1 Introduction↩︎

Let \((M,g)\) be a closed Riemannian manifold and \(W^{k,p}(M)\) the corresponding Sobolev space of order \(k\) with respect to \(L^p(M)\), where \(L^p(M)\) is endowed with a Riemannian volume form \(d\nu(g)\) and \(d\nu(g)=\sqrt{\det(g_{ij})}dx,\) and \(dx\) stands for the Lebesgue volume element of \(\mathbb{R}^n,\) \(n=\dim(M)\) (see Definition 2.3). Herein we find the set of Gâteaux differentiability for the Sobolev norm of \(W^{k,1}(M)\). The results also hold for open sets in the euclidean space. We point out that the case of the spaces \(W^{k,p}\), when \(1<p<\infty\) boils down to the Fréchet differentiability of the \(L^p\) norm. The case of our interest \(p=1\) is more pathological.
The problem of finding the set of differentiability of Lipschitz functions was initially considered by Rademacher in [1], where he states his classical theorem:

More generally, an extension due to Stepanov shows that the same conclusion holds for locally Lipschitz functions \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m\). The Rademacher-Stepanov theorem is a fundamental tool in the study of differentiability and is at the basis of some hallmarks, as the Whitney’s extension theorem (cf. [2], [3], [4]) and its further recent developmets by Fefferman (cf. [5], [6]) and Figalli [7]. The Rademacher-Stepanov theorem can also be deduced from the theory of Sobolev spaces as a consequence of a Calderon’s result. Indeed, if \(f\in W_{loc}^{1,\,p}({\mathbb{R}}^n)\) and \(n< p\leq \infty\), then \(f\) is differentiable almost everywhere and its derivative equals its weak derivative almost everywhere. Consequently, since every locally Lipschitz function \(f\) belongs to \(W_{loc}^{1,\,\infty}({\mathbb{R}}^n)\). Then, \(f\) is differentiable almost everywhere (cf. Chapter 6 of [8].). We would also point out that the Rademacher–Stepanov theorem has been proved in the setting of sub-Riemannian geometry in [9]. On the other hand, an important contribution to the differentiability of Sobolev functions with respect to Sobolev norms has been developed in [10] and [11].
The converse of Rademacher Theorem has been recently established by Preiss and Speight (cf. [12]). Indeed, they prove that: if \(n>1\) then there exists a Lebesgue null set in \(\mathbb{R}^{n}\) containing a point of differentiability of each Lipschitz function \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n-1}\). A more detailed discussion around the Rademacher’s Theorem and some converse versions can be found in [13].
An extension of Rademacher-Stepanov Theorem to functions defined on separable Banach spaces and valued into Banach spaces was established by Phelps in [14]:

Other extensions of Rademacher-Stepanov Theorem to Lipschitz maps between infinite-dimensional Banach spaces have been investigated by several authors e.g. Aronszajn [15], Bongiorno [16], Mankiewicz [17], Phelps [14], Preiss and Zajíček [18], among others). The research on the differentiability on Banach spaces has been a very active field in the last decades. We just quote some few recent examples: In Potapov and Sukachev’s work [19], they answer to a question in the theory of Schatten- von Neumann ideals of whether their norms have the same differentiability properties as the norms of their commutative counterparts. On the other hand Lindenstrauss et al. [20] prove that a real valued Lipschitz function on an Asplund space has points of Fréchet differentiability. In  [21] Goldys and Gozzi apply the Phelps Theorem in the proof of existence and uniqueness of solutions of second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces, obtaining optimal feedback for first order stochastic PDEs that arise in economic theory, the theory of population dynamics and financial models. We also point out that the \(L^1\)-norm has been of recent interest in the analysis of big data due to its major role in minimisations problems [22]. The study of the set of differentiability of Lipschitz functions on Banach spaces has been carried out for instance in Deville et al. [23], Diestel [24], Dore and Maleva [25] among others.
We also point out that on compact Riemannian manifolds, the \(L^p\) spaces being totally ordered by inclusion, the \(L^1(M)\) space arises as a maximal one with respect to \(p\) in the range \(1\leq p\leq\infty\) and the norm \(\|\cdot\|_{L^1(M)}\) as a minimal one. In a future work we will be studying some geometric properties with respect to the \(L^1(M)\) norm and level sets using a geometric characterization as in [26]. It is also reasonable to expect some extensions of our results to compact manifolds with boundary and contact manifolds [27]. Other recent works on Sobolev spaces can be found in [28], [29] and [30].
The main result of this work establishes the set of Gâteaux differentiability of the the Sobolev norm of \(W^{k,1}(M)\) and is stated in Theorem 5, giving also the corresponding formulae for the Gâteaux derivative. These results extend a \(L^1\) version obtained in [31] for a suitable family of measure spaces. Herein we focous on a version on manifolds thinking on potential applications to differential equations or variational calculus on manifolds.

2 Preliminaries↩︎

In this section we briefly recall a basic background. We start with the classical notions of differentiability on Banach spaces. Secondly, we introduce our scale of Sobolev spaces on Riemannian manifolds.

2.1 Gâteaux differentiability on Banach spaces↩︎

The most simple notion of differentiability on Banach spaces is the one of Gâteaux differentiability. Let \(X\), \(Y\) be Banach spaces and \(\Omega\subseteq X\) an open set. A function \(f:\Omega\rightarrow Y\) is Gâteaux differentiable (\(G\)-differentiable) at a point \(a\in\Omega\), if \[\lim\limits_{t\rightarrow 0} \frac{f(a+th)-f(a)}{t}=u \left(h\right)\] exists for all \(h\in X\), and \(u\) defines a bounded linear operator from \(X\) into \(Y\). The operator \(u\) is called the Gâteaux derivative of \(f\) at the point \(a\) and is denoted by \(\partial_{G}f(a)\).
A function \(f:\Omega \rightarrow Y\) is Fréchet differentiable (\(F\)-differentiable) at \(a\), if there is a bounded linear operator \(u\) such that \[f\left(a+h\right)-f\left(a\right)-u\left(h\right)=r\left(h\right)\] where \(\lim\limits_{h\rightarrow 0} \frac{\left\|r\left(h\right)\right\|_{Y}}{\left\| h \right\|_{X}}=0\). The operator \(u\) is called the Fréchet derivative of \(f\) at the point \(a\) and is denoted by \(\partial_{F}f(a)\).
It is not difficult to see that, if \(f\) is \(F\)-differentiable at \(a\) then \(f\) is \(G\)-differentiable at \(a\) and in that case \(\partial_{F}f(a)=\partial_{G}f(a)\). For a given function \(\varphi:X\rightarrow Y\), we say that \(B\) is the set of \(G\)-differentiability or \(F\)-differentiability of \(\varphi\), if respectively for every element of the set \(B\) the function \(\varphi\) is \(G\)-differentiable or \(F\)-differentiable.
On the other hand, a set is called a \(G_{\delta}\) set if it can be expressed as a countable intersection of open sets. This type of set is fundamental in the definition of the F-Asplund and \(G\)-Asplund spaces, since these respectively are Banach spaces in which all convex and continuous function defined in an open and convex subset is \(F\)-differentiable (resp. \(G\)-differentiable) in a dense and \(G_{\delta}\) set. The following theorem shows important examples of \(G\)-Asplund spaces.

Moreover, it is well known that the set of all \(F\)-Asplund spaces is strictly contained in the set of all \(G\)-Asplund spaces. In fact, as proved in Deville et al. [23], the usual norm of \(\ell^{1}(\mathbb{R})\) is not \(F\)-differentiable at any point of the space \(\ell^{1}(\mathbb{R})\). Then \(\ell^{1}(\mathbb{R})\) is not a \(F\)-Asplund space; but \(\ell^{1}(\mathbb{R})\) is a separable Banach space and therefore a \(G\)-Asplund space by Mazur’s theorem.
As we have quoted before, the \(L^p\) norm is \(F\)-differentiable at any point in the case \(1<p<\infty\) and its corresponding derivative is given by 2 . Moreover, since the dual of \(L^{p}(\Omega)\) is \(L^{q}(\Omega)\), for \(1<p<\infty\) and \(\frac{1}{p}+\frac{1}{q}=1\), and they are separable Banach spaces, by Theorem 2.12 in Phelps [32] the space \(L^{p}(\Omega)\) is a \(F\)-Asplund space for \(1<p<\infty\).

2.2 Sobolev spaces on Riemannian manifolds↩︎

Let \((M,g)\) be a Riemannian manifold. For any \(k\in \mathbb{N}\) and for \(u\in C^{\infty}(M),\) \(\nabla^{k}u\) denotes the \(k\)-th covariant derivative of \(u,\) (with \(\nabla^{0}u:=u\)). In order to illustrate this notion, we observe that in local coordinates the components of \(\nabla u\) take the form \((\nabla u)_i=\partial_{i}u,\) while the components of \(\nabla^2 u\) in local coordinates are given by \[(\nabla^2 u)_{ij}=\partial_{ij}u-\Lambda^{k}_{ij}\partial_k u.\] The notation \(\Lambda^{k}_{ij}\) stands for the Christoffel symbols with respect to the (unique) Riemannian connection.
In general, by definition one has that \[|\nabla^k u|^2=g^{i_1j_1}\cdots g^{i_kj_k}(\nabla^k u)_{i_1\cdots i_k}(\nabla^k u)_{j_1\cdots j_k}.\] For every \(k\geq 1,\) we denote by \(C^{p}_{k}(M)\) the spaces of smooth functions \(u\in C^\infty(M),\) such that \(|\nabla u|\in L^p(M)\) for any \(0\leq j\leq k.\) Hence \[C^{p}_{k}(M)=\{u\in C^\infty(M): \int_{M}|\nabla^j u|^pd\nu(g)<\infty,\,0\leq j\leq k\}\]where in local coordinates \(d\nu(g)=\sqrt{\det(g_{ij})}dx,\) and \(dx\) stands for the Lebesgue volume element of \(\mathbb{R}^n,\) \(n=\dim(M).\) If \(M\) is compact one has that \(C^{p}_{k}(M)=C^\infty(M).\) Now, with the notation above, one can introduce in a global way the definition of the Sobolev spaces, see Hebey [34].

In particular, if \((M,g)\) is a compact and without boundary let us follow Taylor [35]. There is a global formulation for the space \(W^{k,p}(M,g)\) in terms of vector fields. In this (exceptional) case, \(M\) being a closed manifold, the space \(W^{k,p}(M)=W^{k,p}(M,g)\) is independent of the metric \(g\) (indeed, if \(M\) is not compact, in general, \((M,g)\) depends of \(g,\) which can be proved if one considers the case of Gaussians and non-Gaussian metrics). The fact of the independence of \(W^{k,p}(M,g)\) with respect to the choice of the Riemmanian metric \(g\) is well-known, but we recall its proof for the convenience of the reader.

On the other hand, one can prove that \(u\in W^{k,p}(M),\) if and only for every arbitrary family of smooth vector fields \(X_{j_i}\in TM,\) where \(1\leq j_i\leq \ell,\) and \(\ell \leq k,\) one has that \[X_{j_1}X_{j_2}\cdots X_{j_\ell} u\in L^{p}(M).\] In such a case one has the equivalent norm \[\Vert u\Vert_{{W}^{k,p}(M)}=\left(\sum_{|\alpha|\leq k}\int_{M}|D^{\alpha} u|^pd\nu(g)\right)^{\frac{1}{p}},\]where we have used the multi-index notation \[\label{alpha} D^{\alpha}:=X_{j_1}^{\alpha_1}\cdots X_{j_\ell}^{\alpha_\ell},\tag{1}\] for an arbitrary choice of \(\alpha=(\alpha_1,\cdots , \alpha_\ell).\)
We recall that in any of the above cases, the elements of Sobolev spaces should be understood as equivalence classes, where \(f\sim g\) if \(\|f-g\|_{W^{k,1}(M)}=0\).
The notion of capacity may lead to some further properties on the set of Gâteaux differentiability and we will be addressing this problem in a next work, as well as the study of the set of Gâteaux differentiability for the norms on Nikol’skii-Besov spaces \(B^s_1(M)\).

3 The \(G\)-differentiability set for the \(W^{k,1}(M)\) norm↩︎

In this section we present our main result. We first recall the formulae for the Fréchet differentiability in the case of \(L^{p}(\Omega)\) spaces, where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and \(1<p<\infty\). Indeed, it is well-known (cf. [24]) that the norm \(\|f\|_{L^{p}} = (\int_{\Omega}|f(w)|^{p}dw)^{1/p}\) is Fréchet differentiable and moreover its derivative is given by \[\partial_{F}\|f\|_{L^{p}}=\frac{|f|^{p-1}\mathop{\mathrm{sign\,}}(f)}{\|f\|_{L^{p}}^{p-1}},\label{formF1q}\tag{2}\] for any \(f\neq 0\). As a consequence, an extension is available for our Sobolev spaces \(W^{k,p}(M)\), with \(1<p<\infty\) obtaining that \(\|\cdot\|_{W^{k,p}(M)}\) is Fréchet differentiable and satisfies \[\partial_{F}\|f\|_{W^{k,p}(M)}=\sum_{|\alpha|\leq k}\frac{|D^{\alpha}f|^{p-1}\mathop{\mathrm{sign\,}}(D^{\alpha}f)}{\|f\|_{W^{k,p}(M)}^{p-1}},\] this for \(D^{\alpha}f\), the multi-index notation of a partial derivative of \(f\), with \(|\alpha|\leq k\) and \(f\neq 0\).
We can now focus in the pathological case \(p=1\) and the Sobolev norm \(\|\cdot\|_{W^{k,1}(M)}\). We start by establishing some preliminary results within a more general context of meausure spaces. Given a measure space \((\Omega, \mathcal{F}, \mu)\), we denote by \(\mathcal{L}^1(\Omega, \mu)\) the space of the real valued integrable functions over \(\Omega\). On \(\mathcal{L}(\Omega, \mu)\) one defines the seminorm \(\varphi(f)=\int_{\Omega}|f(x)|d\mu(x)\). One also defines on \(\mathcal{L}(\mu)\) an equivalence relation \(\mathcal{R}\) by, \(f\mathcal{R}g\Longleftrightarrow f = g\, \mu-a.e.\) The equivalence class of \(f\) is denoted by \(\tilde{f}.\) Thus, the quotient \(\mathcal{L}(\Omega)/ \mathcal{R}\) defines the classical Banach space \(L^{1}(\Omega,\mu)\) endowed with the norm \(\varphi(\tilde{f})=\int_{\Omega}|f(x)|d\mu(x)\).
The signature function will arise in a natural way in our study of the Gâteaux derivative.

In order to prove the main theorem we first recall a mild lemma which will help to obtain a pointwise formulae for the Gâteaux derivative and its proof can be found in Page 97, Lemma 3.1 of [31].

We now consider a measure space \((\Omega, \mathcal{F},\mu)\). As a consequence of the lemma above we obtain the following corollary.

We are now ready to present our main result. Let us fix a compact Riemannian manifold \((M,g).\) The following theorem gives a precise description of the \(G\)-differentiability set for the usual \(W^{k,1}(M)\) norm and the corresponding formulae for the Gâteaux derivative. We shall use the notation \(D^{\alpha}:=X_{j_1}^{\alpha_1}\cdots X_{j_\ell}^{\alpha_\ell},\) in 1 for the arbitrary compositions of smooth vector fields. We will denote by \(\mu\) the measure corresponding to the Riemannian metric \(g\), i.e., \(\mu=\nu(g)\).
Let \(B\) be the subset of \(W^{k,1}(M)\) consisting of equivalence classes \(\tilde{f}\) such that for some representative \(f\) of the class \(\tilde{f}\), one has that \(D^{\alpha}f(x)\neq 0\,\) \(\mu\) a.e. for every \(|\alpha|\leq k\). It is clear that if \(\tilde{f}\in B\), any representative of the class \(\tilde{f}\) satisfies the condition defining \(B\). That is, if \(f_1\) is another representative of the class \(\tilde{f}\), then \(D^{\alpha}f_1(x)\neq 0\,\) \(\mu\) a.e. for every \(|\alpha|\leq k\), due to the definition of the equivalence relationship through the corresponding Sobolev norm of \(W^{k,1}(M)\).

We now deduce a consequence for the mean value theorem for the Gâteaux derivative, where the knowledge on the set of differentiability as described in Theorem 5 will be crucial.

The first author was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). The second author was supported by Vic. Inv. Universidad del Valle CI 71352.
\(\bullet\) Our manuscript has not associated data.
\(\bullet\) No conflict of interest/Competing interests

References↩︎