# Tensorization of Sobolev spaces

Silvia Ghinassi  Department of Mathematics, University of Washington, Seattle,
WA 98195, USA<EMAIL_ADDRESS>,  Vikram Giri  Department of Mathematics,
Princeton University, Princeton, NJ 08544 , USA<EMAIL_ADDRESS>and
Elisa Negrini  IPAM, UCLA, Los Angeles, CA 90095, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that Sobolev spaces on Cartesian and warped products of metric spaces
tensorize, when one of the factor is a doubling space supporting a Poincaré
inequality.

###### Key words and phrases:

Sobolev space, metric space, doubling, cartesian product, warped product

###### 2010 Mathematics Subject Classification:

Primary: 53C23. Secondary: 46E35, 51F30.

E.N. is supported by the Simons Postdoctoral program at IPAM and NSF DMS
1925919. S.G. is partially supported by NSF DMS-FRG-1853993. V.G. has been
supported by NSF DMS-FRG-1854344. The project started while the authors were
participating at the AMS MRC 2020 on Analysis in Metric Spaces. We are
grateful to Nicola Gigli, Nageswari Shanmungalingam, and Luca Capogna for
their helpful guidance. We are also grateful Angela Wu for useful insights in
the early stages of the project.

##  1\. Introduction

The theory of first order Sobolev spaces in abstract metric measure spaces
became an independent field of study the late 90’s. Hajłasz Ha (96) defined
the concept of Hajłasz gradient and subsequently the so-called Sobolev-Hajłasz
spaces. Heinonen and Koskela soon introduced the concept of upper gradient to
replace the distributional gradient in more abstract spaces, HK (95, 96, 98).
Their definition requires an inequality to hold true for every rectifiable
curve in the underlying space. Shanmungalingam Sha (00) introduced the concept
of weak upper gradients, and the relative Newton-Sobolev spaces by allowing
the requirement to fail for a null set of curves (in the sense of modulus of a
curve family).

Another crucial contribution to the field is due to Cheeger Che (99), who used
a relaxation procedure of weak upper gradients to define relaxed upper
gradients. Both Shanmungalingam and Cheeger require the underlying metric
space to be doubling, and supporting some type of Poincaré inequality. Such
spaces possess important geometric properties that are fairly restrictive. A
remarkable result due to Cheeger Che (99) is that in doubling metric measure
spaces supporting a Poincaré inequality Lipschitz functions are differentiable
almost everywhere. Another important point of view for understanding what it
means for a metric space to support a Poincaré inequality is that it is
closely related the existence of “thick” families of curve, a fact much
emphasized by David and Semmes DS (90); Sem (96).

On one hand, Cheeger was interested in understanding the differential
structure of Gromov-Hausdorff limits, while the approach by Shanmungalingam,
Koskela, and Heinonen stemmed from the study of functional properties of
quasiconformal maps. See also HKST (15) for an exposition on many of the
aforementioned results.

More recently, in the interest of studying spaces that satisfy a lower Ricci
curvature bound, Lott and Villani LV (07) and (independently) Sturm Stu06b ;
Stu06a  introduced the celebrated curvature-dimension condition. In order to
further study these spaces, Ambrosio, Gigli, and Savarè AGS14b  introduced the
so-called RCD spaces and a new notion of first order Sobolev spaces, suitable
for a larger class of metric measure spaces. There has been a flurry of
activity in the last few years, due to the interest in studying these general
metric spaces with a lower curvature bound. See the brief surveys Hon (20);
Nab (20) for more information. Their approach is similar in nature to
Shanmungalingam’s, but with a different, and more probabilistic, concept of
null set.

Under sufficient geometric assumptions, the notions of weak upper gradient due
to Cheeger, Shanmungalingam, and Ambrosio-Gigli-Savarè are equivalent AGS14a
(see also AGS (13) for a more focused exposition), while Hajłasz gradients are
not; one should think of them as the Hardy–Littlewood maximal function of
upper gradients. Shanmungalingam Sha (00) proved that the Hajłasz gradient of
a continuous function, up to some modifications on a set of measure zero, is
an upper gradient. On the other hand, it is known that an upper gradient may
not be a Hajłasz gradient, even if the underlying space is well connected.

The tensorization problem asks what is the relation between the Sobolev
functions on two metric measure spaces $X$ and $Y$ and the ones on their
product space (equipped with the product measure, and Cartesian product
metric). The problem, while basic, seems to not have been explored until
recently; in fact it is still open for the general case. Ambrosio, Gigli, and
Savarè AGS14b  first proved the result under some restrictive condition on the
curvature of _both_ spaces (namely the $\mathsf{RCD}(K,\infty)$ condition).
Ambrosio, Pinamonti, and Speight APS (15) proved it for spaces that are
doubling and support a Poincaré inequality, or under certain quadracity
assumptions. Ambrosio, Gigli, and Savarè AGS14b  proved that, for a general
metric space, it is always true that we can control the Euclidean product of
the weak upper gradients in each factor by the weak upper gradient on the
product.

In this paper we prove that the conjecture is true, provided _one_ of the
factors is doubling and has a Poincaré inequality. We also prove that the same
is true if one considers warped products of metric measure spaces. Both
results (and their proofs) generalize the work of Gigli and Han GH (18), who
proved the conjecture when one of the factors is an interval, both for
Cartesian and warped products. We remark that the proof we present is almost
self contained, relying only on earlier results concerning the density of
Lipschitz functions by Ambrosio, Gigli, and Savarè AGS14a .

Independently and concurrently, Eriksson-Bique, Soultanis, and Rajala EBRS22a
proved similar results to ours, in the Cartesian case. Their proof relies on
techniques developed by the first and third author EBS (21). The same authors
EBRS22b  then also proved tensorization where both factors are quasi-
Hilbertian metric spaces.

##  2\. Preliminaries

For basic definitions on metric spaces, we refer the interested reader to Hei
(01). We start with a few definitions and result that will be needed to define
Sobolev functions.

######  Definition 2.1 (Absolutely continuous curves).

Let $p\in[1,+\infty]$ and $\gamma\in C([0,1],X)$ and indicate
$\gamma_{t}=\gamma(t)$ for $t\in[0,1]$. We say that $\gamma\in
AC^{p}([0,1],X)$ if there exists $G\in L^{p}([0,1])$ such that

(2.1) $d(\gamma_{s},\gamma_{t})\leq\int_{s}^{t}G(r)\,dr\quad\forall
t,s\in[0,1],\;s<t$

For $p=1$ the space $AC^{1}([0,1],X)$ is denoted $AC([0,1],X)$ and it is the
space of absolutely continuous curves.

######  Theorem 2.2 (Theorem 1.1.2 in AGS (05) ).

For $\gamma\in AC([0,1],X)$ there exists an a.e. minimal function $G$ which
satisfies (2.1) called the metric derivative which can be computed for a.e.
$t\in[0,1]$ as

$|\dot{\gamma}_{t}|:=\lim_{s\rightarrow
t}\frac{d(\gamma_{t},\gamma_{s})}{|s-t|}$

######  Proposition 2.3 (BBI (22)).

The length of a curve $\gamma\in AC([0,1],X)$ is given by:

$l[\gamma]:=\int_{0}^{1}|\dot{\gamma}_{t}|\,dt$

If $(X,d)$ is a length space then for any $x,y\in X$

$\operatorname{d}(x,y)=\inf\left\\{\;\int_{0}^{1}|\dot{\gamma}_{t}|\,dt;\quad\gamma\in
AC([0,1],X)\text{ connecting }x\text{ and }y\;\right\\}$

######  Definition 2.4 (Metric doubling).

Given a metric measure space $(X,d,\mu)$, we say that $X$ is _metric doubling_
if there exists a constant $C_{X}>1$ such that, for all $r>0$ every ball of
radius $r$ can be covered by $C_{X}$ balls of radius $r/2$.

######  Remark 2.5.

There is another, closely related, notion of doubling spaces; a space is said
to be _measure doubling_ if there exists a constant $C_{X}>1$ such that, for
all $x\in X$ and $r>0$ we have $\mu(B(x,2r))\leq C_{X}\mu(B(x,r))$. Every
metric space that carries a doubling measure is automatically metric doubling,
and every complete metric doubling space is also measure doubling. For a proof
see Hei (01). Because we will only be dealing with complete (and separable)
metric spaces, we will simply refer to spaces as _doubling_ , as the two
notions are equivalent in this setting (and the relative constants are
quantitatively related).

######  Definition 2.6 (Local and global Lipschitz constants).

Given $f\colon X\to[0,+\infty]$, the local Lipschitz constant of $f$ is the
function $\operatorname{lip}(f):X\rightarrow[0,+\infty]$ defined as

(2.2) $\displaystyle\operatorname{lip}(f)(x)=\begin{cases}\limsup_{y\to
x}\frac{|f(x)-f(y)|}{\operatorname{d}_{X}(x,y)}&\quad\text{if }x\text{ is not
isolated}\\\ 0&\quad\text{otherwise}\end{cases}$

Analogously, the global Lipschitz constant is defined as

(2.3) $\operatorname{Lip}(f)=\limsup_{y\neq
x}\frac{|f(x)-f(y)|}{\operatorname{d}_{X}(x,y)},$

and if $X$ is a length space,
$\operatorname{Lip}(f)=\sup_{x}\operatorname{lip}(f)(x)$.

######  Definition 2.7 (Test plan).

Let $(X,d,m)$ be a metric measure space and $\pi\in\mathcal{P}(C([0,1],X))$.
$\pi$ is said to have bounded compression if there exists a constant $C>0$
such that for all $t\in[0,1]$

$(e_{t})_{\\#}\pi\leq Cm$

$\pi$ is said to be a test plan if it has bounded compression, it is
concentrated on $AC^{2}([0,1],X)$ and

$\int_{0}^{1}\;\int\;|\dot{\gamma}_{t}|^{2}\;d\pi(\gamma)\;dt\leq+\infty$

######  Definition 2.8 (Sobolev class).

Let $(X,d,m)$ be a metric measure space. A Borel function
$f:X\rightarrow\mathbb{R}$ belongs to the Sobolev class $S^{2}(X,d,m)$
(respectively $S_{loc}^{2}(X,d,m)$ ) if there exists a non-negative function
$G\in L^{2}(X,m)$ (respectively $G\in L_{loc}^{2}(X,m)$) such that for all
test plans $\pi$

$\int\;|f(\gamma_{1})-f(\gamma_{0})|\;d\pi(\gamma)\leq\int\;\int_{0}^{1}G(\gamma_{s})|\dot{\gamma}_{s}|\;ds\;d\pi(\gamma)$

In this case the function $G$ is called a 2-weak upper gradient of $f$, or
simply a weak upper gradient of $f$.

######  Remark 2.9.

Among all weak upper gradients of $f$ there exists a minimal function $G$ in
the $m$-a.e. sense. Such minimal function is called minimal weak upper
gradient and we denote it by $|Df|$. Notice that if $f$ is Lipschitz, then
$|Df|\leq\operatorname{lip}(f)$ $m$-a.e. since $\operatorname{lip}(f)$ is a
weak upper gradient of $f$.

We will use that minimal weak upper gradients are lower semicontinuous in the
following sense: if $f_{n}\in S^{2}(X,d,m)$ is a sequence converging $m$-a.e.
to some $f$ such that the sequence given by the $|Df_{n}|$’s is bounded in
$L^{2}(X,m)$, then $f\in S^{2}(X,d,m)$ and for all $G$ that are the
$L^{2}$-weak limit of some subsequence of $|Df_{n}|$ we have $|Df|\leq G$.
Finally, we will later use the fact that space $S^{2}_{loc}(X,d,m)\cap
L^{\infty}_{loc}(X,d,m)$ is an algebra and for all $f,g\in
S^{2}_{loc}(X,d,m)\cap L^{\infty}_{loc}(X,d,m)$ the following inequality
holds:

(2.4) $|D(fg)|\leq|f||Dg|+|g||Df|\quad m\text{-a.e.}.$

For additional details on the properties of minimal weak upper gradients, see
AGS14a .

######  Definition 2.10 (Sobolev space).

The Sobolev space $W^{1,2}(X,d,m)$ is defined as

$W^{1,2}(X,d,m):=S^{2}(X,d,m)\cap L^{2}(X,m).$

$W^{1,2}(X,d,m)$ endowed with the norm

$\|f\|_{W^{1,2}(X,d,m)}:=\|f\|_{L^{2}(X,m)}+\||Df|\|_{L^{2}(X,m)}$

is a Banach space, but in general it is not an Hilbert space. If
$W^{1,2}(X,d,m)$ is an Hilbert space, we say that $(X,d,m)$ is infinitesimally
Hilbertian.

######  Lemma 2.11 (Density in energy of Lipschitz functions, AGS14a ).

Let $Y$ be a complete and separable metric measure space, and $f\in
W^{1,2}(Y)$. Then there exists a sequence of Lipschitz functions $f_{n}$ that
converges to $f$ in $L^{2}$ and such that $\operatorname{lip}(f_{n})$
converges in $L^{2}$ to $|Df|$.

######  Assumption 2.12.

We will have, unless otherwise specified, the following set of assumptions for
the metric measure space $(X,\operatorname{d}_{X},m_{X})$:

  * •

$(X,\operatorname{d}_{X})$ is a complete and separable length space,

  * •

$m_{X}$ is a non-negative Borel measure with respect to $\operatorname{d}_{X}$
and it is finite on bounded sets,

  * •

$\operatorname{supp}(m_{X})=X$.

In the following we may denote this space simply by $X$.

######  Assumption 2.13.

We will have, unless otherwise specified, the following set of assumptions for
the metric measure space $(Y,\operatorname{d}_{Y},m_{Y})$:

  * •

$(Y,\operatorname{d}_{Y},m_{Y})$ is complete, separable, and $C_{Y}$-doubling
length space,

  * •

$m_{Y}$ is a non-negative Borel measure with respect to $\operatorname{d}_{Y}$
and it is finite on bounded sets,

  * •

$\operatorname{supp}(m_{Y})=Y$,

  * •

$Y$ supports a $(2,2)$-Poincaré inequality, that is, for every $r>0$, there
exists constants $\lambda\geq 1$ and $C_{p}$, such that for any metric ball
$B\subset Y$ of radius smaller than $r$, we have

$\fint_{B}|u-u_{B}|^{2}\,dm_{Y}\leq
C_{P}\operatorname{rad}(B)^{2}\int_{\lambda B}|g|^{2},$

where $g$ is any weak upper gradient for $u$, and $u_{B}=\fint_{B}u\,dm_{Y}$.

In the following we may denote this space simply by $Y$.

######  Remark 2.14.

The most restrictive assumptions (on $Y$) is the existence of a Poincaré
inequality. This fact will be used exclusively to obtain (3.16) in Lemma 3.9.

##  3\. Cartesian products

The product space $X\times Y$ is given the Euclidean product metric

(3.1) $\operatorname{d}_{X\times
Y}((x,t),(y,s)):=\sqrt{\operatorname{d}_{X}(x,y)^{2}+\operatorname{d}_{Y}(t,s)^{2}}$

and the measure on it is the usual product measure and is denoted simply by
$m$.

######  Definition 3.1 (Beppo Levi space).

The Beppo-Levi space ${\mathsf{BL}}(X,Y)$ is the space of functions $f\in
L^{2}(X\times Y;\mathbb{R})$ such that

  1. (1)

$f(x,\cdot)\in W^{1,2}(Y)$ for $m_{X}\text{ - a.e. }x$

  2. (2)

$f(\cdot,t)\in W^{1,2}(X)$ for $m_{Y}\text{ - a.e. }t$

  3. (3)

the function

(3.2)
$|Df|_{{\mathsf{BL}}}(x,t):=\sqrt{|Df(x,\cdot)|_{Y}(t)^{2}+|Df(\cdot,t)|_{X}(x)^{2}}$

belongs to $L^{2}(X\times Y;\mathbb{R})$

with the Beppo-Levi norm

(3.3)
$\|f\|_{{\mathsf{BL}}}:=\sqrt{\|f\|_{L^{2}}^{2}+\||Df|_{{\mathsf{BL}}}\|_{L^{2}}^{2}}$

We will also write $|Df(x,\cdot)|_{Y}(t)$ as
$|{{\partial}f}/{{\partial}t}|(x,t)$ and $|Df(\cdot,t)|_{X}(x)$ as
$|{{\partial}f}/{{\partial}x}|(x,t)$.
Our main theorem for this section is:

######  Theorem 3.2.

The sets $W^{1,2}(X\times Y)$ and ${\mathsf{BL}}(X,Y)$ coincide and for every
$f\in W^{1,2}(X\times Y)={\mathsf{BL}}(X,Y)$ we have

$|Df|_{{\mathsf{BL}}}\leq|Df|_{X\times Y}\leq
C_{0}|Df|_{{\mathsf{BL}}}\quad\text{$m_{X\times Y}$-a.e.}$

As in the interval case, we only need to show ${\mathsf{BL}}(X,Y)\subset
W^{1,2}(X\times Y)$ as the other inclusion has already been shown in AGS14b :

######  Proposition 3.3 (Prop. 6.18 in AGS14b ).

We have $W^{1,2}(X\times Y)\subset{\mathsf{BL}}(X\times Y)$ and

(3.4) $\int_{X\times Y}|Df|_{{\mathsf{BL}}}^{2}\leq\int_{X\times
Y}|Df|_{X\times Y}^{2}.$

To prove our main theorem we will need the following lemmas.

######  Lemma 3.4.

Let $N>0$ be a fixed natural number. Let $f:X\times Y\to\mathbb{R}$ be of the
form $f(x,t)=\sum_{i=1}^{N}h_{i}(t)g_{i}(x)$ where
$g_{i}\in\operatorname{Lip}(X)$ and $h_{i}\in\operatorname{Lip}(Y)$ for all
$1\leq i\leq N$. Then

(3.5) $\operatorname{lip}_{X\times
Y}(f)^{2}(x,t)\leq\operatorname{lip}_{X}(f(\cdot,t))^{2}(x)+\operatorname{lip}_{Y}(f(x,\cdot))^{2}(t)$

for every $(x,t)\in X\times Y$

This lemma replaces Lemma 3.3 in GH (18), the key difference being the use of
triangle inequality instead of Cauchy-Scwhartz.

###### Proof.

We have

(3.6) $\displaystyle\begin{split}&\operatorname{lip}_{X\times
Y}(f)^{2}(x,t)=\limsup_{(y,s)\to(x,t)}\frac{|f(y,s)-f(x,t)|^{2}}{\operatorname{d}_{X\times
Y}((y,s),(x,t))^{2}}\\\
&\leq\limsup_{(y,s)\to(x,t)}\frac{1}{\operatorname{d}_{X\times
Y}((y,s),(x,t))^{2}}\left(\operatorname{d}_{X}(y,x)^{2}\frac{|f(y,s)-f(x,s)|^{2}}{\operatorname{d}_{X}(y,x)^{2}}+\operatorname{d}_{Y}(s,t)^{2}\frac{|f(x,s)-f(x,t)|^{2}}{\operatorname{d}_{Y}(s,t)^{2}}\right)\\\
&\leq\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}h_{i}(s)(g_{i}(y)-g_{i}(x))\right|^{2}}{\operatorname{d}_{X}(y,x)^{2}}+\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}(h_{i}(s)-h_{i}(t))g_{i}(x)\right|^{2}}{\operatorname{d}_{Y}(s,t)^{2}}\\\
&\leq\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}h_{i}(t)(g_{i}(y)-g_{i}(x))+(h_{i}(s)-h_{i}(t))(g_{i}(y)-g_{i}(x))\right|^{2}}{\operatorname{d}_{X}(x,y)^{2}}\\\
&\qquad+\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}(h_{i}(s)-h_{i}(t))g_{i}(x)\right|^{2}}{\operatorname{d}_{Y}(s,t)^{2}}\\\
&\leq\left(\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}h_{i}(t)(g_{i}(y)-g_{i}(x))\right|}{\operatorname{d}_{X}(x,y)}+\sum_{i=1}^{N}\left|h_{i}(s)-h_{i}(t)\right|\frac{\left|g_{i}(y)-g_{i}(x)\right|}{\operatorname{d}_{X}(y,x)}\right)^{2}\\\
&\qquad+\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}(h_{i}(s)-h_{i}(t))g_{i}(x)\right|^{2}}{\operatorname{d}_{Y}(s,t)^{2}}\\\
&\leq
2\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}h_{i}(t)(g_{i}(y)-g_{i}(x))\right|^{2}}{\operatorname{d}_{X}(y,x)^{2}}+2\limsup_{(y,s)\to(x,t)}\sum_{i=1}^{N}\left|h_{i}(s)-h_{i}(t)\right|^{2}\frac{\left|g_{i}(y)-g_{i}(x)\right|^{2}}{\operatorname{d}_{X}(x,y)^{2}}\\\
&\qquad+\limsup_{(y,s)\to(x,t)}\frac{\left|\sum_{i=1}^{N}(h_{i}(s)-h_{i}(t))g_{i}(x)\right|^{2}}{\operatorname{d}_{Y}(s,t)^{2}}\\\
&=2\operatorname{lip}_{X}(f(\cdot,t))^{2}(x)+\operatorname{lip}_{Y}(f(x,\cdot))^{2}(t),\end{split}$

where in the last step we used that $h$ is continuous. ∎

######  Definition 3.5.

The class $\mathcal{A}\subset{\mathsf{BL}}_{loc}(X\times Y)$ consists of
functions $f$ that can be written as $f(x,t)=\sum_{i=1}^{M}h_{i}(t)g_{i}(x)$
for $g_{i}\in W^{1,2}(X)$, for some _fixed_ $M$ such that $M>C_{Y}^{3}$ and
$h_{i}\in\operatorname{Lip}(Y)$. The class $\tilde{\mathcal{A}}$ will denote
those functions that are locally in class $\mathcal{A}$. Note that
$\tilde{\mathcal{A}}\subset S^{2}_{loc}(X\times Y)$ (see Proposition 2.6 in GH
(18)).

In order to discretize the problem, we need to introduce a suitable partition
on $Y$. One option would be to use generalization of dyadic cubes in metric
measure spaces (such as the ones constructed in Chr (90)). However, since we
will not be using the tree-like structure, we can work with a simpler
construction, such as the one in the following lemma.

######  Lemma 3.6 (Lemma 7.1 in ACDM (15)).

For every $k\in{\mathbb{N}}$ there exists a collection of open subsets of $Y$,
$Q_{i,k}$ and points $t_{i,k}$ (the “centers” of the “cubes”), and $i\in
I_{k}$, where $I_{k}$ is a countable set, such that

  * •

$m_{Y}\left(Y\setminus\bigcup_{i}Q_{i,k}\right)=0$ for all $k\in{\mathbb{Z}}$;

  * •

For every $i,j\in I_{k}$ either $Q_{j,k}=Q_{i,k}$ or $Q_{j,k}\cap
Q_{i,k}=\varnothing$ (i.e. the $Q_{i,k}$’s form a partition);

  * •

if $i\neq j$, then $d_{Y}(t_{i,k},t_{j,k})>\frac{1}{k}$;

  * •

each $Q_{i,k}$ is comparable to a ball centered at $t_{i,k}$ of radius roughly
$\frac{1}{k}$,

$B\left(t_{i,k},\frac{1}{3}\frac{1}{k}\right)\subset Q_{i,k}\subset
B\left(t_{i,k},\frac{5}{4}\frac{1}{k}\right).$

######  Remark 3.7.

We say that $Q_{i,k}$ and $Q_{j,k}$ are neighbors, and we write $Q_{i,k}\sim
Q_{j,k}$, if their distance is less than $\frac{1}{k}$. If that is the case,
then it must be that $d_{Y}(t_{i,k},t_{j,k})\leq\frac{4}{k}$. Given
$Q_{i,k}\sim Q_{j,k}$, we have that their centers are at most
$\frac{10}{4}\frac{1}{k}+\frac{1}{k}=\frac{14}{4}\frac{1}{k}\leq\frac{4}{k}$
where $\frac{5}{4k}$ comes from the fact that both cubes are contained in a
ball with radius $\frac{5}{4k}$ while the $\frac{1}{k}$ is from the definition
of $Q_{i,k}\sim Q_{j,k}$. This implies that each $Q_{i,k}$ has at most
$C_{Y}^{3}$ neighbors. In fact, we can cover $B(t_{i,k},\frac{4}{k})$ with
$C_{Y}^{3}$ balls of radius $\frac{1}{2}\frac{1}{k}$, but the condition
$d_{Y}(t_{i,k},t_{j,k})>\frac{1}{k}$ means each ball can only contain one of
the $t_{i,k}$’s.

######  Lemma 3.8.

There exists a constant $C_{0}>0$ depending only on the doubling constant of
$Y$ such that for every $f\in\tilde{\mathcal{A}}$, we have that

$|Df|_{{\mathsf{BL}}}\leq|Df|\leq C_{0}|Df|_{{\mathsf{BL}}}$

for $(m_{X}\times m_{Y})$-a.e. $(x,t)$.

###### Proof.

Since the statement is local in nature, it suffices to show it for
$f\in\mathcal{A}$ where $f(x,t)=\sum_{l=1}^{M}h_{l}(t)g_{l}(x)$ and the
$h_{l}$’s have compact support. Given the other inclusion, it suffices to
prove that there exists a constant $C_{0}$ such that $|Df|^{2}\leq
C_{0}^{2}|Df|_{{\mathsf{BL}}}^{2}$ for $m$-a.e. $(x,t)$. Thus it suffices to
prove that for any $F\subset Y$ and $E\subset X$ Borel, we have that

(3.7) $\int_{E\times F}|Df|^{2}(x,t)\,dm(x,t)\leq C_{0}\int_{E\times
F}|{{\partial}f}/{{\partial}t}|^{2}(x,t)+|{{\partial}f}/{{\partial}x}|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x).$

For every $k\in{\mathbb{N}}$, let $\\{\chi_{i,k}\\}_{i\in I}$ be a partition
of unity subordinate to the partition of $Y$ by “cubes” $Q_{i,k}$ as above.
That is, $\chi_{i,k}=1$ on $B\left(t_{i,k},\frac{1}{3}\frac{1}{k}\right)$, and
$\operatorname{supp}\chi_{i,k}\subset
B\left(t_{i,k},\frac{5}{4}\frac{1}{k}\right)$. Each $\chi_{i,k}$ is
$c_{1}k$-Lipschitz, where $c_{1}=c_{1}(C_{Y})$. See for instance the
construction in KL (02). Now define

(3.8) $f_{k}(x,t):=\sum_{i\in I_{k}}\chi_{i,k}(t)f_{k,i}(x),$

where $f_{k,i}(x):=\fint_{Q_{i,k}}f(x,t)\,dt$. For a fixed $(x,t)$ at most
$C_{Y}^{3}$ terms in the sum are non-zero, and hence
$f_{k}\in\tilde{\mathcal{A}}$. Moreover,

$\|f_{i,k}\|_{L^{2}(X\times Y)}^{2}\leq\int_{X\times
Y}\fint_{Q_{ik}}|f(x,t)|^{2}=\fint_{Q_{ik}}\int_{X\times
Y}|f(x,t)|^{2}=\|f\|_{L^{2}(X\times Y)}^{2},$

and so

$\displaystyle\begin{split}\|f_{k}\|_{L^{2}(X\times Y)}^{2}=\left\|\sum_{i\in
I_{k}}\chi_{i,k}(t)f_{k,i}(x)\right\|^{2}_{L^{2}(X\times Y)}\leq
C_{Y}^{3}\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\left\|f_{k,i}\right\|^{2}_{L^{2}(X\times Y)}\leq
C_{Y}^{6}\|f\|^{2}_{L^{2}(X\times Y)}.\end{split}$

Consequently, by the dominated convergence theorem, we get

(3.9) $f_{k}\to f\qquad\text{in}\quad L^{2}(X\times Y).$

Since $f_{k,i}\in W^{1,2}(X)$ we have a sequence of Lipschitz functions
$f_{k,i}^{n}\colon X\to\mathbb{R}$ such that $f_{k,i}^{n}\to f_{k,i}$ in
$L^{2}(X,m_{X})$ and $\operatorname{lip}_{X}(f_{k,i}^{n})\to|Df_{k,i}|_{X}$ in
$L^{2}(X,m_{X})$. Now define

(3.10) $F_{k}^{n}(x,t):=\sum_{i\in I_{k}}\chi_{i,k}(t)f_{k,i}^{n}(x).$

Note that $F_{k}^{n}\in\operatorname{Lip}(X\times Y)\cap\tilde{\mathcal{A}}$
and, since $h_{i}(t)$ is continuous,
$\operatorname{lip}_{X}(F_{k}^{n})\to|\partial f_{k}/\partial x|$ in
$L^{2}(X\times Y)$. Now fix $t\in Y$. Recall that $\chi_{j,k}(t)=1-\sum_{i\neq
j}\chi_{i,k}(t)$ for any choice of $j\in{\mathbb{N}}$, and so we have that,
for any $s\in F$,

(3.11) $\displaystyle\begin{split}&F_{k}^{n}(x,t)-F_{k}^{n}(x,s)=\sum_{i\in
I_{k}}\chi_{i,k}(t)f_{k,i}^{n}(x)-\sum_{i\in
I_{k}}\chi_{i,k}(s)f_{k,i}^{n}(x)\\\ &=\sum_{i\neq
j}\chi_{i,k}(t)f_{k,i}^{n}(x)+\chi_{j,k}(t)f_{k,j}^{n}(x)-\chi_{j,k}(s)f_{k,j}^{n}(x)-\sum_{i\neq
j}\chi_{i,k}(s)f_{k,i}^{n}(x)\\\ &=\sum_{i\neq
j}\chi_{i,k}(t)f_{k,i}^{n}(x)-\sum_{i\neq
j}\chi_{i,k}(t)f_{k,j}^{n}(x)+\sum_{i\neq
j}\chi_{i,k}(s)f_{k,j}^{n}(x)-\sum_{i\neq j}\chi_{i,k}(s)f_{k,i}^{n}(x)\\\
&=\sum_{i\neq
j}\left(\chi_{i,k}(t)-\chi_{i,k}(s)\right)(f_{k,i}^{n}(x)-f_{k,j}^{n}(x)).\end{split}$

Note that the sum above is finite, as only finitely many of the $\chi_{i,n}$
are nonzero. Dividing this by $t-s$, and then taking a limit we obtain

(3.12)
$\displaystyle\begin{split}\operatorname{lip}_{Y}(F_{k}^{n}(x))(t)&=\limsup_{s\to
t}\left|\sum_{i\neq
j}\frac{\chi_{i,k}(t)-\chi_{i,k}(s)}{t-s}(f_{k,i}^{n}(x)-f_{k,j}^{n}(x))\right|\\\
&\leq\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\operatorname{Lip}_{Y}(\chi_{i,k})\left|f_{k,i}^{n}(x)-f_{k,j}^{n}(x)\right|,\end{split}$

Note that because $Y$ is doubling the last sum is also finite. We define

(3.13)
$\displaystyle\begin{split}g_{k}^{n}(x,t)&:=\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\operatorname{Lip}_{Y}(\chi_{i,k})\left|f_{k,i}^{n}(x)-f_{k,j}^{n}(x)\right|\quad\text{for
$t\in Q_{j,k}$}\\\ g_{k}(x,t)&:=\sum_{\begin{subarray}{c}i\neq j\\\
Q_{i,k}\sim
Q_{j,k}\end{subarray}}\operatorname{Lip}_{Y}(\chi_{i,k})\left|f_{k,i}(x)-f_{k,j}(x)\right|\quad\text{for
$t\in Q_{j,k}$}\end{split}$

Note that we have $g_{k}^{n}\to g_{k}$ in $L^{2}(X\times Y)$.

Next, we want to show that the $g_{k}$’s are good approximations for the
$Y$-gradient of $f$. Observe that if $Q_{i,k}$ and $Q_{j,k}$ are neighbors,
and $t\in Q_{i,k}$ and $y\in Q_{j,k}$ then
$\operatorname{d}(t,y)\leq(10/4+10/4+1)\frac{1}{k}=\frac{6}{k}$. We have

(3.14)
$\displaystyle\begin{split}\lim_{k\to\infty}g_{k}(x,t)&=\lim_{k\to\infty}\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\operatorname{Lip}_{Y}(\chi_{i,k})\left|f_{k,i}(x)-f_{k,j}(x)\right|\\\
&=\lim_{k\to\infty}\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\operatorname{Lip}_{Y}(\chi_{i,k})\left|\sum_{l=1}^{M}\left[\fint_{Q_{i,k}}h_{l}(t)\,dt-\fint_{Q_{j,k}}h_{l}(t)\,dt\right]g_{l}(x)\right|\\\
&\leq c_{1}\lim_{k\to\infty}\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}k\left|\sum_{l=1}^{M}(h_{l}(\tilde{t}_{i,k})-h_{l}(\tilde{t}_{j,k}))g_{l}(x)\right|\\\
&=c_{1}\lim_{k\to\infty}\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}k\operatorname{d}(\tilde{t}_{i,k},\tilde{t}_{j,k})\left|\sum_{l=1}^{M}\frac{h_{l}(\tilde{t}_{i,k})-h_{l}(\tilde{t}_{j,k})}{\operatorname{d}(\tilde{t}_{i,k},\tilde{t}_{j,k})}g_{l}(x)\right|\\\
&=Cc_{1}\lim_{k\to\infty}\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\left|\sum_{l=1}^{M}\frac{h_{l}(\tilde{t}_{i,k})-h_{l}(\tilde{t}_{j,k})}{\operatorname{d}(\tilde{t}_{i,k},\tilde{t}_{j,k})}g_{l}(x)\right|\\\
&{\leq Cc_{1}C_{Y}^{3}|\partial f/\partial t|.}\end{split}$

where $C$ is a geometric constant such that
$k\operatorname{d}(\tilde{t}_{i,k},\tilde{t}_{j,k})<C$ for neighbouring cubes
$Q_{i,k}\sim Q_{j,k}$. We have used that, by construction,
$\operatorname{Lip}_{Y}(\chi_{i,k})=c_{1}k$.

By Lemma 3.4 we have

(3.15) $|\operatorname{lip}_{X\times Y}(F_{k}^{n})|^{2}(x,t)\leq
2|\operatorname{lip}_{X}(F_{k}^{n}(\cdot,t))|^{2}(x)+|\operatorname{lip}_{Y}(F_{k}^{n}(x,\cdot))|^{2}(t)\qquad\text{$m$-a.e.
$(x,t)$}.$

Now the above with the lower semicontinuity of minimal weak upper gradients
gives us

$\displaystyle\int_{E\times F}|Df|^{2}(x,t)\,dm(x,t)$
$\displaystyle\leq\limsup_{k\to\infty}\int_{F\times
E}|Df_{k}|^{2}(x,t)\,dm(x,t)$
$\displaystyle\leq\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{X\times Y}(F_{k}^{n})^{2}(x,t)\,dm(x,t)$
$\displaystyle\leq\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{Y}(F_{k}^{n})^{2}(x,t)\,dm(x,t)$
$\displaystyle\qquad+2\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{X}(F_{k}^{n})^{2}(x,t)\,dm(x,t).$

Then, to finish the proof, it is enough to prove the following two
inequalities

(3.16) $2\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{X}(F_{k}^{n})^{2}(x,t)\,dm(x,t)\leq C_{0}\int_{E\times
F}|\partial f/\partial x|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x).$

(3.17) $\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{Y}(F_{k}^{n})^{2}(x,t)\,dm(x,t)\leq C_{0}\int_{E\times
F}|\partial f/\partial t|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x);$

To prove (3.16),

(3.18)
$\displaystyle\begin{split}&\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{X}(F_{k}^{n})^{2}(x,t)\,dm(x,t)\\\ &\leq
C_{Y}^{3}\limsup_{k\to\infty}\int_{F}\left(\sum_{i\in
I_{k}}\chi_{i,k}(t)\lim_{n\to\infty}\int_{E}(\operatorname{lip}_{X}f_{k,i}^{n}(x))^{2}\,dm_{X}(x)\right)dm_{Y}(t)\\\
&=C_{Y}^{3}\limsup_{k\to\infty}\int_{F}\left(\sum_{i\in
I_{k}}\chi_{i,k}(t)\int_{E}|\partial f_{k,i}/\partial
x|^{2}(x)\,dm_{X}(x)\right)dm_{Y}(t)\\\
&\leq(C_{Y}^{3})^{2}\limsup_{k\to\infty}\int_{F}\left(\sum_{i\in
I_{k}}\chi_{i,k}(t)\int_{E}\left(\fint_{Q_{ik}}|\partial f/\partial
x|^{2}(x)\,ds\right)\,dm_{X}(x)\right)dm_{Y}(t)\\\
&=C_{Y}^{6}\limsup_{k\to\infty}\int_{E}\int_{F}\left(\sum_{i\in
I_{k}}\chi_{i,k}(t)\fint_{Q_{ik}}|\partial f/\partial
x|^{2}(x)\,ds\right)\,dm(x,t)\end{split}$

Because $Y=\bigcup_{j}Q_{j,k}$, and the sum over $i\in I_{k}$ has a finite
number of non-zero summands, we have

(3.19) $\displaystyle\begin{split}&\int_{Y}\left(\sum_{i\in
I_{k}}\chi_{i,k}(t)\fint_{Q_{ik}}|\partial f/\partial
x|^{2}(x,s)\,dm_{Y}(s)\right)\,dm_{Y}(t)\\\ &\leq\sum_{j\in I_{k}}\sum_{i\in
I_{k}}\int_{Q_{j,k}}\chi_{i,k}(t)N_{i,k}(x)\,dm_{Y}(t),\end{split}$

where we have set

$N_{i,k}(x)=\fint_{Q_{ik}}|\partial f/\partial x|^{2}(x,s)\,dm_{Y}(s).$

The integral inside the two sums is nonzero only if $Q_{i,k}$ and $Q_{j,k}$
are neighbors, or $i=j$. In the latter case we simply have

(3.20) $\displaystyle\begin{split}\sum_{i\in
I_{k}}\int_{Q_{j,k}}\chi_{i,k}(t)N_{i,k}(x)\,dm_{Y}(t)&\leq\sum_{i\in
I_{k}}\int_{Q_{i,k}}N_{i,k}(x)\,dm_{Y}(t)\\\ &=\sum_{i\in
I_{k}}|Q_{i,k}|N_{i,k}(x)\\\ &=\sum_{i\in
I_{k}}|Q_{i,k}|\frac{1}{|Q_{i,k}|}\int_{Q_{i,k}}|\partial f/\partial
x|^{2}(x,s)\,dm_{Y}(s)\\\ &=\sum_{i\in I_{k}}\int_{Q_{i,k}}|\partial
f/\partial x|^{2}(x,s)\,dm_{Y}(s)\\\ &=\int_{Y}|\partial f/\partial
x|^{2}(x,s)\,dm_{Y}(s).\end{split}$

If $Q_{i,k}$ and $Q_{j,k}$ are neighbors (and there is as most $C_{Y}^{3}$
$j$’s such that $Q_{j,k}$ is a neighbor of $Q_{i,k}$), we have

(3.21) $\displaystyle\begin{split}\sum_{j\in I_{k}}\sum_{i\in
I_{k}}\int_{Q_{j,k}}\chi_{i,k}(t)N_{i,k}(x)\,dm_{Y}(t)&=\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim Q_{j,k}\end{subarray}}\sum_{i\in I_{k}}|Q_{j,k}|N_{i,k}(x)\\\
&=\sum_{\begin{subarray}{c}i\neq j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}\sum_{i\in
I_{k}}|Q_{j,k}|\frac{1}{|Q_{i,k}|}\int_{Q_{i,k}}|\partial f/\partial
x|^{2}(x,s)\,dm_{Y}(s)\\\ &\leq C_{Y}^{2}C_{Y}^{3}\sum_{i\in
I_{k}}\int_{Q_{i,k}}|\partial f/\partial x|^{2}(x,s)\,dm_{Y}(s)\\\
&=C_{Y}^{5}\int_{Y}|\partial f/\partial x|^{2}(x,s)\,dm_{Y}(s),\end{split}$

where we used the fact that $\frac{|Q_{j,k}|}{|Q_{i,k}|}\leq C_{Y}^{2}$
(because cubes are comparable to balls). This concludes the proof of (3.16).

To prove (3.17), recall that $\operatorname{lip}_{Y}(F_{k,n})\leq g_{k}^{n}$,
$g_{k}^{n}\to g_{k}$ in $L^{2}(X\times Y)$ and we have proved that
$\lim_{k\to\infty}g_{k}(x,t)\leq 6c_{1}C_{Y}^{3}|\partial f/\partial t|$.

Then we have

$\displaystyle\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}\operatorname{lip}_{Y}(F_{k}^{n})^{2}(x,t)\,dm(x,t)$
$\displaystyle\leq\limsup_{k\to\infty}\lim_{n\to\infty}\int_{E\times
F}(g_{k}^{n})^{2}(x,t)\,dm(x,t)$ $\displaystyle\leq
6c_{1}C_{Y}^{3}\limsup_{k\to\infty}\int_{E\times F}(g_{k})^{2}(x,t)\,dm(x,t)$
$\displaystyle\leq 6c_{1}C_{Y}^{3}\int_{E\times F}|\partial f/\partial
t|^{2}\,dm(x,t)$

By choosing $C_{0}=\max\\{C_{Y}^{6},12c_{1}C_{Y}^{3}\\}$ the desired result
holds. ∎

######  Lemma 3.9.

For any $f\in{\mathsf{BL}}(X,Y)$ there exists a sequence
$(f_{k})\subset\tilde{\mathcal{A}}\cap{\mathsf{BL}}(X,Y)$ with $f_{k}\to f$ in
$L^{2}(X\times Y)$ and such that
$|Df_{k}|_{{\mathsf{BL}}}\to|Df|_{{\mathsf{BL}}}$ in $L^{2}(X\times Y)$ as
$n\to\infty$.

###### Proof.

With a standard truncation and diagonal argument, we can assume that the given
$f$ is bounded with bounded support. Given $f\in{\mathsf{BL}}(X,Y)$, we define
a sequence of function $f_{k}$’s as in the previous lemma, that is

$f_{k}(x,t)=\sum_{i\in I_{k}}\chi_{i,k}(t)f_{k,i}(x)$

Observe that $f_{k}\in\tilde{\mathcal{A}}\cap{\mathsf{BL}}(X,Y)$ and that
(3.9) says that $f_{k}\to f$ in $L^{2}(X\times Y)$.

It remains to show $|Df_{k}|_{{\mathsf{BL}}}\to|Df|_{{\mathsf{BL}}}$. To show
this notice that it suffices to show the following two inequalities for every
$k\in\mathbb{N}$,

(3.22) $\int_{X\times
Y}|{\partial}f_{k}/{\partial}x|^{2}(x,t)\,dm(x,t)\leq\int_{X\times
Y}|{{\partial}f}/{{\partial}x}|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x).$ (3.23)
$\int_{X\times
Y}|{\partial}f_{k}/{\partial}t|^{2}(x,t)\,dm(x,t)\leq\int_{X\times
Y}|{{\partial}f}/{{\partial}t}|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x);$

To prove (3.22), we first notice that by convexity and lower semi-continuity,
we have

$\displaystyle\int_{X}|{\partial}f_{i,k}/{\partial}x|^{2}(x)dm_{X}(x)\leq\int_{X}\fint_{Q_{i,k}}|{\partial}f/{\partial}x|^{2}(x,s)dm_{Y}(s)dm_{X}(x)$

Also, we have

$\displaystyle|{\partial}f_{k}/{\partial}x|^{2}(x,t)$
$\displaystyle=\left|\frac{{\partial}}{{\partial}x}\left(\sum_{i=0}^{\infty}\chi_{i,k}(t)f_{i,k}(x)\right)\right|^{2}$
$\displaystyle\leq\sum_{i=0}^{\infty}\chi_{i,k}(t)|{\partial}f_{i,k}/{\partial}x|^{2}(x)$

Thus, we have

$\displaystyle\int_{X\times Y}|{\partial}f_{k}/{\partial}x|^{2}(x,t)\,dm(x,t)$
$\displaystyle=\int_{X}\int_{Y}|{\partial}f_{k}/{\partial}x|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x)$
$\displaystyle\leq\int_{X}\int_{Y}\sum_{i\in
I_{k}}\chi_{i,k}(t)|{\partial}f_{i,k}/{\partial}x|^{2}(x)\,dm_{Y}(t)\,dm_{X}(x)$
$\displaystyle\leq\int_{X}\sum_{i\in
I_{k}}|Q_{i,k}||{\partial}f_{i,k}/{\partial}x|^{2}(x)\,dm_{X}(x)$
$\displaystyle\leq\int_{X}\sum_{i\in
I_{k}}|Q_{i,k}|\frac{1}{|Q_{i,k}|}\int_{Q_{i,k}}|{\partial}f/{\partial}x|^{2}(x,s)\,dm_{Y}(s)\,dm_{X}(x)$
$\displaystyle\leq C\int_{X\times
Y}|{{\partial}f}/{{\partial}x}|^{2}(x,t)\,dm_{Y}(t)\,dm_{X}(x)\,.$

It remains to show (3.23). Set $B=B(t_{j,k},\frac{6}{k})$, and note that if
$Q_{i,k}$ and $Q_{j,k}$ are neighbors, then they are both contained in $B$. We
have

(3.24)
$\displaystyle\begin{split}&\left|\fint_{Q_{i,k}}f(x,t)\,dm_{Y}(t)-\fint_{Q_{j,k}}f(x,t)\,dm_{Y}(t)\right|^{2}\\\
&\leq\left(\left|\fint_{Q_{i,k}}f(x,t)\,dm_{Y}(t)-\fint_{B}f(x,t)\,dm_{Y}(t)\right|^{2}+\left|\fint_{B}f(x,t)\,dm_{Y}(t)-\fint_{Q_{j,k}}f(x,t)\,dm_{Y}(t)\right|^{2}\right)\\\
&\leq\left(\frac{1}{|Q_{i,k}|}\int_{Q_{i,k}}\left|f(x,t)-f_{B}\right|^{2}\,dm_{Y}(t)+\frac{1}{|Q_{j,k}|}\int_{Q_{j,k}}\left|f(x,t)-f_{B}\right|^{2}\,dm_{Y}(t)\right)\\\
&\leq\left(\frac{1}{|Q_{i,k}|}\int_{B}\left|f(x,t)-f_{B}\right|^{2}\,dm_{Y}(t)+\frac{1}{|Q_{j,k}|}\int_{B}\left|f(x,t)-f_{B}\right|^{2}\,dm_{Y}(t)\right)\\\
&\leq 2C_{Y}^{5}\fint_{B}\left|f(x,t)-f_{B}\right|^{2}\,dm_{Y}(t)\end{split}$

Hence, using Poincaré inequality, we obtain

(3.25) $\displaystyle\begin{split}\int_{X\times
Y}&|{\partial}f_{k}/{\partial}t|^{2}(x,t)\,dm(x,t)\leq
C_{Y}^{3}\int_{X}\sum_{j}\int_{Q_{j,k}}k^{2}\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}|f_{k,j}(x)-f_{k,i}(x)|^{2}\,dm_{Y}(t)\,dm_{X}(x)\\\
&=C_{Y}^{3}\int_{X}\sum_{j}\int_{Q_{j,k}}\sum_{\begin{subarray}{c}i\neq j\\\
Q_{i,k}\sim
Q_{j,k}\end{subarray}}k^{2}\left|\fint_{Q_{i,k}}f(x,s)\,dm_{Y}(s)-\fint_{Q_{j,k}}f(x,s)\,dm_{Y}(s)\right|^{2}\,dm_{Y}(t)\,dm_{X}(x)\\\
&\leq
2C_{Y}^{5}C_{Y}^{3}\int_{X}\sum_{j}\int_{Q_{j,k}}\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}k^{2}\fint_{B}\left|f(x,s)-f_{B}\right|^{2}\,dm_{Y}(s)\,dm_{Y}(t)\,dm_{X}(x)\\\
&\leq
2C_{Y}^{8}C_{P}\int_{X}\sum_{j}\int_{Q_{j,k}}\sum_{\begin{subarray}{c}i\neq
j\\\ Q_{i,k}\sim
Q_{j,k}\end{subarray}}k^{2}\operatorname{rad}(B)^{2}\fint_{\lambda B}|\partial
f/\partial s|^{2}\,dm_{Y}(s)\,dm_{Y}(t)\,dm_{X}(x)\\\ &\leq 2\cdot
6^{2}C_{Y}^{8}C_{P}C_{Y}^{3}\int_{X}\sum_{j}\int_{Q_{j,k}}\fint_{\lambda
B}|\partial f/\partial s|^{2}\,dm_{Y}(s)\,dm_{Y}(t)\,dm_{X}(x)\\\ &\leq 2\cdot
6^{2}C_{Y}^{11}C_{P}\int_{X}\sum_{j}\fint_{\lambda B}\int_{Q_{j,k}}|\partial
f/\partial t|^{2}\,dm_{Y}(t)\,dm_{Y}(s)\,dm_{X}(x)\\\ &\leq 2\cdot
6^{2}C_{Y}^{11}C_{P}\int_{X\times Y}|\partial f/\partial
t|^{2}\,dm(x,t),\end{split}$

which concludes the proof. ∎

We are now ready to prove the main theorem.

###### Proof of Theorem 3.2.

We already observed that $W^{1,2}(X\times Y)\subset{\mathsf{BL}}(X\times Y)$.
Let $f\in{\mathsf{BL}}(X\times Y)$ and let
$\\{f_{k}\\}\subset{\mathsf{BL}}(X\times Y)\cap\tilde{\mathcal{A}}$ be as in
Lemma 3.9. Lemma 3.8 says that

$|Df_{k}|_{X\times Y}=|Df_{k}|_{{\mathsf{BL}}}.$

The right hand side converges to $|Df|_{{\mathsf{BL}}}$ in $L^{2}(X\times Y)$
and because $f_{k}\to f$ in $L^{2}$, the lower semicontinuity of weak upper
gradients implies that if $f\in W^{1,2}(X\times Y)$ and

$|Df|_{X\times Y}\leq|Df|_{{\mathsf{BL}}},$

which together with Lemma 3.3 concludes the proof. ∎

##  4\. Warped products

For this section, we still have Assumptions 2.12 and 2.13.

######  Definition 4.1.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be length spaces, and $w_{d}\colon
Y\to[0,\infty)$ a continuous function. Let $\gamma=(\gamma^{X},\gamma^{Y})$ be
a curve such that $\gamma^{X}$ and $\gamma^{Y}$ are absolutely continuous.
Then the $w_{d}$-length of $\gamma$ is defined as

$\ell_{w}(\gamma)=\lim_{\tau}\sum_{i=1}^{n}\sqrt{d_{Y}^{2}(\gamma^{Y}_{t_{i-1}},\gamma^{Y}_{t_{i}})+w_{d}^{2}(\gamma^{Y}_{t_{i-1}}d_{X}^{2}(\gamma^{X}_{t_{i-1}},\gamma^{X}_{t_{i}})},$

where $\tau$ is a partition of $[0,1]$ and the limit is taken over refinement
ordering of partitions.

The limit exists and

$\ell_{w}(\gamma)=\int_{0}^{1}\sqrt{|\dot{\gamma}^{Y}_{t}|^{2}+w_{d}^{2}(\gamma_{t}^{Y})|\dot{\gamma}^{X}_{t}|^{2}}\,dt$

######  Definition 4.2.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be length spaces, and $w_{d}\colon
Y\to[0,\infty)$ a continuous function. We define a pseudo-metric
$\operatorname{d}_{w}$ on on the space $X\times Y$ by

$\operatorname{d}_{w}(p,q)=\inf\\{\ell_{w}(\gamma)\mid\text{$\gamma^{X}\in
AC([0,1],X)$, $\gamma^{Y}\in AC([0,1],Y)$, and
$\gamma_{0}=p,\gamma_{1}=q$}\\},$

for any $p,q\in X\times Y$.

The pseudo-metric induces an equivalent relation on $X\times Y$ and hence a
metric on the quotient. We denote (slight abuse of notation) the completion of
such quotient by $(X\times_{w}Y,d_{w})$. If both $X$ and $Y$ are separable, so
is $X\times_{w}Y$. Let $\pi\colon X\times Y\to X\times_{w}Y$ be the quotient
map,

######  Definition 4.3.

Let $(X,d_{X},m_{X})$ and $(Y,d_{Y},m_{Y})$ be complete separable and length
metric spaces equipped with non-negative Radon measures. Assume that
$m_{X}(X)<\infty$ and let $w_{d},w_{m}\colon Y\to[0,\infty)$ be continuous
functions. Then the warped product $(X\times_{w}Y,d_{w})$ is defined as above
and the Radon measure $m_{w}$ is defined as

$m_{w}=\pi_{*}((w_{m}m_{Y})\times m_{X}).$

Note that the assumption that $m_{X}$ is a finite measure is needed to ensure
that $m_{w}$ is Radon (it is always Borel).

######  Definition 4.4.

As a set, the Beppo Levi space ${\mathsf{BL}}(X\times_{w}Y)$ is the subset of
$L^{2}(X\times_{w}Y,m_{w})$ of all functions $f$ such that

  * •

for $m_{X}$-a.e. $x\in X$, we have $f^{(x)}=f(x,\cdot)\in
W^{1,2}(Y,w_{m}m_{Y})$;

  * •

for $w_{m}m_{Y}$-a.e. $t\in Y$, we have $f^{(t)}=f(\cdot,t)\in W^{1,2}(X)$;

  * •

the function

$|Df|_{{\mathsf{BL}}_{w}}=\sqrt{w_{d}^{-2}|Df^{(t)}|_{X}^{2}(x)+|Df^{(x)}|_{Y}^{2}(t)}$

belongs to $L^{2}(X\times_{w}Y,m_{w})$.

On ${\mathsf{BL}}(X\times_{w}Y)$ we put the norm

$\|f\|_{{\mathsf{BL}}(X\times_{w}Y)}=\sqrt{\|f\|_{L^{2}}^{2}+\||Df|_{{\mathsf{BL}}_{w}}\|_{L^{2}}^{2}}.$

To handle the warped case we need to introduce an auxiliary space:

######  Definition 4.5.

Let $\mathcal{V}\subset{\mathsf{BL}}_{w}(X,Y)$ be the space of functions $f$
which are identically $0$ on $X\times\Omega$, where $\Omega$ is an open set
that satisfies $\\{w_{m}=0\\}\subset\Omega\subset Y$.
${\mathsf{BL}}_{0,w}(X,Y)\subset{\mathsf{BL}}_{w}(X,Y)$ is defined as the
closure of $\mathcal{V}$ in ${\mathsf{BL}}_{w}(X,Y)$.

We want to compare the Beppo-Levi space with the Sobolev space on the warped
product, and the respective notions of minimal upper gradients. If we merely
assume that $w_{m}$ and $w_{d}$ are continuous and that
$\\{w_{d}=0\\}\subset\\{w_{m}=0\\}$ (the same assumptions in GH (18) for the
analogous case) we want to show that

${\mathsf{BL}}_{w,0}(X,Y)\subset
W^{1,2}(X\times_{w}Y)\subset{\mathsf{BL}}_{w}(X,Y),$

and that for every $f\in W^{1,2}(X\times_{w}Y)\subset{\mathsf{BL}}_{w}(X,Y)$
the inequalities

$|Df|_{{\mathsf{BL}}_{w}}\leq|Df|_{X\times_{w}Y}\leq
C_{0}|Df|_{{\mathsf{BL}}_{w}}$

hold $m_{w}$-a.e., and $C_{0}>0$ is as in Lemma 3.2.

######  Proposition 4.6.

We have $W^{1,2}(X\times_{w}Y)\subset{\mathsf{BL}}_{w}(X,Y)$.

###### Proof.

Let $f\in W^{1,2}(X\times_{w}Y)$. Then by density in energy of Lipschitz
functions, we can find a sequence $f_{n}$ of Lipschitz functions on
$X\times_{w}Y$ such that $f_{n}\rightarrow f$ and
$\operatorname{lip}(f_{n})\rightarrow|Df|_{X\times_{w}Y}$ in
$L^{2}(X\times_{w}Y)$. Passing to a subsequence, which we do not relabel, we
can assume that $\|f_{n}-f\|_{L^{2}(X\times_{w}Y,m_{w})}<n^{-4}$. Then

(4.1)
$\displaystyle\begin{split}&\|\sum_{n=1}^{\infty}\|f_{n}^{(t)}-f^{(t)}\|_{L^{2}(X,m_{X})}\|_{L^{2}(Y,w_{m}m_{Y})}\\\
&=\left(\int_{Y}\;\left(\sum_{n=1}^{\infty}\frac{n}{n}\sqrt{\int_{X}|f_{n}^{(t)}(x)-f^{(t)}(x)|^{2}dm_{X}(x)}\right)^{2}\,w_{m}(t)dm_{Y}(t)\right)^{1/2}\\\
&\leq\left(\sum_{n=1}^{\infty}\frac{1}{n^{2}}\right)\left(\int_{Y}\;\int_{X}\sum_{n=1}^{\infty}n^{2}|f_{n}^{(t)}(x)-f^{(t)}(x)|^{2}dm_{X}(x)\,w_{m}(t)dm_{Y}(t)\right)^{1/2}\\\
&=C\|\sum_{n=1}^{\infty}n^{2}|f_{n}-f|\|_{L^{2}(X\times_{w}Y,m_{w})}\\\ &\leq
C\sum_{n=1}^{\infty}n^{2}\||f_{n}-f|\|_{L^{2}(X\times_{w}Y,m_{w})}<\infty\end{split}$

This shows that for $w_{m}m_{Y}$-a.e. $t\in Y$ we have
$\sum_{n=1}^{\infty}\|f_{n}^{(t)}-f^{(t)}\|_{L^{2}(X,m_{X})}<\infty$ and so in
particular $f_{n}^{(t)}\rightarrow f^{(t)}$ in $L^{2}(X,m_{X})$. Similarly for
$m_{x}$-a.e. $x\in X$, we have $f_{n}^{(x)}\rightarrow f^{(x)}$ in
$L^{2}(Y,w_{m}m_{Y})$.

Now observe that for $(x,t)\in X\times_{w}Y$ we have:

(4.2)
$\displaystyle\begin{split}\operatorname{lip}(f_{n})(x,t)&=\limsup_{(y,s)\rightarrow(x,t)}\frac{|f_{n}(y,s)-f_{n}(x,t)|}{d_{w}((y,s),(x,t))}\\\
&\geq\limsup_{s\rightarrow
t}\frac{|f_{n}(x,s)-f_{n}(x,t)|}{d_{w}((x,s),(x,t))}\\\
&=\limsup_{s\rightarrow
t}\frac{|f_{n}^{(x)}(s)-f_{n}^{(x)}(t)|}{d_{Y}(s,t)}=\operatorname{lip}_{Y}(f_{n}^{(x)})(t)\end{split}$

Then, by Fatou’s lemma:

(4.3)
$\displaystyle\begin{split}&\int_{X}\liminf_{n\rightarrow\infty}\int_{Y}\,\operatorname{lip}_{Y}(f_{n}^{(x)})^{2}(t)w_{m}(t)dm_{Y}(t)dm_{X}(x)\\\
&\leq\liminf_{n\rightarrow\infty}\int_{X\times_{w}Y}\operatorname{lip}_{Y}(f_{n}^{(x)})^{2}(t)dm_{w}(x,t)\\\
&\leq\liminf_{n\rightarrow\infty}\int_{X\times_{w}Y}\operatorname{lip}(f_{n})^{2}(x,t)dm_{w}(x,t)\\\
&=\int_{X\times_{w}Y}|Df|^{2}_{X\times_{w}Y}dm_{w}(x,t)<\infty\end{split}$

Since $f_{n}^{(x)}\rightarrow f^{(x)}$ in $L^{2}(Y,w_{m}m_{Y})$ for
$m_{x}$-a.e. $x\in X$, the last inequality together with the lower
semicontinuity of minimal weak upper gradients gives that $f^{(x)}\in
W^{1,2}(Y,w_{m}m_{Y})$ for $m_{x}$-a.e. $x\in X$ and

(4.4)
$\int_{X_{w}}|Df^{(x)}|_{Y}^{2}(t)dm_{w}(x,t)\;\leq\int_{X_{w}}|Df|^{2}_{X\times_{w}Y}dm_{w}(x,t)$

With an analogous argument we can get conditions on $f^{(t)}$: Starting from
the bound:

(4.5)
$\displaystyle\begin{split}\operatorname{lip}(f_{n})(x,t)&=\limsup_{(y,s)\rightarrow(x,t)}\frac{|f_{n}(y,s)-f_{n}(x,t)|}{d_{w}((y,s),(x,t))}\\\
&\geq\limsup_{y\rightarrow
x}\frac{|f_{n}(y,t)-f_{n}(x,t)|}{d_{w}((y,t),(x,t))}\\\
&=\limsup_{y\rightarrow
x}\frac{|f_{n}^{(t)}(y)-f_{n}^{(t)}(x)|}{w_{d}(t)d_{X}(x,y)}=\frac{1}{w_{d}(t)}\operatorname{lip}_{X}(f_{n}^{(t)})(x)\end{split}$

This inequality, valid for every $t\in Y$ such that $w_{d}(t)>0$, grants that
$f^{(t)}\in W^{1,2}(X)$ for $w_{m}m_{Y}$-a.e. $t\in Y$ (here we are using the
assumption that $\\{w_{d}=0\\}\subset\\{w_{m}=0\\}$) and that:

(4.6)
$\int_{X\times_{w}Y}\frac{|Df^{(t)}|_{X}^{2}(x)}{w_{d}^{2}(t)}dm_{w}(x,t)\;\leq\int_{X\times_{w}Y}|Df|^{2}_{x_{w}}dm_{w}(x,t)$

The bounds 4.4 and 4.6 ensure that $f\in BL(X_{w})$ so that the desired
inclusion is proved. ∎

######  Lemma 4.7 (Lemma 3.11 in GH (18)).

Let $X$ be a set, $d_{1},\,d_{2}$ two distances on $X$ and $m_{1},\,m_{2}$ two
measures. Assume also that $(X,d_{1},m_{1})$ and $(X,d_{2},m_{2})$ are metric
spaces that satisfy Assumption 2.12 and that for some $C>0$ we have $m_{2}\leq
Cm_{1}$ and that for some $L>0$ we have $d_{1}\leq Ld_{2}$. Then, denoting by
$S(X_{1})$ and $S(X_{2})$ the Sobolev classes relative to $(X,d_{1},m_{1})$
and $(X,d_{2},m_{2})$ respectively and by $|Df|_{1}$ and $|Df|_{2}$ the
associated minimal weak upper gradients, we have $S(X_{1})\subset S(X_{2})$
and for every $f\in S(X_{1})$ the inequality $|Df|_{2}\leq L|Df|_{1}$ holds
$m_{2}$-a.e.

######  Proposition 4.8.

Let $f\in W^{1,2}(X\times_{w}Y)\subset{\mathsf{BL}}_{w}(X,Y)$. Then
$|Df|_{{\mathsf{BL}}_{w}}\leq|Df|_{X\times_{w}Y}\leq
C_{0}|Df|_{{\mathsf{BL}}_{w}}$ $m_{w}$-a.e., where $C_{0}$ is as in Lemma 3.2.

###### Proof.

Fix $\varepsilon>0$. Let $t_{0}$ be such that $w_{m}(t_{0})>0$, and hence
$w_{d}(t_{0})>0$. By continuity we can find $\delta>0$ such that

(4.7) $\left|\frac{w_{d}(t)}{w_{d}(s)}\right|\leq 1+\varepsilon\qquad\text{for
all }t,s\in\overline{B(t_{0},3\delta)}.$

Let $\chi\colon Y\to[0,1]$ be a Lipschitz function such that $\chi\equiv 1$ on
$\overline{B(t_{0},\delta)}$, and $\chi\equiv 0$ outside of
$\overline{B(t_{0},3\delta)}$. Define two continuous functions (in order to
have the product being warped only around $t_{0}$) as follows:

(4.8) $\overline{w_{d}}(t)=\begin{cases}w_{d}(t)&\text{if
}\operatorname{d}_{Y}(t,t_{0})\leq 2\delta,\\\
\frac{\operatorname{d}_{Y}(t,t_{0})-2\delta}{\delta}w_{d}(t_{0})+\frac{3\delta-\operatorname{d}_{Y}(t,t_{0})}{\delta}w_{d}(t)&\text{if
}2\delta<\operatorname{d}_{Y}(t,t_{0})<3\delta,\\\ w_{d}(t_{0})&\text{if
}\operatorname{d}_{Y}(t,t_{0})\geq 3\delta,\end{cases}$ (4.9)
$\overline{w_{m}}(t)=\begin{cases}w_{m}(t)&\text{if
}\operatorname{d}_{Y}(t,t_{0})\leq 2\delta,\\\
\frac{\operatorname{d}_{Y}(t,t_{0})-2\delta}{\delta}w_{m}(t_{0})+\frac{3\delta-\operatorname{d}_{Y}(t,t_{0})}{\delta}w_{m}(t)&\text{if
}2\delta<\operatorname{d}_{Y}(t,t_{0})<3\delta,\\\ w_{m}(t_{0})&\text{if
}\operatorname{d}_{Y}(t,t_{0})\geq 3\delta,\end{cases}$

and let
$(X\times_{\overline{w}}Y,\operatorname{d}_{\overline{w}},m_{\overline{w}})$
be the corresponding product space. Consider the function $\overline{f}\colon
X\times_{w}Y\to\mathbb{R}$ defined by $\overline{f}(x,t)=\chi(t)f(t,x)$.
Clearly $\overline{f}\in W^{1,2}(X\times_{w}Y)$, and hence to
${\mathsf{BL}}_{w}(X,Y)$. Since minimal upper gradients are local we know that

$|Df|_{X\times_{w}Y}=|D\overline{f}|_{X\times_{w}Y}\quad\text{and}\quad|Df|_{{\mathsf{BL}}_{w}}=|D\overline{f}|_{{\mathsf{BL}}_{w}}\quad\text{
$m_{w}$-a.e. on $X\times\overline{B(t_{0},\delta)}$}.$

Because $\overline{f}$ is supported on $B(t_{0},3\delta)\times X$, where
$w_{d}$ is positive we can think of $\overline{f}$ as a function on
$X\times_{\overline{w}}Y$. With this identification we have

$|D\overline{f}|_{X\times_{w}Y}=|D\overline{f}|_{X\times_{\overline{w}}Y}\quad\text{and}\quad|D\overline{f}|_{{\mathsf{BL}}_{w}}=|D\overline{f}|_{{\mathsf{BL}}_{\overline{w}}}\quad\text{
$m_{w}$-a.e. on $X\times\overline{B(t_{0},2\delta)}$}.$

We now want to use this “localized warped space” to utilize the results for
the Cartesian product we have obtained in the previous section.

Consider first the space
$\overline{X}:=(X,w_{d}(t_{0})\operatorname{d}_{X},w_{m}(t_{0})m_{X})$. This
is simply our original metric measure space $(X,\operatorname{d}_{X},m_{X})$
which has been re-scaled. Now consider $(\overline{X}\times
Y,\operatorname{d},m)$ where with a slight abuse of notation we denote by
$\operatorname{d}$ and $m$ are the appropriate Cartesian metric and measure,
respectively. Set $c=\min_{\overline{B(t_{0},3\delta)}}w_{m}$ and
$C=\max_{\overline{B(t_{0},3\delta)}}w_{m}$. From the definition of $m_{w}$ we
immediately have

$cm\leq m_{\overline{w}}\leq Cm,$

and, recalling (4.7), we have

(4.10)
$(1+\varepsilon)^{-1}\operatorname{d}\leq\operatorname{d}_{\overline{w}}\leq(1+\varepsilon)\operatorname{d}.$

Let us denote by $\overline{\mathsf{BL}}:={\mathsf{BL}}(\overline{X},Y)$.
Using Lemma 4.7, recalling that $X\times_{\overline{w}}Y$ and
$\overline{X}\times Y$ coincide as sets, we obtain

$(1+\varepsilon)^{-1}|D\overline{f}|_{\overline{X}\times
Y}\leq|D\overline{f}|_{X\times_{\overline{w}}Y}\leq(1+\varepsilon)|D\overline{f}|_{\overline{X}\times
Y}$

and

$(1+\varepsilon)^{-1}|D\overline{f}|_{\overline{\mathsf{BL}}}\leq|D\overline{f}|_{{\mathsf{BL}}_{\overline{w}}}\leq(1+\varepsilon)|D\overline{f}|_{\overline{\mathsf{BL}}}.$

We now exploit the Cartesian results applied to the pair $(\overline{X},Y)$:
by Lemma 3.2, we know that

$|D\overline{f}|_{\overline{\mathsf{BL}}}\leq|D\overline{f}|_{\overline{X}\times
Y}\leq C_{0}|D\overline{f}|_{\overline{\mathsf{BL}}}$

$m$-a.e.. Putting everything together we obtain

$(1+\varepsilon)^{-2}|Df|_{{\mathsf{BL}}_{\overline{w}}}\leq|Df|_{X\times_{\overline{w}}Y}\leq(1+\varepsilon)^{2}C_{0}|Df|_{{\mathsf{BL}}_{\overline{w}}}$

$m_{w}$-a.e. on $X\times B(t_{0},\delta)$. Because $t_{0}$ was arbitrary, and
because every cover of $\\{w_{m}>0\\}\subset Y$ has a countable subcover, the
conclusion holds by letting $\varepsilon\to 0$. ∎

######  Proposition 4.9.

We have ${\mathsf{BL}}_{0,w}(X,Y)\subset W^{1,2}(X\times_{w}Y)$.

###### Proof.

Thanks to Proposition 4.8, it is sufficient to prove that $\mathcal{V}\subset
W^{1,2}(X\times_{x}Y)$, where $\mathcal{V}$ is defined as in Definition 4.5.
Fix $k\in{\mathbb{N}}$, let $t_{0}\in Y$, and let

(4.11) $\psi_{n,k}(t)=\sum_{\begin{subarray}{c}j\\\
\operatorname{supp}\chi_{j,k}\cap
B(t_{0},n)\neq\varnothing\end{subarray}}\chi_{j,k}(t),$

where the $\chi_{i,k}$’s are the partition of unity subordinate to the
covering by “cubes” we defined in the proof of Lemma 3.8. Observe that
$\psi_{k,n}\equiv 1$ inside $B(t_{0},n-1)$. With a slight abuse of notation we
will denote $\psi_{n,k}$ simply as $\psi_{n}$, as $k$ will remained fixed
throughout the proof. For $f\in{\mathsf{BL}}_{w}(X,Y)$ we define
$f_{n}(x,t):=\psi_{n}(t)f(x,t)$, and note that by definition
$f_{n}\in{\mathsf{BL}}_{w}(X,Y)$, while by the dominated convergence theorem
and inequality (2.4) we obtain $f_{n}\rightarrow f$ in
${\mathsf{BL}}_{w}(X,Y)$.

The idea of the proof is as follows: we show that any $f\in\mathcal{V}$ with
support contained in $Y\cap B(t_{1},R)$ for $R>0$ and $t_{1}\in Y$ belongs to
$W^{1,2}(X\times_{w}Y)$. This together with Proposition 4.8, which ensures
${\mathsf{BL}}$-convergence implies $W^{1,2}$-convergence, completes the
proof.

Accordingly, fix such $f\in\mathcal{V}$ and for $r\in(0,1)$ let
$\Omega_{r}\subset Y$ be the r-neighborhood of $\\{w_{m}=0\\}$. Now, find
$r\in(0,1)$ such that $f$ is $m_{w}$-a.e. zero on $X\times\Omega_{2r}$. Then,
recalling that $\\{w_{d}=0\\}\subset\\{w_{m}=0\\}$ and by continuity and
compactness we have that there exist constants $0<c<C<\infty$ such that

$c<w_{d}(t),\;w_{m}(t)<C,\quad\forall t\in Y\cap
B(t_{1},R)\setminus\Omega_{\frac{r}{2}}.$

We now use a comparison argument similar to the one used in Proposition 4.8.
Let $w^{\prime}_{d}$ and $w^{\prime}_{m}$ two continuous functions which agree
with $w_{d}$ and $w_{m}$ on $B(t_{1},R)\setminus\Omega_{\frac{r}{2}}$ and such
that $c<w^{\prime}_{d}(t),\;w^{\prime}_{m}(t)<C$ on the whole $Y$. Consider
now the warped product $(X\times_{w^{\prime}}Y,d_{w^{\prime}},m_{w^{\prime}})$
and Cartesian product of $(X\times Y,d,m)$ of $X$ and $Y$. Then by Lemma 4.7
and the properties of $w^{\prime}_{d},\;w^{\prime}_{m}$ we have the following
equalities of sets:

(4.12) ${\mathsf{BL}}_{w^{\prime}}(X,Y)={\mathsf{BL}}(X,Y)\quad\text{and}\quad
W^{1,2}(X\times_{w^{\prime}}Y)=W^{1,2}(X\times Y).$

Moreover by Theorem 3.2 the following equality of sets also holds:

(4.13) ${\mathsf{BL}}(X,Y)=W^{1,2}(X\times Y).$

Finally, putting together (4.12) and (4.13) we obtain:

(4.14) ${\mathsf{BL}}_{w^{\prime}}(X,Y)={\mathsf{BL}}(X,Y)=W^{1,2}(X\times
Y)=W^{1,2}(X\times_{w^{\prime}}Y).$

By construction of $w^{\prime}_{d},\;w^{\prime}_{m}$ we have that
$f\in{\mathsf{BL}}_{w^{\prime}}(X,Y)$ so that, by equation (4.14), $f\in
W^{1,2}(X\times_{w^{\prime}}Y)$. By density in energy of Lipschitz functions
(Lemma 2.11) there exists a sequence of $d_{w^{\prime}}$-Lipschitz functions
$f_{n}$ that converges to $f$ in $L^{2}(X\times_{w^{\prime}}Y)$ and, denoting
by $\operatorname{lip}^{\prime}$ the local Lipschitz constant computed w.r.t.
the distance $d_{w^{\prime}}$, we also have

$\sup_{n\in\mathbb{N}}\int\operatorname{lip}^{\prime}(f_{n})^{2}\,dm_{w^{\prime}}<\infty$

uniformly bounded in $n$. From now on we assume $f_{n}$ is bounded for every
$n\in\mathbb{N}$. This is possible up to replacing the original $f_{n}$ with
$\min(\max(f_{n},-C_{n}),C_{n})$ for sufficiently large $C_{n}$.

Now similarly to what we did at the beginning of the proof find a Lipschitz
function $\psi\colon Y\rightarrow[0,1]$ which is identically 0 on
$\Omega_{r}\cup(Y\setminus B(t_{1},R+1))$ and identically 1 on $Y\cap
B(t_{1},R)\setminus\Omega_{2r}$ and set
$\tilde{f}_{n}(t,x):=\psi(t)f_{n}(t,x)$. By construction the $\tilde{f}_{n}$’s
are still $d_{w^{\prime}}$-Lipschitz, they converge to $f\in
L^{2}(X\times_{w^{\prime}}Y)$ and satisfy

(4.15)
$\sup_{n\in\mathbb{N}}\int\operatorname{lip}^{\prime}(\tilde{f}_{n})^{2}\,dm_{w^{\prime}}<\infty.$

To conclude the proof we need to show that the $\tilde{f}_{n}$’s are also
$d_{w}$-Lipschitz, converge to $f\in L^{2}(X\times_{w}Y)$ and satisfy

(4.16)
$\sup_{n\in\mathbb{N}}\int\operatorname{lip}(\tilde{f}_{n})^{2}\,dm_{w}<\infty.$

Once this is proved, the conclusion follows from the lower semicontinuity of
weak upper gradients and the bound
$|D\tilde{f}_{n}|_{X\times_{w}Y}\leq\operatorname{lip}(\tilde{f}_{n})$ valid
$m_{w}$-a.e..

First, $\tilde{f}_{n}$ converges to $f\in L^{2}(X\times_{w}Y)$ since the
functions $\tilde{f}_{n}$ and $f$ are concentrated on
$X\times(B(t_{1},R)\setminus\Omega_{r})$ and on this set the measures $m_{w}$
and $m_{w^{\prime}}$ agree by definition. Moreover, since $w_{d}$ and
$w_{d^{\prime}}$ agree on $X\times(B(t_{1},R)\setminus\Omega_{r})$, by
definition we also have

$\lim_{(y,s)\rightarrow(x,t)}\frac{d_{w}((y,s),(x,t))}{d_{w^{\prime}}((y,s),(x,t))}=1\quad\forall\;(x,t)\in
X\times(B(t_{1},R)\setminus\Omega_{r})$

so that, in particular,
$\operatorname{lip}(\tilde{f}_{n})=\operatorname{lip}^{\prime}(\tilde{f}_{n})$.
This together with (4.15) proves (4.16).

It remains to prove that $\tilde{f}_{n}$ are $d_{w}$-Lipschitz, that is, we
need to prove that $\operatorname{Lip}(\tilde{f}_{n})<\infty$. Recall that,
since we are on a length space, the Lipschitz constant of a function is equal
to the supremum of the local Lispchitz constants. Then, by denoting
$\operatorname{Lip}^{\prime}(\tilde{f}_{n})$ the $d_{w^{\prime}}$-Lipschitz
constant and recalling that by construction of $\tilde{f}_{n}$,
$\operatorname{Lip}^{\prime}(\tilde{f}_{n})<\infty$, we have:

$\operatorname{Lip}(\tilde{f}_{n})=\sup_{X\times_{w}Y}\operatorname{lip}(\tilde{f}_{n})=\sup_{X\times_{w^{\prime}}Y}\operatorname{lip}^{\prime}(\tilde{f}_{n})=\operatorname{Lip}^{\prime}(\tilde{f}_{n})<\infty$

which concludes the proof. ∎

By adding a few extra assumptions on the function $w_{m}$ we can improve the
previous result and show that, with these additional assumptions, the
inclusions above are all equalities.

######  Proposition 4.10.

Assume that $w_{m}$ satisfies

(4.17) $\displaystyle\\{w_{m}=0\\}\subset Y\text{ is discrete;}$ (4.18)
$w_{m}$ decays at least linearly near its zeros, i.e.  $\displaystyle
w_{m}(t)\leq C\inf_{s\mid w_{m}(s)=0}\operatorname{d}_{Y}(t,s),\quad\forall
t\in Y.$

Then ${\mathsf{BL}}_{0,w}(X,Y)=W^{1,2}(X\times_{w}Y)={\mathsf{BL}}_{w}(X,Y)$.

###### Proof.

It is easy to see that ${\mathsf{BL}}_{w}(X,Y)\cap L^{\infty}(X\times_{w}Y)$
is dense in ${\mathsf{BL}}_{w}(X,Y)$, using a standard truncation argument, so
that it is enough to show that for any $f\in{\mathsf{BL}}_{w}(X,Y)\cap
L^{\infty}(X\times_{w}Y)$ there is a sequence in $\mathcal{V}$ that converges
to it in ${\mathsf{BL}}_{w}(X,Y)$.

Pick $f\in{\mathsf{BL}}_{w}(X,Y)\cap L^{\infty}(X\times_{w}Y)$ and define
$D(t)=\min_{s\mid w_{m}(s)=0}\operatorname{d}_{Y}(t,s)$. For
$m,n\in{\mathbb{N}}$, with $n>1$, fix $x_{0}\in X$ and $t_{0}\in Y$, and let
$\psi_{n,k}(t)$ be as in (4.11). Moreover define

$\displaystyle\sigma_{m}(x)$
$\displaystyle=\max\\{0,\min\\{m-\operatorname{d}_{X}(x,x_{0}),1\\}\\}$
$\displaystyle\eta_{n}(t)$
$\displaystyle=\max\\{0,\min\\{1-\frac{|\log(D(t))|}{\log n},1\\}\\}.$

Define $f_{n,m,k}(x,t)=\psi_{n,k}(t)\eta_{n}(t)\sigma_{m}(x)f(x,t)$. Because
the product of the three auxiliary functions is Lipschitz and bounded for all
$n,m,k$, $f_{n,m,k}\in{\mathsf{BL}}_{w}(X,Y)$, and because $\eta_{n}$ is $0$
in a neighborhood of $\\{w_{m}=0\\}$, we also have $f_{n,m,k}\in\mathcal{V}$,
for all $n,m,k$.

By construction, the function
$(x,t)\mapsto\psi_{n,k}(t)\eta_{n}(t)\sigma_{m}(x)$ is uniformly bounded by
$1$, and hence by the dominated convergence theorem we have that $f_{n,m,k}\to
f$ in $L^{2}(X\times_{w}Y)$, as $n,m\to\infty$, for every $k$.

Now, recalling (2.4), and because $\sigma_{m}$ is $1$-Lipschitz, for
$m_{w}$-a.e. $(x,t)$

$|\partial/\partial
x(f-f_{n,m,k})(x,t)|\leq|\psi_{n,k}(t)\eta_{n}(t)\sigma_{m}(x)-1||\partial
f/\partial x|(x,t)+|f(x,t)|1_{\\{\operatorname{d}_{X}(\cdot,x_{0})\geq
m-1\\}}(x).$

Applying the dominated convergence theorem again we get that, as
$n,m\to\infty$,

$\int_{X\times Y}|\partial/\partial x(f-f_{n,m,k})(x,t)|^{2}dm_{w}\to 0.$

Using again Lipschitzness and the product rule we have

$\displaystyle|\partial/\partial t(f-f_{n,m,k})(x,t)|$
$\displaystyle\leq|\psi_{n,k}(t)\eta_{n}(t)\sigma_{m}(x)-1||\partial
f/\partial t|(x,t)$
$\displaystyle\qquad+c_{1}C_{Y}^{3}k|f(x,t)|1_{\\{\operatorname{d}_{Y}(\cdot,t_{0})\geq
n-1\\}}(t)$
$\displaystyle\qquad+|f(x,t)|1_{\\{\operatorname{d}_{X}(\cdot,x_{0})\leq
m\\}}(x)1_{\\{\operatorname{d}_{Y}(\cdot,t_{0})\leq
n\\}}(t)|\partial\eta_{n}/\partial t|(t),$

for $m_{w}$-a.e. $(x,t)$. Once again by the dominated convergence theorem, the
first two terms go to $0$ in $L^{2}$ as $n,m\to\infty$. For the last term,
observe first that $|\partial\eta_{n}/\partial
t|(t)\leq\frac{1_{D^{-1}([n^{-1},1])}(t)}{D(t)\log n}$. Now we can use both
the additional assumptions on $w_{m}$: if $y_{1},\dots,y_{N}$ are the finite
number of zeroes of $w_{m}$ inside $\overline{B(t_{0},n-1)}$, and recalling
that $f$ is bounded, we have

$\displaystyle\int_{X\times Y}|f(x,t)|^{2}$ $\displaystyle
1_{\\{\operatorname{d}_{X}(\cdot,x_{0})\leq
m\\}}(x)1_{\\{\operatorname{d}_{Y}(\cdot,t_{0})\leq
n\\}}|\partial\eta_{n}/\partial t|^{2}(t)\,dm_{w}(x,t)$
$\displaystyle\leq\frac{\|f\|^{2}_{L^{\infty}}m_{X}(B(x_{0},m))}{(\log
n)^{2}}\int_{B(t_{0},n)\cap
D^{-1}([n^{-1},1])}\frac{1}{D(t)^{2}}w_{m}(t)\,dm_{Y}(t)$ $\displaystyle\leq
C\frac{\|f\|^{2}_{L^{\infty}}m_{X}(B(x_{0},m))}{(\log
n)^{2}}\int_{B(t_{0},n)\cap D^{-1}([n^{-1},1])}\frac{1}{D(t)}\,dm_{Y}(t)$
$\displaystyle\leq C\frac{\|f\|^{2}_{L^{\infty}}m_{X}(B(x_{0},m))}{(\log
n)^{2}}\sum_{i=1}^{N}\int_{\\{t\mid\operatorname{d}_{Y}(y_{i},t)\in[n^{-1},1]\\}}\frac{1}{\operatorname{d}_{Y}(t,y_{i})}\,dm_{Y}(t)$
$\displaystyle=C\frac{\|f\|^{2}_{L^{\infty}}m_{X}(B(x_{0},m))}{(\log
n)^{2}}\sum_{i=1}^{N}\int_{\frac{1}{n}}^{1}\frac{1}{s}\,ds$
$\displaystyle=NC\frac{\|f\|^{2}_{L^{\infty}}m_{X}(B(x_{0},m))}{\log n}.$

The last term goes to $0$ as $n\to\infty$ for every $m,k\in{\mathbb{N}}$, and
so we have proved the desired result.

∎

## References

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