^{x}=\frac{1}{|I_{2}|}\int_{I_{2}}f(x,y)\,dy$

and

$f_{I_{1}\times I_{2}}=\frac{1}{|I_{1}||I_{2}|}\int_{I_{1}\times
I_{2}}f(x,y)\,dydx.$

We denote $\mathcal{I}_{1}^{(1)}$ the $n_{1}-$dimensional fractional integral
of order 1 and $\mathcal{I}_{1}^{(2)}$ the $n_{2}-$dimensional fractional
integral of order 1. That is, for a function
$f:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\to\mathbb{R}$, we have that

(1.7)
$\mathcal{I}_{1}^{(1)}(f)(x,y)=\int_{\mathbb{R}^{n_{1}}}\frac{f^{y}(\bar{x})}{|x-\bar{x}|^{n_{1}-1}}\
d\bar{x}$

and

(1.8)
$\mathcal{I}_{1}^{(2)}(f)(x,y)=\int_{\mathbb{R}^{n_{2}}}\frac{f^{x}(\bar{y})}{|y-\bar{y}|^{n_{2}-1}}\
d\bar{y},$

where $f^{x}(y)=f(x,y)$ denotes the slice of the function $f$ for a fixed
$x\in I_{1}$ (similarly for $f^{y}$). Then $T$ can be expressed as the
composition

$T=\mathcal{I}_{1}^{(1)}\circ\mathcal{I}_{1}^{(2)}.$

The main relevance for us to consider this multilinear operator is due to its
connection with a special non-standard oscillation introduced, as far as we
know, in [CUMP04] with respect to non-constant average defined by the quantity

$\pi_{I_{1}\times I_{2}}(f):=f_{I_{1}}^{y}+f_{I_{2}}^{x}-f_{I_{1}\times
I_{2}}$

which will play the role of the average over the cube as in the one parameter
case. To be more precise we have in the following lemma a substitute of the
model example (1.3) in the setting of product spaces where the usual gradient
is replaced by the following mixed derivatives matrix $\displaystyle
f\mapsto\nabla_{x}\nabla_{y}f,$ where

$\displaystyle\nabla_{x}\nabla_{y}f=\left(\frac{\partial^{2}f}{\partial
x_{i}\,\partial y_{j}}\right)_{i,j}\quad\text{
and}\quad\displaystyle|\nabla_{x}\nabla_{y}f|=\bigg{(}\sum_{i,j}\Big{|}\frac{\partial^{2}f}{\partial
x_{i}\,\partial y_{j}}\Big{|}^{2}\bigg{)}^{\frac{1}{2}}.$

We don’t know where this object was first introduced but it was considered by
J. M. Wilson in [Wil91, Wil95] when studying spectral type properties of the
two-parameters Schrödinger operators $\Delta_{n_{1}}\circ\Delta_{n_{2}}-V$,
where $V\in L_{loc}^{1}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})$ and where
$\Delta_{n_{i}}$ is the Laplace operator in $\mathbb{R}^{n_{i}}$ where the
multiparameter Harmonic Analysis theory played a central role like in the
present work.

We will rely on the following pointwise inequality, analogous to (1.3)

######  Lemma 1.2.

[CUMP04, Proposition 6.1] Let $R\in\mathfrak{R}$ of the form $R=I_{1}\times
I_{2},$ as before. Then we have the following pointwise estimate for any $f\in
C^{2}(R)$

(1.9) $|f(x,y)-\pi_{R}(f)(x,y)|\leq T(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y),$

for every $(x,y)\in I_{1}\times I_{2}$.

As a consequence of this we have

(1.10)
$\int_{R}|f-\pi_{R}(f)|\,dx\,dy\lesssim\ell(I_{1})\ell(I_{2})\int_{R}|\nabla_{x}\nabla_{y}f|\,dx\,dy,$

although there is a more stronger and hence more useful version in (1.13). A
natural question arising here is whether these two estimates are equivalent or
not, as it happens in the classical setting of cubes mentioned before.

Here and in the reminder of this article we will use the following notation
for (weighted) local $L^{p}$ norms over a rectangle $R$:

$\left\|f\right\|_{L^{p}\left(R,w\right)}:=\left(\int_{R}|f|^{p}wdx\right)^{\frac{1}{p}}.$

Similarly, we will use the standard notation for the weak $(r,\infty)$
(quasi)-norm: for any $0<r<\infty$, measurable $R$ and weight $w$, we define

$\|f\|_{L^{r,\infty}\big{(}R,w\big{)}}:=\sup_{t>0}t\,\left(w(\\{x\in
R:|f(x)|>t\\})\right)^{1/r}.$

As usual, when dealing with the constant weight $w\equiv 1$, we will simply
omit to mention it.

In our context, the natural scenario for the study of weighted inequalities is
the Lebesgue space $L^{p}(w)$, where $w$ is a weight in the class
$A_{p,\mathfrak{R}}$ associated to the family $\mathfrak{R}$. Let us first
introduce its obvious definition adapted to the geometry of such basis of
rectangles. For a weight $w$ in $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$,
$n=n_{1}+n_{2}$, we will say that $w\in A_{p,\mathfrak{R}}$ if

(1.11)
$[w]_{A_{p,\mathfrak{R}}}:=\sup_{R\in\mathfrak{R}}\left(\frac{1}{|R|}\int_{R}w(x)\,dx\right)\left(\frac{1}{|R|}\int_{R}w(x)^{-\frac{1}{p-1}}\,dx\right)^{p-1}<\infty,$

and in the case $p=1$, for a finite constant $c$

(1.12) $\frac{1}{|R|}\int_{R}w(x)\,dx\leq c\,\inf_{R}w\qquad R\in\mathfrak{R}$

and the smallest of the constants $c$ is denoted by
$[w]_{A_{1,\mathfrak{R}}}$.

Our first result is about a quantitative weighted Poincaré inequality for the
$A_{p,\mathfrak{R}}$ class.

######  Theorem 1.3.

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. Define $R=I_{1}\times
I_{2}$. Then the following local weak type Poincaré inequality holds for any
$f\in C^{2}(R),$

$\|f-\pi_{R}(f)\|_{L^{p,\infty}(R,w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}+\frac{1}{p-1}}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}.$

Note that we only obtain here the weak norm, since the maximal operator is not
bounded in $L^{1}$. This is also reflected in the exponent in the $A_{p}$
constant, as a sum of $\frac{1}{p}$ from the weak bound used in the first step
and $\frac{1}{p-1}$ from the strong bound that we are forced to use as a last
step.

We remark here that in the classical setting of cubes, a sort of “weak implies
strong” argument (often called the “truncation” method) can be applied to
obtain the $(p,p)$-Poincaré inequality from the weak estimate. Here, since we
are not in position to claim that such an argument works for mixed gradients
$\nabla_{x}\nabla_{y}$, we can only propose the following result as a
conjecture

######  Conjecture 1.4.

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. Then for any rectangle
of the form $R=I_{1}\times I_{2}$ and any $f\in C^{2}(R)$, we have

$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p-1}+\frac{1}{p}}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}.\\\
$

Another natural question is to find a Sobolev-Poincaré type inequality like
(1.1). Naturally, $T$, as defined in (1.6), should play the role of
$\mathcal{I}_{1}$ in the one-parameter case. However we failed doing this.
Further, the method developed in [CMPR], which avoids the use of fractional
operators, cannot be applied here since the initial Poincaré (1.10) is not, as
already mentioned, the standard one and hence a new theory must be developed.

As we mentioned above the ideas presented in the proof of Theorem 1.3 can not
be extended to the case $p=1$. We can, however, prove the following result.

######  Proposition 1.5.

Let $w$ be a weight in $A_{1,\mathfrak{R}}$. Then the following $(1,1)$
Poincaré inequality holds uniformly in $R=I_{1}\times I_{2}\in\mathfrak{R}$
for $f\in C^{2}(R)$.

(1.13)
$\int_{R}|f-\pi_{R}(f)|\,w\,dx\,dy\lesssim\ell(I_{1})\ell(I_{2})[w]_{A_{1,\mathfrak{R}}}^{2}\int_{R}|\nabla_{x}\nabla_{y}f|\,w\,dx\,dy.$

As a consequence, the following theorem shows that the $(p,p)$-Poincaré
inequality can be obtained directly but with larger bounds.

######  Theorem 1.6.

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. Then, for any
$R=I_{1}\times I_{2}$ in $\mathfrak{R}$ any $f\in C^{2}(R)$ we have that

$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{\frac{2}{p-1}}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}.$

Observe that the behaviour of the exponent as $p\to 1$ is not the expected one
in view of (1.13). We remedy the situation by including here a variation of
both Theorem 1.3 and Theorem 1.6 by means of an extrapolation type argument.

######  Theorem 1.7.

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. Then, for any
$R=I_{1}\times I_{2}$ in $\mathfrak{R}$ and any $f\in C^{2}(R)$ we have that

(1.14)
$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{\min\\{4,\frac{2}{p-1}\\}}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}.$

This theorem will be a consequence of a more precise result from Theorem 3.1
below involving a “dual maximal operator” presented in Section 3.

###  1.2. Fractional Integrals

We include here the needed definitions about fractional integrals and $A_{p}$
classes of weights. For $0<\alpha<n$, the fractional integral operator or
Riesz potential $\mathcal{I}_{\alpha}$ is defined by

$\mathcal{I}_{\alpha}f(x)=\int_{R^{n}}\frac{f(y)}{|x-y|^{n-\alpha}}dy.$

A key ingredient in our proofs is the pointwise estimate in Lemma 1.8
involving the fractional integral and the Hardy-Littlewood maximal function
$M$ defined by

$Mf(x)=\sup_{Q\ni
x}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|f(y)|\
dy,$

where the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}$ with sides
parallel to the coordinate axes containing the point $x$.

######  Lemma 1.8.

For every cube $Q\subset\mathbb{R}^{n}$ we have

$\mathcal{I}_{1}(g\chi_{Q})(x)\leq C_{n}\ell(Q)M(\chi_{Q}g)(x),\qquad x\in Q.$

In addition, we need to recall the following very well know result about the
sharp weighted bound for the Hardy-Littlewood maximal function. To that end,
we recall here the definition of Muckenhoupt weights $A_{p}$ for cubes in
$\mathbb{R}^{n}$.

######  Definition 1.9.

For a given $p\in(1,\infty)$, the Muckenhoupt $A_{p}$ of weights is defined by
the condition

(1.15) $[w]_{A_{p}}:=\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}w(y)\
dy\right)\left(\frac{1}{|Q|}\int_{Q}w(y)^{1-p^{\prime}}\
dy\right)^{p-1}<\infty,$

where the supremum is taken over all the cubes $Q$ in $\mathbb{R}^{n}$. The
limiting case of (1.15) when $p=1$, defines the class $A_{1}$; that is, the
set of weights $w$ such that

$[w]_{A_{1}}:=\sup_{Q}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}w\,dx\bigg{)}\operatorname*{ess\,sup}_{Q}(w^{-1})<+\infty.$

This is equivalent to $w$ having the property

$Mw(x)\leq[w]_{A_{1}}w(x)\qquad\text{ a.e. }x\in\mathbb{R}^{n}.$

We need to consider, in our bi-parametric setting, the $A_{p}$ property for
the slices $w^{x}$ and $w^{y}$. We remark here that according to [GCRdF85,
Lemma 6.2] we have that $w^{x}\in A_{p}(\mathbb{R}^{n_{2}})$ and moreover
$[w^{x}]_{A_{p}(\mathbb{R}^{n_{2}})}\leq[w]_{A_{p,\mathfrak{R}}}$. The same
holds for $w^{y}=w(x,y)$ with $y\in\mathbb{R}^{n_{2}}$.

We recall now Buckley’s result on sharp weighted $L^{p}$ norms.

######  Theorem 1.10.

[Buc93] If $1<p<\infty$ and $w\in A_{p}$ then

(1.16) $\|Mf\|_{L^{p,\infty}(w)}\lesssim[w]_{A_{p}}^{1/p}\|f\|_{L^{p}(w)}$
(1.17) $\|Mf\|_{L^{p}(w)}\lesssim[w]^{\frac{1}{p-1}}_{A_{p}}\|f\|_{L^{p}(w)}$

where in each case the exponents are best possible.

##  2\. Proofs for Biparameter Poincaré inequalities

We include here the proofs of the main results.

###### Proof of Theorem 1.3.

According to inequality (1.9) and using Lemma 1.8 twice on each direction
$n_{i}$, we have

$\displaystyle\|f-\pi_{R}(f)\|_{L^{p,\infty}(R,w)}$ $\displaystyle\leq$
$\displaystyle
c_{n}\|T(|\nabla_{x}\nabla_{y}f\chi_{R}|)\|_{L^{p,\infty}(R,w)}$
$\displaystyle\leq$ $\displaystyle c_{n}\ell(I_{1})\ell(I_{2})\|M^{n_{2}}\circ
M^{n_{1}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)\|_{L^{p,\infty}(R,w)}$

where we denote $M^{n_{1}}$ and $M^{n_{2}}$ the $n_{1}$-dimensional and
$n_{2}$-dimensional Hardy-Littlewood maximal operator respectively.

Now, for a fixed $\lambda>0$, we denote

$\Omega_{\lambda}:=\\{(x,y)\in R:M^{n_{2}}\circ
M^{n_{1}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y)>\lambda\\}.$

Also, for each fixed $x\in I_{1}$ we put

$\Omega_{\lambda}^{x}:=\\{(x,y):y\in I_{2},M^{n_{2}}\circ
M^{n_{1}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y)>\lambda\\}.$

Then

$w(\Omega_{\lambda})=\int_{\Omega_{\lambda}}w(x,y)\,dx\,dy=\int_{I_{1}}w^{x}(\Omega_{\lambda}^{x})\,dx.$

Here, we use again the standard notation of $w^{x}(y)=w(x,y)$ to denote the
slice of the function $w$ for a fixed $x\in I_{1}$. Now we apply Theorem 1.10
as follows. The first step is to use the sharp weak type bound for $M^{n_{2}}$
with the weight $w^{x}(y)$. Then,

$\displaystyle w(\Omega_{\lambda})$ $\displaystyle\leq
c_{n_{2}}\frac{[w]_{A_{p,\mathfrak{R}}}^{p}}{\lambda^{p}}\int_{I_{1}}M^{n_{1}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y)^{p}\,dxdy$
$\displaystyle\leq
c_{n,p}\frac{[w]_{A_{p,\mathfrak{R}}}^{p}[w]_{A_{p,\mathfrak{R}}}^{p^{\prime}}}{\lambda^{p}}\int_{I_{1}}|\nabla_{x}\nabla_{y}f\chi_{R}(x,y)|^{p}\,dxdy$

by applying the sharp $(p,p)$ strong bound for $M^{n_{1}}$ considering now the
weight $w^{y}(x)\in A_{p}(\mathbb{R}^{n_{1}})$. This concludes the proof
immediately. ∎

######  Remark 2.1.

Note that, as we mentioned before, this argument can not be applied to the
case $p=1$, given that the Hardy-Littlewood operator is not bounded in
$L^{1}$.

We include now the proof of the strong bound for $p>1$ stated in Theorem 1.6.

###### Proof of Theorem 1.6.

Let’s write $I=\int_{R}|f-\pi_{R}(f)|^{p}w\,dx\,dy$ and recall that $w^{x}$ is
a function on $y$. Then

$\displaystyle I$ $\displaystyle\lesssim$
$\displaystyle\ell(I_{1})^{p}\ell(I_{2})^{p}\int_{R}[M^{n_{1}}M^{n_{2}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y)]^{p}w(x,y)\,dxdy$
$\displaystyle\approx$
$\displaystyle\ell(I_{1})^{p}\ell(I_{2})^{p}\int_{I_{1}}\int_{I_{2}}M^{n_{1}}(M^{n_{2}}(|\nabla_{x}\nabla_{y}f\chi_{R}|))(x,y)^{p}w^{x}(y)\,dy\,dx$
$\displaystyle\lesssim$
$\displaystyle\ell(I_{1})^{p}\ell(I_{2})^{p}[w^{x}]^{\frac{p}{p-1}}_{A_{p}(\mathbb{R}^{n_{1}})}\int_{I_{1}}\int_{I_{2}}[M^{n_{2}}(|\nabla_{x}\nabla_{y}f\chi_{R}|)(x,y)]^{p}w^{y}(x)\,dxdy$
$\displaystyle\lesssim$
$\displaystyle\ell(I_{1})^{p}\ell(I_{2})^{p}[w]_{A_{p,\mathfrak{R}}}^{\frac{p}{p-1}}[w^{y}]^{\frac{p}{p-1}}_{A_{p}(\mathbb{R}^{n_{2}})}\int_{I_{2}}\int_{I_{1}}|\nabla_{x}\nabla_{y}f(x,y)|^{p}w(x,y)\,dy\,dx$
$\displaystyle\approx$
$\displaystyle\ell(I_{1})^{p}\ell(I_{2})^{p}[w]_{A_{p,\mathfrak{R}}}^{\frac{2p}{p-1}}\int_{I_{2}}\int_{I_{1}}|\nabla_{x}\nabla_{y}f(x,y)|^{p}w(x,y)\,dy\,dx$

where in first step we use estimate (1.9) and Lemma 1.8. Finally, we apply the
sharp bound of the Hardy-Litllewood operator twice, first in
$L^{p}_{w^{x}}(\mathbb{R}^{n_{2}})$ and then in
$L_{w^{y}}^{p}(\mathbb{R}^{n_{1}})$. ∎

Now we present the proof of the natural substitute valid for $p=1$.

###### Proof of Proposition 1.5.

Let’s call $I$ to the LHS on (1.13). Then, by Lemma 1.2 and recalling that

$Tf(x,y)=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{1}}}\frac{f(\bar{x},\bar{y})}{|x-\bar{x}|^{n_{1}-1}|y-\bar{y}|^{n_{2}-1}}\,d\bar{x}\,d\bar{y}.$
$\displaystyle I$ $\displaystyle\lesssim$
$\displaystyle\int_{R}T(|\nabla_{x}\nabla_{y}f\chi_{R}|)w\,dxdy$
$\displaystyle\lesssim$
$\displaystyle\int_{R}\int_{R}\frac{|\nabla_{x}\nabla_{y}f(\bar{x},\bar{y})|}{|x-\bar{x}|^{n-1}|y-\bar{y}|^{m-1}}\,d\bar{x}d\bar{y}\,wdxdy$
$\displaystyle\lesssim$
$\displaystyle\int_{R}|\nabla_{x}\nabla_{y}f(\bar{x},\bar{y})|\left[\int_{I_{1}}\frac{1}{|x-\bar{x}|^{n-1}}\int_{I_{2}}\frac{w^{x}(y)}{|y-\bar{y}|^{m-1}}\,dy\,dx\right]\,d\bar{x}d\bar{y}$
$\displaystyle\lesssim$
$\displaystyle\ell(I_{2})[w]_{A_{1,\mathfrak{R}}}\int_{R}|\nabla_{x}\nabla_{y}f(\bar{x},\bar{y})|\int_{I_{1}}\frac{w^{\bar{y}}(x)}{|x-\bar{x}|^{n-1}}\,dx\,d\bar{x}\,d\bar{y}$
$\displaystyle\lesssim$
$\displaystyle[w]_{A_{1,\mathfrak{R}}}^{2}\ell(I_{1})\ell(I_{2})\int_{R}|\nabla_{x}\nabla_{y}f(\bar{x},\bar{y})|w(\bar{x},\bar{y})\,d\bar{x}\,d\bar{y},$

where we have used the following inequalities

$\displaystyle\int_{I_{2}}\frac{w^{x}(y)}{|y-\bar{y}|^{m-1}}\,dy$
$\displaystyle\lesssim$
$\displaystyle\ell(I_{2})\,[w]_{A_{1,\mathfrak{R}}}\,w(x,\bar{y}),$
$\displaystyle\int_{I_{1}}\frac{w^{\bar{y}}(x)}{|x-\bar{x}|^{n-1}}\,dx$
$\displaystyle\lesssim$
$\displaystyle\ell(I_{1})\,[w]_{A_{1,\mathfrak{R}}}\,w(\bar{x},\bar{y})\,.$

∎

##  3\. non-standard Poincaré inequalities and Extrapolation

In this section we will prove Theorem 1.7 using ideas from extrapolation.

Define, as usual, the maximal operator:

(3.1) $M_{\mathfrak{R}}f(x):=\sup_{x\in
R\in\mathfrak{R}}\frac{1}{|R|}\int_{R}|f(y)|\,dy.$

Using the know weighted estimate on each factor, we obtain the estimate for
any $p\in(1,\infty)$:

(3.2)
$\|M_{\mathfrak{R}}\|_{L^{p}(w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{\frac{2}{p-1}}.$

We also need the “dual” operator

(3.3) $M_{\mathfrak{R}}^{\prime}f:=\frac{M_{\mathfrak{R}}(fw)}{w}.$

Using that $w^{1-p^{\prime}}\in A_{p^{\prime},\mathfrak{R}}$ and also that $M$
is bounded on $L^{p^{\prime}}(w^{1-p^{\prime}})$, we conclude that
$M^{\prime}$ is bounded on $L^{p^{\prime}}(w)$ and

(3.4)
$\|M^{\prime}_{\mathfrak{R}}\|_{L^{p^{\prime}}(w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{2}.$

We have the following intermediate result:

######  Theorem 3.1.

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. Then, for any
$R=I_{1}\times I_{2}$ in $\mathfrak{R}$ and any $f\in C^{2}(R)$ we have that

(3.5)
$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\lesssim\|M_{\mathfrak{R}}^{\prime}\|_{L^{p^{\prime}}(w)}^{2}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}$

###### Proof.

Fix $p>1$, and let $w\in A_{p,\mathfrak{R}}$.

Therefore, we can define the following Rubio de Francia type iteration
algorithm:

$\mathcal{R}^{\prime}h(x)=\sum_{k=0}^{\infty}\frac{(M_{\mathfrak{R}}^{\prime})^{k}h(x)}{2^{k}\|M_{\mathfrak{R}}^{\prime}\|^{k}_{L^{p^{\prime}}(w)}}.$

where $M_{\mathfrak{R}}^{\prime}\,$ is the “dual” operator (3.3) and
$(M_{\mathfrak{R}}^{\prime})^{0}=I_{d}$. Then we have that:
(1) $h\leq\mathcal{R}^{\prime}(h)$
(2) $\|\mathcal{R}^{\prime}(h)\|_{L^{p^{\prime}}(w)}\leq
2\,\|h\|_{L^{p^{\prime}}(w)}$
(3) $[\mathcal{R}^{\prime}(h)\,w]_{A_{1,\mathfrak{R}}}\leq
2\,\|M_{\mathfrak{R}}^{\prime}\|_{L^{p^{\prime}}(w)}$

By duality there exists a non-negative function $h\in L^{p^{\prime}}(w)$,
$\|h\|_{L^{p^{\prime}}(w)}=1$, supported in $R$, such that,

$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}=\int_{R}|f-\pi_{R}(f)|\,hw\,dx\leq\int_{R}|f-\pi_{R}(f)|\,\mathcal{R}^{\prime}h\,w\,dx$

and since $\mathcal{R}^{\prime}(h)\,w\in A_{1,\mathfrak{R}}$ with
$[\mathcal{R}^{\prime}(h)\,w]_{A_{1,\mathfrak{R}}}\leq
2\,\|M_{\mathfrak{R}}^{\prime}\|_{L^{p^{\prime}}(w)}$ we can apply Proposition
1.5 to obtain

$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\leq
c\,\ell(I_{1})\ell(I_{2})\|M_{\mathfrak{R}}^{\prime}\|_{L^{p^{\prime}}(w)}^{2}\,\int_{R}|\nabla_{x}\nabla_{y}f|\mathcal{R}^{\prime}h\,w\,dx.$

Let us focus now in the last integral and apply Hölder inequality and the
properties of $\mathcal{R}^{\prime}$:

$\displaystyle\int_{R}|\nabla_{x}\nabla_{y}f|\mathcal{R}^{\prime}h\,w\,dx$
$\displaystyle\leq$
$\displaystyle\left(\int_{R}|\nabla_{x}\nabla_{y}f|^{p}\,w\,dx\right)^{\frac{1}{p}}\,\left(\int_{\mathbb{R}^{n}}(\mathcal{R}^{\prime}h)^{p^{\prime}}\,w\,dx\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{R}|\nabla_{x}\nabla_{y}f|^{p}\,w\,dx\right)^{\frac{1}{p}}\,\left(\int_{R}h^{p^{\prime}}\,w\,dx\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\left(\int_{R}|\nabla_{x}\nabla_{y}f|^{p}\,w\,dx\right)^{\frac{1}{p}}.$

∎

We are now able to present the proof of Theorem 1.7 using, on one hand, the
estimate from (3.5) combined with inequality (3.4) to obtain

$\|f-\pi_{R}(f)\|_{L^{p}(R,w)}\lesssim[w]_{A_{p,\mathfrak{R}}}^{4}\ell(I_{1})\ell(I_{2})\,\|\nabla_{x}\nabla_{y}f\|_{L^{p}(R,w)}.$

On the other hand, we already had the estimate provided in Theorem 1.6.
Combining the two results, we get the desired result stated in (1.14).

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