# Partial derivatives in the nonsmooth setting

Nicola Gigli Chiara Rigoni SISSA, Trieste. Email<EMAIL_ADDRESS>für
Angewandte Mathematik, Universität Bonn. Email<EMAIL_ADDRESS>
###### Abstract

We study partial derivatives on the product of two metric measure structures,
in particular in connection with calculus via modules as proposed by the first
named author in [12].

Our main results are:

  * i)

The extension to this non-smooth framework of Schwarz’s theorem about symmetry
of mixed second derivatives

  * ii)

A quite complete set of results relating the property $f\in W^{2,2}({\sf
X}\times{\sf Y})$ on one side with that of $f(\cdot,y)\in W^{2,2}({\sf X})$
and $f(x,\cdot)\in W^{2,2}({\sf Y})$ for a.e. $y,x$ respectively on the other.
Here ${\sf X},{\sf Y}$ are ${\sf RCD}$ spaces so that second order Sobolev
spaces are well defined.

These results are in turn based upon the study of Sobolev regularity, and of
the underlying notion of differential, for a map with values in a Hilbert
module: we mainly apply this notion to the map $x\mapsto{\mathrm{d}}_{\\_}{\sf
y}f(x,\cdot)$ in order to build, under the appropriate regularity
requirements, its differential ${\mathrm{d}}_{\\_}{\sf
x}{\mathrm{d}}_{\\_}{\sf y}f$.

###### Contents

  1. 1 Introduction
  2. 2 Notation and preliminary results
    1. 2.1 Sobolev spaces for locally integrable objects
    2. 2.2 Calculus tools on product spaces
    3. 2.3 Other differential operators in the product space
  3. 3 Partial first order derivatives on the product space
    1. 3.1 Partial first order derivatives for functions in two variables
    2. 3.2 Sobolev maps with values in a Hilbert module
    3. 3.3 Schwarz’s theorem on symmetry of mixed second derivatives
  4. 4 Second order partial derivatives on the product space
    1. 4.1 Basic material
      1. 4.1.1 Reminders about calculus on ${\sf RCD}$ spaces
      2. 4.1.2 Product of ${\sf RCD}$ spaces
      3. 4.1.3 Setting up the problem
    2. 4.2 Hessian on the product space
      1. 4.2.1 From regularity on the factors to regularity on the product
      2. 4.2.2 From regularity on the product to regularity on the factors
    3. 4.3 Covariant derivative on the product space
      1. 4.3.1 From regularity on the factors to regularity on the product
      2. 4.3.2 From regularity on the product to regularity on the factors

##  1 Introduction

Given two smooth manifolds $M,N$ and points $p\in M$, $q\in N$, it is a
classical fact in differential geometry that

$T_{\\_}{(p,q)}(M\times N)\cong T_{\\_}pM\times T_{\\_}qN$ (1.0.1)

where the isomorphism is given by
$({\mathrm{d}}\pi_{\\_}N,{\mathrm{d}}\pi_{\\_}M)$, with
$\pi_{\\_}N,\pi_{\\_}M$ being the projection onto the respective coordinates
and ${\mathrm{d}}\pi_{\\_}N,{\mathrm{d}}\pi_{\\_}M$ denote their
differentials. This being valid for any $(p,q)\in M\times N$ we can regard any
vector field $v$ on $M\times N$ via its components, i.e. we can write
$v=(v_{\\_}x,v_{\\_}y)$ where $v_{\\_}x$ is a vector field on $M$ parametrized
by $q\in N$ (or, better, a section of the pullback of the tangent bundle of
$M$ via $\pi_{\\_}M$) and symmetrically $v_{\\_}y$ is a vector field on $N$
parametrized by $p\in M$.

In the nonsmooth setting the analogue of (1.0.1) has been investigated in our
earlier work [16]. Here the concept of tangent space is interpreted in terms
of the language of normed modules introduced in [12] and is tightly linked to
Sobolev calculus. In this sense it should not surprise that the validity of
(the analogue of) (1.0.1) on metric measure spaces is related to the validity
of appropriate tensorization properties of Sobolev spaces (this latter topic
has been investigated only relatively recently: the first studies we are aware
of have been conducted in [4], see also [6] and [13] for more recent
contributions). Without entering into details here, let us just say that these
tensorization properties of Sobolev spaces we are alluding to are always
satisfied if the spaces ${\sf X},{\sf Y}$ are ${\sf RCD}$ spaces (see
Proposition 4.1) and that in this case the appropriate analogue of (1.0.1)
holds (see Theorem 2.8). This has been established in our earlier work [16].

In the present manuscript we push the investigation further, in particular in
connection with higher order differentiations (in [16] only first order
derivatives have been considered). Our main results are:

  * i)

A generalization to this setting of the classical theorem by Schwarz about
symmetry of second order derivatives: we shall see in Theorem 3.15 that under
fairly general assumptions the identity

${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f={\mathrm{d}}_{\\_}{\sf
y}{\mathrm{d}}_{\\_}{\sf x}f$ (1.0.2)

holds even in this framework

  * ii)

A thorough study of the relation between second order regularity in the
product of two ${\sf RCD}$ spaces and second order regularity in the factors
which-among other things-shows that the expected formula

${\rm Hess}(f)=\Bigg{(}\begin{array}[]{cc}{\rm Hess}_{\\_}{\sf
x}(f)&{\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f\\\
{\mathrm{d}}_{\\_}{\sf y}{\mathrm{d}}_{\\_}{\sf x}f&{\rm Hess}_{\\_}{\sf
y}(f)\end{array}\Bigg{)}$ (1.0.3)

holds true (see Theorems 4.7, 4.12 and Propositions 4.8, 4.13). The discussion
here is complicated by the fact that on ${\sf RCD}$ spaces there is no single
‘second order Sobolev space’ as it is not clear whether the $`H$’ version
(obtained by completion of the space of ‘smooth’ functions) coincides with the
$`W$’ one (obtained via integration by parts). Along similar lines we also
investigate differentiability properties of vector fields in relation to that
of their components (see Theorems 4.15, 4.18 and Proposition 4.19).

A crucial aspect of our analysis, and perhaps the most important advance to
the theory among those given in the current manuscript, is in the possibility
of speaking of Sobolev regularity for functions with values in a (Hilbert)
module, and in the related concept of differential. Notice indeed that in
order to just state the identities (1.0.2) and (1.0.3), it is crucial to know
what ${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f$ is and given that
for a function $f$ in two variables the object ${\mathrm{d}}_{\\_}{\sf y}f$
can be interpreted as the map sending $x$ to ${\mathrm{d}}_{\\_}{\sf
y}f(x,\cdot)\in L^{0}(T^{*}{\sf Y})$, it is imperative to be able to
differentiate this sort of maps. This analysis is carried out in Section 3.2.

Acknowledgement: The second author gratefully acknowledges support by the
European Union through the ERC-AdG 694405 RicciBounds.

##  2 Notation and preliminary results

For us a metric measure space $({\sf X},\mathsf{d},\mathfrak{m})$ will always
be a complete and separable metric space $({\sf X},\mathsf{d})$ equipped with
a reference non-negative (and non-zero) Borel measure $\mathfrak{m}$ which is
finite on bounded sets.

###  2.1 Sobolev spaces for locally integrable objects

In this section we briefly recall some basic definitions of Sobolev-related
objects, with particular focus on quantities which are only integrable on
bounded sets. For the definition and properties of Sobolev functions and
minimal weak upper gradients we refer to [7] (see also [20] and the more
recent [3, 4] whose presentation we are going to follow) while for what
concerns $L^{\infty}$ and $L^{0}$ normed modules we refer to [12, 16].

Given a metric measure space $({\sf X},\mathsf{d},\mathfrak{m})$, by
$L^{2}_{\\_}\mathsf{loc}({\sf X})$ we mean the space of (equivalence classes
w.r.t. $\mathfrak{m}$-a.e. equality of) Borel functions $f:{\sf
X}\to\mathbb{R}$ such that ${\raise 1.29167pt\hbox{$\chi$}}_{\\_}Bf\in
L^{2}({\sf X})$ for every bounded Borel set $B\subset{\sf X}$. A sequence
$(f_{\\_}n)\subset L^{2}_{\\_}\mathsf{loc}({\sf X})$ converges to $f$ in
$L^{2}_{\\_}\mathsf{loc}({\sf X})$ provided ${\raise
1.29167pt\hbox{$\chi$}}_{\\_}Bf_{\\_}n\to{\raise
1.29167pt\hbox{$\chi$}}_{\\_}Bf$ in $L^{2}({\sf X})$ for every bounded Borel
set $B\subset{\sf X}$.

A test plan $\pi$ is a Borel probability measure on $C([0,1],{\sf X})$ such
that

$\begin{split}({\rm{e}}_{\\_}t)_{\\_}*{\mbox{\boldmath$\pi$}}&\leq
C\mathfrak{m},\qquad\forall t\in[0,1],\\\
\iint_{\\_}0^{1}|\dot{\gamma}_{\\_}t|\,{\mathrm{d}}t\,{\mathrm{d}}{\mbox{\boldmath$\pi$}}(\gamma)&<\infty,\end{split}$

for some $C>0$.

The Sobolev class ${S}^{2}({\sf X})$ (resp. ${S}^{2}_{\\_}\mathsf{loc}({\sf
X})$) is the space of all Borel functions $f:{\sf X}\to\mathbb{R}$ for which
there exists $G\in\mathit{L}^{2}(\mathfrak{m})$ (resp.
$G\in\mathit{L}^{2}_{\\_}\mathsf{loc}(\mathfrak{m})$) non-negative, called
weak upper gradient, such that for every test plan $\bm{\pi}$ it holds

$\int|f(\gamma_{\\_}1)-f(\gamma_{\\_}0)|\,{\mathrm{d}}\pi(\gamma)\leq\iint_{\\_}0^{1}G(\gamma_{\\_}t)|\dot{\gamma}_{\\_}t|\,{\mathrm{d}}t\,{\mathrm{d}}\bm{\pi}(\gamma).$

It can be proved that $f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X})$ and $G$ is a
weak upper gradient if and only if for every test plan $\pi$ we have that for
$\pi$-a.e. $\gamma$ the map $t\mapsto f(\gamma_{\\_}t)$ is in $W^{1,1}(0,1)$
and

$\Big{|}\frac{{\mathrm{d}}}{{\mathrm{d}}t}f(\gamma_{\\_}t)\Big{|}\leq
G(\gamma_{\\_}t)|\dot{\gamma}_{\\_}t|\qquad a.e.\ t\in[0,1].$

In particular, this characterization implies the existence of a minimal weak
upper gradient in the $\mathfrak{m}$-a.e. sense: we call it minimal weak upper
gradient and denote it by $|\mathrm{D}f|$.

The Sobolev space $\mathit{W}^{1,2}({\sf X})$ (resp.
$\mathit{W}^{1,2}_{\\_}\mathsf{loc}({\sf X})$) is defined as
$\mathit{L}^{2}\cap{S}^{2}({\sf X})$ (resp.
$\mathit{L}^{2}_{\\_}\mathsf{loc}\cap{S}^{2}_{\\_}\mathsf{loc}({\sf X})$). It
can be proved that $f\in\mathit{W}^{1,2}_{\\_}\mathsf{loc}({\sf X})$ if and
only if $\eta f\in\mathit{W}^{1,2}({\sf X})$ for every $\eta$ Lipschitz with
bounded support. We recall that $\mathit{W}^{1,2}({\sf X})$ is a Banach space
when endowed with the norm

$\|f\|_{\\_}{\mathit{W}^{1,2}({\sf
X})}^{2}:=\|f\|_{\\_}{\mathit{L}^{2}(\mathfrak{m})}^{2}+\||\mathrm{D}f|\|_{\\_}{\mathit{L}^{2}(\mathfrak{m})}^{2}.$

We say ([11]) that $({\sf X},\mathsf{d},\mathfrak{m})$ is infinitesimally
Hilbertian provided $\mathit{W}^{1,2}({\sf X})$ is a Hilbert space.

It is useful to recall that minimal weak upper gradients have the following
important locality property:

$|\mathrm{D}f|=|\mathrm{D}g|\qquad\mathfrak{m}-a.e.\text{ on
}\\{f=g\\}\qquad\qquad\forall f,g\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}).$

From the notion of minimal weak upper gradient it is possible to extract the
one of differential via the following result:

######  Theorem/Definition 2.1.

There exists a unique couple $(\mathit{L}^{0}(T^{*}{\sf X}),\mathrm{d})$,
where $\mathit{L}^{0}(T^{*}{\sf X})$ is a $\mathit{L}^{0}({\sf X})$-normed
module and $\mathrm{d}:{S}^{2}_{\\_}\mathsf{loc}({\sf
X})\to\mathit{L}^{0}(T^{*}{\sf X})$ is a linear map, such that

  * i)

$|\mathrm{d}f|=|\mathrm{D}f|$ $\mathfrak{m}$-a.e. for every
$f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X})$,

  * ii)

$\mathit{L}^{0}(T^{*}{\sf X})$ is generated by
$\\{\mathrm{d}f:f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X})\\}$, i.e.
$\mathit{L}^{0}$-linear combinations of objects of the form $\mathrm{d}f$ are
dense in $\mathit{L}^{0}(T^{*}{\sf X})$.

Uniqueness is intended up to unique isomorphism, i.e. if
$(\mathscr{M},\mathrm{d}^{\prime})$ is another such couple, then there is a
unique isomorphism $\Phi\colon\mathit{L}^{0}(T^{*}{\sf X})\to\mathscr{M}$ such
that $\Phi(\mathrm{d}f)=\mathrm{d}^{\prime}f$ for every
$f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X})$.

The space of vector fields $\mathit{L}^{0}(T{\sf X})$ is defined as the dual
of the $\mathit{L}^{0}$-normed module $\mathit{L}^{0}(T^{*}{\sf X})$. It can
be equivalently characterized as the $\mathit{L}^{0}$-completion of the dual
$\mathit{L}^{2}(T{\sf X})$ of the $\mathit{L}^{2}$-normed module
$\mathit{L}^{2}(T^{*}{\sf X})$ (see [12], [8]).
$\mathit{L}^{2}_{\\_}\mathsf{loc}(T{\sf X})\subset\mathit{L}^{0}(T{\sf X})$ is
the space of $X$’s such that
$|X|\in\mathit{L}^{2}_{\\_}\mathsf{loc}(\mathfrak{m})$.

We say that $X\in\mathit{L}^{2}_{\\_}\mathsf{loc}(T{\sf X})$ has divergence in
$\mathit{L}^{2}_{\\_}\mathsf{loc}$, and write
$X\in\mathrm{D}({\rm{div}}_{\\_}{\mathsf{loc}})$, if there exists
$h\in\mathit{L}^{2}_{\\_}\mathsf{loc}({\sf X})$ such that

$\displaystyle\int fh\ \mathrm{d}\mathfrak{m}=-\int\mathrm{d}f(X)\
\mathrm{d}\mathfrak{m},\quad\text{for every}\,\,\,f\in\mathit{W}^{1,2}({\sf
X})\,\,\,\text{with bounded support}.$

In this case we call $h$ (which is unique by the density of
$\mathit{W}^{1,2}({\sf X})$ in $\mathit{L}^{2}(\mathfrak{m})$) the divergence
of $X$, and denote it by ${\rm div}(X)$.

Let us now assume that $({\sf X},\mathsf{d},\mathfrak{m})$ is infinitesimally
Hilbertian, so that the pointwise norms in $\mathit{L}^{0}(T^{*}{\sf X})$ and
in $\mathit{L}^{0}(T{\sf X})$ induce pointwise scalar products. In this case
the modules $\mathit{L}^{0}(T^{*}{\sf X})$ and $\mathit{L}^{0}(T{\sf X})$ are
canonically isomorphic via the Riesz (musical) isomorphism

$\flat\colon\mathit{L}^{0}(\mathit{T}{\sf
X})\rightarrow\mathit{L}^{0}(\mathit{T}^{\ast}{\sf X})\qquad\
\text{and}\qquad\ \sharp\colon\mathit{L}^{0}(\mathit{T}^{\ast}{\sf
X})\rightarrow\mathit{L}^{0}(\mathit{T}{\sf X})$

defined by

$X^{\flat}(Y):=\left<X,Y\right>\qquad\text{and}\qquad\left<\omega^{\sharp},X\right>:=\omega(X)$

for every $X,Y\in\mathit{L}^{0}(T{\sf X})$ and
$\omega\in\mathit{L}^{0}(T^{*}{\sf X})$. The gradient of a function
$f\in\mathit{W}^{1,2}_{\\_}\mathsf{loc}({\sf X})$ is defined as $\nabla
f:=(\mathrm{d}f)^{\sharp}\in\mathit{L}^{2}_{\\_}\mathsf{loc}(T{\sf X})$.

We say that $f\in\mathit{W}^{1,2}_{\\_}\mathsf{loc}({\sf X})$ has Laplacian in
$\mathit{L}^{2}_{\\_}\mathsf{loc}(\mathfrak{m})$, namely
$f\in\mathrm{D}(\Delta_{\\_}\mathsf{loc})$, if there exists
$h\in\mathit{L}^{2}_{\\_}\mathsf{loc}(\mathfrak{m})$ such that it holds

$\displaystyle\int gh\,\mathrm{d}\mathfrak{m}=-\int\left<\nabla f,\nabla
g\right>\mathrm{d}\mathfrak{m},\quad\text{for
every}\,\,\,g\in\mathit{W}^{1,2}({\sf X})\,\,\,\text{with bounded support}$

(this is the same as requiring that $\nabla
f\in\mathrm{D}({{\rm{div}}}_{\\_}\mathsf{loc})$ with ${{\rm{div}}}(\nabla
f)=h$). In this case we call $h$ the Laplacian of $f$, and we denote it by
$\Delta f$. If $f,h\in L^{2}({\sf X})$ we shall write $f\in D(\Delta)$ instead
of $f\in D(\Delta_{\\_}\mathsf{loc})$, and in this case the Laplacian is
equivalently defined as infinitesimal generator of the Dirichlet form

$\mathsf{E}(f):=\left\\{\begin{array}[]{ll}\dfrac{1}{2}\displaystyle{\int|{\mathrm{d}}f|^{2}\,{\mathrm{d}}\mathfrak{m}}&\qquad\text{
if }f\in\mathit{W}^{1,2}({\sf X}),\\\ +\infty&\qquad\text{
otherwise}.\end{array}\right.$ (2.1.1)

From the properties of the minimal upper gradient we deduce that $\mathsf{E}$
is convex, lower semicontinuous and with dense domain, namely
$\\{f:\mathsf{E}(f)<\infty\\}$ is dense in $L^{2}(\mathfrak{m})$. Hence, the
classical theory of gradient flows of convex functions on Hilbert spaces
ensures existence and uniqueness of a 1-parameter semigroup $({{\sf
h}}_{\\_}t)_{\\_}{t\geq 0}$ of continuous operators from $L^{2}(\mathfrak{m})$
to itself such that for every $f\in L^{2}(\mathfrak{m})$ the curve
$t\mapsto{{\sf h}}_{\\_}t(f)\in L^{2}(\mathfrak{m})$ is continuous on
$[0,\infty)$, absolutely continuous on $(0,\infty)$ and satisfies

$\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}{{\sf h}}_{\\_}t(f)=\Delta
f,\qquad\text{for a.e. }t>0,$

where it is part of the statement the fact that ${{\sf
h}}_{\\_}t(f)\in\mathrm{D}(\Delta)$ for every $f\in L^{2}(\mathfrak{m})$ and
$t>0$ (see [2], and the references therein). We remark that in the case in
which $({\sf X},\mathsf{d},\mathfrak{m})$ is infinitesimally Hilbertian, the
Laplacian and the operators ${{\sf h}}_{\\_}t$ are linear.

###  2.2 Calculus tools on product spaces

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces. The product space ${\sf X}\times{\sf Y}$ will be always implicitly
endowed with the product measure and the distance

$({\sf d}_{\\_}{\sf X}\otimes{\sf d}_{\\_}{\sf
Y})^{2}\big{(}(x_{\\_}1,y_{\\_}1),(x_{\\_}2,y_{\\_}2)\big{)}:={\sf
d}_{\\_}{\sf X}^{2}(x_{\\_}1,x_{\\_}2)+{\sf d}_{\\_}{\sf
Y}^{2}(y_{\\_}1,y_{\\_}2).$

In the following we will denote by $\pi_{\\_}{\sf X}\colon{\sf X}\times{\sf
Y}\to{\sf X}$, the canonical projection on the first coordinate. Observe that
this is a map of local bounded deformation, meaning that for every bounded set
$B\subset{\sf X}\times{\sf Y}$ there is a constant $C(B)>0$ such that
$\pi_{\\_}{\sf X}\lower 3.0pt\hbox{$|_{B}$}$ is $C(B)$-Lipschitz and
$(\pi_{\\_}{\sf X})_{\\_}*(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}\lower 3.0pt\hbox{$|_{B}$})\leq
C(B)\mathfrak{m}_{\\_}{\sf X}$: this in particular allows to construct a
pullback module over ${\sf X}\times{\sf Y}$ starting from a $L^{0}({\sf
X})$-module $\mathscr{M}$ over ${\sf X}$ (see [16, Section 3.1] for the
definition of such pullback). Given the particular structure of the projection
map, a very explicit construction can be given to such pullback, as we briefly
discuss now.

Let us consider a $L^{0}({\sf X})$-module $\mathscr{M}$ over ${\sf X}$. We
have on one side the pullback $([\pi_{\\_}{\sf
X}^{\ast}]\mathscr{M},[\pi_{\\_}{\sf X}^{\ast}])$ of $\mathscr{M}$ through
$\pi_{\\_}{\sf X}$ (see [16, Theorem/Definition 3.2] and notice that in
particular $[\pi_{\\_}{\sf X}^{\ast}]\mathscr{M}$ is a $L^{0}({\sf
X}\times{\sf Y})$-module) and on the other the module $L^{0}({\sf
Y},\mathscr{M})$ that we are going to define now. $L^{0}({\sf Y},\mathscr{M})$
is the space of all the equivalence classes up to $\mathfrak{m}_{\\_}{\sf
Y}$-a.e. equality of strongly measurable (i.e., Borel and essentially
separably valued) functions from ${\sf Y}$ to $\mathscr{M}$. This space
canonically carries the structure of a $L^{0}({\sf X}\times{\sf Y})$-module.
Indeed:

  * -

the multiplication of an element of $L^{0}({\sf Y},\mathscr{M})$ by a function
$f\in L^{0}({\sf X}\times{\sf Y})$ is defined as the map ${\sf Y}\ni y\mapsto
f(\cdot,y)v(\cdot,y)\in\mathscr{M}$. Recalling that $L^{0}({\sf X}\times{\sf
Y})\sim L^{0}({\sf Y};L^{0}({\sf X}))$ and approximating $f\in L^{0}({\sf
X}\times{\sf Y})$ with functions with finite range as maps from ${\sf Y}$ to
$L^{0}({\sf X})$, it is not hard to see that $y\mapsto f(\cdot,y)v(\cdot,y)$
has essentially separable range, provided that $y\mapsto v(\cdot,y)$ does;

  * -

the pointwise norm of $v\in L^{0}({\sf Y},\mathscr{M})$ is obtained by
composing the map $y\mapsto v(\cdot,y)\in\mathscr{M}$ with the pointwise norm
on $\mathscr{M}$. Again the isomorphism $L^{0}({\sf Y},L^{0}({\sf X}))\sim
L^{0}({\sf X}\times{\sf Y})$ ensures that the so-defined map takes values in
$L^{0}({\sf X}\times{\sf Y})$.

It is then clear that this pointwise norm induces a complete distance on
$L^{0}({\sf Y},\mathscr{M})$ via the formula

${\sf d}_{\\_}{L^{0}}(v,w):=\int 1\wedge|v-w|\,{\mathrm{d}}\mathfrak{n},$

where $\mathfrak{n}\in\mathscr{P}({\sf X}\times{\sf Y})$ has the same
negligible sets of $\mathfrak{m}_{\\_}{\sf X}\times\mathfrak{m}_{\\_}{\sf Y}$
(the topology induced by ${\sf d}_{\\_}{L^{0}}$ is independent on the choice
of the particular $\mathfrak{n}$) and thus that, as claimed, $L^{0}({\sf
Y},\mathscr{M})$ is a $L^{0}({\sf X}\times{\sf Y})$-module.

By construction, $L^{0}({\sf Y},\mathscr{M})$ is generated by constant maps
and for any $v\in\mathscr{M}$ the map $\hat{v}\in L^{0}({\sf Y},\mathscr{M})$
constantly equal to $v$ satisfies $|\hat{v}|=|v|\circ\pi_{\\_}{\sf X}$. In
other words, by the characterization of pullback of modules, the module
$L^{0}({\sf Y},\mathscr{M})$ and the map $v\mapsto\hat{v}$ can be identified
with $([\pi_{\\_}{\sf X}^{\ast}]\mathscr{M},[\pi_{\\_}{\sf X}^{\ast}])$,
meaning that there is a unique isomorphism $\Phi:L^{0}({\sf
Y},\mathscr{M})\to[\pi_{\\_}{\sf X}^{\ast}]\mathscr{M}$ such that
$\Phi(\hat{v})=[\pi_{\\_}{\sf X}^{\ast}]v$ for any $v\in\mathscr{M}$.

In what will come next, we shall often implicitly use this identification in
the case in which $\mathscr{M}=L^{0}(T^{*}{\sf X})$ , namely $L^{0}({\sf
Y},L^{0}(T^{*}{\sf X}))\sim[\pi_{\\_}{\sf X}^{*}]L^{0}(T^{*}{\sf X})$.

The fact that $\pi_{\\_}{\sf X}$ is of local bounded deformation ensures that
it can be used to pullback 1-forms (see [12] and [16, Section 3.1.2]): for any
$f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}_{\\_}1)$ we have that

$f\circ\pi_{\\_}{\sf X}\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}\times{\sf
Y})\,\,\,\text{with}\,\,\,\lvert{\mathrm{d}}(f\circ\pi_{\\_}{\sf
X})\rvert=\lvert{\mathrm{d}}f\rvert\circ\pi_{\\_}{\sf
X},\quad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}\text{-a.e.},$ (2.2.1)

and from this fact it follows that there exists a unique linear and continuous
map $\pi_{\\_}{\sf X}^{\ast}\colon L^{0}(\mathit{T}^{\ast}{\sf X})\to
L^{0}(\mathit{T}^{\ast}({\sf X}\times{\sf Y}))$ with the property that

$\begin{split}\pi_{\\_}{\sf
X}^{\ast}({\mathrm{d}}f)&={\mathrm{d}}(f\circ\pi_{\\_}{\sf
X}),\,\,\qquad\forall f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}),\\\ \pi_{\\_}{\sf
X}^{\ast}(g\omega)&=g\circ\pi_{\\_}{\sf X}\,\pi_{\\_}{\sf
X}^{\ast}\omega,\,\,\quad\forall g\in L^{0}({\sf X}),\omega\in
L^{0}(\mathit{T}^{\ast}{\sf X}),\\\ \lvert\pi_{\\_}{\sf
X}^{\ast}\omega\rvert&=\lvert\omega\rvert\circ\pi_{\\_}{\sf
X},\,\,\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}\text{-a.e.},\,\forall\omega\in L^{0}(\mathit{T}^{\ast}{\sf X}).\end{split}$

All these constructions can be repeated with the roles of ${\sf X}$ and ${\sf
Y}$ inverted.

By the universal property of the pullback of modules it is easy to see that
the map $\pi_{\\_}{\sf X}^{*}$ just described splits through the pullback map
from $L^{0}(T{\sf X})$ to $L^{0}({\sf Y},L^{0}(T{\sf X}))$ and a module
morphism $\Phi_{\\_}{\sf X}:L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))\to
L^{0}(T^{*}({\sf X}\times{\sf Y}))$. More precisely we have the following
proposition (see [16, Proposition 3.7] for the proof):

######  Proposition 2.2.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces. There exists a unique $L^{0}({\sf X}\times{\sf Y})$-linear and
continuous map $\Phi_{\\_}{\sf x}$ from $L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))$
to $L^{0}(T^{*}({\sf X}\times{\sf Y}))$ such that

$\Phi_{\\_}{\sf x}(\widehat{{\mathrm{d}}g})={\mathrm{d}}(g\circ\pi_{\\_}{\sf
X})\qquad\forall g\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}),$

where $\widehat{{\mathrm{d}}g}:{\sf Y}\to L^{0}(T^{*}{\sf X})$ is the function
identically equal to ${\mathrm{d}}g$. Such map preserves the pointwise norm.

Similarly, there is a unique $L^{0}({\sf X}\times{\sf Y})$-linear and
continuous map $\Phi_{\\_}{\sf y}:L^{0}({\sf X},L^{0}(T^{*}{\sf Y}))\to
L^{0}(T^{*}({\sf X}\times{\sf Y}))$ such that

$\Phi_{\\_}{\sf y}(\widehat{{\mathrm{d}}h})={\mathrm{d}}(h\circ\pi_{\\_}{\sf
Y})\qquad\forall h\in{S}^{2}_{\\_}\mathsf{loc}({\sf Y}),$

where $\widehat{{\mathrm{d}}h}:{\sf X}\to L^{0}(T^{*}{\sf Y})$ is the function
identically equal to ${\mathrm{d}}h$, and such map preserves the pointwise
norm.

A way to think at the above is the following. Say that we have two smooth
manifolds $M_{\\_}1$ and $M_{\\_}2$ and a map assigning to every $x_{\\_}2\in
M_{\\_}2$ a 1-form $\omega(x_{\\_}2)$ on $M_{\\_}1$. Then we might think at
such map as the 1-form $\Omega$ on $M_{\\_}1\times M_{\\_}2$ which at the
point $(x_{\\_}1,x_{\\_}2)$ has value $(\omega(x_{\\_}2)(x_{\\_}1),0)$. Here a
way of thinking at $(\omega(x_{\\_}2)(x_{\\_}1),0)$ as element of the
cotangent space at $(x_{\\_}1,x_{\\_}2)$ is the following: say that
$\omega(x_{\\_}2)(x_{\\_}1)={\mathrm{d}}_{\\_}{x_{\\_}1}f$ for some smooth
function $f\colon M_{\\_}1\to\mathbb{R}$ ($f$ depends on $x_{\\_}2$, but such
dependence is not emphasized here). Then we can think/define
$(\omega(x_{\\_}2)(x_{\\_}1),0)$ as the differential of
$f\circ\pi_{\\_}1\colon M_{\\_}1\times M_{\\_}2\to\mathbb{R}$ at the point
$(x_{\\_}1,x_{\\_}2)$. In the setting of metric measure spaces the assignment
$\omega_{\\_}\cdot\mapsto\Omega$ is the map $\Phi_{\\_}{\sf x}$ defined by
Proposition 2.2 above.

Without further informations on the structure of ${\sf X},{\sf Y}$ it seems
hard to find other relations between calculus on the base spaces and calculus
on the product. In particular, one would expect the map $\Phi_{\\_}{\sf
x}\oplus\Phi_{\\_}{\sf y}$ (see below for the precise definition and in
particular Theorem 2.8) to be an isomorphism of modules: an investigation of
this fact, carried out in [16], shows that it depends on the validity of the
‘Assumption 2.5’ below. The quotation marks are due because we don’t know of
any example for which such assumption is not satisfied, so perhaps there is a
chance that Assumption 2.5 is rather a theorem; for our purposes it will be
sufficient to know that we can actually prove that Assumption 2.5 holds if the
given spaces are ${\sf RCD}$, see Proposition 4.1 below.

In what follows we shall denote by ${\mathrm{d}}_{\\_}{\sf
x},{\mathrm{d}}_{\\_}{\sf y},{\mathrm{d}}$ the differentials in ${\sf X},{\sf
Y},{\sf X}\times{\sf Y}$ respectively.

######  Definition 2.3 (Tensorization of the Cheeger energy).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces. We say that they have the property of tensorization of the Cheeger
energy provided for any $f\in\mathit{L}^{2}({\sf X}\times{\sf Y})$ the
following holds: $f\in\mathit{W}^{1,2}({\sf X}\times{\sf Y})$ if and only if

  * -

for $\mathfrak{m}_{\\_}{\sf X}$-a.e. $x\in{\sf X}$ it holds
$f(x,\cdot)\in\mathit{W}^{1,2}({\sf Y})$ with
$\displaystyle\int\lvert{\mathrm{d}}_{\\_}{\sf
y}f(x,\cdot)\rvert^{2}(y)\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})(x,y)<\infty$

  * -

for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ it holds
$f(\cdot,y)\in\mathit{W}^{1,2}({\sf X})$ with
$\displaystyle\int\lvert{\mathrm{d}}_{\\_}{\sf
y}f(\cdot,y)\rvert^{2}(x)\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})(x,y)<\infty$

and, in this case, we have

$\lvert{\mathrm{d}}f\rvert^{2}=\lvert{\mathrm{d}}_{\\_}{\sf
x}f\rvert^{2}+\lvert\mathrm{d}_{\\_}{\sf
y}f\rvert^{2}\qquad\mathfrak{m}_{\\_}1\otimes\mathfrak{m}_{\\_}2\text{-a.e.}.$
(2.2.2)

######  Definition 2.4 (Strong measurability of the sections).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces. We say that they have the property of the strong measurability of the
sections if for any $f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$ the
maps ${\sf Y}\ni y\mapsto{\mathrm{d}}_{\\_}{\sf y}f\in
L^{0}(\mathit{T}^{\ast}{\sf X})$ and ${\sf X}\ni
x\mapsto{\mathrm{d}}_{\\_}{\sf x}f\in L^{0}(\mathit{T}^{\ast}{\sf Y})$ are
essentially separably valued.

We shall be interested in spaces ${\sf X},{\sf Y}$ satisfying the following:

######  Assumption 2.5.

$({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ are two metric measure spaces for
which both the tensorization of Cheeger energy 2.3 and the strong
measurability of the sections 2.4 hold.

For our purposes it is important to recall that if ${\sf X},{\sf Y}$ are ${\sf
RCD}$ spaces, then they satisfy such assumption, see Proposition 4.1.

######  Remark 2.6.

We will refer to [16] for the proofs of the forthcoming results. Notice that
there the assumption made involves the density of a suitable class of
functions in $\mathit{W}^{1,2}({\sf X}\times{\sf Y})$, called “product
algebra”, in the strong topology of $\mathit{W}^{1,2}({\sf X}\times{\sf Y})$.
However, the reason why the density of the product algebra is needed in [16]
is exactly to show the strong measurability of the sections, therefore all the
results proved in [16] are still true once we work under Assumption 2.5.
$\blacksquare$

The following lemma provides a link from differentials on ${\sf X}\times{\sf
Y}$ to differentials on ${\sf X}$ and ${\sf Y}$, thus going in the opposite
direction of Proposition 2.2:

######  Lemma 2.7 ([16, Lemma 3.12]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces satisfying Assumption 2.5.

Then for every $f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$ we have
that $f(\cdot,y)\in{S}^{2}_{\\_}\mathsf{loc}({\sf X})$ for
$\mathfrak{m}_{\\_}2$-a.e. $y$ and the map $y\mapsto\mathrm{d}_{\\_}{\sf
x}f(\cdot,y)$ (that below we shall simply denote by ${\mathrm{d}}_{\\_}{\sf
x}f$) belongs to $\mathit{L}^{0}({\sf Y},\mathit{L}^{0}(\mathit{T}^{\ast}{\sf
X}))$. Moreover, for $(f_{\\_}n)\subset{S}^{2}_{\\_}\mathsf{loc}({\sf
X}_{\\_}1\times{\sf X}_{\\_}2)$ we have

$\mathrm{d}f_{\\_}n\to\mathrm{d}f\text{ in }\mathit{L}^{0}(T^{*}({\sf
X}\times{\sf Y}))\qquad\Rightarrow\qquad\mathrm{d}_{\\_}{\sf
x}f_{\\_}n\to\mathrm{d}_{\\_}{\sf x}f\text{ in }\mathit{L}^{0}({\sf
Y},\mathit{L}^{0}(T^{*}{\sf X})).$

Similarly for the roles of ${\sf X}$ and ${\sf Y}$ inverted. Finally, the
identity (2.2.2) holds for any $f\in{S}^{2}_{\\_}\mathsf{loc}({\sf
X}\times{\sf Y})$.

We now turn to the ‘full’ relation between forms on ${\sf X},{\sf Y}$ and
forms on ${\sf X}\times{\sf Y}$. To this aim, notice that for given
$L^{0}$-normed modules $\mathscr{M}_{\\_}1,\mathscr{M}_{\\_}2$ on the same
space ${\rm Z}$, the product $\mathscr{M}_{\\_}1\times\mathscr{M}_{\\_}2$ is
canonically a $L^{0}$-normed module on ${\rm Z}$ once it is endowed with the
product topology, the multiplication by $L^{0}$-functions given by
$f(v_{\\_}1,v_{\\_}2):=(fv_{\\_}1,fv_{\\_}2)$ and the pointwise norm defined
as

$|(v_{\\_}1,v_{\\_}2)|^{2}:=|v_{\\_}1|^{2}+|v_{\\_}2|^{2}.$

In particular, $L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))\times L^{0}({\sf
X},L^{0}(T^{*}{\sf Y}))$ is a $L^{0}({\sf X}\times{\sf Y})$-normed module and
we can define $\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf y}$ as

$\begin{array}[]{cccc}\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf
y}\colon&\mathit{L}^{0}({\sf Y},\mathit{L}^{0}(\mathit{T}^{\ast}{\sf
X}))\times\mathit{L}^{0}({\sf X},\mathit{L}^{0}(\mathit{T}^{\ast}{\sf
Y}))&\rightarrow&\mathit{L}^{0}(\mathit{T}^{\ast}({\sf X}\times{\sf Y}))\\\
&(\omega,\sigma)&\mapsto&\Phi_{\\_}{\sf x}(\omega)+\Phi_{\\_}{\sf
y}(\sigma)\end{array}$

We then have the following result:

######  Theorem 2.8 ([16, Theorem 3.13]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces satisfying Assumption 2.5.

Then $\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf y}$ is an isomorphism of modules,
i.e. it is $L^{0}({\sf X}\times{\sf Y})$-linear, continuous, surjective and
for every $\omega\in\mathit{L}^{0}({\sf
Y},\mathit{L}^{0}(\mathit{T}^{\ast}{\sf X}))$ and
$\sigma\in\mathit{L}^{0}({\sf X},\mathit{L}^{0}(\mathit{T}^{\ast}{\sf Y}))$
the following identity holds

$|\Phi_{\\_}{\sf x}(\omega)+\Phi_{\\_}{\sf
y}(\sigma)|^{2}=|\omega|^{2}+|\sigma|^{2}\quad\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e..$

Moreover, for every $f\in{S}^{2}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$ it
holds:

$\mathrm{d}f=\Phi_{\\_}{\sf x}(\mathrm{d}_{\\_}{\sf x}f)+\Phi_{\\_}{\sf
y}(\mathrm{d}_{\\_}{\sf y}f).$

In other words, and in line with the discussion made after Proposition 2.2,
this last theorem provides a decomposition of $L^{0}(T^{\ast}({\sf
X}\times{\sf Y}))$ in two submodules, the image of $\Phi_{\\_}{\sf x}$ and the
image of $\Phi_{\\_}{\sf y}$, and we shall think at these as the decomposition
of a 1-form on ${\sf X}\times{\sf Y}$ into its components cotangent to ${\sf
X}$ and ${\sf Y}$ respectively: for brevity, given $\omega\in L^{0}(T^{*}({\sf
X}\times{\sf Y}))$ we shall write

$\omega=(\omega_{\\_}{\sf x},\omega_{\\_}{\sf y})$ (2.2.3)

meaning that $\omega=\Phi_{\\_}{\sf x}(\omega_{\\_}{\sf x})+\Phi_{\\_}{\sf
y}(\omega_{\\_}{\sf y})$. Theorem 2.8 above ensures that both
$\omega_{\\_}{\sf x}$ and $\omega_{\\_}{\sf y}$ are uniquely determined by
$\omega$. In the smooth case, given a 1-form $\omega$ on the product of two
manifolds $M_{\\_}1,M_{\\_}2$ and $(x_{\\_}1,x_{\\_}2)\in M_{\\_}1\times
M_{\\_}2$, the 1-form $\omega_{\\_}{\sf x}(x_{\\_}2)$ at the point $x_{\\_}1$
is the restriction of $\omega(x_{\\_}1,x_{\\_}2)$ to the kernel of the
differential of $\pi_{\\_}{M_{\\_}2}$ in
$T_{\\_}{(x_{\\_}1,x_{\\_}2)}(M_{\\_}1\times M_{\\_}2)$ (this kernel being
isomorphic to $T_{\\_}{x_{\\_}1}M_{\\_}1$ via the differential of
$\pi_{\\_}{M_{\\_}1}$).

Now recall that if $L^{0}(T{\sf X})$ is separable, then the dual of
$L^{0}(Y,L^{0}(T^{*}{\sf X}))$ be canonically identified with
$L^{0}(Y,L^{0}(T{\sf X}))$ via the coupling

$\left.\begin{array}[]{ll}L^{0}(Y,L^{0}(T^{*}{\sf X}))\ni\omega\\\
L^{0}(Y,L^{0}(T^{*}{\sf X}))\ni v\end{array}\right\\}\qquad\Rightarrow\qquad
y\,\,\mapsto\,\,\omega(y)(v(y))\in L^{0}({\sf X}).$

Below we shall assume that $L^{0}(T{\sf X})$ is separable (recall that this is
always the case if ${\sf X}$ is infinitesimally Hilbertian, see [1]) and
constantly identify $L^{0}(Y,L^{0}(T{\sf X}))$ with $L^{0}(Y,L^{0}(T^{*}{\sf
X}))^{*}$ via the above isomorphism. Same assumption and identification with
the roles of ${\sf X},{\sf Y}$ swapped.

With this said, the $L^{0}({\sf X}\times{\sf Y})$-linear map $\Phi_{\\_}{\sf
x}\colon L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))\to L^{0}(T^{*}({\sf X}\times{\sf
Y}))$ has the adjoint

$\Phi_{\\_}{\sf x}^{\ast}\colon L^{0}(T({\sf X}\times{\sf Y}))\to L^{0}({\sf
Y},L^{0}(T{\sf X}))$

characterized by the property that for any $v\in L^{0}(T({\sf X}\times{\sf
Y}))$ and $\omega\in L^{0}({\sf Y},L^{0}(T^{\ast}{\sf X}))$ it holds

$\omega\big{(}\Phi^{\ast}_{\\_}{\sf x}(v)\big{)}=\big{(}\Phi_{\\_}{\sf
x}(\omega)\big{)}(v).$

Similarly, the operator $\Phi_{\\_}{\sf y}^{\ast}\colon L^{0}(T({\sf
X}\times{\sf Y}))\to L^{0}({\sf X},L^{0}(T{\sf Y}))$, adjoint of
$\Phi_{\\_}{\sf y}$ is characterized by the fact that for any $v\in
L^{0}(T({\sf X}\times{\sf Y}))$ and $\eta\in L^{0}({\sf X},L^{0}(T^{\ast}{\sf
Y}))$ we have

$\eta\big{(}\Phi^{\ast}_{\\_}{\sf y}(v)\big{)}=\big{(}\Phi_{\\_}{\sf
y}(\eta)\big{)}(v).$

Since the adjoint of $\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf y}$ is

$(\Phi_{\\_}{\sf x}^{*},\Phi_{\\_}{\sf y}^{*}):L^{0}(T({\sf X}\times{\sf
Y}))\to L^{0}({\sf Y},L^{0}(T{\sf X}))\times L^{0}({\sf X},L^{0}(T{\sf Y})),$

from Theorem 2.8, we deduce that $(\Phi_{\\_}{\sf x}^{*},\Phi_{\\_}{\sf
y}^{*})$ is an isomorphism of modules.

Given $v\in L^{0}(T({\sf X}\times{\sf Y}))$, we shall write $v_{\\_}{\sf
x},v_{\\_}{\sf y}$ in place of $\Phi_{\\_}{\sf x}^{*}(v),\Phi_{\\_}{\sf
y}^{*}(v)$ respectively and also often write

$v=(v_{\\_}{\sf x},v_{\\_}{\sf y})$ (2.2.4)

thus implicitly identifying $L^{0}(T({\sf X}\times{\sf Y}))$ with $L^{0}({\sf
Y},L^{0}(T{\sf X}))\times L^{0}({\sf X},L^{0}(T{\sf Y}))$ through
$(\Phi_{\\_}{\sf x}^{*},\Phi_{\\_}{\sf y}^{*})$. Notice that in the smooth
setting $v_{\\_}{\sf x},v_{\\_}{\sf y}$ are the components of the vector field
$v$ along ${\sf X},{\sf Y}$ respectively.

We conclude recalling that if ${\sf X},{\sf Y}$ are infinitesimally
Hilbertian, then also ${\sf X}\times{\sf Y}$ is so and the decomposition given
in Theorem 2.8 is orthogonal:

######  Proposition 2.9 ([16, Proposition 3.14]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces infinitesimally Hilbertian and satisfying Assumption 2.5.

Then ${\sf X}\times{\sf Y}$ is also infinitesimally Hilbertian and for every
$\omega\in\mathit{L}^{0}({\sf Y},\mathit{L}^{0}(\mathit{T}^{*}{\sf X}))$ and
$\eta\in\mathit{L}^{0}({\sf X},\mathit{L}^{0}(\mathit{T}^{*}{\sf Y}))$ we have

$\left<\Phi_{\\_}{\sf x}(\omega),\Phi_{\\_}{\sf
y}(\eta)\right>=0\quad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}-a.e..$ (2.2.5)

###  2.3 Other differential operators in the product space

Up to now we have seen how the differential behaves under products of spaces.
We shall now investigate other differentiation operators (divergence and
Laplacian) and for simplicity we shall stick to the case of infinitesimally
Hilbertian spaces.

As for the case of differentials, we shall denote by ${\rm div}_{\\_}{\sf
x},\Delta_{\\_}{\sf x}$ the divergence and Laplacian in the space ${\sf X}$
(and similarly for ${\sf Y}$) while we keep the un-labeled versions ${\rm
div},\Delta$ for the operators in the product space.

######  Proposition 2.10 ([16, Proposition 3.15]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces infinitesimally Hilbertian and satisfying Assumption 2.5.

Then $v\in D({\rm div}_{\\_}{\mathsf{loc}},{\sf X})$ if and only if
$(\hat{v},0)\in D({\rm{div}}_{\\_}{\mathsf{loc}},{{\sf X}\times{\sf Y}})$,
where $\hat{v}\in\mathit{L}^{0}({\sf Y},\mathit{L}^{0}(T{\sf X}))$ is the
function identically equal to $v$ (and we are adopting the convention in
(2.2.4)), and in this case

${\rm div}\big{(}(\hat{v},0)\big{)}={\rm div}_{\\_}{\sf
x}(v)\circ\pi_{\\_}{\sf X}.$

######  Proposition 2.11 ([16, Proposition 3.16]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces infinitesimally Hilbertian and satisfying Assumption 2.5.

Let $v=(v_{\\_}{\sf x},v_{\\_}{\sf y})\in\mathit{L}^{2}(\mathit{T}({\sf
X}\times{\sf Y}))$ be such that:

  * -

$v_{\\_}{\sf x}\in D({\rm div}_{\\_}{\sf x},{\sf X})$ for
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ with $\int|{{\rm div}_{\\_}{\sf
x}(v_{\\_}{\sf x})}|^{2}\ \mathrm{d}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})<\infty$,

  * -

$v_{\\_}{\sf y}\in D({\rm div}_{\\_}{\sf y},{\sf Y})$ for
$\mathfrak{m}_{\\_}{\sf X}$-a.e. $x\in{\sf X}$ with $\int|{{\rm div}_{\\_}{\sf
y}(v_{\\_}{\sf y})}|^{2}\ \mathrm{d}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})<\infty$.

Then $v\in D({\rm{div}},{\sf X}\times{\sf Y})$ and

${\rm div}(v)={\rm div}_{\\_}{\sf x}(v_{\\_}{\sf x})\circ\pi_{\\_}{\sf X}+{\rm
div}_{\\_}{\sf y}(v_{\\_}{\sf y})\circ\pi_{\\_}{\sf Y}.$ (2.3.1)

######  Remark 2.12.

The viceversa of Proposition 2.11 \- obviously - does not hold, indeed the
vector field on $\mathbb{R}^{2}$ defined by

$X(x_{\\_}1,x_{\\_}2)=\text{sgn}(x_{\\_}1-x_{\\_}2)(\mathbf{e}_{\\_}1+\mathbf{e}_{\\_}2),$

where sgn is the sign function, is easily seen to have null divergence, while
the distributional divergence of its components are Dirac masses (that get
canceled out in considering the divergence in the product space).

In other words, the problem is that if one has integrability for the two
quantities on the right hand side of (2.3.1), then the left hand side has the
same integrability properties, but from the integrability of the latter one
cannot deduce informations on the integrability of both addends in the right
hand side.  $\blacksquare$

A direct application of Proposition 2.10 and Proposition 2.11 gives the
following result on the Laplacian on the product space:

######  Proposition 2.13 ([16, Corollary 3.17]).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf
Y},{\sf d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure
spaces infinitesimally Hilbertian and satisfying Assumption 2.5. Then:

  * i)

$f\in D(\Delta_{\\_}\mathsf{loc},{\sf X})$ if and only if $f\circ\pi_{\\_}{\sf
X}\in D(\Delta_{\\_}\mathsf{loc},{\sf X}\times{\sf Y})$ and in this case

$\Delta(f\circ\pi_{\\_}{\sf X})=(\Delta_{\\_}{\sf x}f)\circ\pi_{\\_}{\sf X}.$

  * ii)

Let $f\in\mathit{W}^{1,2}({\sf X}\times{\sf Y})$ be such that

    * –

for $\mathfrak{m}_{\\_}{\sf X}$-a.e. $x\in{\sf X}$, $f^{x}:=f(x,\cdot)\in
D(\Delta,{\sf Y})$ with $\int|\Delta_{\\_}{\sf
y}f^{x}|^{2}(y)\,\mathrm{d}(\mathfrak{m}_{\\_}{\sf
X}\times\mathfrak{m}_{\\_}{\sf Y})(x,y)<\infty$,

    * –

for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$, $f_{\\_}{y}:=f(\cdot,y)\in
D(\Delta,{\sf X})$ with $\int|\Delta_{\\_}{\sf
x}f_{\\_}{y}|^{2}(x)\,\mathrm{d}(\mathfrak{m}_{\\_}{\sf
X}\times\mathfrak{m}_{\\_}{\sf Y})(x,y)<\infty$.

Then $f\in D(\Delta,{\sf X}\times{\sf Y})$ and

$\Delta f(x,y)=\Delta_{\\_}{\sf x}f_{\\_}{y}(x)+\Delta_{\\_}{\sf
y}f^{x}(y)\qquad\mathfrak{m}_{\\_}1\times\mathfrak{m}_{\\_}2-a.e.\ (x,y).$

##  3 Partial first order derivatives on the product space

###  3.1 Partial first order derivatives for functions in two variables

Aim of this section is to review, in preparation of the subsequent results,
the properties of those functions of two variables which are differentiable
with respect one of them. We start with the following:

######  Definition 3.1 (The space $L^{0}({\sf
Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ and the operator
${\mathrm{d}}_{\\_}{\sf x}$).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces. Then
$L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))\subset L^{0}({\sf
Y},L^{0}({\sf X}))\sim L^{0}({\sf X}\times{\sf Y})$ is the space of functions
$f\in L^{0}({\sf X}\times{\sf Y})$ such that $f(\cdot,y)\in
S^{2}_{\\_}\mathsf{loc}({\sf X})$ for $\mathfrak{m}_{\\_}{\sf Y}$-a.e.
$y\in{\sf Y}$, and the map ${\sf Y}\ni y\mapsto{\mathrm{d}}_{\\_}{\sf x}f\in
L^{0}(T^{*}{\sf X})$ belongs to $L^{0}({\sf Y};L^{0}(T^{*}{\sf X}))$.

We then define ${\mathrm{d}}_{\\_}{\sf x}:L^{0}({\sf
Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))\to L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))$
by ${\mathrm{d}}_{\\_}{\sf x}f(y):={\mathrm{d}}_{\\_}{\sf x}f(\cdot,y)$ for
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$.

We shall also denote by $L^{2}({\sf Y},S^{2}({\sf X}))\subset L^{0}({\sf
Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ the space of functions $f$ such that
$\int|{\mathrm{d}}_{\\_}{\sf x}f|^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})<\infty$.

The definition above guarantees that ${\mathrm{d}}_{\\_}{\sf x}f$ is a well
defined element of $L^{0}({\sf Y},L^{0}(T^{*}{\sf X}))$ (i.e. that it is Borel
and essentially separably valued).

It is worth noticing that if the space $W^{1,2}({\sf X})$ is separable (as it
is the case if it is Hilbert - recall [1, Proposition 7.6]), then elements of
$L^{0}({\sf Y},S^{2}_{\\_}\mathsf{loc}({\sf X}))$ can be easily singled out:

######  Proposition 3.2.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces with
$W^{1,2}({\sf X})$ separable and $f\in L^{0}({\sf X}\times{\sf Y})$.

Then $f\in L^{0}({\sf Y},S^{2}_{\\_}\mathsf{loc}({\sf X}))$ if and only if for
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ we have $f(\cdot,y)\in
S^{2}_{\\_}\mathsf{loc}({\sf X})$.

Proof. We start with the ‘only if’ and to this aim define
$\eta^{m}:=1\wedge(m-{\sf d}_{\\_}{\sf X}(x,\bar{x}))^{+}$ where
$\bar{x}\in{\sf X}$ is a fixed point. Then it follows from our assumption that
for every $n,m\in\mathbb{N}$ we have that for $\mathfrak{m}_{\\_}{\sf Y}$-a.e.
$y\in{\sf Y}$ the function $x\mapsto\eta^{m}(x)f^{n}(x,y)$, where
$f^{n}:=n\wedge f\vee(-n)$, belongs to $W^{1,2}({\sf X})$. Fix $y\in{\sf Y}$
for which this holds and let first $m\to\infty$ and then $n\to\infty$ to
deduce that $x\mapsto f(x,y)$ belongs to $S^{2}_{\\_}\mathsf{loc}({\sf X})$,
giving the claim.

For the ‘if’ we notice that by the very definition of $L^{0}({\sf
Y},S^{2}_{\\_}\mathsf{loc}({\sf X}))$ and up to replace $f$ by $\eta^{m}f^{n}$
with $\eta^{m},f^{n}$ as above and letting first $m\to\infty$ and then
$n\to\infty$, it is sufficient to deal with the case $f\in L^{2}({\sf
X}\times{\sf Y})$ with $f(\cdot,y)\in W^{1,2}({\sf X})$ for
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$.

In this case to conclude it is sufficient to prove that $f\in L^{0}({\sf
Y},W^{1,2}({\sf X}))$ that is to say, as $W^{1,2}({\sf X})$ is separable, that
$y\mapsto f(\cdot,y)\in W^{1,2}({\sf X})$ is (the equivalence class up to
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. equality of) a Borel map. Since $y\mapsto
f(\cdot,y)\in L^{2}({\sf X})$ is Borel by (2.2.1) and Assumption 2.5, to
conclude it is sufficient to show that the Borel structure on $W^{1,2}({\sf
X})$ induced by the $W^{1,2}$-distance coincides with that induced by the
$L^{2}$-distance. Since $\|f\|_{\\_}{L^{2}}\leq\|f\|_{\\_}{W^{1,2}}$ it is
clear that $L^{2}$-open sets are also $W^{1,2}$-open, whence the same is true
for Borel ones. For the opposite inclusion we notice that since the
$W^{1,2}$-norm is $L^{2}$-lower semicontinuous, we have that a
$W^{1,2}$-closed ball is a $L^{2}$-closed set, thus a $W^{1,2}$-open ball is
the union of a countable number of $L^{2}$-closed sets, hence $L^{2}$-Borel.
The conclusion follows.  $\square$

######  Remark 3.3 (Partial differentiation as example of $D$-structure).

In [17] the authors present an axiomatic approach to the theory of Sobolev
spaces over abstract metric measure spaces, introducing the notion of
$D$-structure. Given a metric measure space $({\sf X},{\sf d},\mathfrak{m})$
and an exponent $p\in(1,\infty)$, a $D$-structure on $({\sf X},{\sf
d},\mathfrak{m})$ is any map $D$ associating to any function $u\in
L^{p}_{\\_}\mathsf{loc}(\mathfrak{m})$ a family of non-negative Borel
functions $D[u]$, called pseudo-gradients, such that a proper list of axioms
is fulfilled (see [17, Section 1.2]). Intuitively, these pseudo-gradients
provide a control from above on the variation of $u$, in a suitable sense.

This allows to define the space $W^{p}({\sf X},{\sf d},\mathfrak{m},D)$ as the
set of all functions in $L^{p}(\mathfrak{m})$ admitting a pseudo-gradient in
$L^{p}(\mathfrak{m})$. Moreover, using standard techniques of functional
analysis, to any Sobolev function $u\in W^{p}({\sf X},{\sf d},\mathfrak{m},D)$
it can be associated a uniquely determined minimal object $\underline{D}u\in
D[u]\cap L^{p}(\mathfrak{m})$, called minimal pseudo-gradient of $u$.

In [15], the authors prove that, under suitable locality assumptions, these
$D$-structures give rise to a first-oder differential structure, namely to a
natural notion of cotangent module $L^{p}(T^{\ast}{\sf X};D)$, whose
properties are analogous to the ones of the cotangent module
$L^{2}(T^{\ast}{\sf X})$: the main result of such paper shows the existence of
an abstract differential ${\mathrm{d}}\colon W^{1,p}({\sf X},{\sf
d},\mathfrak{m},D)\to L^{p}(T^{\ast}{\sf X};D)$, which is a linear operator
such that for any $u\in W^{1,p}({\sf X},{\sf d},\mathfrak{m},D)$ the pointwise
norm $\lvert{\mathrm{d}}u\rvert\in L^{p}(\mathfrak{m})$ of ${\mathrm{d}}u$
coincides with $\underline{D}u$ in the $\mathfrak{m}$-a.e. sense.

We point out that an example of $D$-structure satisfying all the locality
conditions as investigated in [15] is the one that assigns to any $u\in
L^{2}({\sf Y},W^{1,2}({\sf X}))\subset L^{2}({\sf X}\times{\sf Y})$ the set
$D[u]:=\\{G\in L^{2}({\sf X}\times{\sf Y}):G\geq|{\mathrm{d}}_{\\_}{\sf x}u|\
\mathfrak{m}_{\\_}{\sf X}\times\mathfrak{m}_{\\_}{\sf Y}-a.e.\\}$.
$\blacksquare$

The operator ${\mathrm{d}}_{\\_}{\sf x}$ defined on $L^{0}({\sf
Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ trivially inherits all the calculus
rules of ${\mathrm{d}}_{\\_}{\sf x}:S^{2}_{\\_}\mathsf{loc}({\sf X})\to
L^{0}(T^{*}{\sf X})$:

######  Proposition 3.4 (Calculus rules).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces. Then
the following hold:

  * i)

_Closure_. Let $(f_{\\_}n)\subset L^{2}({\sf Y};S^{2}({\sf X}))$ be
$\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e. converging to
$f_{\\_}\infty$. Assume also that for some open sets $\Omega^{k}\subset{\sf
X}$ (resp. Borel subsets $B^{k}\subset{\sf Y}$) with
$\cup_{\\_}k\Omega^{k}={\sf X}$ (resp. $\mathfrak{m}_{\\_}{\sf Y}({\sf
Y}\setminus\cup_{\\_}kB^{k})=0$) and $\omega\in L^{0}({\sf Y};L^{0}(T^{*}{\sf
X}))$ we have that $({\raise 1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}{\mathrm{d}}_{\\_}{\sf x}f_{\\_}n)$
converges to ${\raise 1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}\omega$ in the weak topology of the Banach
space $L^{2}({\sf Y},L^{2}(T^{*}{\sf X}))$ for any $k\in\mathbb{N}$ (notice
that this holds in particular if ${\mathrm{d}}_{\\_}{\sf
x}f_{\\_}n\rightharpoonup\omega$ in $L^{2}({\sf Y},L^{2}(T^{*}{\sf X}))$).

Then $f_{\\_}\infty\in L^{2}({\sf Y};S^{2}({\sf X}))$ and
${\mathrm{d}}_{\\_}{\sf x}f_{\\_}\infty=\omega$.

The same conclusion holds if $(f_{\\_}n)\subset L^{2}({\sf Y};W^{1,2}({\sf
X}))$ converges to $f_{\\_}\infty$ in the weak topology of $L^{2}({\sf
X}\times{\sf Y})$.

  * ii)

_Locality_. For any $f,g\in L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$
we have

${\mathrm{d}}_{\\_}{\sf x}f={\mathrm{d}}_{\\_}{\sf
x}g\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e.\ on\
\\{f=g\\}.$ (3.1.1)

  * iii)

_Chain rule_. Let $f\in L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ and
$N\subset\mathbb{R}$ Borel and $\mathcal{L}^{1}$-negligible. Then
${\mathrm{d}}_{\\_}{\sf x}f=0$ $\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e. on $f^{-1}(N)$. Moreover, for
$\varphi:\mathbb{R}\to\mathbb{R}$ Lipschitz we have $\varphi\circ f\in
L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ and

${\mathrm{d}}_{\\_}{\sf x}(\varphi\circ f)=\varphi^{\prime}\circ
f{\mathrm{d}}_{\\_}{\sf x}f,$

(notice that by what just claimed, the actual definition of
$\varphi^{\prime}\circ f$ on the set $f^{-1}(\\{\text{non differentiability
points of }\varphi\\})$ is irrelevant because on such set
${\mathrm{d}}_{\\_}{\sf x}f$ is zero).

  * iv)

_Leibniz rule_. Let $f,g\in L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf
X}))\cap L^{\infty}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$. Then $fg\in
L^{0}({\sf Y},S^{2}_{\\_}{\mathsf{loc}}({\sf X}))$ as well with

${\mathrm{d}}_{\\_}{\sf x}(fg)=f{\mathrm{d}}_{\\_}{\sf
x}g+g{\mathrm{d}}_{\\_}{\sf x}f.$

If $g$ only depends on the $y$ variable, then the above holds without assuming
that $f\in L^{\infty}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$.

Proof. Locality, Chain rule and Leibniz rule follow directly from the
definition of ${\mathrm{d}}_{\\_}{\sf x}$ and the analogous properties of
functions depending solely on the $x$ variable. For the closure we notice that
first using Mazur’s lemma, then passing to subsequences and finally with a
diagonal argument we can assume that for any $k\in\mathbb{N}$ we have ${\raise
1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}n(y)\to{\raise 1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}\omega(y)$ in
$L^{2}(T^{*}{\sf X})$ for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$. Thus
the claim follows from the closure of the differential operator on functions
depending solely on the $x$ variable and on the fact that the result does not
depend on the subsequence chosen. The second claim about the closure follows
along similar lines by using Mazur’s lemma to reduce to strong convergence in
$L^{2}({\sf X}\times{\sf Y})$ and then passing to a subsequence to achieve
$\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e. convergence.
$\square$

The definitions given allow to reformulate Theorem 2.8 in the following way:

######  Proposition 3.5.

Let $({\sf X},{\sf d},\mathfrak{m}_{\\_}{\sf X})$ and $({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces
satisfying Assumption 2.5 and let $f\in S^{2}_{\\_}\mathsf{loc}({\sf
X}\times{\sf Y})$.

Then $y\mapsto f(\cdot,y)$ is in $L^{0}({\sf Y},S^{2}_{\\_}\mathsf{loc}({\sf
X}))$ and, symmetrically, $x\mapsto f(x,\cdot)$ is in $L^{0}({\sf
X},S^{2}_{\\_}\mathsf{loc}({\sf Y}))$, Moreover we have

${\mathrm{d}}f=\Phi_{\\_}{\sf x}({\mathrm{d}}_{\\_}{\sf x}f)+\Phi_{\\_}{\sf
y}({\mathrm{d}}_{\\_}{\sf y}f).$ (3.1.2)

Proof. This is simply a restatement of Theorem 2.8.  $\square$

In what follows we shall adopt the more compact (and more similar to the one
adopted in the smooth setting and in (2.2.3)) notation

${\mathrm{d}}f=({\mathrm{d}}_{\\_}{\sf x}f,{\mathrm{d}}_{\\_}{\sf y}f)$
(3.1.3)

in place of (3.1.2): this should hopefully clarify that we are dealing with
partial derivatives and that ${\mathrm{d}}_{\\_}{\sf
x}f,{\mathrm{d}}_{\\_}{\sf y}f$ are the two components of ${\mathrm{d}}f$.
Recalling the definition of $\Phi_{\\_}{\sf x}$, this means that for $f\in
S^{2}_{\\_}\mathsf{loc}({\sf X})$ we have

${\mathrm{d}}(f\circ\pi_{\\_}{\sf X})=({\mathrm{d}}_{\\_}{\sf x}f,0).$

Similarly, if the spaces under consideration are also infinitesimally
Hilbertian (as it often will be the case) then we shall adopt a similar
notation for the gradients to the one in (2.2.4), so that $\nabla
f=(\nabla_{\\_}{\sf x}f,\nabla_{\\_}{\sf y}f)$ and, for $f\in
S^{2}_{\\_}\mathsf{loc}({\sf X})$, $\nabla(f\circ\pi_{\\_}{\sf
X})=(\nabla_{\\_}{\sf x}f,0)$.

In this direction it will be useful to notice that what just said and
Proposition 2.11 allow to write

${\rm div}(h\nabla(g\circ\pi_{\\_}{\sf X}))={\rm div}_{\\_}{\sf
x}(h\nabla_{\\_}{\sf x}g)\qquad\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e.$ (3.1.4)

for any $h\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf X}\times{\sf Y})$
and $g\in D(\Delta_{\\_}{\sf X})$.

We now show that under appropriate integrability assumptions for both the
function and the differential, belonging to $L^{0}({\sf
Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$ can be checked via integration by parts.
Notice that this requires also that Assumption 2.5 holds and that ${\sf X}$ is
infinitesimally Hilbertian (this latter point can be slightly weakened, but we
won’t push in this direction).

######  Proposition 3.6.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces
satisfying Assumption 2.5 and with ${\sf X}$ being infinitesimally Hilbertian.
Also, let $f\in L^{2}({\sf X}\times{\sf Y})$. Then the following are
equivalent:

  * i)

$f\in L^{2}({\sf Y},W^{1,2}({\sf X}))$,

  * ii)

There is $A\in L^{0}({\sf Y};L^{0}(T^{*}{\sf X}))$ with $|A|\in L^{2}({\sf
X}\times{\sf Y})$ such that

$-\int f{\rm div}(h\nabla(g\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})=\int h\left<A,{\mathrm{d}}_{\\_}{\sf
x}g\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})$ (3.1.5)

holds for any $h\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf X}\times{\sf
Y})$ and $g\in D(\Delta_{\\_}{\sf X})$.

Moreover, if this is the case the choice $A={\mathrm{d}}_{\\_}{\sf x}f$ is the
only one for which $(ii)$ holds.

Proof.
$(i)\Rightarrow(ii)$ From (3.1.4) and the definition of ${\rm div}_{\\_}{\sf
x}$ we obtain

$\begin{split}-\int f{\rm div}(h\nabla(g\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})&=-\int\Big{(}\int f(\cdot,y){\rm div}_{\\_}{\sf
x}(h(\cdot,y)\nabla_{\\_}{\sf x}g)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\Big{)}{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}(y)\\\ &=\int\Big{(}\int
h\left<{\mathrm{d}}_{\\_}{\sf x}f,{\mathrm{d}}_{\\_}{\sf
x}g\right>\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\Big{)}{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y},\end{split}$

which proves $(ii)$ with $A:={\mathrm{d}}_{\\_}{\sf x}f$.

$(ii)\Rightarrow(i)$ For every $n\in\mathbb{N}$, let $(A^{n}_{\\_}i)$ be a
Borel partition of ${\sf Y}$ made of at most countable sets, with
$\mathfrak{m}_{\\_}{\sf Y}(A^{n}_{\\_}i)\in(0,\infty)$, ${\rm
diam}(A^{n}_{\\_}i)\leq\frac{1}{n}$ and so that $(A^{n+1}_{\\_}i)$ is a
refinement of $(A^{n}_{\\_}i)$. Put $f^{n}_{\\_}i:=\mathfrak{m}_{\\_}{\sf
Y}(A^{n}_{\\_}i)^{-1}\int_{\\_}{A^{n}_{\\_}i}f(\cdot,y)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
Y}(y)\in L^{2}({\sf X})$ and $f^{n}(x,y):=\sum_{\\_}i{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{A^{n}_{\\_}i}(y)f^{n}_{\\_}i(x)\in L^{2}({\sf
X}\times{\sf Y})$. It is clear that $f^{n}\to f$ in $L^{2}({\sf X}\times{\sf
Y})$ and thus the lower semicontinuity of the Cheeger-Dirichlet energy ${\sf
E}_{\\_}{\sf X}$ on ${\sf X}$ easily gives

$\int{\sf E}_{\\_}{\sf X}(f(\cdot,y))\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
Y}(y)\leq\varliminf_{\\_}{n\to\infty}\int{\sf E}_{\\_}{\sf
X}(f^{n}(\cdot,y))\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}(y).$ (3.1.6)

For every $n,i$ let $(h^{n}_{\\_}{i,m})\subset\mathop{\rm
Lip}\nolimits_{\\_}{\sf bs}({\sf X}\times{\sf Y})$ be a sequence of functions
1-Lipschitz in the $x$ variable, with values in $[0,1]$ and such that
$(h^{n}_{\\_}{i,m}),(|{\mathrm{d}}_{\\_}{\sf x}h^{n}_{\\_}{i,m}|)$ converge
$\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e. to ${\raise
1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times A^{n}_{\\_}i}$ and $0$,
respectively, as $m\to\infty$. Also, for $t>0$ let us put
$g^{n}_{\\_}{i,t}:={{\sf h}}_{\\_}{{\sf X},t}(f^{n}_{\\_}i)\in
D(\Delta_{\\_}{\sf X})$. Then an application of the dominate convergence
theorem shows that for every $n,i,t$ we have

$\begin{split}-\int f{\rm div}_{\\_}{\sf x}(h^{n}_{\\_}{i,m}\nabla_{\\_}{\sf
x}(g^{n}_{\\_}{i,t}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})&\quad\to\quad-\int_{\\_}{{\sf X}\times
A^{n}_{\\_}i}f\Delta_{\\_}{\sf
X}g^{n}_{\\_}{i,t}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})\\\ \int
h^{n}_{\\_}{i,m}\left<A,{\mathrm{d}}_{\\_}{\sf
x}g^{n}_{\\_}{i,t}\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})&\quad\to\quad\int\left<A,{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times
A^{n}_{\\_}i}{\mathrm{d}}_{\\_}{\sf
x}g^{n}_{\\_}i\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})\end{split}$

as $m\to\infty$, so that taking into account our assumption and the identity
(3.1.4) we obtain

$-\int_{\\_}{{\sf X}\times A^{n}_{\\_}i}f\Delta_{\\_}{\sf
X}g^{n}_{\\_}{i,t}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})=\int\left<A,{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times
A^{n}_{\\_}i}{\mathrm{d}}_{\\_}{\sf
x}g^{n}_{\\_}i\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}).$ (3.1.7)

Now observe that from the closure of the differential it is easy to justify
the following computation:

$\begin{split}-\int_{\\_}{{\sf X}\times A^{n}_{\\_}i}f\Delta_{\\_}{\sf
X}g^{n}_{\\_}{i,t}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})&=-\int_{\\_}{A^{n}_{\\_}i}\int_{\\_}{\sf
X}f(\cdot,y)\Delta_{\\_}{{\sf X}}{{\sf h}}_{\\_}{{\sf
X},t}(f^{n}_{\\_}i)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}(y)\\\
&=-\int_{\\_}{A^{n}_{\\_}i}\int_{\\_}{\sf X}{{\sf h}}_{\\_}{{\sf
X},t/2}(f(\cdot,y))\Delta_{\\_}{{\sf X}}{{\sf h}}_{\\_}{{\sf
X},t/2}(f^{n}_{\\_}i)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}(y)\\\ &=\int_{\\_}{\sf
X}\int_{\\_}{A^{n}_{\\_}i}\left<{\mathrm{d}}_{\\_}{\sf x}{{\sf h}}_{\\_}{{\sf
X},t/2}(f(\cdot,y)),{\mathrm{d}}_{\\_}{\sf x}{{\sf h}}_{\\_}{{\sf
X},t/2}(f^{n}_{\\_}i)\right>\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
Y}(y)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf X}\\\ &=\mathfrak{m}_{\\_}{\sf
Y}(A^{n}_{\\_}i)\int_{\\_}{\sf X}|{\mathrm{d}}_{\\_}{\sf x}{{\sf
h}}_{\\_}{{\sf X},t/2}(f^{n}_{\\_}i)|^{2}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\\\ &=2\mathfrak{m}_{\\_}{\sf Y}(A^{n}_{\\_}i){\sf E}_{\\_}{\sf
X}\big{(}{{\sf h}}_{\\_}{{\sf X},t/2}(f^{n}_{\\_}i)\big{)}.\end{split}$

On the other hand by Young’s inequality and the fact that ${\sf E}_{\\_}{\sf
X}$ is decreasing along the heat flow we have

$\begin{split}\int\left<A,{\raise 1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times
A^{n}_{\\_}i}{\mathrm{d}}_{\\_}{\sf
x}g^{n}_{\\_}i\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})&\leq\frac{1}{2}\int_{\\_}{{\sf X}\times
A^{n}_{\\_}i}|A|^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})+\mathfrak{m}_{\\_}{\sf
Y}(A^{n}_{\\_}i){\sf E}_{\\_}{\sf X}(g^{n}_{\\_}i)\\\
&\leq\frac{1}{2}\int_{\\_}{{\sf X}\times
A^{n}_{\\_}i}|A|^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})+\mathfrak{m}_{\\_}{\sf
Y}(A^{n}_{\\_}i){\sf E}_{\\_}{\sf X}\big{(}{{\sf h}}_{\\_}{{\sf
X},t/2}(f^{n}_{\\_}i)\big{)}.\end{split}$

Coupling these last two (in)equalities with (3.1.7) we deduce that

$\mathfrak{m}_{\\_}{\sf Y}(A^{n}_{\\_}i){\sf E}_{\\_}{\sf
X}(f^{n}_{\\_}i)=\lim_{\\_}{t\downarrow 0}\mathfrak{m}_{\\_}{\sf
Y}(A^{n}_{\\_}i){\sf E}_{\\_}{\sf X}\big{(}{{\sf h}}_{\\_}{{\sf
X},t/2}(f^{n}_{\\_}i)\big{)}\leq\frac{1}{2}\int_{\\_}{{\sf X}\times
A^{n}_{\\_}i}|A|^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}).$

Summing over $i$ and recalling (3.1.6) we conclude that

$\int_{\\_}{\sf Y}{\sf E}_{\\_}{\sf
X}(f(\cdot,y))\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
Y}(y)\leq\frac{1}{2}\int_{\\_}{{\sf X}\times{\sf
Y}}|A|^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}),$

showing that $f\in L^{2}({\sf Y},W^{1,2}({\sf X}))$ (recall Proposition 3.2),
as desired.

To conclude that $A={\mathrm{d}}_{\\_}{\sf x}f$ is the only choice for which
(3.1.5) holds, it is enough to show that the vector space generated by
elements of the form $h{\mathrm{d}}_{\\_}{\sf x}g$ with $h,g$ as in the
statement is dense in $L^{2}({\sf Y};L^{2}(T^{*}{\sf X}))$. To see this,
notice that arguing as we just did the closure of such space contains all the
elements of the form ${\raise 1.29167pt\hbox{$\chi$}}_{\\_}{B\times
C}{\mathrm{d}}_{\\_}{\sf x}g$ for $B,C$ Borel subsets of ${\sf X},{\sf Y}$
respectively and $g\in W^{1,2}({\sf X})$. Then the conclusion follows from the
fact that $L^{2}(T^{*}{\sf X})$ is generated by differentials of
$W^{1,2}$-functions and the fact that piecewise constant maps from ${\sf Y}$
to $L^{2}(T^{*}{\sf X})$ are dense in $L^{2}({\sf Y};L^{2}(T^{*}{\sf X}))$.
$\square$

###  3.2 Sobolev maps with values in a Hilbert module

In this section we study the differential of a function on ${\sf X}$ with
values in a Hilbert module $\mathscr{H}$ over ${\sf Y}$. The prototype case we
want to cover is that of $\mathscr{H}:=L^{2}(T^{*}{\sf Y})$ as we aim to give
a meaning, in the non-smooth context, to the Sobolev regularity of
$x\mapsto{\mathrm{d}}_{\\_}{\sf y}f$ and to its differential
${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f$.

For the purpose of the present discussion, a Hilbert module on ${\sf Y}$ is a
$L^{2}$-normed $L^{\infty}$-module on ${\sf Y}$. Given such a module
$\mathscr{H}$ and its dimensional decomposition
$(E_{\\_}i)_{\\_}{i\in\mathbb{N}\cup\\{\infty\\}}$ (see [12]), a local Hilbert
base $(e_{\\_}i)_{\\_}{i\in\mathbb{N}}$ is a collection of elements of the
$L^{0}$-completion of $\mathscr{H}$ such that for every $n\in\mathbb{N}$ we
have $\left<e_{\\_}i,e_{\\_}j\right>=\delta_{\\_}{ij}$ on $E_{\\_}{n}$ for
every $i,j<n$.

######  Definition 3.7 (The space $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces, with
${\sf X}$ infinitesimally Hilbertian. Moreover, let $\mathscr{H}$ be a
separable Hilbert module over ${\sf Y}$, and
$(e_{\\_}i)_{\\_}{i\in\mathbb{N}}$ be a local Hilbert base of $\mathscr{H}$
with $|e_{\\_}i|\in L^{\infty}({\sf Y})$ for any $i\in\mathbb{N}$.

Then the space $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})\subset L^{0}({\sf X};\mathscr{H})$ is the
space of functions $f$ such that:

  * i)

the map $y\mapsto\left<f(\cdot),e_{\\_}i\right>(y)$ is in $L^{0}({\sf
Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$, for every $i\in\mathbb{N}$,

  * ii)

the function $|{\mathrm{d}}_{\\_}{\sf x}f|\colon{\sf X}\times{\sf
Y}\to[0,\infty]$ defined by

$|{\mathrm{d}}_{\\_}{\sf x}f|^{2}:=\sum_{\\_}i|{\mathrm{d}}_{\\_}{\sf
x}(\Phi_{\\_}i\circ f)|^{2}$ (3.2.1)

belongs to $L^{0}({\sf Y};L^{2}_{\\_}\mathsf{loc}({\sf X}))$ (i.e. is such
that $|{\mathrm{d}}_{\\_}{\sf x}f|(\cdot,y)\in L^{2}_{\\_}\mathsf{loc}({\sf
X})$ for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$), where
$\Phi_{\\_}i:\mathscr{H}\to L^{0}({\sf Y})$ is given by
$\Phi_{\\_}i(v):=\left<v,e_{\\_}i\right>$.

By $W^{1,2}({\sf X};\mathscr{H})\subset S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ we denote the space of functions $f\in
S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$ with
$|f|,|{\mathrm{d}}_{\\_}{\sf x}f|\in L^{2}({\sf X}\times{\sf Y})$.

######  Remark 3.8.

Picking $\mathscr{H}:=L^{2}({\sf Y})$ and comparing the above with Definition
3.1 we see that $S^{2}_{\\_}\mathsf{loc}({\sf X};L^{2}_{\\_}\mathsf{loc}({\sf
Y}))=L^{0}({\sf Y},S^{2}_{\\_}\mathsf{loc}({\sf X}))$.

Then, taking into account that the infinitesimal Hilbertianity of ${\sf X}$
yields the separability of $W^{1,2}({\sf X})$ and Proposition 3.2 it is clear
that $W^{1,2}({\sf X};L^{2}({\sf Y}))\sim L^{2}({\sf Y};W^{1,2}({\sf X}))$ as
both spaces are made of those functions $f\in L^{2}({\sf X}\times{\sf Y})$
such that $f(\cdot,y)\in W^{1,2}({\sf X})$ for $\mathfrak{m}_{\\_}{\sf
Y}$-a.e. $y$ and so that $|{\mathrm{d}}_{\\_}{\sf x}f|\in L^{2}({\sf
X}\times{\sf Y})$.  $\blacksquare$

######  Remark 3.9.

A priori, Definition 3.7 depends on the particular fixed local Hilbert base
$(e_{\\_}i)_{\\_}{i\in\mathbb{N}}$ but we shall see in Proposition 3.10 below
that this is not the case. Such independence is also based on the assumption
that ${\sf X}$ is infinitesimally Hilbertian, which is needed in order to
ensure that $\lvert{\mathrm{d}}_{\\_}{\sf x}f\rvert$ is well defined (i.e.
that it does not depend on the particular base chosen). In fact, the proof of
Proposition 3.10 below is based on the fact that we can make use of the
parallelogram identity in the chain of equalities (3.2.2), and this is
actually guaranteed by the fact that ${\sf X}$ is infinitesimally Hilbertian.
$\blacksquare$

######  Proposition 3.10.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces, with
${\sf X}$ infinitesimally Hilbertian and let $\mathscr{H}$ be a separable
Hilbert module over ${\sf Y}$.

Then the space $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ does not depend on the particular local
base $(e_{\\_}i)$ of $\mathscr{H}$ chosen and for $f\in
S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$ the function
$|{\mathrm{d}}_{\\_}{\sf x}f|$ defined by (3.2.1) also does not depend on the
base.

Moreover the following holds:

  * i)

_Locality._ For any $f\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ we have

$|{\mathrm{d}}_{\\_}{\sf x}f|=0\qquad\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e.\ on\ \\{f=0\\}.$

  * ii)

_Subadditivity._ For any $f,g\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ and $\alpha,\beta\in\mathbb{R}$ we have

$|{\mathrm{d}}_{\\_}{\sf x}(\alpha f+\beta
g)|\leq|\alpha||{\mathrm{d}}_{\\_}{\sf x}f|+|\beta||{\mathrm{d}}_{\\_}{\sf
x}g|\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e..$

  * iii)

_Lower semicontinuity._ Let $(f_{\\_}n)\subset\mathit{W}^{1,2}({\sf
X};\mathscr{H})$ be weakly $L^{2}({\sf X};\mathscr{H})$-converging to $f\in
L^{2}({\sf X};\mathscr{H})$ and such that $|\mathrm{D}_{\\_}{\sf
x}f_{\\_}n|\rightharpoonup G$ in the weak topology of $L^{2}({\sf X}\times{\sf
Y})$ for some $G$. Then $f\in\mathit{W}^{1,2}({{\sf X};\mathscr{H}})$ and
$|\mathrm{D}_{\\_}{\sf x}f|\leq G$ $\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e..

Proof. Let $\\{e_{\\_}i\\}$ be a local Hilbert base of $\mathscr{H}$ such that
$f\in S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$ in the
sense of the Definition 3.7 above, and let $\\{\tilde{e}_{\\_}j\\}$ be another
local Hilbert base of $\mathscr{H}$. Put $f_{\\_}n:=\sum_{\\_}{i\leq
n}e_{\\_}i\Phi^{i}\circ f$ so that $f_{\\_}n\to f$ $\mathfrak{m}_{\\_}{\sf
X}$-a.e. as $n\to\infty$ and, as direct consequence of the definitions,
$f_{\\_}n\in S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$.

We have $\langle f_{\\_}n,\tilde{e}_{\\_}j\rangle=\sum_{\\_}{i\leq
n}\langle\tilde{e}_{\\_}j,e_{\\_}i\rangle\Phi^{i}\circ f_{\\_}n$, which is
trivially an element in $L^{0}({\sf Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$
(notice that $\Phi^{i}\circ f_{\\_}n\in S^{2}_{\\_}\mathsf{loc}({\sf X})$ by
assumption and $\langle\tilde{e}_{\\_}j,e_{\\_}i\rangle\in L^{0}({\sf Y})$,
hence both functions can be seen as elements of $L^{0}({\sf
Y},S^{2}_{\\_}\mathsf{loc}({\sf X}))$ and then apply the Leibniz rule in
Proposition 3.4), and

$\begin{split}\sum_{\\_}j|{\mathrm{d}}_{\\_}{\sf x}\langle
f_{\\_}n,\tilde{e}_{\\_}j\rangle|^{2}&=\sum_{\\_}j\sum_{\\_}{i,k\leq
n}\langle\tilde{e}_{\\_}j,e_{\\_}i\rangle\langle\tilde{e}_{\\_}j,e_{\\_}k\rangle\,\left<{\mathrm{d}}_{\\_}{\sf
x}(\Phi^{i}\circ f_{\\_}n),{\mathrm{d}}_{\\_}{\sf x}(\Phi^{k}\circ
f_{\\_}n)\right>\\\ &=\sum_{\\_}{i,k\leq n}\left<{\mathrm{d}}_{\\_}{\sf
x}(\Phi^{i}\circ f_{\\_}n),{\mathrm{d}}_{\\_}{\sf x}(\Phi^{k}\circ
f_{\\_}n)\right>\underbrace{\sum_{\\_}j\langle\tilde{e}_{\\_}j,e_{\\_}i\rangle\langle\tilde{e}_{\\_}j,e_{\\_}k\rangle}_{\\_}{=\left<e_{\\_}i,e_{\\_}k\right>=\delta_{\\_}{ik}}=\sum_{\\_}{i\leq
n}|{\mathrm{d}}_{\\_}{\sf x}(\Phi^{i}\circ f_{\\_}n)|^{2},\end{split}$ (3.2.2)

where the order of summation can be swapped because $i,k$ run over a finite
set. In particular we have

$\sum_{\\_}j|{\mathrm{d}}_{\\_}{\sf x}\langle
f_{\\_}n,\tilde{e}_{\\_}j\rangle|^{2}\leq\sum_{\\_}{i}|{\mathrm{d}}_{\\_}{\sf
x}(\Phi^{i}\circ f)|^{2}\quad\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e.\ \qquad\forall n\in\mathbb{N}.$
(3.2.3)

It follows that for every bounded Borel subset $B\subset{\sf X}$ we have that
for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ the sequence of functions
$({\raise 1.29167pt\hbox{$\chi$}}_{\\_}B|{\mathrm{d}}_{\\_}{\sf x}\langle
f_{\\_}n,\tilde{e}_{\\_}j\rangle|(\cdot,y))$ is bounded in $L^{2}({\sf X})$.
Also, for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ the functions
$(\langle f_{\\_}n(\cdot),\tilde{e}_{\\_}j\rangle(y))$ converge to $\langle
f(\cdot),\tilde{e}_{\\_}j\rangle(y)$ $\mathfrak{m}_{\\_}{\sf X}$-a.e., hence
what just said and the lower semicontinuity of minimal weak upper gradients
gives that $\langle f(\cdot),\tilde{e}_{\\_}j\rangle(y)\in
S^{2}_{\\_}\mathsf{loc}({\sf X})$ with $|{\mathrm{d}}_{\\_}{\sf x}\langle
f,\tilde{e}_{\\_}j\rangle|(\cdot,y)\lower 3.0pt\hbox{$|_{B}$}\leq
G_{\\_}j(\cdot,y)$ $\mathfrak{m}_{\\_}{\sf X}$-a.e., where $G_{\\_}j$ is any
weak $L^{2}$-limit of some subsequence of $({\raise
1.29167pt\hbox{$\chi$}}_{\\_}B|{\mathrm{d}}_{\\_}{\sf x}\langle
f_{\\_}n,\tilde{e}_{\\_}j\rangle|(\cdot,y))$. Then from Lemma 3.11 below,
(3.2.3) and the arbitrariness of $B$ we deduce that

$\sum_{\\_}j|{\mathrm{d}}_{\\_}{\sf x}\langle
f,\tilde{e}_{\\_}j\rangle|^{2}\leq\sum_{\\_}{i}|{\mathrm{d}}_{\\_}{\sf
x}(\Phi^{i}\circ f)|^{2}\quad\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e..$

Swapping the roles of the bases $(e_{\\_}i)$ and $(\tilde{e}_{\\_}j)$ we
conclude.

Finally, properties $(i),(ii),(iii)$ are direct consequences of the
definitions and the analogous properties for $L^{2}({\sf Y})$-valued functions
(for $(iii)$ we also use Lemma 3.11 below).  $\square$

######  Lemma 3.11.

Let $({\sf X},{\sf d},\mathfrak{m})$ be a metric measure space and for every
$i\in\mathbb{N}$ let $(g^{i}_{\\_}n)\subset L^{2}({\sf X})$ be a sequence of
non-negative functions such that
$\sqrt{\sum_{\\_}i|g^{i}_{\\_}n|^{2}}\rightharpoonup G$ in $L^{2}({\sf X})$
for some $G\geq 0$ as $n\to\infty$. Also, let $g^{i}\in L^{2}({\sf X})$ be
non-negative and such that $g^{i}\leq G^{i}$ $\mathfrak{m}$-a.e. for any weak
$L^{2}$-limit $G^{i}$ of $(g^{i}_{\\_}n)$. Then

$\sqrt{\sum_{\\_}n|g^{i}|^{2}}\leq G\qquad\mathfrak{m}-a.e..$

Proof. With a diagonalization argument, up to pass to a subsequence we can
assume that $g^{i}_{\\_}n\rightharpoonup G^{i}$ for some $(G^{i})\subset
L^{2}({\sf X})$. Now let $A\subset\ell_{\\_}2$ be countable and dense in the
set of non-negative sequences of $\ell_{\\_}2$-norm $\leq 1$ and notice that

$\sqrt{\sum_{\\_}i|h_{\\_}i|^{2}}=\sup_{\\_}{(a_{\\_}i)\in
A}\sum_{\\_}ia_{\\_}ih_{\\_}i\quad\text{for any sequence of non-negative
numbers $h_{\\_}i$}.$ (3.2.4)

For any $(a_{\\_}i)\in A$, $\mathfrak{m}$-a.e. we have

$\begin{split}\sum_{\\_}ia_{\\_}ig^{i}\leq\sum_{\\_}ia_{\\_}iG^{i}=\lim_{\\_}{N\to\infty}\sum_{\\_}{i=1}^{N}a_{\\_}iG^{i}=\lim_{\\_}{N\to\infty}\operatorname*{\text{weak-}L^{2}\text{-lim}}_{\\_}{n\to\infty}\sum_{\\_}{i=1}^{N}a_{\\_}ig^{i}_{\\_}n\leq\operatorname*{\text{weak-}L^{2}\text{-lim}}_{\\_}{n\to\infty}\sum_{\\_}{i}a_{\\_}ig^{i}_{\\_}n.\end{split}$

Now observe that since for every $n\in\mathbb{N}$ we have that
$\mathfrak{m}$-a.e. it holds
$\sum_{\\_}{i}a_{\\_}ig^{i}_{\\_}n\leq\sqrt{\sum_{\\_}i|g^{i}_{\\_}n|^{2}}$,
the same relation is in place for the respective weak $L^{2}$-limits, thus
from the above we get $\sum_{\\_}ia_{\\_}ig^{i}\leq G$ $\mathfrak{m}$-a.e..
Then the arbitrariness of $(a_{\\_}i)\in A$ and (3.2.4) give the conclusion.
$\square$

To the Sobolev space $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ we can canonically associate a
differentiation operator:

######  Theorem 3.12 (Module-valued partial derivatives).

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces, with
${\sf X}$ infinitesimally Hilbertian and let $\mathscr{H}$ be a separable
Hilbert module over ${\sf Y}$.

Then there exists a unique couple $(L^{0}(T^{*}{\sf
X};\mathscr{H}),{\mathrm{d}}_{\\_}{\sf x})$ where $L^{0}(T^{*}{\sf
X};\mathscr{H})$ is a $L^{0}({\sf X}\times{\sf Y})$-normed module,
${\mathrm{d}}_{\\_}{\sf x}:S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})\to L^{0}(T^{*}{\sf X};\mathscr{H}^{0})$ is
linear and satisfies

  * i)

for any $f\in S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$
the pointwise norm of ${\mathrm{d}}_{\\_}{\sf x}f$ coincides with
$|{\mathrm{d}}_{\\_}{\sf x}f|$ $\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e.,

  * ii)

$L^{0}({\sf X}\times{\sf Y})$-linear combinations of elements of the form
${\mathrm{d}}_{\\_}{\sf x}f$ for $f\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ are dense in $L^{0}(T^{*}{\sf
X};\mathscr{H})$.

Uniqueness here is intended up to unique isomorphism, i.e. if
$(\tilde{\mathscr{M}},\tilde{\mathrm{d}}_{\\_}{\sf x})$ is another such
couple, then there is a unique isomorphism $\Phi:L^{0}(T^{*}{\sf
X};\mathscr{H})\to\tilde{\mathscr{M}}$ such that
$\Phi\circ{\mathrm{d}}_{\\_}{\sf x}=\tilde{\mathrm{d}}_{\\_}{\sf x}$.

An explicit example of couple as above is given by the module

$L^{0}(T^{*}{\sf X};\mathscr{H}):=L^{0}({\sf Y};L^{0}(T^{*}{\sf X}))\otimes
L^{0}({\sf X};\mathscr{H})$ (3.2.5)

and the operator

$S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})\ni
f\quad\mapsto\quad{\mathrm{d}}_{\\_}{\sf
x}f:=\sum_{\\_}i{\mathrm{d}}_{\\_}{\sf x}(\Phi_{\\_}i\circ
f)\otimes\hat{e}_{\\_}i,$ (3.2.6)

where $(e_{\\_}i)\subset\mathscr{H}$ is a local Hilbert base of $\mathscr{H}$,
$\hat{e}_{\\_}i\in L^{0}({\sf X};\mathscr{H})$ is the function constantly
equal to $e_{\\_}i$ and $\Phi_{\\_}i:\mathscr{H}\to L^{0}({\sf Y})$ is given
by $\Phi_{\\_}i(\cdot):=\left<\cdot,e_{\\_}i\right>$, for any
$i\in\mathbb{N}$.

Proof. Uniqueness is a simple consequence of the definitions, see e.g. the
arguments in [16, Theorem/Definition 3.2] and notice that they can be easily
adapted - we omit the details. Existence also follows by mimicking the
construction in [12], [16] or, alternatively, by the explicit construction we
provide below.

To check that the couple made by the module $L^{0}({\sf Y};L^{0}(T^{*}{\sf
X}))\otimes L^{0}({\sf X};\mathscr{H}^{0})$ and the operator
${\mathrm{d}}_{\\_}{\sf x}$ as defined by (3.2.6) satisfies the requirements,
we must first check that ${\mathrm{d}}_{\\_}{\sf x}$ is well defined, i.e.
that the series in (3.2.6) converges in $L^{0}({\sf Y};L^{0}(T^{*}{\sf
X}))\otimes L^{0}({\sf X};\mathscr{H}^{0})$. To see this, notice that the
addends are pointwise orthogonal and that
$\\{\hat{e}_{\\_}i=0\\}\subset\\{\Phi^{i}\circ f=0\\}$, thus for any
$N,M\in\mathbb{N}$, $N<M$ we have

$\Big{|}\sum_{\\_}{i=N}^{M}{\mathrm{d}}_{\\_}{\sf x}(\Phi_{\\_}i\circ
f)\otimes\hat{e}_{\\_}i\Big{|}^{2}=\sum_{\\_}{i=N}^{M}|{\mathrm{d}}_{\\_}{\sf
x}(\Phi_{\\_}i\circ f)|^{2},$ (3.2.7)

and since for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$ and bounded Borel
set $B\subset{\sf X}$ the series $\sum_{\\_}i|{\mathrm{d}}_{\\_}{\sf
x}(\Phi_{\\_}i\circ f)|^{2}(\cdot,y){\raise 1.29167pt\hbox{$\chi$}}_{\\_}B$ is
convergent in $L^{2}({\sf X})$, this is sufficient to conclude that the series
in (3.2.6) converges in $L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes L^{0}({\sf
X};\mathscr{H}^{0})$.

Then the fact that ${\mathrm{d}}_{\\_}{\sf x}$ is linear is obvious from the
definition, while the fact that $(i)$ holds follows from (3.2.7) above.

To check $(ii)$, notice that for any $f\in L^{0}({\sf
Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$ and $i\in\mathbb{N}$, the function
$F:=fe_{\\_}i\colon{\sf X}\to\mathscr{H}$, defined by
$F(x)=f(x,\cdot)e_{\\_}i$ for $\mathfrak{m}_{\\_}{\sf X}$-a.e. $x\in{\sf X}$,
satisfies $\Phi^{j}\circ F=0$ for $j\neq i$ and $\Phi^{i}\circ F=f{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times\\{e_{\\_}i\neq 0\\}}$. In
particular, directly from Definition 3.1 we have that $\Phi^{i}\circ F\in
L^{0}({\sf Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$ with ${\mathrm{d}}_{\\_}{\sf
x}(\Phi^{i}\circ F)={\raise 1.29167pt\hbox{$\chi$}}_{\\_}{{\sf
X}\times\\{e_{\\_}i\neq 0\\}}{\mathrm{d}}_{\\_}{\sf x}f$. It is then clear, by
the very Definition 3.7, that $F\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ with

${\mathrm{d}}_{\\_}{\sf x}F=({\raise 1.29167pt\hbox{$\chi$}}_{\\_}{{\sf
X}\times\\{e_{\\_}i\neq 0\\}}{\mathrm{d}}_{\\_}{\sf
x}f)\otimes\hat{e}_{\\_}i={\mathrm{d}}_{\\_}{\sf x}f\otimes({\raise
1.29167pt\hbox{$\chi$}}_{\\_}{{\sf X}\times\\{e_{\\_}i\neq
0\\}}\hat{e}_{\\_}i)={\mathrm{d}}_{\\_}{\sf x}f\otimes\hat{e}_{\\_}i.$

The conclusion follows from the fact that differentials of functions in
$L^{0}({\sf Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$ generate $L^{0}({\sf
Y};T^{*}{\sf X}))$ (trivial consequence of the fact that differentials of
functions on $S^{2}_{\\_}\mathsf{loc}({\sf X})$ generate $L^{0}(T^{*}{\sf
X})$) and the family $(\hat{e}_{\\_}i)$ generates $L^{0}({\sf
X};\mathscr{H})$.  $\square$

The explicit representation as in (3.2.6) allows some manipulation of the
object ${\mathrm{d}}_{\\_}{\sf x}f$. For instance, if $\mathscr{H}^{\prime}$
is another Hilbert module over ${\sf Y}$ and $T\colon L^{0}({\sf
X};\mathscr{H})\to L^{0}({\sf X};\mathscr{H}^{\prime})$ is a module morphism,
then $T$ induces a map, still denoted by $T$ (with a slight abuse of notation,
as perhaps ${\rm Id}\otimes T$ would be the canonical choice), from
$L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes L^{0}({\sf X};\mathscr{H})$ to
$L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes L^{0}({\sf
X};\mathscr{H}^{\prime})$ by acting on the ‘second factors’. More precisely,
the map

$L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes L^{0}({\sf
X};\mathscr{H})\ni\sum_{\\_}{i=1}^{n}\omega_{\\_}i\otimes
v_{\\_}i\quad\mapsto\quad\sum_{\\_}{i=1}^{n}\omega_{\\_}i\otimes(Tv_{\\_}i)\in
L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes L^{0}({\sf X};\mathscr{H}^{\prime})$

which is trivially well defined by the $L^{0}({\sf X}\times{\sf Y})$-linearity
of $T$, can be shown (see below) to be continuous, and thus can be uniquely
extended to a continuous map from $L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes
L^{0}({\sf X};\mathscr{H})$ to $L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes
L^{0}({\sf X};\mathscr{H}^{\prime})$. To check the continuity one possibility
is to notice that we can always rewrite a finite sum such as
$\sum_{\\_}{i=1}^{n}\omega_{\\_}i\otimes v_{\\_}i$ so that the
$\omega_{\\_}i$’s are pointwise orthogonal and once this is done we have

$\Big{|}\sum_{\\_}{i=1}^{n}\omega_{\\_}i\otimes(Tv_{\\_}i)\Big{|}^{2}=\sum_{\\_}{i=1}^{n}|\omega_{\\_}i|^{2}|Tv_{\\_}i|^{2}\leq|T|^{2}\sum_{\\_}{i=1}^{n}|\omega_{\\_}i|^{2}|v_{\\_}i|^{2}=|T|^{2}\,\Big{|}\sum_{\\_}{i=1}^{n}\omega_{\\_}i\otimes
v_{\\_}i\Big{|}^{2}.$

Thus for $T$ as above it makes sense to speak about $T({\mathrm{d}}_{\\_}{\sf
x}f)$ for $f\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$.

We shall mostly apply this construction in two cases: either when - as in
formula (3.2.9) - $T:\mathscr{H}\to\mathscr{H}^{\prime}$ is a module morphism
which induces by post-composition a module morphism from $L^{0}({\sf
X};\mathscr{H})$ to $L^{0}({\sf X};\mathscr{H}^{\prime})$, or when - as in
formula (3.2.8) - a given element $w\in L^{0}({\sf X};\mathscr{H})$ is
considered and $T:L^{0}({\sf X};\mathscr{H})\to L^{0}({\sf X}\times{\sf Y})$
is given by $T(v):=\left<v,w\right>$.

We now collect some properties of the newly defined differentiation operator:

######  Proposition 3.13.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces, with
${\sf X}$ infinitesimally Hilbertian and let $\mathscr{H}$ be a separable
Hilbert module over ${\sf Y}$. Then:

  * i)

_Closure._ Let $(f_{\\_}n)\subset S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc}))$ be $\mathfrak{m}_{\\_}{\sf X}$-a.e.
converging to some $f\in L^{0}({\sf X};\mathscr{H})$. Assume also that for
some open sets $\Omega^{k}\subset{\sf X}$ (resp. Borel subsets
$B^{k}\subset{\sf Y}$) with $\cup_{\\_}k\Omega^{k}={\sf X}$ (resp.
$\mathfrak{m}_{\\_}{\sf Y}({\sf Y}\setminus\cup_{\\_}kB^{k})=0$) and
$\omega\in L^{0}(T^{*}{\sf X};\mathscr{H})$ we have that $({\raise
1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}{\mathrm{d}}_{\\_}{\sf x}f_{\\_}n)$
converges to ${\raise 1.29167pt\hbox{$\chi$}}_{\\_}{\Omega^{k}}{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}\omega$ in the weak topology of
$L^{2}(T^{*}{\sf X};\mathscr{H})$ (this happens or instance if
$({\mathrm{d}}_{\\_}{\sf x}f_{\\_}n\rightharpoonup\omega)$ in $L^{2}(T^{*}{\sf
X};\mathscr{H})$).

Then $f\in S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc}))$
with ${\mathrm{d}}_{\\_}{\sf x}f=\omega$.

  * ii)

_Locality._ For any $f,g\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ we have

${\mathrm{d}}_{\\_}{\sf x}f={\mathrm{d}}_{\\_}{\sf
x}g\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf Y}-a.e.\ on\
\\{f=g\\}.$

  * iii)

_Leibniz rule._ For any $f,g\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$ with $|f|,|g|\in
L^{\infty}_{\\_}\mathsf{loc}({\sf X}\times{\sf Y})$ we have
$\left<f,g\right>\in S^{2}_{\\_}\mathsf{loc}({\sf
X};L^{0}_{\\_}\mathsf{loc}({\sf Y}))$ with

${\mathrm{d}}_{\\_}{\sf x}\left<f,g\right>=\left<{\mathrm{d}}_{\\_}{\sf
x}f,g\right>+\left<f,{\mathrm{d}}_{\\_}{\sf x}g\right>.$ (3.2.8)

  * iv)

_Chain rule._ Let $\mathscr{H}^{\prime}$ be another Hilbert module over ${\sf
Y}$, $T:\mathscr{H}\to\mathscr{H}^{\prime}$ a module morphism with
$|\mathit{T}|\in L^{\infty}({\sf Y})$ and $f\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc})$. Then $T\circ f\in
S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}^{\prime}_{\\_}\mathsf{loc})$ and

${\mathrm{d}}_{\\_}{\sf x}(T\circ f)=T({\mathrm{d}}_{\\_}{\sf x}f).$ (3.2.9)

Proof. The closure property $(i)$ is a direct consequence of the the very
definition (3.2.6) and of the closure of the differential of functions in
$L^{0}({\sf Y};S^{2}_{\\_}\mathsf{loc}({\sf X}))$ established in point $(i)$
of Proposition 3.4. Similarly, point $(ii)$ is a direct consequence of the
analogous property (3.1.1) and Definition (3.2.6).

For what concerns $(iii)$, let $(e_{\\_}i)$ be a local Hilbert base of
$\mathscr{H}$, write $f_{\\_}i:=\Phi^{i}\circ f$, $g_{\\_}i:=\Phi^{i}\circ g$
for brevity and notice that $\left<f,g\right>=\sum_{\\_}if_{\\_}ig_{\\_}i$
with the series converging $\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}$-a.e.. Then observe that from the Leibniz
rule proved in point $(iv)$ of Proposition 3.4 we have that

${\mathrm{d}}_{\\_}{\sf
x}\sum_{\\_}{i=1}^{N}f_{\\_}ig_{\\_}i=\sum_{\\_}{i=1}^{N}g_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i+\sum_{\\_}{i=1}^{N}f_{\\_}i{\mathrm{d}}_{\\_}{\sf x}g_{\\_}i,$

thus by the closure of the differential the conclusion will follow if we prove
that for any $\Omega\subset{\sf X}$ open and bounded there is a sequence
$k\mapsto B^{k}\subset{\sf Y}$ of Borel sets with $\mathfrak{m}_{\\_}{\sf
Y}({\sf Y}\setminus\cup_{\\_}kB^{k})=0$ such that ${\raise
1.29167pt\hbox{$\chi$}}_{\\_}\Omega{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}(\sum_{\\_}{i=1}^{N}g_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i+\sum_{\\_}{i=1}^{N}f_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}i)\to{\raise 1.29167pt\hbox{$\chi$}}_{\\_}\Omega{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}(\left<{\mathrm{d}}_{\\_}{\sf
x}f,g\right>+\left<f,{\mathrm{d}}_{\\_}{\sf x}g\right>)$ in $L^{2}(T^{*}{\sf
X};L^{2}({\sf Y}))$ as $N\to\infty$ for every $k\in\mathbb{N}$. We pick
$B^{k}:=\\{y\in{\sf Y}\ :\ \int_{\\_}\Omega|{\mathrm{d}}_{\\_}{\sf
x}f|^{2}(\cdot,y)+|{\mathrm{d}}_{\\_}{\sf
x}g|^{2}(\cdot,y)\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf X}<k\\}\cap
B_{\\_}k(\bar{y})$ for some fixed $\bar{y}\in{\sf Y}$, and notice that the
assumption $f,g\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc}))$ gives $\mathfrak{m}_{\\_}{\sf Y}({\sf
Y}\setminus\cup_{\\_}kB^{k})=0$. Then we observe that

$\begin{split}\int_{\\_}{\Omega\times
B^{k}}\Big{|}\sum_{\\_}{i=N}^{M}g_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\Big{|}^{2}\,{\mathrm{d}}(\mathfrak{m}_{\\_}X\times\mathfrak{m}_{\\_}{\sf
Y})&\leq\int_{\\_}{\Omega\times
B^{k}}{\sum_{\\_}{i=N}^{M}|g_{\\_}i|^{2}}{\sum_{\\_}{i=N}^{M}|{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i|^{2}}\,{\mathrm{d}}(\mathfrak{m}_{\\_}X\times\mathfrak{m}_{\\_}{\sf
Y})\\\ &\leq\||g|\|_{\\_}{L^{\infty}(\Omega\times
B^{k})}\int_{\\_}{\Omega\times B^{k}}{\sum_{\\_}{i\geq
N}|{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i|^{2}}\,{\mathrm{d}}(\mathfrak{m}_{\\_}X\times\mathfrak{m}_{\\_}{\sf
Y})\end{split}$

and that by construction it holds

$\int_{\\_}{\Omega\times B^{k}}{\sum_{\\_}{i\geq 0}|{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i|^{2}}\,{\mathrm{d}}(\mathfrak{m}_{\\_}X\times\mathfrak{m}_{\\_}{\sf
Y})=\int_{\\_}{\Omega\times B^{k}}{\sum_{\\_}{i\geq 0}|{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i|^{2}}\,{\mathrm{d}}(\mathfrak{m}_{\\_}X\times\mathfrak{m}_{\\_}{\sf
Y})\leq k\,\mathfrak{m}_{\\_}{\sf X}(\Omega)\mathfrak{m}_{\\_}{\sf
Y}(B_{\\_}k(\bar{y}))<\infty.$

These show that ${\raise 1.29167pt\hbox{$\chi$}}_{\\_}\Omega{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}\sum_{\\_}{i=1}^{N}g_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\to{\raise 1.29167pt\hbox{$\chi$}}_{\\_}\Omega{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{B^{k}}\sum_{\\_}{i=1}^{\infty}g_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i$ in $L^{2}(T^{*}{\sf X};L^{2}({\sf Y}))$ as $N\to\infty$ for every
$k\in\mathbb{N}$, so that the conclusion follows noticing that
$\left<{\mathrm{d}}_{\\_}{\sf
x}f,g\right>=\sum_{\\_}ig_{\\_}i{\mathrm{d}}_{\\_}{\sf x}f_{\\_}i$ (from
${\mathrm{d}}_{\\_}{\sf x}f=\sum_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\otimes\hat{e}_{\\_}i$ and $g=\sum_{\\_}ig_{\\_}i\hat{e}_{\\_}i$)
and arguing similarly for the other addend.

We pass to $(iv)$ and start claiming that for $f\in
S^{2}_{\\_}\mathsf{loc}({\sf X};L^{0}_{\\_}\mathsf{loc}({\sf Y}))$ and
$v\in\mathscr{H}$ with $|v|\in L^{\infty}({\sf Y})$, the function ${\sf X}\ni
x\mapsto f(x,\cdot)v\in\mathscr{H}$ belongs to $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}_{\\_}\mathsf{loc}))$ with ${\mathrm{d}}_{\\_}{\sf
x}(fv)={\mathrm{d}}_{\\_}{\sf x}f\otimes v$. To see this, notice that we can
write $v=|v|e_{\\_}1$, where $e_{\\_}1\in\mathscr{H}$ is the first element of
some local Hilbert base, use the fact that $f|v|\in
S^{2}_{\\_}\mathsf{loc}({\sf X};L^{0}_{\\_}\mathsf{loc}({\sf Y}))$ with
${\mathrm{d}}_{\\_}{\sf x}(f|v|)=|v|{\mathrm{d}}_{\\_}{\sf x}f$ (point $(iii)$
of Proposition 3.4) and conclude recalling the very definitions of
$S^{2}_{\\_}\mathsf{loc}({\sf X};\mathscr{H}_{\\_}\mathsf{loc})$ and of
${\mathrm{d}}_{\\_}{\sf x}$ given in (3.2.6).

Now denote by $(e_{\\_}i),(e^{\prime}_{\\_}j)$ local Hilbert basis of
$\mathscr{H},\mathscr{H}^{\prime}$ respectively and put for brevity
$f_{\\_}i:=\Phi^{i}\circ f\in S^{2}_{\\_}\mathsf{loc}({\sf
X};L^{0}_{\\_}\mathsf{loc}({\sf Y}))$ and $a_{\\_}{ij}:=\langle
Te_{\\_}i,e^{\prime}_{\\_}j\rangle\in L^{\infty}({\sf Y})$. Then what we just
proved ensures that for any $i,j$ we have
$f_{\\_}ia_{\\_}{ij}e^{\prime}_{\\_}j\in S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}^{\prime}_{\\_}\mathsf{loc})$ with ${\mathrm{d}}_{\\_}{\sf
x}(f_{\\_}ia_{\\_}{i,j}e^{\prime}_{\\_}j)={\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\otimes(a_{\\_}{ij}e^{\prime}_{\\_}j)$. Adding up in $j$ and using
the closure of the differential proved in point $(i)$ above (notice that
$\sum_{\\_}j|a_{\\_}{ij}|^{2}=|e_{\\_}i|^{2}\in L^{\infty}({\sf Y})$) we
deduce that
$T(f_{\\_}ie_{\\_}i)=f_{\\_}iT(e_{\\_}i)=\sum_{\\_}jf_{\\_}ia_{\\_}{ij}e^{\prime}_{\\_}j$
belongs to $S^{2}_{\\_}\mathsf{loc}({\sf
X};\mathscr{H}^{\prime}_{\\_}\mathsf{loc})$ with

${\mathrm{d}}_{\\_}{\sf
x}\big{(}T(f_{\\_}ie_{\\_}i)\big{)}=\sum_{\\_}j{\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\otimes(a_{\\_}{ij}e^{\prime}_{\\_}j)={\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\otimes\Big{(}\sum_{\\_}ja_{\\_}{ij}e^{\prime}_{\\_}j\Big{)}={\mathrm{d}}_{\\_}{\sf
x}f_{\\_}i\otimes T(e_{\\_}i)=T\big{(}{\mathrm{d}}_{\\_}{\sf x}f_{\\_}i\otimes
e_{\\_}i\big{)},$

where the last identity comes from the very definition of the action of $T$ on
$L^{0}(T^{*}{\sf X};\mathscr{H})=L^{0}(T^{*}{\sf X};L^{0}({\sf Y}))\otimes
L^{0}({\sf X};\mathscr{H})$. The conclusion (3.2.9) now follows adding up in
$i$ and using again the closure of the differential.  $\square$

We conclude the section with a statement similar to Proposition 3.6 which
allows to check whether $f$ belongs to $W^{1,2}({\sf X};\mathscr{H})$ via
integration by parts:

######  Proposition 3.14.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two metric measure spaces
satisfying Assumption 2.5 with ${\sf X}$ being infinitesimally Hilbertian,
$\mathscr{H}$ an Hilbert module over ${\sf Y}$ and $f\in L^{2}({\sf
X};\mathscr{H})$. Then the following are equivalent:

  * i)

$f\in W^{1,2}({\sf X};\mathscr{H})$,

  * ii)

There is $A\in L^{0}(T^{*}{\sf X};\mathscr{H})$ with $|A|_{\\_}{\lower
1.29167pt\hbox{\scriptsize{\sf HS}}}\in L^{2}({\sf X}\times{\sf Y})$ and a
generating set $D\subset\mathscr{H}$ made of bounded elements such that

$-\int\left<f,v\right>{\rm div}(h\nabla(g\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})=\int h\left<A,{\mathrm{d}}_{\\_}{\sf x}g\otimes
v\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})$ (3.2.10)

holds for any $v\in D$, $h\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf
X}\times{\sf Y})$ and $g\in D(\Delta_{\\_}{\sf X})$.

Moreover, if this is the case the choice $A={\mathrm{d}}_{\\_}{\sf x}f$ is the
only one for which $(ii)$ holds and (3.2.10) holds for every bounded element
$v\in\mathscr{H}$.

Proof.
$(i)\Rightarrow(ii)$ Let $v\in\mathscr{H}$ be with $|v|\in L^{\infty}$ and
$T:\mathscr{H}\to L^{0}({\sf Y})$ be given by
$T(\cdot):=\left<\cdot,v\right>$. Then point $(iv)$ in Proposition 3.13 gives
that $\left<f,v\right>\in W^{1,2}({\sf X};L^{2}({\sf Y}))$ with
${\mathrm{d}}_{\\_}{\sf x}(\left<f,v\right>)=\left<{\mathrm{d}}_{\\_}{\sf
x}f,v\right>\in L^{2}({\sf Y};L^{2}(T^{*}{\sf X}))$. Thus we can apply (3.1.5)
of Proposition 3.6 to $\left<f,v\right>$ with $g,h$ as in the statement to get

$\begin{split}-\int\left<f,v\right>{\rm div}(h\nabla(g\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})&=\int h\langle\left<{\mathrm{d}}_{\\_}{\sf
x}f,v\right>,{\mathrm{d}}_{\\_}{\sf
x}g\rangle\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})\\\ &=\int h\left<{\mathrm{d}}_{\\_}{\sf
x}f,{\mathrm{d}}_{\\_}{\sf x}g\otimes
v\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}),\end{split}$

which is the claim with $A={\mathrm{d}}_{\\_}{\sf x}f$.

$(ii)\Rightarrow(i)$ Applying Proposition 3.6 to $\left<f,v\right>$ for $v\in
D$ we obtain that $\left<f,v\right>\in W^{1,2}({\sf X};L^{2}({\sf Y}))$ with
${\mathrm{d}}_{\\_}{\sf x}\left<f,v\right>=\left<A,v\right>$ (this expression
meaning that $\left<{\mathrm{d}}_{\\_}{\sf
x}\left<f,v\right>,\omega\right>=\left<A,\omega\otimes v\right>$ for any
$\omega\in L^{0}({\sf Y};L^{0}(T^{*}{\sf X}))$). By the linearity of the
differential and taking also into account the second claim in the Leibniz rule
in point $(iv)$ of Proposition 3.4 we deduce that for $v_{\\_}i\in D$ and
$a_{\\_}i\in L^{\infty}({\sf Y})$ we have
$\left<f,\sum_{\\_}ia_{\\_}iv_{\\_}i\right>\in W^{1,2}({\sf X};L^{2}({\sf
Y}))$ with

${\mathrm{d}}_{\\_}{\sf
x}\big{\langle}f,\big{(}\sum_{\\_}ia_{\\_}iv_{\\_}i\big{)}\big{\rangle}=\sum_{\\_}ia_{\\_}i{\mathrm{d}}_{\\_}{\sf
x}\langle f,v_{\\_}i\rangle=\sum_{\\_}ia_{\\_}i\left<A,v_{\\_}i\right>=\langle
A,\sum_{\\_}ia_{\\_}iv_{\\_}i\rangle,$

where the sums here are finite. From the fact that $D$ generates $\mathscr{H}$
it is easy to deduce that for any $h\in\mathscr{H}$ with $|h|\in
L^{\infty}({\sf Y})$ there exists a sequence of $L^{\infty}$-linear
combinations of elements of $D$ whose pointwise norm is uniformly bounded and
$\mathfrak{m}_{\\_}{\sf Y}$-a.e. converging to $h$. Hence from what we proved
above and the closure of the differential we conclude that for any
$h\in\mathscr{H}$ with $|h|\in L^{\infty}$ we have

$\left<f,h\right>\in W^{1,2}({\sf X};L^{2}({\sf Y}))\quad\text{ with
}\quad{\mathrm{d}}_{\\_}{\sf x}\left<f,h\right>=\langle A,h\rangle.$ (3.2.11)

In particular this holds for $h=e_{\\_}i$ with $(e_{\\_}i)$ a local Hilbert
base of $\mathscr{H}$. Hence from the very definition of $W^{1,2}({\sf
X};\mathscr{H})$, the identity
$\sum_{\\_}i|\left<A,e_{\\_}i\right>|^{2}=|A|_{\\_}{\lower
1.29167pt\hbox{\scriptsize{\sf HS}}}^{2}$ and the assumption $|A|_{\\_}{\lower
1.29167pt\hbox{\scriptsize{\sf HS}}}\in L^{2}({\sf X},\times{\sf Y})$ we
conclude that $f\in W^{1,2}({\sf X};\mathscr{H})$.

To obtain that $A={\mathrm{d}}_{\\_}{\sf x}f$ notice that from (3.2.6) we have
$\left<{\mathrm{d}}_{\\_}{\sf x}f,e_{\\_}i\right>={\mathrm{d}}_{\\_}{\sf
x}\left<f,e_{\\_}i\right>$ and recall (3.2.11) to deduce that
${\mathrm{d}}_{\\_}{\sf x}\left<f,h_{\\_}i\right>=\left<A,e_{\\_}i\right>$ for
any $i$. The claim follows.  $\square$

###  3.3 Schwarz’s theorem on symmetry of mixed second derivatives

Let $\mathscr{H}^{1},\mathscr{H}^{2}$ be two Hilbert modules over the same
space (in our case that would be ${\sf X}\times{\sf Y}$). Then there is a
natural _transposition_ map $A\mapsto A^{\rm tr}$ from
$\mathscr{H}^{1}\otimes\mathscr{H}^{2}$ to
$\mathscr{H}^{2}\otimes\mathscr{H}^{1}$, defined as the only
$L^{\infty}$-linear and continuous extension of the map sending
$v_{\\_}1\otimes v_{\\_}2$ to $v_{\\_}2\otimes v_{\\_}1$ for any
$v_{\\_}i\in\mathscr{H}^{i}$, $i=1,2$. It is trivial to check that the
transposition is well defined by this as well as the fact that transposing
twice returns the identity.

Now consider infinitesimally Hilbertian spaces ${\sf X},{\sf Y}$ and a
function $f\in L^{2}({\sf Y};W^{1,2}({\sf X}))$. Then in principle we have
${\mathrm{d}}_{\\_}{\sf x}f\in L^{2}({\sf Y};L^{2}(T^{*}{\sf X}))$. Now assume
that in fact we have more regularity and that in fact it holds
${\mathrm{d}}_{\\_}{\sf x}f\in W^{1,2}({\sf Y};L^{2}(T^{*}{\sf X}))$. Then
recalling (3.2.5) we see that the differential ${\mathrm{d}}_{\\_}{\sf
y}{\mathrm{d}}_{\\_}{\sf x}f$ is in

$L^{2}(T^{*}{\sf Y};L^{2}(T^{*}{\sf X}))\subset L^{0}(T^{*}{\sf
Y};L^{0}(T^{*}{\sf X}))\cong L^{0}({\sf X};L^{0}(T^{*}{\sf Y}))\otimes
L^{0}({\sf Y};L^{0}(T^{*}{\sf X})).$ (3.3.1)

In these circumstances and assuming also that $f\in L^{2}({\sf X};W^{1,2}({\sf
Y}))$ (see Example 3.16 below), it is natural to wonder if we also have
${\mathrm{d}}_{\\_}{\sf y}f\in W^{1,2}({\sf X};L^{2}(T^{*}{\sf Y}))$ and
whether mixed derivatives commute. Inspecting the spaces where
${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f$ and
${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f$ belong, we see that the
correct way to phrase this question is to ask whether

${\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f=({\mathrm{d}}_{\\_}{\sf
y}{\mathrm{d}}_{\\_}{\sf x}f)^{\rm tr}.$ (3.3.2)

holds, where here the transposition is acting on $L^{0}({\sf
X};L^{0}(T^{*}{\sf Y}))\otimes L^{0}({\sf Y};L^{0}(T^{*}{\sf X}))$ (recall
(3.3.1)).

In the following theorem we prove that this is actually the case under natural
assumptions. Before stating it let us remark that for ${\sf X},{\sf Y}$
infinitesimally Hilbertian we have

$W^{1,2}({\sf X}\times{\sf Y})=W^{1,2}({\sf X};L^{2}({\sf Y}))\cap
W^{1,2}({\sf Y};L^{2}({\sf X})),$

as can be seen from Proposition 3.5 together with the equivalence
$W^{1,2}({\sf X};L^{2}({\sf Y}))\sim L^{2}({\sf Y};W^{1,2}({\sf X}))$ pointed
out in Remark 3.8. In particular, for $f\in W^{1,2}({\sf X}\times{\sf Y})$
both ${\mathrm{d}}_{\\_}{\sf x}f$ and ${\mathrm{d}}_{\\_}{\sf y}f$ are defined
and these objects belong to $L^{2}({\sf Y};L^{2}(T^{*}{\sf X}))$ and
$L^{2}({\sf X};L^{2}(T^{*}{\sf Y}))$ respectively.

In the course of the forthcoming proof and also in subsequent part of the
paper, given $f_{\\_}1\in L^{0}({\sf X})$ and $f_{\\_}2\in L^{0}({\sf Y})$ we
shall denote by $f_{\\_}1\otimes f_{\\_}2\in L^{0}({\sf X}\times{\sf Y})$ the
function given by

$f_{\\_}1\otimes f_{\\_}2(x,y):=f_{\\_}1(x)f_{\\_}2(y).$

We then have:

######  Theorem 3.15.

Let $({\sf X},{\sf d}_{\\_}{\sf X},\mathfrak{m}_{\\_}{\sf X}),({\sf Y},{\sf
d}_{\\_}{\sf Y},\mathfrak{m}_{\\_}{\sf Y})$ be two infinitesimally Hilbertian
metric measure spaces satisfying Assumption 2.5. Let $f\in W^{1,2}({\sf
X}\times{\sf Y})$ and assume that ${\mathrm{d}}_{\\_}{\sf x}f\in W^{1,2}({\sf
Y};L^{2}(T^{*}{\sf X}))$.

Then ${\mathrm{d}}_{\\_}{\sf y}f\in W^{1,2}({\sf X};L^{2}(T^{*}{\sf Y}))$ as
well and (3.3.2) holds.

Proof. Let $g_{\\_}1\in D(\Delta_{\\_}{\sf x})\cap\mathop{\rm
Lip}\nolimits({\sf X})$ (resp. $g_{\\_}2\in D(\Delta_{\\_}{\sf
y})\cap\mathop{\rm Lip}\nolimits({\sf Y})$) and $h=h_{\\_}1\otimes
h_{\\_}2\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf X}\times{\sf Y})$ for
$h_{\\_}1\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf X})$ and
$h_{\\_}2\in\mathop{\rm Lip}\nolimits_{\\_}{\sf bs}({\sf Y})$. Then from
Proposition 3.14 applied with the roles of ${\sf X},{\sf Y}$ swapped and
$\mathscr{H}:=L^{2}(T^{*}{\sf X})$ together with the assumption
${\mathrm{d}}_{\\_}{\sf x}f\in W^{1,2}({\sf Y};L^{2}(T^{*}{\sf X}))$ we
conclude that

$-\int\langle{\mathrm{d}}_{\\_}{\sf x}f,{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1\rangle\,{\rm div}(h\nabla(g_{\\_}2\circ\pi_{\\_}{\sf
Y}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})=\int h\langle{\mathrm{d}}_{\\_}{\sf y}{\mathrm{d}}_{\\_}{\sf
x}f,{\mathrm{d}}_{\\_}{\sf y}g_{\\_}2\otimes{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1\rangle\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y}).$ (3.3.3)

Recalling (3.1.4) we get

${\rm div}(h\nabla(g_{\\_}2\circ\pi_{\\_}{\sf Y}))={\rm div}_{\\_}{\sf
y}(h\nabla_{\\_}{\sf y}g_{\\_}2)=h\Delta_{\\_}{\sf
y}g_{\\_}2+\left<{\mathrm{d}}_{\\_}{\sf y}h,{\mathrm{d}}_{\\_}{\sf
y}g_{\\_}2\right>=h\Delta_{\\_}{\sf
y}g_{\\_}2+h_{\\_}1\left<{\mathrm{d}}_{\\_}{\sf
y}h_{\\_}2,{\mathrm{d}}_{\\_}{\sf y}g_{\\_}2\right>,$

where in the last step we used the identity ${\mathrm{d}}_{\\_}{\sf
y}h=h_{\\_}1{\mathrm{d}}_{\\_}{\sf y}h_{\\_}2$ which follows directly from
Definition 3.1. It is clear that the rightmost side in the above belongs to
$L^{2}({\sf Y};W^{1,2}({\sf X}))$ with differential ${\mathrm{d}}_{\\_}{\sf
x}$ given by $h_{\\_}2\Delta_{\\_}{\sf y}g_{\\_}2\,{\mathrm{d}}_{\\_}{\sf
x}h_{\\_}1+\left<{\mathrm{d}}_{\\_}{\sf y}h_{\\_}2,{\mathrm{d}}_{\\_}{\sf
y}g_{\\_}2\right>{\mathrm{d}}_{\\_}{\sf x}h_{\\_}1$, thus we have

$\begin{split}-\int\langle{\mathrm{d}}_{\\_}{\sf x}f,{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1\rangle&\,{\rm div}(h\nabla(g_{\\_}2\circ\pi_{\\_}{\sf
Y}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})\\\ &=-\int\int\langle{\mathrm{d}}_{\\_}{\sf x}f,{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1\rangle\,\big{(}h\Delta_{\\_}{\sf
y}g_{\\_}2+h_{\\_}1\left<{\mathrm{d}}_{\\_}{\sf
y}h_{\\_}2,{\mathrm{d}}_{\\_}{\sf
y}g_{\\_}2\right>\big{)}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}\\\ &=\int\int
f\big{(}h\Delta_{\\_}{\sf x}g_{\\_}1\Delta_{\\_}{\sf
y}g_{\\_}2+h_{\\_}1\Delta_{\\_}{\sf x}g_{\\_}1\left<{\mathrm{d}}_{\\_}{\sf
y}h_{\\_}2,{\mathrm{d}}_{\\_}{\sf y}g_{\\_}2\right>\\\
&\qquad\qquad+h_{\\_}2\Delta_{\\_}{\sf y}g_{\\_}2\left<{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1,{\mathrm{d}}_{\\_}{\sf
x}h_{\\_}1\right>+\left<{\mathrm{d}}_{\\_}{\sf
y}h_{\\_}2,{\mathrm{d}}_{\\_}{\sf
y}g_{\\_}2\right>\left<{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1,{\mathrm{d}}_{\\_}{\sf
x}h_{\\_}1\right>\big{)}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf Y}\\\ &=\cdots\text{(by the symmetry of
the last expression)}\\\ &=-\int\langle{\mathrm{d}}_{\\_}{\sf
y}f,{\mathrm{d}}_{\\_}{\sf y}g_{\\_}2\rangle\,{\rm
div}(h\nabla(g_{\\_}1\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}).\end{split}$

Hence from (3.3.3) we get

$-\int\langle{\mathrm{d}}_{\\_}{\sf y}f,{\mathrm{d}}_{\\_}{\sf
y}g_{\\_}2\rangle\,{\rm div}(h\nabla(g_{\\_}1\circ\pi_{\\_}{\sf
X}))\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y})=\int h\langle{\mathrm{d}}_{\\_}{\sf y}{\mathrm{d}}_{\\_}{\sf
x}f,{\mathrm{d}}_{\\_}{\sf y}g_{\\_}2\otimes{\mathrm{d}}_{\\_}{\sf
x}g_{\\_}1\rangle\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\otimes\mathfrak{m}_{\\_}{\sf Y})$

and the claim follows by the arbitrariness of $g_{\\_}1,g_{\\_}2,h$, by
applying Proposition 3.14 again (with $D:=\\{\nabla_{\\_}{\sf
y}g_{\\_}2:g_{\\_}2\in D(\Delta_{\\_}{\sf y})\cap\mathop{\rm
Lip}\nolimits({\sf Y})\\}$).  $\square$

######  Example 3.16.

Let ${\sf X}={\sf Y}=[0,1]$ with the Euclidean distance and measure and
consider the function $f:={\raise
1.29167pt\hbox{$\chi$}}_{\\_}{[0,1]\times[0,1/2]}\in L^{2}([0,1]\times[0,1])$.
It trivially belongs to $L^{2}({\sf Y};W^{1,2}({\sf X}))$ with
${\mathrm{d}}_{\\_}{\sf x}f=0$, thus evidently ${\mathrm{d}}_{\\_}{\sf x}f\in
W^{1,2}({\sf Y};L^{2}(T^{*}{\sf X}))$ with ${\mathrm{d}}_{\\_}{\sf
y}{\mathrm{d}}_{\\_}{\sf x}f=0$. Yet, obviously $f\notin L^{2}({\sf
X};W^{1,2}({\sf Y}))$, showing that the assumption $f\in L^{2}({\sf
X};W^{1,2}({\sf Y}))$ is necessary in Theorem 3.15.  $\blacksquare$

##  4 Second order partial derivatives on the product space

###  4.1 Basic material

####  4.1.1 Reminders about calculus on ${\sf RCD}$ spaces

Here we briefly recall those notions about calculus on ${\sf RCD}$ spaces that
will be used in the rest of the manuscript. These will be used throughout the
text without further notice.

From now on we fix a metric measure space $({\sf X},\mathsf{d},\mathfrak{m})$,
satisfying the ${\sf RCD}(K,\infty)$ condition for some $K\in\mathbb{R}$,
meaning that it is an infinitesimally Hilbertian metric measure space
satisfying the ${\sf CD}(K,\infty)$ condition of Lott-Villani [18] and Sturm
[21]. This in particular means that the energy functional introduced in
(2.1.1) is a quadratic form: we call heat flow $({{\sf h}}_{\\_}t)$ its
gradient flow in $\mathit{L}^{2}(\mathfrak{m})$, which is linear and self-
adjoint. It is useful to recall the following standard a priori estimates:

$\displaystyle\mathsf{E}({{\sf h}}_{\\_}tf)$
$\displaystyle\leq\frac{1}{4t}\|f\|^{2}_{\\_}{\mathit{L}^{2}(\mathfrak{m})},$
(4.1.1) $\displaystyle\|\Delta{{\sf
h}}_{\\_}tf\|^{2}_{\\_}{\mathit{L}^{2}(\mathfrak{m})}$
$\displaystyle\leq\frac{1}{2t^{2}}\|f\|^{2}_{\\_}{\mathit{L}^{2}(\mathfrak{m})},$
(4.1.2)

valid for every $t>0$ and every $f\in\mathit{L}^{2}(\mathfrak{m})$. Moreover,
it holds

$\|{{\sf
h}}_{\\_}tf\|_{\\_}{L^{2}(\mathfrak{m})}\leq\|f\|_{\\_}{L^{2}(\mathfrak{m})},\qquad\forall
t\geq 0,\,\forall f\in L^{2}(\mathfrak{m}).$ (4.1.3)

Finally, a crucial property of the heat flow which is strongly related to the
lower curvature bound is the Bakry-Émery contraction estimate (see [14] and
[4]):

$\lvert{\mathrm{d}}{{\sf h}}_{\\_}tf\rvert^{2}\leq e^{-2Kt}{{\sf
h}}_{\\_}t(\lvert{\mathrm{d}}f\rvert^{2}),\qquad\mathfrak{m}\text{-a.e.},\quad\forall
t\geq 0,\quad\forall f\in\mathit{W}^{1,2}({\sf X})$ (4.1.4)

from which it follows the $L^{\infty}-\mathop{\rm Lip}\nolimits$
regularization estimate

$\||{\mathrm{d}}{{\sf
h}}_{\\_}tf|\|_{\\_}{L^{\infty}}\leq\frac{1}{\sqrt{2I_{\\_}{2K}(t)}}\|f\|_{\\_}{L^{\infty}},\qquad\forall
f\in L^{2}(\mathfrak{m}),\ t>0,$ (4.1.5)

where $I_{\\_}{K}(t):=\int_{\\_}0^{t}e^{Ks}\,{\mathrm{d}}s$.

Then the space of test functions ${\rm TestF}({\sf X})$ on ${\sf X}$ is
defined as

${\rm TestF}({\sf X}):=\Big{\\{}f\in L^{\infty}\cap W^{1,2}({\sf X})\cap
D(\Delta)\ :\ |{\mathrm{d}}f|\in L^{\infty}({\sf X}),\ \Delta f\in
W^{1,2}({\sf X})\Big{\\}}.$

It turns out that ${\rm TestF}({\sf X})$ is an algebra dense in $W^{1,2}({\sf
X})$ (in particular gradients of test functions generate the whole
$L^{2}(T{\sf X})$) and that $|{\mathrm{d}}f|^{2}\in W^{1,2}({\sf X})$ for any
$f\in{\rm TestF}({\sf X})$ (see [19]). Another direct consequence of the Ricci
curvature lower bound is the following regularity result, which allows to pass
from a Sobolev information to a metric one (see [4], [9], [10]):

$\text{any}\,f\in{\rm TestF}({\sf X})\,\text{ has a Lipschitz representative
}\,\bar{f}\colon{\sf X}\to\mathbb{R}\,\text{ with }\,\mathop{\rm
Lip}\nolimits(\bar{f})\leq\|\lvert\nabla
f\rvert\|_{\\_}{L^{\infty}(\mathfrak{m})}.$ (4.1.6)

Moreover, it is also useful to know that

$f\in L^{2}\cap L^{\infty}(\mathfrak{m})\qquad\Rightarrow\qquad{{\sf
h}}_{\\_}tf\in{\rm TestF}({\sf X}),\ \forall t>0,$ (4.1.7)

which in particular guarantees the density of ${\rm TestF}({\sf X})$ in
$W^{1,2}({\sf X})$, and that

$\forall B\subset{\sf X}\text{ bounded there is $f\in{\rm TestF}({\sf X})$
with bounded support identically 1 on $B$.}$ (4.1.8)

The space $W^{2,2}_{\\_}\mathsf{loc}({\sf X})$ is the space of $f\in
W^{1,2}_{\\_}\mathsf{loc}({\sf X})$ for which there exists $A\in
L^{2}_{\\_}\mathsf{loc}((T^{*})^{\otimes 2}{\sf X})$ such that

$\begin{split}2\int&hA(\nabla g,\nabla\tilde{g})\,{\mathrm{d}}\mathfrak{m}\\\
&=-\int\left<\nabla f,\nabla g\right>{\rm div}(h\nabla\tilde{g})+\left<\nabla
f,\nabla\tilde{g}\right>{\rm div}(h\nabla g)+h\left<\nabla
f,\nabla(\left<\nabla
g,\nabla\tilde{g}\right>)\right>\,{\mathrm{d}}\mathfrak{m}\end{split}$ (4.1.9)

for every $g,\tilde{g},h\in{\rm TestF}({\sf X})$ with bounded support. In this
case we call $A$ the Hessian of $f$ and denote it by ${\rm Hess}f$. If
$f\in\mathit{W}^{1,2}({\sf X})$ and ${\rm Hess}f\in L^{2}((T^{*})^{\otimes
2}{\sf X})$ we say that $f\in W^{2,2}({\sf X})$ (it is not hard to check that
this definition coincides with the one proposed in [12]). The space
$W^{2,2}({\sf X})$ is a separable Hilbert space when endowed the norm

$\|f\|_{\\_}{W^{2,2}({\sf X})}^{2}:=\|f\|_{\\_}{L^{2}({\sf
X})}^{2}+\||\mathrm{D}f|\|_{\\_}{L^{2}({\sf X})}^{2}+\||{\rm
Hess}f|_{\\_}{\lower 1.29167pt\hbox{\scriptsize{\sf HS}}}\|_{\\_}{L^{2}({\sf
X})}^{2}.$

We have $D(\Delta)\subset W^{2,2}({\sf X})$ with

$\int|{\rm Hess}(f)|_{\\_}{\lower 1.29167pt\hbox{\scriptsize{\sf
HS}}}^{2}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf X}\leq\int|\Delta
f|^{2}-K|{\mathrm{d}}f|^{2}\,{\mathrm{d}}\mathfrak{m}_{\\_}{\sf
X}\qquad\forall f\in D(\Delta).$ (4.1.10)

The space $H^{2,2}({\sf X})$ is the $W^{2,2}({\sf X})$-closure of $D(\Delta)$
(or, equivalently, of ${\rm TestF}({\sf X})$) and, similarly,
$H^{2,2}_{\\_}\mathsf{loc}({\sf X})$ as the $W^{2,2}_{\\_}\mathsf{loc}({\sf
X})$-closure of ${\rm TestF}({\sf X})$, i.e.: $f\in
H^{2,2}_{\\_}\mathsf{loc}({\sf X})\subset W^{2,2}_{\\_}\mathsf{loc}({\sf X})$
provided there exists a sequence $(f_{\\_}n)\subset{\rm TestF}({\sf X})$ such
that $f_{\\_}n,{\mathrm{d}}f_{\\_}n,{\rm Hess}{f_{\\_}n}$ converge to
$f,{\mathrm{d}}f,{\rm Hess}f$ in $L^{2}_{\\_}\mathsf{loc}({\sf
X}),L^{2}_{\\_}\mathsf{loc}(T^{*}{\sf
X}),L^{2}_{\\_}\mathsf{loc}((T^{*})^{\otimes 2}{\sf X})$ respectively.

We shall also often use the identity

$2{\rm Hess}f(\nabla g,\nabla
h)={\mathrm{d}}\left<\left<{\mathrm{d}}f,{\mathrm{d}}g\right>,{\mathrm{d}}h\right>+{\mathrm{d}}\left<\left<{\mathrm{d}}f,{\mathrm{d}}h\right>,{\mathrm{d}}g\right>-{\mathrm{d}}\left<{\mathrm{d}}f,\left<{\mathrm{d}}g,{\mathrm{d}}h\right>\right>$

valid for any $f,g,h\in{\rm TestF}({\sf X})$.

The space of Sobolev vector fields $W^{1,2}_{\\_}{C,\mathsf{loc}}(T{\sf X})$
is defined as the space of $v\in L^{2}_{\\_}\mathsf{loc}(T{\sf X})$ for which
there is $T\in L^{2}_{\\_}\mathsf{loc}(T^{\otimes 2}{\sf X})$ such that

$\int hT:(\nabla
g\otimes\nabla\tilde{g})\,{\mathrm{d}}\mathfrak{m}=\int-\left<v,\nabla\tilde{g}\right>{\rm
div}(h\nabla g)+h{\rm Hess}{\tilde{g}}(X,\nabla g)\,{\mathrm{d}}\mathfrak{m}$

for every $h,g,\tilde{g}\in{\rm TestF}({\sf X})$ with bounded support. In this
case we call $T$ the covariant derivative of $v$ and denote it by
$\nabla_{\\_}Cv$. If $|v|,|\nabla_{\\_}Cv|_{\\_}{\lower
1.29167pt\hbox{\scriptsize{\sf HS}}}\in L^{2}({\sf X})$ we shall say that
$v\in W^{1,2}_{\\_}C(T{\sf X})$; this definition of $W^{1,2}_{\\_}C(T{\sf X})$
coincides with the one given in [12]. Vector fields of the form $g\nabla f$
for $f,g\in{\rm TestF}({\sf X})$ are in $W^{1,2}_{\\_}C(T{\sf X})$: we denote
by ${\rm{Test}}V({\sf X})$ the linear span of such vector fields, and by
$H^{1,2}_{\\_}{C}(T{\sf X})$ its $W^{1,2}_{\\_}C$-closure . The space
$H^{1,2}_{\\_}{C,\mathsf{loc}}(T{\sf X})$ is then equivalently defined either
as the subspace of $L^{2}_{\\_}\mathsf{loc}(T{\sf X})$ made of vectors $v$ of
such that $fv\in H^{1,2}_{\\_}C(T{\sf X})$ for every $f\in{\rm TestF}({\sf
X})$ with bounded support or as the $W^{1,2}_{\\_}{C,\mathsf{loc}}$-closure of
$H^{1,2}_{\\_}C(T{\sf X})$, i.e. as the space of vector fields $v\in
W^{1,2}_{\\_}{C,\mathsf{loc}}(T{\sf X})$ such that there is $(v_{\\_}n)\subset
H^{1,2}_{\\_}C(T{\sf X})$ such that $v_{\\_}n\to v$ and
$\nabla_{\\_}Cv_{\\_}n\to\nabla_{\\_}Cv$ in $L^{2}_{\\_}\mathsf{loc}(T{\sf
X})$ and $L^{2}_{\\_}\mathsf{loc}(T^{\otimes 2}{\sf X})$ as $n\to\infty$.

A useful result we are going to use later on is the following (we refer to
[12, Theorem 3.4.2-(iv) and (v)] for its proof):

$f\in\mathit{W}^{2,2}_{\\_}\mathsf{loc}({\sf X})\quad\Rightarrow\quad\nabla
f\in\mathit{W}^{1,2}_{\\_}{C,\mathsf{loc}}(\mathit{T}{\sf X}),\text{ with
}\nabla(\nabla f)=(\mathrm{Hess}(f))^{\sharp}.$

Moreover, ${\rm{Test}}V({\sf X})\subset\mathit{W}^{1,2}_{\\_}C(\mathit{T}{\sf
X})$ with

$\nabla v=\sum_{\\_}i\nabla g_{\\_}i\otimes\nabla
f_{\\_}i+g_{\\_}i(\mathrm{Hess}(f_{\\_}i))^{\sharp},\qquad\text{for}\,\,v=\sum_{\\_}ig_{\\_}i\nabla
f_{\\_}i.$

####  4.1.2 Product of ${\sf RCD}$ spaces

We start recalling the following fact, which allows to use the results
obtained in Sections 2.2 and 2.3 in the case of interest for us:

######  Proposition 4.1.

Let ${\sf X},{\sf Y}$ be two ${\sf RCD}(K,\infty)$ spaces. Then they satisfy
Assumption 2.5 and the product ${\sf X}\times{\sf Y}$ is ${\sf RCD}(K,\infty)$
as well.

Proof. See [5, Theorem 5.1].  $\square$

A direct consequence of the tensorization of the Cheeger energy is that the
heat flow tensorizes as well. In what follows we will denote by ${{\sf
h}}_{\\_}t^{\sf X},{{\sf h}}_{\\_}t^{\sf Y},{{\sf h}}_{\\_}t$ the heat flows
on ${\sf X},{\sf Y},{\sf X}\times{\sf Y}$ respectively. We will also apply the
operator ${{\sf h}}_{\\_}t^{\sf X}$ to functions in two variables: in this
case we mean that ‘$y$’ variable is ‘frozen’, i.e.:

${{\sf h}}_{\\_}t^{\sf X}f(x,y):={{\sf h}}_{\\_}t^{\sf
X}(f(\cdot,y))(x)\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}-a.e.\ x,y.$

We then have:

######  Corollary 4.2.

Let ${\sf X},{\sf Y}$ be two ${\sf RCD}(K,\infty)$ spaces. Then for every
$f\in L^{2}({\sf X}\times{\sf Y})$ we have

${{\sf h}}_{\\_}tf={{\sf h}}_{\\_}t^{\sf X}({{\sf h}}_{\\_}t^{\sf Y}f)={{\sf
h}}_{\\_}t^{\sf Y}({{\sf h}}_{\\_}t^{\sf X}f).$ (4.1.11)

In particular, for $f=f_{\\_}1\otimes f_{\\_}2$ with $f_{\\_}1\in L^{2}({\sf
X})$ and $f_{\\_}2\in L^{2}({\sf Y})$ we have

${{\sf h}}_{\\_}tf={{\sf h}}_{\\_}t^{\sf X}f_{\\_}1\otimes{{\sf
h}}_{\\_}t^{\sf Y}f_{\\_}2.$ (4.1.12)

Proof. See [4, Section 6.4] and [5, Section 5.1].  $\square$

####  4.1.3 Setting up the problem

Let us fix two ${\sf RCD}(K,\infty)$ spaces ${\sf X},{\sf Y}$. Much like in
the previous chapter we shall put subscripts ${\sf x},{\sf y}$ to
differentiation operators to specify that they act on the $x,y$ variable
respectively. Thus, for instance, with some abuse of notation we shall denote
by ${\rm Hess}_{\\_}{\sf x}(f)$ both the Hessian of a function $f\in
W^{2,2}({\sf X})$ and the map $y\mapsto{\rm Hess}_{\\_}{\sf x}(f(\cdot,y))$,
if $f$ is defined on ${\sf X}\times{\sf Y}$ and is such that $f(\cdot,y)\in
W^{2,2}({\sf X})$ for $\mathfrak{m}_{\\_}{\sf Y}$-a.e. $y\in{\sf Y}$. It may
also occur that for $f\in W^{2,2}({\sf X})$ we write ${\rm Hess}_{\\_}{\sf
x}(f)$ to denote the constant map $y\mapsto{\rm Hess}_{\\_}{\sf x}(f)\in
L^{2}((T^{*})^{\otimes 2}{\sf X})$: this choice will alleviate the notational
burden, and it will be clear from the context when we will be doing so.

With this said, in Section 4.2 we prove that the Hessian of a function $f$ on
${\sf X}\times{\sf Y}$ admits a decomposition of the form

${\rm Hess}(f)=\Bigg{(}\begin{array}[]{cc}{\rm Hess}_{\\_}{\sf
x}(f)&{\mathrm{d}}_{\\_}{\sf x}{\mathrm{d}}_{\\_}{\sf y}f\\\
{\mathrm{d}}_{\\_}{\sf y}{\mathrm{d}}_{\\_}{\sf x}f&{\rm Hess}_{\\_}{\sf
y}(f)\end{array}\Bigg{)}$

meaning that the right hand side is a well defined tensor in $L^{2}$ if and
only if so it is the left hand side, and similarly in Section 4.3 we prove
that the Covariant Derivative of a vector field $v=(v_{\\_}{{\sf
x}},v_{\\_}{{\sf y}})$ in the tangent module over ${\sf X}\times{\sf Y}$
admits a decomposition of the form

$\nabla_{\\_}Cv=\Bigg{(}\begin{array}[]{cc}\nabla_{\\_}{C,{\sf x}}v_{\\_}{{\sf
x}}&\quad{\mathrm{d}}_{\\_}{\sf x}v_{\\_}{{\sf y}}\\\ {\mathrm{d}}_{\\_}{\sf
y}v_{\\_}{\sf x}&\quad\nabla_{\\_}{C,{\sf y}}v_{\\_}{{\sf
y}}\end{array}\Bigg{)}.$

To make this rigorous we start from Theorem 2.8 and notice that
$\Phi_{\\_}{\sf x}\colon L^{0}({\sf Y};L^{0}(\mathit{T}^{*}{\sf X}))\to
L^{0}(T^{*}({\sf X}\times{\sf Y}))$ induces a map, denoted by $\Phi_{\\_}{\sf
x}^{\otimes 2}$, from $L^{0}({\sf Y};L^{0}(\mathit{T}^{*}{\sf X}))\otimes
L^{0}({\sf Y};L^{0}(\mathit{T}^{*}{\sf X}))$ to $L^{0}(T^{*}({\sf X}\times{\sf
Y}))\otimes L^{0}(T^{*}({\sf X}\times{\sf Y}))=L^{0}((T^{*})^{\otimes 2}({\sf
X}\times{\sf Y}))$ by acting component-wise. More precisely, the $L^{0}({\sf
X}\times{\sf Y})$-linear extension of the map

$\omega_{\\_}1\otimes\omega_{\\_}2\quad\mapsto\quad\Phi_{\\_}{\sf
x}(\omega_{\\_}1)\otimes\Phi_{\\_}{\sf
x}(\omega_{\\_}2)\qquad\forall\omega_{\\_}1,\omega_{\\_}2\in L^{0}({\sf
Y};L^{0}(\mathit{T}^{*}{\sf X}))$

(which is easily seen to be well defined) can be extended by continuity to a
norm-preserving map. This is a consequence of the fact that $\Phi_{\\_}{\sf
x}$ itself is norm-preserving (see Proposition 2.2), so that we have

$\begin{split}\Big{|}\sum_{\\_}i\Phi_{\\_}{\sf
x}(\omega_{\\_}{i,1})\otimes\Phi_{\\_}{\sf
x}(\omega_{\\_}{i,2})\Big{|}^{2}&=\sum_{\\_}{i,j}\left<\Phi_{\\_}{\sf
x}(\omega_{\\_}{i,1}),\Phi_{\\_}{\sf
x}(\omega_{\\_}{j,1})\right>\left<\Phi_{\\_}{\sf
x}(\omega_{\\_}{i,2}),\Phi_{\\_}{\sf x}(\omega_{\\_}{j,2})\right>\\\
&=\sum_{\\_}{i,j}\left<\omega_{\\_}{i,1},\omega_{\\_}{j,1}\right>\left<\omega_{\\_}{i,2},\omega_{\\_}{j,2}\right>=\Big{|}\sum_{\\_}i\omega_{\\_}{i,1}\otimes\omega_{\\_}{i,2}\Big{|}^{2}\end{split}$

for any finite choice of $\omega_{\\_}{i,1},\omega_{\\_}{i,2}\in L^{0}({\sf
Y};L^{0}(\mathit{T}^{*}{\sf X}))$.

In a similar way we can define the maps

$\begin{split}\Phi_{\\_}{\sf y}^{\otimes 2}&:L^{0}({\sf
X};L^{0}(\mathit{T}^{*}{\sf Y}))\otimes L^{0}({\sf X};L^{0}(\mathit{T}^{*}{\sf
Y}))\quad\to\quad L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))\\\
\Phi_{\\_}{\sf x}\otimes\Phi_{\\_}{\sf y}&:L^{0}({\sf
Y};L^{0}(\mathit{T}^{*}{\sf X}))\otimes L^{0}({\sf X};L^{0}(\mathit{T}^{*}{\sf
Y}))\quad\to\quad L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))\\\
\Phi_{\\_}{\sf y}\otimes\Phi_{\\_}{\sf x}&:L^{0}({\sf
X};L^{0}(\mathit{T}^{*}{\sf Y}))\otimes L^{0}({\sf Y};L^{0}(\mathit{T}^{*}{\sf
X}))\quad\to\quad L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))\end{split}$

and prove that they are $L^{0}({\sf X}\times{\sf Y})$-linear and norm
preserving.

Then, from the fact that $\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf y}$ is an
isomorphism of modules, we deduce that

$(\Phi_{\\_}{\sf x}\oplus\Phi_{\\_}{\sf y})^{\otimes 2}=(\Phi_{\\_}{\sf
x}^{\otimes 2})\oplus(\Phi_{\\_}{\sf x}\otimes\Phi_{\\_}{\sf
y})\oplus(\Phi_{\\_}{\sf y}\otimes\Phi_{\\_}{\sf x})\oplus(\Phi^{\otimes
2}_{\\_}{\sf y})$

is also an isomorphism of modules, in the sense made precise by the following
statement:

######  Proposition 4.3.

Let ${\sf X},{\sf Y}$ be ${\sf RCD}(K,\infty)$ spaces. Then, with the notation
introduced above, the following holds.

For any $A\in L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))$ there are
unique $A_{\\_}{{\sf x}{\sf x}},A_{\\_}{{\sf x}{\sf y}},A_{\\_}{{\sf y}{\sf
x}},A_{\\_}{{\sf y}{\sf y}}\in L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf
Y}))$ in the images of $\Phi_{\\_}{\sf x}^{\otimes 2},\Phi_{\\_}{\sf
x}\otimes\Phi_{\\_}{\sf y},\Phi_{\\_}{\sf y}\otimes\Phi_{\\_}{\sf
x},\Phi^{\otimes 2}_{\\_}{\sf y}$, respectively, such that

$A=A_{\\_}{{\sf x}{\sf x}}+A_{\\_}{{\sf x}{\sf y}}+A_{\\_}{{\sf y}{\sf
x}}+A_{\\_}{{\sf y}{\sf y}}.$ (4.1.13)

Moreover, we have $\left<A_{\\_}{ij},A_{\\_}{i^{\prime}j^{\prime}}\right>=0$
whenever $(i,j)\neq(i^{\prime},j^{\prime})$ and thus

$|A|^{2}=|A_{\\_}{{\sf x}{\sf x}}|^{2}+|A_{\\_}{{\sf x}{\sf
y}}|^{2}+|A_{\\_}{{\sf y}{\sf x}}|^{2}+|A_{\\_}{{\sf y}{\sf
y}}|^{2}\qquad\mathfrak{m}_{\\_}{\sf X}\otimes\mathfrak{m}_{\\_}{\sf
Y}\text{-a.e.}.$ (4.1.14)

Proof. We start proving that the images of $\Phi_{\\_}{\sf x}^{\otimes 2}$ and
$\Phi_{\\_}{\sf x}\otimes\Phi_{\\_}{\sf y}$ are pointwise orthogonal. Since we
know that these maps are continuous, it is sufficient to check that
$\left<\Phi_{\\_}{\sf x}^{\otimes 2}(V),\Phi_{\\_}{\sf x}\otimes\Phi_{\\_}{\sf
y}(W)\right>=0$ for arbitrary $V,W$ in dense subsets of the respective source
spaces. We thus pick $V=\sum_{\\_}iv_{\\_}{1,i}\otimes v_{\\_}{2,i}$ and
$W=\sum_{\\_}jv_{\\_}{3,j}\otimes w_{\\_}j$ with
$v_{\\_}{1,i},v_{\\_}{2,i},v_{\\_}{3,j}\in L^{0}({\sf
Y};L^{0}(\mathit{T}^{*}{\sf X}))$ and $w_{\\_}j\in L^{0}({\sf
X};L^{0}(\mathit{T}^{*}{\sf Y}))$. Then

$\begin{split}\left<\Phi_{\\_}{\sf x}^{\otimes 2}(V),\Phi_{\\_}{\sf
x}\otimes\Phi_{\\_}{\sf y}(W)\right>&=\sum_{\\_}{i,j}\left<\Phi_{\\_}{\sf
x}(v_{\\_}{1,i}){\otimes}\Phi_{\\_}{\sf x}(v_{\\_}{2,i}),\Phi_{\\_}{\sf
x}(v_{\\_}{3,j})\otimes\Phi_{\\_}{\sf y}(w_{\\_}{j})\right>\\\
&=\sum_{\\_}{i,j}\left<\Phi_{\\_}{\sf x}(v_{\\_}{1,i}),\Phi_{\\_}{\sf
x}(v_{\\_}{3,j})\right>\underbrace{\left<\Phi_{\\_}{\sf
x}(v_{\\_}{2,i}),\Phi_{\\_}{\sf y}(w_{\\_}{j})\right>}_{\\_}{\equiv
0\atop\text{ by \eqref{eq:orto}}}=0\end{split}$

as desired. Similarly for other couple of images. This is enough to prove that
whenever (4.1.13) holds, the identity (4.1.14) holds as well and in turn this
proves uniqueness of the $A_{\\_}{ij}$’s as in the statement.

Thus it remains to prove the existence of
$A_{\\_}{11},A_{\\_}{12},A_{\\_}{21}$, and $A_{\\_}{22}$. Since (4.1.14) shows
that the maps $A\mapsto A_{\\_}{ij}$, $i,j\in\\{{\sf x},{\sf y}\\}$, are
continuous, we can validate such existence just for $A$’s running in a dense
subset of $L^{0}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))$. We thus pick $A$
of the form $A=\sum_{\\_}i\omega_{\\_}i\otimes\omega_{\\_}i^{\prime}$ for
$\omega_{\\_}i,\omega_{\\_}i^{\prime}\in L^{0}(T^{*}({\sf X}\times{\sf Y}))$,
and recall that, from the surjectivity part of the statement of Theorem 2.8,
for any $i$ we have $\omega_{\\_}i=\Phi_{\\_}{\sf x}(\omega_{\\_}{i,{\sf
x}})+\Phi_{\\_}{\sf y}(\omega_{\\_}{i,{\sf y}})$, and similarly
$\omega_{\\_}i^{\prime}=\Phi_{\\_}{\sf x}(\omega^{\prime}_{\\_}{i,{\sf
x}})+\Phi_{\\_}{\sf y}(\omega^{\prime}_{\\_}{i,{\sf y}})$, for appropriate
$\omega_{\\_}{i,{\sf x}},\omega^{\prime}_{\\_}{i,{\sf x}}\in L^{0}({\sf
Y};L^{0}(\mathit{T}^{*}{\sf X}))$ and $\omega_{\\_}{i,{\sf
y}},\omega^{\prime}_{\\_}{i,{\sf y}}\in L^{0}({\sf X};L^{0}(\mathit{T}^{*}{\sf
Y}))$.

It is then clear from the definitions that with the choices

$\begin{array}[]{rll}&A_{\\_}{{\sf x}{\sf x}}:=\Phi_{\\_}{\sf x}^{\otimes
2}\Big{(}\sum_{\\_}i\omega_{\\_}{i,{\sf x}}\otimes\omega^{\prime}_{\\_}{i,{\sf
x}}\Big{)},&A_{\\_}{{\sf x}{\sf y}}:=\Phi_{\\_}{\sf x}\otimes\Phi_{\\_}{\sf
y}\Big{(}\sum_{\\_}i\omega_{\\_}{i,{\sf x}}\otimes\omega^{\prime}_{\\_}{i,{\sf
y}}\Big{)},\\\ &A_{\\_}{{\sf y}{\sf x}}:=\Phi_{\\_}{\sf
y}\otimes\Phi_{\\_}{\sf x}\Big{(}\sum_{\\_}i\omega_{\\_}{i,{\sf
y}}\otimes\omega^{\prime}_{\\_}{i,{\sf x}}\Big{)},&A_{\\_}{{\sf y}{\sf
y}}:=\Phi_{\\_}{\sf y}^{\otimes 2}\Big{(}\sum_{\\_}i\omega_{\\_}{i,{\sf
y}}\otimes\omega^{\prime}_{\\_}{i,{\sf y}}\Big{)},\\\ \end{array}$

the equality (4.1.13) holds, thus giving the conclusion.  $\square$

###  4.2 Hessian on the product space

####  4.2.1 From regularity on the factors to regularity on the product

In this section we prove that if a function of two variables is such that the
functions obtaining by freezing one variable are in $W^{2,2}$ and with
appropriate integrability assumptions, then the function itself belongs to
$W^{2,2}({\sf X}\times{\sf Y})$, see Theorem 4.7 for the precise statement
(and Proposition 4.8 for a suboptimal version about the space $H^{2,2}({\sf
X}\times{\sf Y})$).

It will be useful to consider the following algebra of functions:

######  Definition 4.4 (The space ${\rm{Test}}_{\\_}\times\mathrm{F}({\sf
X}\times{\sf Y})$).

Let ${\sf X},{\sf Y}$ be two ${\sf RCD}(K,\infty)$ spaces.

Then ${\rm{Test}}_{\\_}\times\mathrm{F}({\sf X}\times{\sf Y})$ is the space of
finite linear combinations of functions of the form $g_{\\_}1\otimes
g_{\\_}2$, with $g_{\\_}1\in{\rm TestF}({\sf X})$ and $g_{\\_}2\in{\rm
TestF}({\sf Y})$.

From the tensorization of the Cheeger energy in Definition 2.3 (which holds in
this setting because of Proposition 4.1), and the tensorization of the
Laplacian given in Corollary 2.13, it is easy to see that
${\rm{Test}}_{\\_}\times({\sf X}\times{\sf Y})\subset{\rm TestF}({\sf
X}\times{\sf Y})$. This newly defined space of functions will be useful in
proving the main result of this section, Theorem 4.7, because, as we will see
in Proposition 4.6, to check whether a function is in $W^{2,2}({\sf
X}\times{\sf Y})$ it is sufficient to verify the defining property of the
Hessian only for functions in the smaller space ${\rm{Test}}_{\\_}\times({\sf
X}\times{\sf Y})$, where computations are easier. In turn, this latter fact
will be a direct consequence of the following density result:

######  Lemma 4.5.

Let ${\sf X},{\sf Y}$ be two ${\sf RCD}(K,\infty)$ spaces and $g\in L^{2}\cap
L^{\infty}({{\sf X}\times{\sf Y}})$ (this in particular holds if $g\in{\rm
TestF}(X\times{\sf Y})$). Then we can find functions
$g_{\\_}{n,t}\in{\rm{Test}}_{\\_}\times\mathrm{F}({\sf X}\times{\sf Y})$,
parametrized by $n\in\mathbb{N}$ and $t>0$, such that for every $t>0$ it
holds:

  * i)

$\sup_{\\_}{n}\|g_{\\_}{n,t}\|_{\\_}{L^{\infty}}<\infty$,

  * ii)

$\sup_{\\_}n\mathop{\rm Lip}\nolimits(g_{\\_}{n,t})<\infty$,

  * iii)

$(g_{\\_}{n,t}),(\Delta g_{\\_}{n,t})$ converge to ${{\sf
h}}_{\\_}tg,\Delta{{\sf h}}_{\\_}tg$ respectively in $L^{2}({\sf X}\times{\sf
Y})$ as $n\to\infty$,

  * iv)

$(\nabla g_{\\_}{n,t}),(\nabla(|\nabla g_{\\_}{n,t}|^{2}))$ converge to
$\nabla{{\sf h}}_{\\_}tg,\nabla(|\nabla{{\sf h}}_{\\_}tg|^{2})$ respectively
in $L^{2}(T({\sf X}\times{\sf Y}))$ as $n\to\infty$.

Proof. It is easy to see that we can find a sequence of equibounded functions
$(g_{\\_}n)$ $L^{2}$-converging to $g$ of the form
$g_{\\_}n=\sum_{\\_}{i}^{N_{\\_}n}\alpha_{\\_}i{\raise
1.29167pt\hbox{$\chi$}}_{\\_}{A_{\\_}{n,i}\times B_{\\_}{n,i}}$, where the
sets $A_{\\_}i\subset{\sf X}$ and $B_{\\_}i\subset{\sf Y}$ are Borel and
bounded and $\alpha_{\\_}{i}\in\real$. We then put $g_{\\_}{n,t}:={{\sf
h}}_{\\_}t(g_{\\_}n)$ and notice that Corollary 4.2 and (4.1.7) ensures that
$g_{\\_}{n,t}\in{\rm{Test}}_{\\_}\times\mathrm{F}({\sf X}\times{\sf Y})$ for
any $n\in\mathbb{N}$, $t>0$.

Then property $(i)$ follows from (4.1.3) and $(ii)$ from (4.1.5), (4.1.6) and
$(i)$. The first convergence in $(iii)$ follows from the continuity of the
heat flow as a map from $L^{2}$ into itself and the second by also taking into
account the a priori estimate (4.1.2). The fact that $\nabla
g_{\\_}{n,t}\to\nabla{{\sf h}}_{\\_}tg$ in $L^{2}(T({\sf X}\times{\sf Y}))$
follows from (4.1.1), which ensures that ${{\sf h}}_{\\_}t:L^{2}\to W^{1,2}$
is a continuous operator.

It remains to prove $L^{2}$-convergence of $(\nabla(|\nabla
g_{\\_}{n,t}|^{2}))$ to $\nabla(|\nabla{{\sf h}}_{\\_}tg|^{2})$. To see this,
start observing that $\nabla(|\nabla f|^{2})=2{\rm Hess}(f)(\nabla f)$ for any
$f\in{\rm TestF}({\sf X}\times{\sf Y})$, and notice that what already proved
together with (4.1.10) show that ${\rm Hess}(g_{\\_}{n,t})\to{\rm Hess}({{\sf
h}}_{\\_}tg)$ in $L^{2}((T^{*})^{\otimes 2}({\sf X}\times{\sf Y}))$ as
$n\to\infty$ for every $t>0$.

Now put $g_{\\_}t:={{\sf h}}_{\\_}tg$ and observe that

$\begin{split}&\|{\rm Hess}(g_{\\_}{n,t})(\nabla g_{\\_}{n,t})-{\rm
Hess}(g_{\\_}t)(\nabla g_{\\_}t)\|_{\\_}{L^{2}}\\\ &\leq\|{\rm
Hess}(g_{\\_}{n,t}-g_{\\_}t)(\nabla g_{\\_}{n,t}-\nabla
g_{\\_}t)\|_{\\_}{L^{2}}+\|{\rm Hess}(g_{\\_}t)(\nabla g_{\\_}{n,t}-\nabla
g_{\\_}t)\|_{\\_}{L^{2}}+\|{\rm Hess}(g_{\\_}{n,t}-g_{\\_}t)(\nabla
g_{\\_}t)\|_{\\_}{L^{2}}.\end{split}$

As $n\to\infty$ we see from $(ii)$ and the $L^{2}$-convergence of Hessians
that the first and third addends in the right hand side go to 0. For the
second we use the $L^{2}$-convergence of gradients and the fact that they are
equibounded to conclude with the dominated convergence theorem.  $\square$

######  Proposition 4.6.

Let ${\sf X},{\sf Y}$ be two ${\sf RCD}(K,\infty)$ spaces,
$f\in\mathit{W}^{1,2}({\sf X}\times{\sf Y})$ and $A\in L^{2}((T^{*})^{\otimes
2}({\sf X}\times{\sf Y}))$ symmetric.

Then $f\in W^{2,2}({\sf X}\times{\sf Y})$ with ${\rm Hess}(f)=A$ if and only
if we have

$-\int\left<{\mathrm{d}}f,{\mathrm{d}}g\right>\,{\rm div}(h\nabla
g)+\tfrac{1}{2}h\langle{\mathrm{d}}f,{\mathrm{d}}{|{\mathrm{d}}g|^{2}}\rangle\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf
X}\times\mathfrak{m}_{\\_}Y)=\int h\left<A,{\mathrm{d}}g\otimes
dg\right>\,{\mathrm{d}}(\mathfrak{m}_{\\_}{\sf X}\times\mathfrak{m}_{\\_}Y)$
(4.2.1)

for every $g,h\in{\rm{Test}}_{\\_}{\times}\mathrm{F}({\sf X}\times{\sf Y})$.

Proof. The ‘only if’ follows from the definitions of $W^{2,2}$ and Hessian and
the inclusion ${\rm{Test}}_{\\_}\times\mathrm{F}({\sf X}\times{\sf