Vector calculus on weighted reflexive Banach spaces
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Vector calculus on weighted reflexive Banach spaces © The Author(s) 2024 Published version Pasqualetto, Enrico; Rajala, Tapio Pasqualetto, E., & Rajala, T. (2024). Vector calculus on weighted reflexive Banach spaces. Mathematische Zeitschrift, 308, Article 48. https://doi.org/10.1007/s00209-024-03605-6 2024
Mathematische Zeitschrift (2024) 308:48 https://doi.org/10.1007/s00209-024-03605-6 Mathematische Zeitschrift Vector calculus on weighted reflexive Banach spaces Enrico Pasqualetto 1·Tapio Rajala 1 Received: 14 June 2023 / Accepted: 11 September 2024 © The Author(s) 2024 Abstract We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles. Keywords Sobolev space ·Weighted Banach space ·Tangent bundle ·Test plan Mathematics Subject Classification 53C23 ·46E35 ·18F15 ·49J52 List of symbols L1Lebesgue measure restricted to [0,1];see(2.1). (B,μ) A weighted Banach space; see Definition 2.1. C(B)Shorthand notation for C([0,1]; B)×[0,1];see(2.2). e The evaluation map e:C(X)→X, given by e(γ, t)=et(γ ) := γt. Der Derivative map; see (2.4). |df|B∗The function Bx→dxfB∗∈Rfor f∈C1(B);see(2.6). Comp(π)Compression constant of a q-test plan π; see Definition 2.2. q(B,μ) q-test plans on the weighted Banach space (B,μ); see Definition 2.2. ˆ πShorthand notation for ˆ π:= π⊗L1;see(2.8). {ˆ πx}x∈BConditional probabilities of the disintegration ˆ π=ˆ πxd(e#ˆ π)(x). W1,p(B,μ) Metric p-Sobolev space on a weighted Banach space (B,μ); see Definition 2.5. |Dμf|The minimal p-weak upper gradient of f∈W1,p(B,μ); see Definition 2.5. SπThe ‘support’ of a q-test plan π;see(2.11). Dq(divμ)Domain of the distributional divergence; see Definition 2.8. divμThe distributional divergence of v∈Dq(divμ); see Definition 2.8. Dμ(B)Space of μ-a.e. defined measurable B-bundles; see Definition 2.10. q(E)q-section space of a measurable B-bundle E∈Dμ(B);see(2.14). The natural partial order on Dμ(B). TμBq-tangent bundle of (B,μ); see Definition 2.11. BTapio Rajala [email protected] Enrico Pasqualetto [email protected] 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014 Jyväskylä, Finland 0123456789().: V,-vol 123
48 Page 2 of 21 E. Pasqualetto, T. Rajala p(E∗ w∗)Space of weakly∗measurable sections of the dual of E. |ω|E∗μ-a.e. equivalence class of X x→ω(x)E(x)∗∈Rfor ω∈p(E∗ w∗). Lp w∗(X,μ;B∗)The dual space of Lq(X,μ;B). ω|EThe restriction of ω∈p(E∗ w∗)to a bundle E∈Dμ(B); see Definition 2.14. d|EThe restricted Fréchet differential C1(B)∩LIPb(B)f→ df|E;see (2.15). [ω]E∗The pointwise seminorm of ω∈Lp(X,μ;B∗)induced by E∈Dμ(B);see (2.16). πEThe ‘restriction’ operator Lp(X,μ;B∗)ω→ ω|E∈p(E∗ w∗);see (2.17). ϕ∗EThe pullback bundle; see (2.18). LfThe functional Lf∈Dq(divμ)∗associated to f∈W1,p(B,μ);see(3.1). WDp μ(f;A)(A,p)-weak differentials of f∈W1,p(B,μ); see Theorem 3.1 c). dμThe p-weak differential operator dμ:W1,p(B,μ)→q(TμB)∗;see(3.2). vπVector field induced by a q-test plan π; see Lemma 3.6. SπBMultivalued map sending x∈Bto the support of Der#ˆ πx;see(3.13). VπBThe bundle obtained as the closure of the span of SπB; see Lemma 3.7. DerπEquivalence class of Der in Lq(C(B), ˆ π;B); see the proof of Theorem 3.3. πVelocity field of a q-test plan π; see Remark 4.3. 1 Introduction During the past couple of decades, Sobolev spaces W1,p(X,μ)have been extensively studied for metric measure spaces (X,d,μ). In spaces satisfying a local Poincaré inequality and measure doubling, called PI-spaces, the strong density of Lipschitz functions [8] implies quite straightforwardly the equivalence of different approaches to Sobolev spaces. In [4], it was noticed that the density in energy of Lipschitz functions, valid without the PI-assumption, is enough for showing the equivalence of notions of upper gradients. In general metric measure spaces one can also introduce an abstract first-order differentiable structure [20]. With more assumptions on the space, the structure is given via Lipschitz charts [8](seealso[15]). If the underlying metric structure is linear, a natural question is to ask if we can connect the abstract differentiable structures with the linear one. In Euclidean spaces with arbitrary reference measure this was addressed in [27]and[31](seealso[18] for the BV case) by starting from the notions of Sobolev space given in [7]and[36]. Although weighted Euclidean spaces need not be PI-spaces, smooth functions are still strongly dense in the Sobolev space defined on them [23]. In the current work we consider the infinite-dimensional linear case of reflexive separable Banach spaces (and, in some results, the larger class of separable Banach spaces having the Radon–Nikodým property). In this context, it is known from [34](seealso[17, 35]) that suitable algebras Aof smooth functions on Bare dense in energy in the Sobolev space W1,p(B,μ). One advantage of working with smooth functions is that they have an everywhere-defined differential, which allows to transfer valuable information from the ambient space to the Sobolev space with respect to arbitrary reference measures. Notice that even though our results are stated in the infinite-dimensional case, they are new already on Euclidean spaces since we allow the exponent p= 2inW1,p. Indeed, only the case p=2 was handled in [31] since there the arguments relied heavily on the Hilbertian structure of the 123
Vector calculus... Page 3 of 21 48 Sobolev space W1,2. Similarly, it would be simpler to prove versions of our results in cases where one could use the Hilbertian structure, such as for W1,2on infinite-dimensional Hilbert spaces, or for the cases where one can renorm the Sobolev space to be Hilbert (compare to [14]). The paper contains three main results, which are valid for weighted Banach spaces (as in Definition 2.1) that are reflexive (or some weaker assumptions, as the Radon–Nikodým property): •In Theorem 3.1 we prove that the metric notion of Sobolev space via upper gradients (see Definition 2.5) coincides with other two approaches (in terms of vector fields having distributional divergence, and via weak differentials) that are tailored to the Banach setting. •In Theorem 3.2 we show that also the minimal weak upper gradients corresponding to the three above approaches do coincide. •Theorem 3.3 connects different approaches to identifying the directions in the Banach space that are analytically relevant for W1,p(B,μ). The first approach is via divergence inspired by Bouchitté–Buttazzo–Seppecher [7] while the other two are via test plans. This result partly generalises the Euclidean one [31, Theorem 3.16]. 2 Preliminaries Throughout the whole paper, whenever an exponent p∈(1,∞)is given, we tacitly denote by q:= p p−1∈(1,∞) its conjugate exponent, and vice versa. Moreover, letting L1be the Lebesgue measure, we shorten L1:= L1|[0,1].(2.1) In this paper, we focus on the family of weighted Banach spaces, which we define as follows: Definition 2.1 (Weighted Banach space) We say that a couple (B,μ)is a weighted Banach space if Bis a separable Banach space and μ≥0 is a finite Borel measure on B. We underline that in the above definition we assume that Bis separable and μis finite. 2.1 Classical notions on Banach spaces GivenanormedspaceV,wedenotebyV∗its dual Banach space. If Bis the Banach space obtained as the completion of V,thenwehavethatB∗∼ =V∗,meaningthatB∗can be canonically identified with V∗. The duality pairing between ω∈V∗and v∈Vwill be denoted by ω,v∈R. Absolutely continuous curves Let Bbe a separable Banach space. We denote by C([0,1]; B)the space of all continuous curves γ:[0,1]→B. It is a complete and separable metric space if endowed with the 123
48 Page 4 of 21 E. Pasqualetto, T. Rajala following distance: dC([0,1];B)(γ, σ ) := max t∈[0,1]γt−σtBfor every γ,σ ∈C([0,1]; B). We then define the complete and separable metric space (C(B), dC(B))as C(B):= C([0,1]; B)×[0,1](2.2) together with the distance dC(B)((γ, t), (σ, s)) := dC([0,1];B)(γ, σ ) +|t−s|for every (γ, t), (σ, s)∈C(B). The evaluation map e:C(B)→Bis defined as e(γ, t)=et(γ ) := γtfor every (γ, t)∈ C(B). Notice that e:C(B)→Bis continuous and et:C([0,1]; B)→Bis 1-Lipschitz for every t∈[0,1].Acurveγ∈C([0,1]; B)is said to be q-absolutely continuous (for some q∈[1,∞]) if there exists a function g∈Lq(0,1)such that g≥0and γt−γsB≤t s g(r)drfor every 0 ≤s<t≤1.(2.3) In the case where q=1, we write ‘absolutely continuous’ instead of ‘1-absolutely continuous’. We denote by ACq([0,1]; B)the family of all q-absolutely continuous curves in B.It holds that ACq([0,1]; B)is a Borel subset of C([0,1]; B). We write AC([0,1]; B)instead of AC1([0,1]; B). Radon–Nikodým property and Asplund spaces A Banach space Bis said to have the Radon–Nikodým property provided every absolutely continuous curve γ:[0,1]→Bis L1-a.e. differentiable, which means that ˙γt:= lim h→0 γt+h−γt h∈Bexists for L1-a.e. t∈[0,1]. By an Asplund space we mean a Banach space Bwhose dual B∗has the Radon–Nikodým property. Recall that every reflexive Banach space is Asplund and has the Radon–Nikodým property. The converse can fail: there exist (separable) Asplund spaces having the Radon– Nikodým property that are not reflexive, e.g. James’ space [26]. See [16] for a thorough treatment of these topics. Given a Banach space Bhaving the Radon–Nikodým property, we define Der :C(B)→B as Der(γ, t):= ˙γt 0B if γ∈AC([0,1]; B)and ˙γtexists, otherwise. (2.4) Then Der :C(B)→Bis a Borel map. For any γ∈AC([0,1]; B)the function Der(γ, ·)B, which is called the metric speed of γ,istheL1-a.e. minimal g∈L1(0,1)with g≥0 satisfying (2.3). Fréchet differential and smooth functions Given a Banach space Band a function f:B→R, we say that fis Fréchet differentiable at x∈Bif there exists an element dxf∈B∗, called the Fréchet differential of fat x,such that lim Bv→0 |f(x+v) −f(x)−dxf,v| vB =0.(2.5) 123
Vector calculus... Page 5 of 21 48 Notice that (2.5) determines uniquely dxfand implies that fis continuous at x. Moreover, we say that fis of class C1if it is Fréchet differentiable at every point of Band Bx→ dxf∈B∗is continuous. We denote by C1(B)the space of real-valued functions of class C1 defined on B. Given any f∈C1(B), we define the Borel function |df|B∗:B→[0,+∞) as |df|B∗(x):= dxfB∗for every x∈B.(2.6) One can readily check that the function fis locally Lipschitz and it holds that |df|B∗= lip(f),wheretheslope lip(f):B→[0,+∞)is defined as lip(f)(x):= 0ifx∈Bis an isolated point and lip(f)(x):= lim sup By→x |f(x)−f(y)| x−yB if x∈Bis an accumulation point. Lebesgue–Bochner spaces Given a finite measure space (X,,μ)and an exponent p∈[1,∞],wedenoteby(Lp(μ), · Lp(μ))the Lebesgue space of exponent p. Recall that Lp(μ) is a Riesz space if endowed with the natural partial order relation: given any two functions f,g∈Lp(μ), we declare that f≤gif and only if f(x)≤g(x)holds for μ-a.e. x∈X. Recall that the Riesz space Lp(μ) is Dedekind complete, which means that every non-empty subset of Lp(μ) that is bounded above has a supremum. Namely, given a set {fi}i∈I⊆Lp(μ) and g∈Lp(μ) such that fi≤gfor every i∈I, then the supremum f:= i∈I fi∈Lp(μ) exists. This means that f≥fifor every i∈I,andthat f≤˜ fwhenever ˜ f∈Lp(μ) satisfies ˜ f≥fifor every i∈I. In a similar way, one can define the infimum i∈Ifi∈Lp(μ).See e.g. [6]. We assume the reader is familiar with the basics of Bochner integration; we refer to [25]and the references therein for a detailed account of this theory. Let us only recall some notation and results. Given a finite measure space (X,,μ), a Banach space B, and an exponent q∈(1,∞),wedenotebyLq(X,μ;B)the q-Lebesgue–Bochner space from (X,,μ)to B. The following hold: •Lq(X,μ;B)is a Banach space and a module over the commutative ring L∞(μ). •Lp(X,μ;B∗)is isomorphic to a subspace of Lq(X,μ;B)∗. •Lq(X,μ;B)∗∼ =Lp(X,μ;B∗)if and only if Bis Asplund. •Lq(X,μ;B)is reflexive if and only if Bis reflexive. •Lq(X,μ;B)is uniformly convex if and only if Bis uniformly convex. Given any v∈Lq(X,μ;B),theμ-a.e. equivalence class |v|Bof the function X x→ v(x)Bbelongs to Lq(μ).If(X, X,μ X)and (Y, Y,μ Y)are finite measure spaces, then each measurable map ϕ:X→Y satisfying ϕ#μX≤CμYfor some C>0 induces a pullback operator ϕ∗:Lq(Y,μ Y;B)→Lq(X,μ X;B)(2.7) for every Banach space Band q∈(1,∞). Namely, we set ϕ∗v:= v◦ϕfor all v∈ Lq(Y,μ Y;B). 123
48 Page 6 of 21 E. Pasqualetto, T. Rajala Given a weighted Banach space (B,μ) and f∈C1(B)∩LIP(B),themapBx→ dxf∈B∗is Bochner integrable (as it is continuous and bounded), thus we can consider its equivalence class df∈Lp(B,μ;B∗)for every p∈(1,∞). For any v∈Lq(B,μ;B),theμ-a.e. equivalence class d f(v) of Bx→dxf,v(x)∈R is in L1(μ). 2.2 Sobolev calculus on weighted Banach spaces Test plans Following [5], we give the ensuing definition of a q-test plan (over a weighted Banach space): Definition 2.2 (Test plan) Let (B,μ)be a weighted Banach space such that Bhas the Radon– Nikodým property and let q∈(1,∞). Then a Borel probability measure πon C([0,1]; B) is said to be a q-test plan on (B,μ)provided the following two requirements are met: (i)There exists a constant C>0 such that (et)#π≤Cμfor every t∈[0,1]. The minimal such Cis called the compression constant of πand denoted by Comp(π). (ii)The measure πis concentrated on ACq([0,1]; B)and has finite kinetic q-energy, i.e. 1 0 ˙γtq Bdtdπ(γ ) < +∞. We denote by q(B,μ)the family of all q-test plans on (B,μ). We introduce the shorthand notation ˆ π:= π⊗L1for every π∈q(B,μ). (2.8) Observe that e#ˆ π≤Comp(π)μ, so that in particular e#ˆ πμ,foreveryπ∈q(B,μ). We will occasionally consider the disintegration ˆ π=ˆ πxd(e#ˆ π)(x)of ˆ πalong e, which means that: •{ ˆ πx}x∈Bare Borel probability measures on C(B)such that ˆ πx(C(B)\e−1(x)) =0fore #ˆ π-a.e. x∈B. •Bx→ ˆ πxis measurable, i.e. Bx→ ˆ πx(E)is Borel for every E⊆C(B)Borel. •ˆ π(E)=ˆ πx(E)d(e#ˆ π)(x)for every E⊆C(B)Borel. The family {ˆ πx}x∈Bis e#ˆ π-a.e. unique. For a proof of its existence, see e.g. [3, Theorem 5.3.1]. Compatible algebras For any Banach space B, we call LIPb(B)the algebra of real-valued bounded Lipschitz functions on B. Following [34, Definition 2.1.17], we give the ensuing definition of compatible algebra: 123
Vector calculus... Page 7 of 21 48 Definition 2.3 (Compatible algebra) Let Bbe a Banach space. Then we say that a set Ais a compatible subalgebra of LIPb(B)provided it is a unital subalgebra of LIPb(B)such that x−yB=sup |f(x)−f(y)|f∈LIPb(B)is 1-Lipschitzfor every x,y∈B. Distinguished examples of compatible subalgebras are LIPb(B)itself, C1(B)∩LIPb(B), and the smaller space of smooth cylindrical functions on B(defined in [34,Example 2.1.19]). A useful fact concerning compatible algebras, which is proved in [34, Lemma 2.1.27], states the following: Lemma 2.4 Let (B,μ)be a weighted Banach space. Let Abe a compatible subalgebra of LIPb(B).ThenAis strongly dense in L p(μ) for every p ∈[1,∞), and is weakly∗dense in L∞(μ). Metric Sobolev spaces Let us recall the definition of the metric Sobolev space via test plans introduced in [5]: Definition 2.5 (Metric Sobolev space) Let (B,μ) be a weighted Banach space such that B has the Radon–Nikodým property and let p∈(1,∞).Thenwesaythat f∈Lp(μ) is a p-Sobolev function provided there exists G∈Lp(μ) with G≥0 such that f(γ1)−f(γ0)dπ(γ ) ≤1 0 G(γt)˙γtBdtdπ(γ ) for every π∈q(B,μ). (2.9) The μ-a.e. minimal Gverifying (2.9) is called the minimal p-weak upper gradient of f and is denoted by |Dμf|∈Lp(μ). The space of all p-Sobolev functions on (B,μ)is denoted by W1,p(B,μ). The Sobolev space W1,p(B,μ)is a Banach space if endowed with the following norm: fW1,p(B,μ) := fp Lp(μ) +|Dμf|p Lp(μ)1/p for every f∈W1,p(B,μ). It holds that LIPb(B)⊆W1,p(B,μ)and |Dμf|≤lip(f)for every f∈LIPb(B). Moreover, the minimal p-weak upper gradient |Dμf|of any given function f∈W1,p(B,μ) can be equivalently characterised as the μ-a.e. minimal G∈Lp(μ) with G≥0 satisfying the following property: for every π∈q(B,μ),wehavethat f◦γ∈W1,1(0,1)holds for π-a.e. γ∈C([0,1]; B)and |(f◦γ) t|≤G(γt)˙γtBfor ˆ π-a.e. (γ, t)∈C(B). (2.10) The following approximation result (which closes the gap with Cheeger’s approach to metric Sobolev spaces [8], based on a relaxation procedure) was proved in [34, Theorem 5.2.7] after [4]: Theorem 2.6 (Density in energy of compatible algebras) Let (B,μ)be a weighted Banach space such that Bhas the Radon–Nikodým property and let p ∈(1,∞).LetAbe a compatible subalgebra of LIPb(B).LetW1,p(B,μ)be a given function. Then there exists a sequence (fn)n⊆Asuch that fn→f,|dfn|B∗→|Dμf|strongly in L p(μ). 123
48 Page 8 of 21 E. Pasqualetto, T. Rajala Another proof of the density in energy of smooth cylindrical functions (and thus of every subalgebra of LIPb(B)containing them) has been recently obtained in [28]. Whereas the results of [4,34] are for arbitrary metric measure spaces and rely on metric tools, the arguments in [28] are tailored for weighted Banach spaces and are based on a purely smooth analysis. It is unknown whether LIPb(B)is strongly dense in W1,p(B,μ). A sufficient condition for the strong density of all compatible algebras is the reflexivity of W1,p(B,μ),see[2, Proposition 42]. Examples of non-reflexive Sobolev spaces are known (see [2, Proposition 44] and [24, Section 12.5]). We also point out that if W1,p(B,μ) is reflexive, then it is separable (see again [2, Proposition 42]) and every compatible subalgebra of LIPb(B)is strongly dense in it. Remark 2.7 It is shown in [13] that minimal p-weak upper gradients depend on p,inthe sense that for f∈W1,p(B,μ)∩W1,˜p(B,μ) with p=˜pit can happen that the minimal p-weak upper gradient of fdiffers from its minimal ˜p-weak upper gradient. Nevertheless, in our notation |Dμf|we do not specify the exponent p, since the latter will be always clear from the context. Master test plans Let (B,μ)be a weighted Banach space such that Bhas the Radon–Nikodým property. Fix an exponent q∈(1,∞).Toanyq-test plan π∈q(B,μ)we associate the Borel set Sπ⊆B given by Sπ:= x∈B d(e#ˆ π) dμ(x)>0.(2.11) The set Sπis uniquely determined up to μ-a.e. null sets, and μ|Sπe#ˆ πand (e#ˆ π)|B\Sπ=0. Itisprovedin[32, Theorem 2.6] that one can always find a master q-test plan πon (B,μ), i.e. a q-test plan having the following property: for any f∈W1,p(B,μ), the function |Dμf|∈Lp(μ) is the minimal G∈Lp(μ) with G≥0 such that (2.10) holds. See also [21, Theorem A.2] for an alternative proof of the existence of a master q-test plan. Moreover, it is proved in [33, Proposition 2] that a given π∈q(B,μ)is a master q-test plan if and only if for every function f∈W1,p(B,μ) it holds that |Dμf|=0μ-a.e. on B\Sπand |Dμf|(x)=ess sup ˆ πx-a.e. (γ,t) 1{Der=0}(γ, t)|(f◦γ) t| ˙γtB for e#ˆ π-a.e. x∈B,(2.12) where ˆ π=ˆ πxd(e#ˆ π)(x)denotes the disintegration of ˆ πalong e. Distributional divergence Testing against smooth functions, we can define the space of vector fields with μ-divergence: Definition 2.8 (Distributional μ-divergence) Let (B,μ) be a weighted Banach space and let q∈(1,∞).ThenwedefineDq(divμ)⊆Lq(B,μ;B)as the space of all vector fields v∈Lq(B,μ;B)for which there exists a function divμ(v) ∈Lq(μ), called the μ-divergence of v, such that df(v) dμ=−fdivμ(v) dμfor every f∈C1(B)∩LIPb(B). 123
Vector calculus... Page 15 of 21 48 3.1 Proof of Theorems 3.1 and 3.2 First, we show that each test plan on a weighted Banach space with the Radon–Nikodým property induces a vector field. The proof is inspired by [10, Proposition 2.4] and [9, Proposition 7.2.3]. Lemma 3.6 (Vector field induced by a test plan) Let (B,μ)be a weighted Banach space such that Bhas the Radon–Nikodým property. Let q ∈(1,∞)and π∈q(B,μ)be given. Let us define vπ(x):= d(e#ˆ π) dμ(x)˙γtdˆ πx(γ, t)∈Bfor μ-a.e. x ∈B.(3.8) Then it holds that vπ∈Lq(B,μ;B)and h|vπ|Bdμ≤1 0 h(γt)˙γtBdtdπ(γ ) for every h :B→[0,+∞)Borel. (3.9) Moreover, it holds that vπ∈Dq(divμ)and divμ(vπ)=d(e0)#π dμ−d(e1)#π dμ.(3.10) Proof First of all, the map vπ:B→Bis μ-measurable thanks to the measurability of x→ ˆ πx. Moreover, given any Borel function h:B→[0,+∞),wehavethat h|vπ|Bdμ≤h(x)d(e#ˆ π) dμ(x)˙γtBdˆ πx(γ, t)dμ(x)=1 0 h(γt)˙γtBdtdπ(γ ), which gives (3.9), thus in particular vπ∈Lq(B,μ;B)by Hölder’s inequality. Finally, for any given function f∈C1(B)∩LIPb(B)we can compute df(vπ)dμ=dγtf,˙γtdˆ πx(γ, t)d(e#ˆ π)(x)=1 0 (f◦γ) tdtdπ(γ ) =f(γ1)−f(γ0)dπ(γ ) =fd(e1)#π dμ−d(e0)#π dμdμ, which shows that vπ∈Dq(divμ)and that the identity in (3.10) holds. The proof is complete. Proof of Theorems 3.1 and 3.2 Step 1: proof of a)⇒b)and |dμf|(TμB)∗≤|Dμf|. Assume that f∈W1,p(B,μ). Pick any sequence (fn)n⊆C1(B)∩LIPb(B)such that fn→fand |dfn|B∗→|Dμf|in Lp(μ).Thenwehavethat fdivμ(v) dμ =lim n→∞ fndivμ(v) dμ ≤lim n→∞ |dfn|B∗|v|Bdμ=|Dμf||v|Bdμ (3.11) for every v∈Dq(divμ). Letting μv:= |Dμf||v|Bμ,wealsodefineTv:C1(B)∩LIPb(B)→ Ras Tv(h):= − fdivμ(hv)dμfor every h∈C1(B)∩LIPb(B). 123
48 Page 16 of 21 E. Pasqualetto, T. Rajala By (2.13), the map Tvis linear. Also, (3.11)gives|Tv(h)|≤hL1(μv)for all h∈C1(B)∩ LIPb(B).SinceC1(B)∩LIPb(B)is dense in L1(μv)and the dual of L1(μv)is L∞(μv),there exists a unique function θv∈L∞(μv)such that |θv|≤1andTv(h)=hθvdμvfor every h∈C1(B)∩LIPb(B). Therefore, letting Lf(v) := θv|Dμf||v|B∈L1(μ), we conclude that |Lf(v)|≤|Dμf||v|Band hLf(v) dμ=−fdivμ(hv)dμfor every v∈Dq(divμ)and h∈C1(B)∩LIPb(B). (3.12) Choosing h:= 1, we obtain that Lfsatisfies (3.1), while the weak∗density of C1(B)∩ LIPb(B)in L∞(μ) ensures that Lfis uniquely determined. Since Lfis linear (thanks to (3.12) and to the linearity of divμ), we conclude that Lf∈Dq(divμ)∗and |dμf|(TμB)∗= |Lf|(TμB)∗≤|Dμf|, proving the inequality ≥in (3.3). The linearity of dμ:W1,p(B,μ)→ Dq(divμ)∗follows from (2.13). Step 2: proof of b)⇒a)and |Dμf|≤|dμf|(TμB)∗. Assume that there exists an element Lf∈Dq(divμ)∗satisfying (3.1). Fix any π∈q(B,μ) and consider the vector field vπ∈Lq(B,μ;B)induced by πas in (3.8). Then we can estimate f(γ1)−f(γ0)dπ(γ ) =fd(e1)#π−fd(e0)#π(3.10) =− fdivμ(vπ)dμ =Lf(vπ)dμ ≤|Lf|(TμB)∗|vπ|Bdμ(3.9) ≤1 0 |Lf|(TμB)∗(γt)˙γtBdtdπ(γ ). Hence, f∈W1,p(B,μ)and |Dμf|≤|Lf|(TμB)∗=|dμf|(TμB)∗, thus the proof of (3.3)is complete. Step 3: proof of (3.4). Given that hLf(v) dμ=− fdivμ(hv) dμ=hdf(v) dμholds for every f,h∈ C1(B)∩LIPb(B)and v∈Dq(divμ),wehavethatLf(v) =df(v) for every f∈C1(B)∩ LIPb(B)and v∈Dq(divμ),sothatd μf=Lf=d|TμBffor every f∈C1(B)∩LIPb(B). In particular, by taking also the identity (3.3) into account, for any f∈C1(B)∩LIPb(B)we obtain that |Dμf|=|d|TμBf|(TμB)∗= v∈Dq(divμ) 1{v=0} df(v) |v|B , thus proving the validity of (3.4). Step 4: proof of a)⇔c)and (3.5). Let us now assume in addition that Bis reflexive. Let f∈W1,p(B,μ) be given. Pick a sequence (fn)n⊆Asuch that fn→fand |dfn|B∗→|Dμf|in Lp(μ).Since Lp(B,μ;B∗)is reflexive, up to a non-relabelled subsequence we have that d fnω weakly in Lp(B,μ;B∗)for some ω∈Lp(B,μ;B∗),sothatω∈WDp μ(f;A). Notice also that we have that |ω|B∗≤|Dμf|.Conversely,let f∈Lp(μ) with WDp μ(f;A)= ∅be given. Take any ω∈WDp μ(f;A)and (fn)n⊆Asuch that fnfweakly in Lp(μ) and d fnω weakly in Lp(B,μ;B∗). Thanks to Mazur’s lemma, we can find a sequence (gn)nof convex combinations of (fn)nso that gn→fstrongly in Lp(μ) and dgn→ωstrongly in 123
Vector calculus... Page 17 of 21 48 Lp(B,μ;B∗). In particular, it holds that |dgn|B∗→|ω|B∗strongly in Lp(μ), which implies that f∈W1,p(B,μ)and |Dμf|≤|ω|B∗. All in all, the property (3.5) is proved. Notice that the implication WDp μ(f;A)= ∅⇒f∈W1,p(B,μ) holds for every B separable. 3.2 Proof of Theorem 3.3 Before passing to the verification of Theorem 3.3, we prove an auxiliary result: Lemma 3.7 Let (B,μ) be a weighted Banach space such that Bhas the Radon–Nikodým property. Let q ∈(1,∞)and π∈q(B,μ)be given. Let us define the multivalued mapping VπB:BBas VπB(x):= clB(span SπB(x)) for μ-a.e. x ∈B,whereweset SπB(x):= spt(Der#ˆ πx) {0B} for e#ˆ π-a.e. x ∈B, for μ-a.e. x ∈B\Sπ.(3.13) Then it holds that SπB:BBis weakly measurable and VπB∈Dμ(B). Proof First, notice that VπB(x)is a closed vector subspace of Bfor μ-a.e. x∈B. Moreover, {x∈B|SπB(x)∩U= ∅}={x∈B|ˆ πx(Der−1(U)) > 0}for every U⊆Bopen, thus the measurability of x→ ˆ πxensures that the multivalued mapping SπBis weakly measurable, whence it follows that clB(SπB)is weakly measurable. Thanks to Proposition 2.9, we can find a sequence (vk)k∈Nof Borel maps vk:B→Bsuch that clB({vk(x):k∈ N})=clB(SπB(x)) holds for μ-a.e. x∈B.Sinceforμ-a.e. x∈Bwe have that VπB(x)=clB n i=1 qivki(x) n∈N,q1,...,qn∈Q,k1,...,kn∈N, we deduce from Proposition 2.9 that VπBis weakly measurable, thus VπB∈Dμ(B). Proof of Theorem 3.3 Step 1: proof of Theorem 3.3 i). Given any π∈q(B,μ),wedenotebyDer π∈Lq(C(B), ˆ π;B)the equivalence class of the mapping Der :C(B)→B.WedefineEπ∈Dˆ π(B)as Eπ(γ, t):= RDerπ(γ, t)for ˆ π-a.e. (γ, t)∈C(B).SinceBis Asplund, we have that Lp w∗(B,μ;B∗)∼ =Lp(B,μ;B∗).Forany f∈A:= C1(B)∩LIPb(B), |(e∗df)|Eπ|(Eπ)∗(γ, t)=1{Derπ=0}(γ, t)|(e∗df)(Derπ)(γ, t)| |Derπ|B(γ, t)=1{Derπ=0}(γ, t)|(f◦γ) t| ˙γtB ≤|Dμf|(γt)=(|dμf|(TμB)∗◦e)(γ, t)=|(e∗df)|e∗TμB|(e∗TμB)∗(γ, t) for ˆ π-a.e. (γ, t)∈C(B).Since{e∗df:f∈A}generates Lp(C(B), ˆ π;B∗)thanks to Lemma 2.17 and (2.19), by using Lemma 2.16 we deduce that Eπe∗TμB, whence the property (3.6) follows. We now prove that TμBis the minimal element of (Dμ(B), )with this property. Let E∈Dμ(B)be such that Derπ∈q(e∗E)for every π∈q(B,μ). Fix a master q-test plan 123
48 Page 18 of 21 E. Pasqualetto, T. Rajala π∈q(B,μ).Then |d|TμBf|(TμB)∗(x)=|Dμf|(x)(2.12) =1Sπ(x)ess sup ˆ πx-a.e. (γ,t) 1{Der=0}(γ, t)|(f◦γ) t| ˙γtB =1Sπ(x)ess sup ˆ πx-a.e. (γ,t) 1{Der=0}(γ, t)|(e∗df)(Derπ)|(γ, t) |Derπ|B(γ, t) ≤1Sπ(x)ess sup ˆ πx-a.e. (γ,t) |(e∗df)|e∗E|(e∗E)∗(γ, t)≤|d|Ef|E∗(x) holds for μ-a.e. x∈B, for every given function f∈A.Since{df:f∈A}generates Lp(B,μ;B∗), we finally conclude that TμBEthanks to Lemma 2.16. The validity of Theorem 3.3 i) follows. De facto, the above proof shows that for any master q-test plan πon (B,μ), the bundle TμBis the unique minimal element of (Dμ(B), )such that ˙γt∈TμB(γt)holds for ˆ π-a.e. (γ, t)∈C(B). Step 2: proof of Theorem 3.3 ii). Let VπBbe as in Lemma 3.7. Then to prove Theorem 3.3 ii) amounts to showing that VπB=TμBfor every master q-test plan πon (B,μ). First, let us prove that VπBTμB.LetSπBbe as in Lemma 3.7. By Proposition 2.9,we can find (vk)k∈N⊆q(VπB)such that clB({vk(x):k∈N})=clB(SπB(x)) for μ-a.e. x∈B. To prove that VπBTμB, it suffices to show that vk∈q(TμB)for every k∈N. We argue by contradiction: suppose there exist k0∈N,aBorelsetE⊆B,andδ>0such that e#ˆ π(B)>0andvk0(x)−TμB(x)B≥δfor e#ˆ π-a.e. x∈E. By the definition of SπB we have ˆ πx(γ, t)∈C(B)˙γt−vk0(x)B<δ >0fore #ˆ π-a.e. x∈E. By taking the property (3.6) into account, it thus follows that e#ˆ πx∈Evk0(x)−TμB(x)B<δ ≥ˆ π(γ, t)∈e−1(E)˙γt−vk0(γt)B<δ >0. This leads to a contradiction with our choice of k0,E,δ. Therefore, we deduce that VπB TμB. We now pass to the verification of TμBVπB.Fore #ˆ π-a.e. x∈B, it holds that ˙γt=Der(γ, t)∈spt(Der#ˆ πx)=SπB(γt)⊆VπB(γt)for ˆ πx-a.e. (γ, t)∈C(B). Therefore, we have that ˙γt∈VπB(γt)for ˆ π-a.e. (γ, t)∈C(B),sothatTμBVπBthanks to the last paragraph of Step 1 of this proof. 4 Consistency with the metric vector calculus The aim of this conclusive section is to check the consistency of the vector calculus we develop in this paper with the differential structure for metric measure spaces introduced by Gigli in [20]. Let (B,μ)be a weighted Banach space, q∈(1,∞),andE∈Dμ(B). Then the q-section space q(E)is a (complete) Lq(μ)-normed L∞(μ)-module, in the sense of [20, Definition 123
Vector calculus... Page 19 of 21 48 1.2.10]. Its dual q(E)∗∼ =p(E∗ w∗)is an Lq(μ)-normed L∞(μ)-module, which coincides with the dual q(E)in the sense of [20, Definition 1.2.6 and Proposition 1.2.14]. Notice that the notion of ‘generating vector subspace’ of q(E)or p(E∗ w∗)we introduced in Definition 2.13 is consistent with the one of [20, Definition 1.4.2]. We refer to [29] for a more detailed discussion on these topics. In [20], the language of Lp(μ)-normed L∞(μ)-modules was used to develop a vector calculus for arbitrary metric measure spaces. In this regard, an object playing a fundamental role is the cotangent module, introduced in [20, Definition 2.2.1] (see also [19, Theorem/Definition 2.8] for another axiomatisation and [22, Theorem 3.2] for the case p= 2). The cotangent module is canonically associated with an abstract differential operator. In the next result, we show the consistency of our machinery with Gigli’s notions of cotangent module and abstract differential. Theorem 4.1 (Cotangent module) Let (B,μ) be a weighted Banach space such that Bis an Asplund space having the Radon–Nikodým property. Let p ∈(1,∞).Then q(TμB)∗is isomorphic to the cotangent module of (B,μ)and the associated differential is dμ:W1,p(B,μ)→q(TμB)∗. Proof First, {d|TμBf:f∈C1(B)∩LIPb(B)}generates q(TμB)∗by Lemma 2.17, thus a fortiori {dμf:f∈W1,p(B,μ)}generates q(TμB)∗. Also, |dμf|(TμB)∗=|Dμf|for every f∈W1,p(B,μ)by (3.3). The cotangent module and the abstract differential are uniquely determined (up to a unique isomorphism) by this property (see [22, Theorem 3.2]), thus the statement is proved. Under the assumptions of Theorem 4.1,thetangent module of (B,μ)(which was defined in [20, Definition 2.3.1] as the dual of its cotangent module) can be different from q(TμB). Indeed, the q-section space q(TμB)is the predual of the cotangent module instead. However: Corollary 4.2 Let (B,μ)be a weighted Banach space with Breflexive and p ∈(1,∞).Then the cotangent module of (B,μ)is reflexive and q(TμB)is isomorphic to the tangent module of (B,μ). Proof Since Lq(B,μ;B)is reflexive, its subspace q(TμB)is reflexive, so also the cotangent module q(TμB)∗is reflexive. In particular, the tangent module q(TμB)∗∗ is isomorphic to q(TμB). We point out that – as far as we know – the reflexivity of the cotangent module might not follow directly from that of W1,p(B,μ). It is known that the reflexivity of the cotangent module implies that of the Sobolev space [20, Proposition 2.2.10], but whether the converse implication holds is still an open problem, cf. with [20, Remark 2.2.11]. Finally, another consistency check: Remark 4.3 (Velocity of a test plan)Let (B,μ)be a weighted Banach space with Breflexive and let q∈(1,∞). Given any q-test plan π∈q(B,μ), one can consider the velocity π∈q(e∗TμB)of πin the sense of [32, Theorem 1.21] (see [20, Theorem 2.3.18] for the original definition, under extra assumptions). Letting Derπbe the ˆ π-a.e. equivalence class of Der :C(B)→B, we claim that Derπ=πfor every π∈q(B,μ). (4.1) 123
48 Page 20 of 21 E. Pasqualetto, T. Rajala Indeed, Step 1 of the proof of Theorem 3.3 shows that Derπ∈q(e∗TμB), and an application of the dominated convergence theorem ensures that for every f∈W1,p(B,μ)it holds that lim h→0 f◦et+h−f◦et h−(e∗df)(Derπ)(·,t) L1(π) =0forL1-a.e. t∈[0,1], whence the claimed identity (4.1) follows by the uniqueness part of [32, Theorem 1.21]. Acknowledgements The authors thank the anonymous referee, whose comments and suggestions led to a significant improvement of the presentation. The first named author has been supported by the MIURPRIN 202244A7YL project “Gradient Flows and Non-Smooth Geometric Structures with Applications to Optimization and Machine Learning”. Funding Open Access funding provided by University of Jyväskylä (JYU). Data availability Not applicable. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Studies in Economic Theory, Springer, Berlin (1999) 2. Ambrosio, L., Colombo, M., Di Marino, S.: Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. Variational methods for evolving objects, pp. 1–58 (2015) 3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008) 4. Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29, 969–996 (2013) 5. Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195, 289–391 (2014) 6. Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007) 7. Bouchitte, G., Buttazzo, G., Seppecher, P.: Energies with respect to a measure and applications to lowdimensional structures. Calc. Var. Partial Differ. Equ. 5, 37–54 (1997) 8. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999) 9. Di Marino, S.: Recent advances on BV and Sobolev spaces in metric measure spaces. PhD thesis, Scuola Normale Superiore (Pisa) (2014) 10. Di Marino, S.: Sobolev and BV spaces on metric measure spaces via derivations and integration by parts. Preprint arXiv:1409.5620 (2014) 11. Di Marino, S., Gigli, N., Pasqualetto, E., Soultanis, E.: Infinitesimal Hilbertianity of locally CAT(κ)- spaces. J. Geom. Anal. 31, 7621–7685 (2021) 12. Di Marino, S., Luˇci´c, D., Pasqualetto, E.: Representation theorems for normed modules. Preprint arXiv:2109.03509 (2021) 13. Di Marino, S., Speight, G.: The p-weak gradient depends on p. Proc. Am. Math. Soc. 143, 5239–5252 (2015) 14. Eriksson-Bique, S., Rajala, T., Soultanis, E.: Tensorization of quasi-Hilbertian Sobolev spaces. Rev. Mat. Iberoam. 40, 565–580 (2024) 15. Eriksson-Bique, S., Soultanis, E.: Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential. Anal. PDE 17, 455–498 (2024) 123
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