1 Introduction
In the recent, very interesting, paper [Reference Erikkson-Bique and SoultanisES21], the authors provided a general construction of charts on metric measure spaces, key features of their notion being: the compatibility with Sobolev calculus (and thus in particular with the differential calculus as developed by Cheeger in [Reference CheegerChe99] and by the second author in [Reference GigliGig15]), a very general existence result, and notable consequences in terms of the structure of the Sobolev spaces (see also [Reference Erikkson-Bique, Rajala and SoultanisERS22a, Reference Erikkson-Bique, Rajala and SoultanisERS22b]). An example in this latter direction is the proof that the space
$W^{1,p}(\textsf {X})$
,
$p\in (1,\infty )$
, is reflexive as soon as the space
$\textsf {X}$
can be covered by a countable number of sets with finite Hausdorff measure (the “previous best” result appeared in [Reference Alberti and MarcheseACD14] and required the metric to be locally doubling).
A crucial step in [Reference Erikkson-Bique and SoultanisES21] is the proof that if
$\varphi :E\subset \textsf {X}\to {\mathbb R}^n$
is a “p-independent weak chart,” then n is bounded from above by the Hausdorff dimension of E: more precisely, the authors prove the following.
Proposition 1.1 Suppose
$\varphi \in \operatorname {\mathrm {Lip}}(\textsf {X},{{\mathbb R}}^n)$
is p-independent on U. Then
$n\leq {\mathrm dim}_{\mathcal {H}}(U)$
.
For the precise meaning of “p-independent weak chart,” we refer to Definition 2.23; for the purpose of this introduction, we shall limit ourselves to point out that in the smooth setting, this would be equivalent to requiring the image of the differential of
$\varphi $
at every point to span the whole tangent space of
${{\mathbb R}}^d$
. Starting from this result, the existence of actual charts is obtained via a suitable maximality argument.
Interestingly, this upper bound is proved via means that have, in principle, little to do with analysis in nonsmooth setting: key ingredients are indeed the elliptic regularity result in [Reference De Philippis and RindlerDR] and the study of the structure of the set of nondifferentiability points of Lipschitz functions in [Reference Ambrosio, Gigli and SavaréAM16].
This sort of procedure has a recent analog in the theory of
$\text {RCD}$
spaces. Let us recall indeed that, in [Reference Mondino and NaberMN14], it has been proved that finite-dimensional
$\text {RCD}$
spaces admit bi-Lipschitz charts covering almost all the space. In [Reference De Philippis, Marchese and RindlerDMR, Reference Kell and MondinoKM18, Reference Mondino and NaberMN14], no information about the behavior of the reference measure with respect to these charts has been provided: this topic has been later studied in [Reference Kell and MondinoKM18] where, relying in a way or another on [Reference De Philippis and RindlerDR] and [Reference Ambrosio, Gigli and SavaréAM16], it has been proved that
$\varphi _*({{\mathfrak m}}_{|E})\ll \mathcal {{\mathscr L}}^n$
for a Mondino–Naber chart
$\varphi :E\to {{\mathbb R}}^n$
.
Of particular interest for the discussion here is the fact that in [Reference Gigli and PasqualettoGP21] only the results in [Reference De Philippis and RindlerDR] have been used, whereas in [Reference Kell and MondinoKM18] also those in [Reference Ambrosio, Gigli and SavaréAM16] were necessary. Comparing this with the results in [Reference Erikkson-Bique and SoultanisES21], it is natural to wonder whether the use of [Reference Ambrosio, Gigli and SavaréAM16] is really crucial or can be avoided: this is the question motivating the present note. Of course, there is nothing wrong in using a well-established result in doing research; our study is simply motivated by the desire of better understanding the interesting construction done in [Reference Erikkson-Bique and SoultanisES21]. The result of our investigation is that [Reference Ambrosio, Gigli and SavaréAM16] is not really needed and the line of thought presented here simplifies not only some of the steps done in [Reference Erikkson-Bique and SoultanisES21], but also some of those in [Reference Gigli and PasqualettoGP21] (see Section 3).
Another remark that we make, consequence of the studies in [Reference Erikkson-Bique and SoultanisES21], is that the dimension of the (co)tangent module (in the sense of [Reference GigliGig15]) on a subset
$E\subset \textsf {X}$
is bounded from above from the Hausdorff dimension of E (see Remark 3.6).
2 Preliminaries
2.1 Test plans and Sobolev functions
In this section, we shall recall the definition of Sobolev space following the approach in [Reference Ambrosio, Gigli and SavaréAGS14]. We say that a triple
$(\textsf {X},{\text {d}},{{\mathfrak m}})$
is a metric measure space if
$(\textsf {X},{\text {d}})$
is a complete and separable metric measure space and
${{\mathfrak m}}$
is a Radon measure which is finite on balls. For the rest of the paper,
$p,q$
will be conjugate exponents, namely,
$\frac {1}{p}+\frac {1}{q}=1$
.
Definition 2.1 We say that a probability measure
$\pi $
on
$C([0,1];\textsf {X})$
is a q-test plan if it is concentrated on
$AC([0,1];\textsf {X})$
and the following two conditions are met:
-
(1)
$\exists \ C=C(\pi )>0$
such that
$e_{t\sharp }\pi \leq C{{\mathfrak m}}$
, where
${\mathfrak m}$
is the reference measure on
$\textsf {X}$
and
$e_t:C([0,1];\textsf {X})\to \textsf {X}$
is the evaluation map
$e_t(\gamma )=\gamma _t$
. -
(2) The following quantity, called kinetic energy, is finite:
$$ \begin{align*} \text{K.E.}(\pi)=\int\int_0^1|\dot{\gamma}_t|^q{{\,\mathrm d}} t{{\,\mathrm d}}\pi(\gamma), \end{align*} $$
where
$|\dot {\gamma }_t|=\lim _{h\to 0}\frac {{{\,\mathrm d}}(\gamma _{t+h},\gamma _t)}{h}$
is the metric derivative of the curve
$\gamma $
.
With this notion at hand, we can introduce the Sobolev space
$\text {W}^{1,p}(\textsf {X},{{\,\mathrm d}},{{\mathfrak m}})$
.
Definition 2.2 We say that a function
$f:\textsf {X}\to {\mathbb R}$
belongs to the Sobolev space
$W^{1,p}(\textsf {X},{{\,\mathrm d}},{{\mathfrak m}})$
if
$f\in {\mathrm {L}}^p({{\mathfrak m}})$
and if
$$ \begin{align} \int|f(\gamma_1)-f(\gamma_0)|{{\,\mathrm d}}\pi(\gamma)\leq\int\int_0^1G(\gamma_t)|\dot{\gamma}_t|{{\,\mathrm d}} t{{\,\mathrm d}}\pi(\gamma)\quad\forall\pi\;\text{q-test plan}, \end{align} $$
with
$G:\textsf {X}\to {{\mathbb R}}_+$
being a Borel function belonging to
${\mathrm {L}}^p({\mathfrak m})$
.
Remark 2.3 It is easy to see that the set of functions G satisfying (2.1) is a closed convex set; hence, it admits an element of minimal norm: we will call such an element p-weak upper gradient and we will denote it by
$|Df|_p$
. With a little bit of work, it is possible to prove that the function
$|Df|_p$
is such that
$|Df|_p\leq G \ {{\mathfrak m}}$
-a.e. for every other G satisfying (2.1).
2.2 The language of
$L^p$
-normed
$L^{\infty }$
-modules
We now switch our attention to the theory of
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-modules developed by the second author in [Reference GigliGig18]: the following material can be found there, unless otherwise stated.
Definition 2.4 (
${\mathrm {L}}^p({{\mathfrak m}})$
-normed module)
We say that a Banach space
$(\mathcal {M}, \left \lVert {\cdot } \right \rVert _{\mathcal {M}})$
is an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module if there exists a bilinear continuous map
$\cdot :\mathrm {L}^{\infty }({{\mathfrak m}})\times \mathcal {M}\to \mathcal {M}$
which makes
$\mathcal {M}$
a module with unity over the ring of
$\mathrm {L}^{\infty }({{\mathfrak m}})$
functions and another map
$|\cdot |:\mathcal {M}\longrightarrow {\mathrm {L}}^p({{\mathfrak m}})$
with nonnegative values such that
$$ \begin{align} \left\lVert {|v|} \right\rVert _{{\mathrm{L}}^p({{\mathfrak m}})}&= \left\lVert {v} \right\rVert _{\mathcal{M}}, \qquad\qquad\ \qquad\end{align} $$
$$ \begin{align} |f\cdot v|&=|f||v|\qquad{{\mathfrak m}}-\text{a.e.} \end{align} $$
for all
$v\in \mathcal {M}$
,
$f\in \mathrm {L}^{\infty }({{\mathfrak m}})$
. We call
$\cdot $
the multiplication and
$|\cdot |$
the pointwise norm.
Remark 2.5 Note that the pointwise norm is continuous thanks to the triangular inequality; in fact,
$$ \begin{align*} \left\lVert {|v|-|w|} \right\rVert _{{\mathrm{L}}^p({\mathfrak m})}\leq \left\lVert {|v-w|} \right\rVert _{{\mathrm{L}}^p({\mathfrak m})}= \left\lVert {v-w} \right\rVert _{\mathcal{M}}. \end{align*} $$
Moreover, with a little bit of abuse of notation, we will write
$fv$
instead of
$f\cdot v$
and write
${\mathrm {L}}^p({\mathfrak m})$
-normed module instead of
${\mathrm {L}}^p({\mathfrak m})$
-normed
$\mathrm {L}^{\infty }({\mathfrak m})$
-module.
A related interesting concept is the one of localization of a module; indeed, it is easy to see that the following object
$$ \begin{align*} \mathcal{M}_{|E}:=\{\chi_E v:\;v\in\mathcal{M}\} \end{align*} $$
is a submodule of
$\mathcal {M}$
and it clearly inherits the normed structure from
$\mathcal {M}$
.
Definition 2.6 (
$\textbf {Local independence}$
)
Let
$\mathcal {M}$
be an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module and
$A\in {{\mathscr {B}}}(X)$
with
${{\mathfrak m}}(A)>0$
, and we say that a family
$v_1,...,v_n\in \mathcal {M}$
is independent on A if, for every
$f_1,...,f_n\in \mathrm {L}^{\infty }({{\mathfrak m}})$
,
$$ \begin{align} \sum_{i=1}^n f_i v_i = 0\quad{{\mathfrak m}}-\text{a.e.}\;\text{on}\;A\implies f_i=0\quad{{\mathfrak m}}-\text{a.e.}\;\text{on}\;A\quad\forall i=1,...,n. \end{align} $$
In the spirit of linear algebra, we shall also define what is the span of a set of vectors.
Definition 2.7 (
$\textbf {Span}$
)
Let
$\mathcal {M}$
be an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module,
$V\subset \mathcal {M}$
a subset, and
$A\in {{\mathscr {B}}}(X)$
. We denote with
$\text {Span}_A(V)$
the closure in
$\mathcal {M}$
of the
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-linear combinations of elements of V. Moreover, we say that
$\text {Span}_A(V)$
is the space generated by V on A.
After this definition, the one of basis and of dimension for an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
arise naturally.
Definition 2.8 We say that a finite family
$v_1,...,v_n\in \mathcal {M}$
is a basis on
$A\in {{\mathscr {B}}}(X)$
if it is independent on A and
$\text {Span}_A\{v_1,...,v_n\}=\mathcal {M}_{|A}$
. If the above happens, we say that the local dimension of
$\mathcal {M}$
on A is n and in case
$\mathcal {M}$
has not dimension k for any
$k\in {{\mathbb N}}$
, we say that it has infinite dimension.
It can be proved that the notion of dimension is well posed, namely, if we have
$v_1,...,v_n$
generating
$\mathcal {M}$
on a set A and
$w_1,...,w_m$
are independent on A, then
$n\geq m$
. Ultimately, this means that two different bases must have the same cardinality.
Building over these tools, we have the following proposition.
Proposition 2.9 Let
$\mathcal {M}$
be an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module. Then there is a unique partition
$\{E_i\}_{i\in {{\mathbb N}}\cup \{\infty \}}$
of
$\textsf {X}$
, up to
${{\mathfrak m}}$
-a.e. equality, such that:
-
(1) for every
$i\in {{\mathbb N}}$
such that
${{\mathfrak m}}(E_i)>0$
,
$\mathcal {M}$
has dimension i on
$E_i$
, -
(2) for every
$E\subset E_{\infty }$
with
${{\mathfrak m}}(E)>0$
,
$\mathcal {M}$
has infinite dimension on E.
2.3 Pullback of a normed module
We now introduce the notion of pullback module which, roughly speaking, is nothing but a module over a space
$\textsf {X}$
obtained by pulling back a module on another space
${\textsf {Y}}$
via a certain map.
Definition 2.10 (Pullback)
Let
$(\textsf {X},{\text {d}}_{\textsf {X}},{{\mathfrak m}}_{\textsf {X}})$
and
$({\textsf {Y}},{\text {d}}_{\textsf {Y}},{{\mathfrak m}}_{\textsf {Y}})$
be metric measure spaces,
$\varphi : \textsf {X}\longrightarrow {\textsf {Y}}$
a map of bounded compression, and
$\mathcal {M}$
an
${\mathrm {L}}^p({{\mathfrak m}}_{\textsf {Y}})$
-normed module. Then there exists a unique, up to unique isomorphism, couple
$(\varphi ^*\mathcal {M},\varphi ^*)$
with
$\varphi ^*\mathcal {M}$
being an
${\mathrm {L}}^p({{\mathfrak m}}_{\textsf {X}})$
-normed module and
$\varphi ^*:\mathcal {M}\longrightarrow \varphi ^*\mathcal {M}$
being a linear and continuous operator such that:
-
(1)
$|\varphi ^*v|=|v|\circ \varphi $
holds
${{\mathfrak m}}_{\textsf {X}}$
-a.e., for every
$v\in \mathcal {M}$
, -
(2) the set
$\{\varphi ^*v{{\ :\ }} v\in \mathcal {M}\}$
generates
$\varphi ^*\mathcal {M}$
as a module.
At this point, one can try to understand what is the relation between the dimension of a module and the one of its pullback via the map
$\varphi $
and in order to do so we need to introduce a sort of left inverse of the pullback operator
$\varphi ^*$
. To do so, let us assume that
$\varphi _{\sharp }{{\mathfrak m}}_{\textsf {X}}={{\mathfrak m}}_Y$
to simplify the exposition.
For
$f\in {\mathrm {L}}^p({{\mathfrak m}}_{\textsf {X}})$
nonnegative, we put
$$ \begin{align} {\mathrm{Pr}}_{\varphi}(f):=\frac{{{\,\mathrm d}}\varphi_{\sharp}(f{{\mathfrak m}}_{\textsf{X}})}{{{\,\mathrm d}}{{\mathfrak m}}_{\textsf{Y}}}, \end{align} $$
and in a natural way, we set
${\mathrm {Pr}}_{\varphi }(f):={\mathrm {Pr}}_{\varphi }(f^+)-{\mathrm {Pr}}_{\varphi }(f^-)$
for general
$f\in {\mathrm {L}}^p({{\mathfrak m}}_{\textsf {X}})$
.
For the next proposition, we need to recall the classical Disintegration theorem. The statement below is taken from [Reference Ambrosio, Colombo and Di MarinoAGS08, Theorem 5.3.1] (see also [Reference FremlinFre06, Chapter 452] and [Reference BogachevBog07, Chapter 10.6]).
Theorem 2.11 (Disintegration)
Let
$\textsf {X},{\textsf {Y}}$
be complete and separable metric spaces, let
$\mu \in \mathcal {P}(\textsf {X})$
, let
$\pi :\textsf {X}\to {\textsf {Y}}$
be a Borel map, and let
$\nu =\pi _{\sharp }\mu \in \mathcal {P}({\textsf {Y}})$
. Then there exists a
$\nu $
-a.e. uniquely determined Borel family of probability measures
$\{\mu _y\}_{y\in {\textsf {Y}}}\subseteq \mathcal {P}(\textsf {X})$
such that
$\mu _x(\textsf {X}\backslash \pi ^{-1}(\{y\}))=0$
for
$\nu $
-a.e.
$y\in \textsf {X}$
and
$$ \begin{align} \int_{\textsf{X}}f{{\,\mathrm d}}\mu = \int_{\textsf{Y}}\biggl(\int_{\pi^{-1}(\{y\})}f{{\,\mathrm d}}\mu_y\biggr){{\,\mathrm d}}\nu(y) \end{align} $$
for every Borel map
$f:\textsf {X}\to [0,+\infty ]$
.
Remark 2.12 Two remarks are in order here: the first one is that the above theorem in [Reference Ambrosio, Colombo and Di MarinoAGS08] is stated for Radon separable metric space, but in our setting, it suffices to state it for complete and separable ones (which in particular are Radon), and the second is that the result easily extends to any
$f:\textsf {X}\to {{\mathbb R}}$
Borel provided, for example, that
$f\in L^1(\mu )$
.
We now recall some properties of the map
$\Pr _{\varphi }$
.
Proposition 2.13 The operator
${\mathrm {Pr}}_{\varphi }: {\mathrm {L}}^p({{\mathfrak m}}_{\textsf {X}})\longrightarrow {\mathrm {L}}^p({{\mathfrak m}}_Y)$
is linear, continuous, and
$$ \begin{align} {\mathrm{Pr}}_{\varphi}(f)(y) = \int_{\textsf{X}} f(x){{\,\mathrm d}}{{\mathfrak m}}_y(x)\quad{{\mathfrak m}}_{\textsf{Y}}-\text{a.e.},\quad\forall f\in {\mathrm{L}}^p({{\mathfrak m}}_{\textsf{X}}), \end{align} $$
where
$y\mapsto m_y$
denotes the disintegration of
${{\mathfrak m}}_{\textsf {X}}$
with respect to the map
$\varphi $
. Finally, it holds
$$ \begin{align} |{\mathrm{Pr}}_{\varphi}(f)|\leq{\mathrm{Pr}}_{\varphi}(|f|)\quad{{\mathfrak m}}_{\textsf{Y}}-\text{a.e.} \end{align} $$
Proof Linearity is a consequence of the linearity of the integral. Formula (2.8) is also trivial, while for (2.7), we have, for any
$A\in {{\mathscr {B}}}({\textsf {Y}})$
,
$$ \begin{align*} \int_{A}{\mathrm{Pr}}_{\varphi}(f)(y){{\,\mathrm d}}{{\mathfrak m}}_{\textsf{Y}} = \int_A{{\,\mathrm d}}\varphi_{\sharp}(f{{\,\mathrm d}}{{\mathfrak m}}_{\textsf{X}})=\int_{\varphi^{-1}(A)}f(x){{\,\mathrm d}}{{\mathfrak m}}_{\textsf{X}}, \end{align*} $$
and by the properties of the disintegration, we have
$$ \begin{align*} \int_{\varphi^{-1}(A)}f(x){{\,\mathrm d}}{{\mathfrak m}}_{\textsf{X}}=\int_{\textsf{Y}}\int_{\varphi^{-1}(A)}f(x){{\,\mathrm d}}{{\mathfrak m}}_y(x){{\,\mathrm d}}{{\mathfrak m}}_{\textsf{Y}}(y) = \int_A\int_{\textsf{X}}f(x){{\,\mathrm d}}{{\mathfrak m}}_y(x){{\,\mathrm d}}{{\mathfrak m}}_{\textsf{Y}}(y), \end{align*} $$
therefore proving (2.7).
To prove continuity, note that the case
$p=\infty $
is due to formula (2.8), while continuity in
${\mathrm {L}}^p({{\mathfrak m}})$
for every
$p\in [1,+\infty )$
follows from the following:
$$ \begin{align*} \int_{{\textsf{Y}}}|{\mathrm{Pr}}_{\varphi}|^p{{\,\mathrm d}}{{\mathfrak m}}_{{\textsf{Y}}}=\int_{{\textsf{Y}}}\biggl|\int_{\textsf{X}}f(x){{\,\mathrm d}}{{\mathfrak m}}_y(x)\biggr|^p{{\,\mathrm d}}{{\mathfrak m}}_{{\textsf{Y}}}(y)\leq\int_{{\textsf{Y}}}\int_{\textsf{X}}|f(x)|^p{{\,\mathrm d}}{{\mathfrak m}}_y(x){{\,\mathrm d}}{{\mathfrak m}}_{{\textsf{Y}}}(y)= \left\lVert {f} \right\rVert ^p_{{\mathrm{L}}^p({{\mathfrak m}})}, \end{align*} $$
where we used Jensen’s inequality and the properties of the disintegration.
In the case of a general
${\mathrm {L}}^p({{\mathfrak m}}_{\textsf {X}})$
-normed module, the continuous operator
${\mathrm {Pr}}_{\varphi }:\varphi ^*\mathcal {M}:\longrightarrow \mathcal {M}$
can be characterized by the following properties:
$$ \begin{align} g{\mathrm{Pr}}_{\varphi}(v)={\mathrm{Pr}}_{\varphi}(g\circ\varphi v),\quad\forall v\in\mathcal{M}\quad\forall g\in \mathrm{L}^{\infty}({{\mathfrak m}}_{\textsf{X}}), \end{align} $$
$$ \begin{align} \ \,{\mathrm{Pr}}_{\varphi}(g\varphi^*v) = {\mathrm{Pr}}_{\varphi}(g)v,\quad\forall v\in\mathcal{M}\quad\forall g\in \mathrm{L}^{\infty}({{\mathfrak m}}_{\textsf{X}}), \end{align} $$
with the bound
$|{\mathrm {Pr}}_{\varphi }(V)|\leq {\mathrm {Pr}}_{\varphi }(|V|)$
still holding
${{\mathfrak m}}_{\textsf {Y}}$
-a.e. for every
$V\in \varphi ^*\mathcal {M}$
.
With these objects, we are now able to describe the structure of the pullback module; in particular (as one can expect by reasoning via pre-composition), the pullback of an n-dimensional module
$\mathcal {M}$
over E is an n-dimensional module over
$\varphi ^{-1}(E)$
(see also [Reference PasqualettoPas18]).
Proposition 2.14 Let
$\mathcal {M}$
be an
${\mathrm {L}}^p({{\mathfrak m}}_{\textsf {Y}})$
-normed module over the m.m.s.
$({\textsf {Y}},d_{\textsf {Y}},\mu )$
, and let
$E\in \mathcal {B}(Y)$
be a Borel set where
$\mathcal {M}$
has dimension n, with
$\{v_1,...,v_n\}$
being a basis. Let
$(\textsf {X},d_{\textsf {X}},{{\mathfrak m}})$
be another m.m.s., and let
$\varphi :\textsf {X}\to {\textsf {Y}}$
be a Borel map such that
$\varphi _{\sharp }{{\mathfrak m}}_{\textsf {X}} = {{\mathfrak m}}_{\textsf {Y}}$
, then
$\{\varphi ^{*}v_1,...,\varphi ^{*}v_n\}$
is a basis of
$\varphi ^{*}\mathcal {M}$
over
$\varphi ^{-1}(E)$
.
Proof We first prove that
$\{\varphi ^*v_1,...,\varphi ^*v_n\}$
generate
$\varphi ^*\mathcal {M}$
over
$\varphi ^{-1}(E)$
.
First recall that
$\varphi ^*\mathcal {M}$
is generated (as module) by
$\{\varphi ^{*}v: v\in \mathcal {M}\}=:V$
. Let us show that
$V\subseteq \text {Span}_{\varphi ^{-1}(E)}\{\varphi ^{*}v_1,...,\varphi ^{*}v_n\}$
: pick
$w\in V$
, then there exists
$v\in \mathcal {M}$
such that
$w=\varphi ^{*}v$
so that there exists
$(A_j)_j\subseteq {{\mathscr {B}}}(\textsf {X})$
partition of E and
$(g_i^j)_{j\in {{\mathbb N}}}\subset \mathrm {L}^{\infty }({{\mathfrak m}}_{\textsf {Y}}) \ \forall i=1,...,n$
such that
$$ \begin{align*} \chi_{A_j}v = \sum_{i=1}^n g_i^j v_i\qquad\forall j\in{{\mathbb N.}} \end{align*} $$
Using the linearity of the pullback map and the fact that
$\varphi ^*(g v)=g\circ \varphi \varphi ^{*}v$
for all
$v\in \mathcal {M}$
,
$g\in \mathrm {L}^{\infty }({{\mathfrak m}}_{\textsf {Y}})$
, we get
$$ \begin{align*} \chi_{\varphi^{-1}(A_j)}w = \sum_{i=1}^n g_i^j\circ\varphi\varphi^{*}v_i. \end{align*} $$
Finally, since the pullback module has a natural structure of
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module, we get that
$\text {Span}_{\varphi ^{-1}(E)}\{\varphi ^{*}v_1,...,\varphi ^{*}v_n\}$
is closed, proving the first result.
We now turn to local independence: assume by contradiction
$\{\varphi ^*v_1,...,\varphi ^{*}v_n\}$
are not independent on
$\varphi ^{-1}(E)$
, then there exist
$f_1,...,f_n\in \mathrm {L}^{\infty }({{\mathfrak m}}_{\textsf {X}})$
such that
$\sum _{i=1}^n f_i\varphi ^{*}v_i=0 \ {{\mathfrak m}}$
-a.e. with (upon relabeling indexes)
$|f_1|>0 \ {{\mathfrak m}}$
-a.e. on some subset
$\tilde {E}$
of positive measure. Without loss of generality, possibly considering a smaller set, we shall assume
$f_1>0 \ {{\mathfrak m}}$
-a.e. so that
$$ \begin{align*} \sum_{i=1}^n f_i\varphi^*v_i=0\quad{{\mathfrak m}}-\text{a.e. on}\;\tilde{E}\implies\sum_{i=1}^n \text{Pr}_{\varphi}(f_i) v_i=0\quad{{\mathfrak m}}-\text{a.e. on}\;\tilde{E}. \end{align*} $$
However, note that
$\text {Pr}_{\varphi }(f_1)>0 $
on some set of positive
${{\mathfrak m}}_{\textsf {Y}}$
measure, contradicting the independence of the
$v_i$
s.
Definition 2.15 We say that the space of
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-linear and continuous maps
$L:\mathcal {M}\to L^1({{\mathfrak m}})$
is the dual module of the module
$\mathcal {M}$
, and we shall denote this space by
$\mathcal {M}^*$
.
Remark 2.16 Being
$\mathcal {M} \ {\mathrm {L}}^p({{\mathfrak m}})$
-normed, we can endow
$\mathcal {M}^*$
with a natural structure of
${\mathrm {L}}^q({{\mathfrak m}})$
-normed module.
2.4 The cotangent and tangent modules
We are now in position to speak about the differential of a Sobolev function as the following proposition shows.
Proposition 2.17 Let
$(\textsf {X},{{\,\mathrm d}},{{\mathfrak m}})$
be a metric measure space, then there exists a unique (up to unique isomorphism) couple
$({\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X}),{{\,\mathrm d}}_p)$
where
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
is an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed
$\mathrm {L}^{\infty }({{\mathfrak m}})$
-module and
${{\,\mathrm d}}_p: W^{1,p}(\textsf {X})\to {\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
is a linear and continuous operator such that:
-
(1)
$|{{\,\mathrm d}}_p f|=|Df|_p \ {{\mathfrak m}}$
-a.e. for every
$f\in W^{1,p}(\textsf {X})$
, -
(2) the set
$\{{{\,\mathrm d}} f:\; f\in W^{1,p}(\textsf {X})\}$
generates
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
.
Remark 2.18 We will call 1-forms the elements of
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
, in analogy with the section of the cotangent bundle on a Riemannian manifold.
Definition 2.19 We denote with
${\mathrm {L}}^q({\mathrm {T}}\textsf {X})$
the dual module of
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
, and we call its elements vector fields or vectors.
Besides the differential of a Sobolev function introduced in Proposition 2.17, one can give another definition which exploits the fact that the map is Lipschitz and such that
$\varphi _{\sharp }{{\mathfrak m}}_{\textsf {X}}\leq C{{\mathfrak m}}_{\textsf {Y}}$
for some C¿0 (namely a map of bounded compression): this class of maps is that of bounded deformation. In this direction, we need to recall the notion of pullback of forms: in order to distinguish it from the pullback of a module, we shall proceed denoting with
$\omega \mapsto [\varphi ^*\omega ]$
the pullback map and with
$\varphi ^*$
the pullback of 1-forms which is the following.
Definition 2.20 Let
$\varphi :\textsf {X}\to {\textsf {Y}}$
be a map of bounded deformation, then we define
$\varphi ^*:{\mathrm {L}}^p({\mathrm {T}}^*{\textsf {Y}})\to {\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
to be the linear map such that
$\varphi ^*({{\,\mathrm d}} f)={{\,\mathrm d}}(f\circ \varphi )$
for all
$f\in W^{1,p}({\textsf {Y}})$
and
$\varphi ^*(g\omega )=g\circ \varphi \varphi ^*\omega $
for all
$g\in \mathrm {L}^{\infty }({\textsf {Y}})$
and
$\omega \in {\mathrm {L}}^p({\mathrm {T}}^*{\textsf {Y}})$
.
Remark 2.21 It is easy to see that, thanks to the regularity properties of
$\varphi $
, the pullback of 1-forms
$\varphi ^*$
is well defined.
Definition 2.22 Given
$\varphi :\textsf {X}\longrightarrow {\textsf {Y}}$
of bounded deformation, we define for all
$p\geq 1$
its p-differential as an operator
$\underline {{{\,\mathrm d}}_p\varphi }: {\mathrm {L}}^q({\mathrm {T}}\textsf {X})\longrightarrow \varphi ^*\bigl ({\mathrm {L}}^p({\mathrm {T}}^*{\textsf {Y}})\bigr )^*$
such that
$$ \begin{align} [\varphi^*\omega](\underline{{{\,\mathrm d}}_p\varphi}(v))=\varphi^*\omega(v)\quad\forall v\in {\mathrm{L}}^q({\mathrm{T}}\textsf{X}),\;\;\forall\omega\in {\mathrm{L}}^p({\mathrm{T}}^*{\textsf{Y}}). \end{align} $$
In the recent work [Reference Erikkson-Bique and SoultanisES21], the authors provide some “charts” over Borel sets
$(E_i)_{i\in {{\mathbb N}}}$
partitioning the metric measure space
${{\mathfrak m}}$
-a.e.: we will briefly recall here the definition.
Definition 2.23 We say that
$\varphi :\textsf {X}\to {{\mathbb R}}^N$
is an EBS chart over the Borel set E if it is a Lipschitz map with the following properties:
-
(1) (p-independence)
${\text {ess}\inf }_{v\in \mathbb {S}^{N-1}}|D(v\cdot \varphi )|_p>0 \ {{\mathfrak m}}$
-a.e on E. -
(2) (maximality) There is no other Lipschitz map
$\varphi :\textsf {X}\to {{\mathbb R}}^M$
with
$M>N$
which is p-independent on a subset of E of positive measure.
The authors proved that the condition of p-independence over a set E is equivalent to the fact that the
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})$
module over E is generated by the differentials of the components of the chart: in other words,
$\{{{\,\mathrm d}}_p\varphi ^1,...,{{\,\mathrm d}}_p\varphi ^N\}$
is a basis for
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})_{|E}$
(see Lemma 6.3 in [Reference Erikkson-Bique and SoultanisES21]), and as a consequence of Theorem 1.4.7 in [Reference GigliGig18], we are able to deduce that
${\mathrm {L}}^q({\mathrm {T}}\textsf {X})_{|E}$
is also an N-dimensional normed module.
3 Main result
In this section, we give an alternative proof to Proposition 4.13 in [Reference Erikkson-Bique and SoultanisES21]. First, we remark that with
$\underline {{{\,\mathrm d}}_p\varphi }$
, we will denote the differential of a map of bounded deformation in the sense of Definition 2.22, whereas with
${{\,\mathrm d}}_p f$
, we denote the differential in the sense of Proposition 2.17. Lastly, let us assume that
${{\mathfrak m}}$
is a finite measure: we can do so because of the inner regularity of the measure
${{\mathfrak m}}$
. Indeed, if for a Borel map
$\psi :\textsf {X}\to {{\mathbb R}}^n$
we have
$\psi _{\sharp }({{\mathfrak m}}_{|E_k})<<{{\mathscr L}}^n$
for every
$k\in {{\mathbb N}}$
with
$(E_k)_k$
compact, such that
$E_{k}\subseteq E_{k+1}$
and
${{\mathfrak m}}(E\backslash \cup _k E_k)=0$
, then
$\psi _{\sharp }({{\mathfrak m}}_{|E})<<{{\mathscr L}}^n$
.
We begin with the following simple lemma, which follows standard arguments in linear algebra.
Lemma 3.1 Let
$\mathcal {M}$
be an
${\mathrm {L}}^p({{\mathfrak m}})$
-normed module, and let
$\mathcal {M}^{*}$
be its dual module. Assume that
$\mathcal {M}$
has dimension n over E: then
$\{v_1,...,v_n\}$
and
$\{\omega _1,...,\omega _n\}$
are basis of
$\mathcal {M}^*$
and
$\mathcal {M}$
(respectively) over E if and only if
$\text {det}[ \omega _i(v_j)]_{ij}>0 \ {{\mathfrak m}}$
-a.e. on E.
Proof Define
$A_{ij}:=[ \omega _i(v_j)]_{ij}$
, and let us assume first that
$\det A>0 \ {{\mathfrak m}}$
-a.e. It is clearly sufficient to prove the independence: assume by contradiction that
$\sum _{i=1}^n g_iv_i=0 \ {{\mathfrak m}}$
-a.e. on some subset B of positive measure, for some
$g_1,...,g_n$
which are not all zero on B (in the measure theoretic sense). Then consider
and note that
-a.e. on B because of the condition on the determinant. However,
=0
${{\mathfrak m}}$
-a.e. on B for every
$i=1,...,n$
, which is clearly a contradiction. This argument trivially applies for
$\{\omega _1,...,\omega _n\}$
as well by considering the transpose of A.
Assume now that
$\{\omega _1,...,\omega _n\}$
and
$\{v_1,...,v_n\}$
are basis over E of
$\mathcal {M}$
and
$\mathcal {M}^*$
, respectively, and by contradiction, let
$\det A = 0 \ {{\mathfrak m}}$
-a.e. on a Borel subset C of positive measure. Then there exists a further measurable subset (which we will not relabel) C of positive measure and
for which
and
-a.e. on C. The latter system of equations means that we have
$$ \begin{align} v_i\biggl(\sum_{j=1}^n g_j\omega_j\biggr) = 0\quad{{\mathfrak m}}-\text{a.e.}\; \text{on}\; C,\;\forall i=1,...,n. \end{align} $$
Set
$\tilde {\omega }=\sum _{j=1}^n g_j\omega _j$
and suppose that
$|\tilde {\omega }|\neq 0 \ {{\mathfrak m}}$
-a.e. on C, then there exists a nonzero continuous functional
$\ell \in \mathcal {M}^{\prime }$
(which is the Banach dual) such that
$\ell (\chi _C\tilde {\omega })=||\chi _C\tilde {\omega }||_{\mathcal {M}}$
and there exists
$L\in \mathcal {M}^*$
(see Proposition 1.2.13 in [Reference GigliGig18]) such that
$$ \begin{align*} \ell(\omega) = \int_{\textsf{X}} L(\omega){{\,\mathrm d}}{{\mathfrak m}}\quad\forall\omega\in\mathcal{M}. \end{align*} $$
In our case, this means that
$ \left \lVert {\chi _C\tilde {\omega }} \right \rVert _{\mathcal {M}}=\int _{C}L(\tilde {\omega }){{\,\mathrm d}}{{\mathfrak m}}> 0$
, so that there must be a Borel set of positive measure where
$\chi _C L(\tilde {\omega })>0$
, which contradicts (3.1) since there exists
$D\subset C$
with
${{\mathfrak m}}(D)>0$
such that
$\chi _D L=\sum _{i=1}^n f_i v_i$
for some
$f_1,...,f_n\in \mathrm {L}^{\infty }({{\mathfrak m}})$
.
Lemma 3.2 Let
$\varphi $
be an EBS chart over the Borel set E, and let
$\{v_1,...,v_n\}\in {\mathrm {L}}^p({\mathrm {T}}\textsf {X})$
be independent over E, then
$\{\underline {{{\,\mathrm d}}_p\varphi }(v_1),...,\underline {{{\,\mathrm d}}_p\varphi }(v_n)\}\in \varphi ^*{\mathrm {L}}^p_{\mu }(T{{\mathbb R}}^n)$
are independent over the same set, where
$\mu =\varphi _{\sharp }({{\mathfrak m}}_{|E})$
and
$L^p_{\mu }(T{{\mathbb R}}^n)$
is the tangent module built over
$({{\mathbb R}}^n,{\text {d}}_{{\mathrm eucl}},\mu )$
.
Proof Consider
$f_1,...,f_n\in \mathrm {L}^{\infty }({{\mathfrak m}})$
such that
$$ \begin{align*} \sum_{i=1}^n f_i\underline{{{\,\mathrm d}}_p\varphi}(v_i)=0\quad{{\mathfrak m}}-\text{a.e.}\;\;\text{on}\;\; E, \end{align*} $$
then set
$v:=\sum _{i=1}^n f_iv_i$
. Note that the maps
$\Pi ^j:{{\mathbb R}}^n\longrightarrow {{\mathbb R}}$
being the projection on the jth component are all 1-Lipschitz with respect to the Euclidean distance, and for this reason, they belong to
$W^{1,p}({{\mathbb R}}^n,\text {d}_{\text {eucl}},\mu )$
: following equation (2.11), we have that, for every
$j=1,...,n$
and choosing
$\omega ={{\,\mathrm d}}_p\Pi _j$
,
$$ \begin{align*} 0 = {{\,\mathrm d}}_p\varphi^j(v)=\sum_{i=1}^n f_i{{\,\mathrm d}}_p\varphi^j(v_i)\quad{{\mathfrak m}}-\text{a.e.}\;\;\text{on}\;\; E, \end{align*} $$
where
$\varphi ^j$
is the jth component of the map
$\varphi $
.
Being the matrix
$A=(A_{ij})_{ij}=\langle {{\,\mathrm d}}_p\varphi ^j,v_i\rangle $
such that
$\det A>0 \ {{\mathfrak m}}$
-a.e., the equations above can be rewritten as
a.e. on E with
, meaning
thanks to Lemma 3.1.
The following result is borrowed from [Reference Lučić, Pasqualetto and RajalaLPR21, Proposition 4.5] where only the metric measure space
$({{\mathbb R}}^n,{{\,\mathrm d}}_{\text {eucl}},\mu )$
is considered.
Proposition 3.3 Assume that there exists a Borel set E such that
$\dim {\mathrm {L}}^p_{\mu }({\mathrm {T}}^*{{\mathbb R}}^n)_{|E}=n$
for some
$p\in (1,+\infty )$
, then
$\mu _{|E}<<{{\mathscr L}}^n$
.
Remark 3.4 It is in the proof of the latter proposition that the results contained in [Reference De Philippis and RindlerDR] are used.
Now we are in place to apply Proposition 3.3 to prove the following.
Theorem 3.5 Let
$\varphi :\textsf {X}\to {{\mathbb R}}^N$
be a p-independent weak chart over a Borel set E of positive measure and with
$p\geq 1$
, then
$\mu =\varphi _{\sharp }({{\mathfrak m}}_{|E})<<{{\mathscr L}}^N$
and
$N\leq \text {dim}_H (E)$
.
Proof For the moment, assume that
$p\in (1,+\infty )$
, and without loss of generality, assume that E to be compact. Thanks to Lemma 3.2, we deduce that
$\varphi ^*{\mathrm {L}}^p_{\mu }({\mathrm {T}}^*{{\mathbb R}}^N)$
has dimension N over the set E, meaning that
${\mathrm {L}}^p_{\mu }({\mathrm {T}}^*{{\mathbb R}}^N)$
has dimension N over the set
$\varphi (E)$
. Being the latter module top dimensional, by Proposition 3.3, we have that
$\mu <<{{\mathscr L}}^N$
, which is the first part of the statement. The second part is immediate since if we had
$N>\text {dim}_H(E)$
we would get
$\mathcal {H}^N(E)=0$
and since the map
$\varphi $
is Lipschitz this implies
$\mathcal {H}^N(\varphi (E))={{\mathscr L}}^N(\varphi (E))\leq C\cdot 0=0$
, so that by absolute continuity
$\mu (\varphi (E))={{\mathfrak m}}(E)=0$
, which is clearly a contradiction.
For the case
$p=1$
, note that, since the measure
${{\mathfrak m}}$
is finite, we have
$|D(v\cdot \varphi )|_1\leq |D(v\cdot \varphi )|_p \ {{\mathfrak m}}$
-a.e. and for every
$v\in \mathbb {S}^{N-1}$
, meaning that
$\varphi $
is also p-independent and the same argument applies.
Remark 3.6 By virtue of the latter theorem, one can see that a control on the Hausdorff dimension l of a subset E of a metric measure space grants that the dimension of
${\mathrm {L}}^p({\mathrm {T}}^{*}\textsf {X})_{|E}$
is bounded by l; hence, the cotangent module is finite-dimensional there. Moreover, the proof presented here simplifies the one in [Reference Gigli and PasqualettoGP21] since there the authors needed to build independent vector fields in
${\mathrm {L}}^2({\mathrm {T}}\textsf {X})$
with
${\mathrm {L}}^2({{\mathfrak m}})$
-integrable divergence and push them to
${{\mathbb R}}^n$
keeping them independent and regular: to do so, they had to use additional properties of the map
${\mathrm {Pr}}_{\varphi }$
and the bi-Lipschitz regularity of their chart
$\varphi $
was essential. Here, instead, we mainly exploit the properties of
${{\mathbb R}}^n$
.
Acknowledgment
We wish to thank Elefterios Soultanis for the numerous conversations we had with him while working on this manuscript.
