Giovanni Brigati
email: brigati@ceremade.dauphine.fr

Ivailo Hartarsky
email: hartarsky@ceremade.dauphine.fr

The normal contraction property for non-bilinear Dirichlet forms

Keywords: Primary 31C45; Secondary 47H20, 31C25, 46E36, 35K55 Non-bilinear Dirichlet form, Dirichlet form, nonlinear semigroup, gradient flow, normal contraction, Cheeger's energy

We analyse the class of convex functionals E over L 2 (X, m) for a measure space (X, m) introduced by Cipriani and Grillo [17] and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if E(φ • f ) E(f ) for all f ∈ L 2 (X, m), and all 1-Lipschitz functions φ : R → R with φ(0) = 0. We prove that normal contraction holds if and only if E is symmetric in the sense E(-f ) = E(f ), for all f ∈ L 2 (X, m). An auxiliary result, which may be of independent interest, states that it suffices to establish the normal contraction property only for a simple two-parameter family of functions φ.

Introduction

Background

Bilinear Dirichlet forms

Bilinear Dirichlet forms are a well-established topic, related with the theory of Markov processes and semigroups, see [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF][START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF][START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF]. In that context, they 1 are usually defined as a bilinear and positive semi-definite form Λ : L 2 (X, m) × L 2 (X, m) → R, represented as Λ(f, g) = (Af, g), ∀f, g ∈ L 2 (X, m), for a linear, symmetric, and positive semi-definite operator A : L 2 (X, m) → L 2 (X, m). Then, the form Λ is called a closed one if A is a closed operator in the sense of [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF], namely, for all sequences (f n ) n ⊂ L 2 (X, m), such that f n → f and A(f n ) → g, for some limits f, g ∈ L 2 (X, m), we have A(f ) = g. The bilinear form Λ is called a (bilinear) Dirichlet form if

Λ(1 ∧ f ∨ 0, 1 ∧ f ∨ 0) Λ(f, f ), ∀f ∈ L 2 (X, m).
By extension, the term Dirichlet form also refers to the quadratic form

E(f ) = 1 2 Λ(f, f ),
associated with a bilinear Dirichlet form Λ. This functional turns out to be always non-negative, convex (since it is quadratic), and l.s.c. (which follows by the closure of the operator A).

Aside their interest in probability, for which we refer to the bibliography of the three books quoted above, bilinear Dirichlet forms are also well-linked with linear diffusion equations and semigroups, see [START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF]. This link gave fruitful results in the theory of metric measure spaces, allowing for an intrinsic/Eulerian approach towards Ricci curvature bounds, [START_REF]Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds[END_REF]. Under mild hypotheses, the authors of [START_REF]Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds[END_REF] could represent any bilinear Dirichlet form E as a quadratic Cheeger's energy on the base space X. One important point is that Ambrosio, Gigli, and Savaré were able to create an appropriate notion of distance d E directly from the Dirichlet form E. Then, via a condition à la Bakry-Emery, on the carré du champ associated with the quadratic form E, the authors give a sense to notions such as Bochner's inequality or a lower bound on the Ricci curvature. Their approach is equivalent to that of Lott and Villani [START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF] and Sturm [START_REF] Sturm | On the geometry of metric measure spaces. I[END_REF][START_REF]On the geometry of metric measure spaces[END_REF], based on optimal transport. The creation of a distance from a bilinear form is a technique present also in [START_REF] Biroli | Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces[END_REF]. Bilinear Dirichlet forms also play a role in potential and capacity theory, see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF][START_REF] Schmidt | Energy forms[END_REF].

Historically, bilinear Dirichlet forms have been introduced by Beurling and Deny in [START_REF] Beurling | Dirichlet spaces[END_REF]. One motivation behind their definition was the fact that being a bilinear Dirichlet form was sufficient to have the normal contraction property (see Eq. ( 3)). The fact that controlling one normal contraction is necessary and sufficient to control all of them is nowadays known as the Beurling-Deny criterion. To prove such a property, one usually approximates the function f with weighted sums of characteristic functions. The normal contraction property is a cornerstone for many purposes. For instance, for the development of a differential calculus [START_REF]Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds[END_REF] and the classification of linear Markov semigroups [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF], both based on bilinear Dirichlet forms.

Non-bilinear Dirichlet forms

Generalising the concept of Dirichlet form to a non-bilinear setting is a more recent problem, started with the two works [START_REF] Biroli | Strongly local nonlinear Dirichlet functionals and forms[END_REF][START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF]. A different kind of generalisation is that of [START_REF] Jost | Nonlinear Dirichlet forms[END_REF], but we will not focus on it, since its purpose is different. Using instruments from [START_REF] Barthélemy | Invariance d'un convexe fermé par un semi-groupe associé à une forme non-linéaire[END_REF][START_REF] Bénilan | Quelques aspects non linéaires du principe du maximum[END_REF][START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF], Cipriani and Grillo [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF] provided two equivalent definitions of a non-bilinear Dirichlet form relevant to us, which will be discussed in further detail in Sections 1.2 and 2. In [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF], a number of properties of the class of non-bilinear Dirichlet forms are given, in particular with respect to Γ-convergence (see [START_REF] Maso | An introduction to Γ-convergence[END_REF]).

Two recent works on the topic are [START_REF] Claus | Nonlinear Dirichlet Forms[END_REF][START_REF]Energy spaces, Dirichlet forms and capacities in a nonlinear setting[END_REF], where Claus recovers many structural properties for non-bilinear Dirichlet forms, among which we find a nonlinear Beurling-Deny principle, see [START_REF] Claus | Nonlinear Dirichlet Forms[END_REF]Theorem 2.39]. In the following sections, he develops a nonlinear theory of capacity. Furthermore, in [START_REF] Claus | Nonlinear Dirichlet Forms[END_REF]Corollary 2.40] (also see [START_REF]Energy spaces, Dirichlet forms and capacities in a nonlinear setting[END_REF]Theorem 3.22]), the normal contraction property is proved for non-bilinear Dirichlet forms, but only for non-decreasing normal contractions and additionally assuming that the form is 0 at 0 (we avert the reader that in [18, Definition 2.31] non-decreasing normal contractions are named simply normal contractions).

Examples Let us mention two classes of basic examples, which generalise corresponding families of local and nonlocal bilinear Dirichlet forms. These lie at the core of the functionals analysed in the references quoted at the end of the section. Let Ω be an open subset of R d and f : Ω × R d → R be a function. Let

E(u) = Ω f (x, Du) dx u ∈ W 1,2 loc (Ω), +∞ otherwise. ( 1
)
We have that E is a non-bilinear Dirichlet form if f is non-negative, measurable in the first argument and l.s.c. in the second one. Note the we do not need to impose convexity, since it is already implied by the l.s.c. [START_REF] Maso | An introduction to Γ-convergence[END_REF].

In addition,

E is symmetric if f (•, -v) = f (•, v), for all v ∈ R d .
Finally, E is always local, due to the locality of Du and the fact that E is an integral functional. Among local forms, we can consider the following.

Example 1.1. Let Ω = R. Let f (x, v) = v ∨ 0.
Then, the integral functional E associated to f by Eq. ( 1) is a non-symmetric non-bilinear Dirichlet form, which does not satisfy the normal contraction property Eq. (3).

In this class of local functionals we also have the distinguished subclass of Finsler metrics, where

f (x, •) = • x , ∀x ∈ Ω.
The form is bilinear if and only if, for all x ∈ Ω, the norm • x satisfies the parallelogram identity, see [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF]Chapter 5].

Some non-local non-bilinear Dirichlet forms appear in [START_REF] Creo | Fractional (s, p)-Robin-Venttsel' problems on extension domains[END_REF], for example. In general we can say that any functional E of the form

E(u) = Ω 2 ψ(u(x) -u(y)) dx dy, ∀u ∈ L 2 (Ω, dx).
is a non-bilinear Dirichlet form for non-negative, l.s.c., convex ψ such that ψ(0) = 0.

In [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF], some interesting examples are developed in detail, ranging from functionals from the calculus of variations to Sobolev seminorms in the context of C ⋆ -monomodules. The theory of [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF] can be applied to nonlinear diffusion equations (see [START_REF] Creo | Fractional (s, p)-Robin-Venttsel' problems on extension domains[END_REF][START_REF] Feo | Anisotropic p-Laplacian evolution of fast diffusion type[END_REF] and the references therein), analysis on graphs [START_REF] Hofmann | Spectral theory, clustering problems and differential equations on metric graphs[END_REF][START_REF] Mugnolo | Semigroup methods for evolution equations on networks, Understanding Complex Systems[END_REF], and analysis on spaces with a very irregular geometry [START_REF] Hinz | Sobolev spaces and calculus of variations on fractals[END_REF][START_REF] Meinert | Partial differential equations on fractals. Existence, uniqueness and approximation results[END_REF]. Furthermore, Cheeger's energies on extended metric spaces are known to be non-bilinear Dirichlet forms [START_REF]Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below[END_REF]. We refer to [START_REF]Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces[END_REF][START_REF]Metric measure spaces with Riemannian Ricci curvature bounded from below[END_REF][START_REF]Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds[END_REF] for this theory, which originates from [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF][START_REF] Shanmugalingam | Newtonian spaces: an extension of Sobolev spaces to metric measure spaces[END_REF]. See also [START_REF] Kell | q-heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space[END_REF][START_REF] Luise | Contraction and regularizing properties of heat flows in metric measure spaces[END_REF] for more estimates and contraction properties of Cheeger's energies.

Main results

Let X be a nonempty set, let F be a σ-algebra over X, and take a σ-finite measure m :

F → [0, ∞]. Let E : L 2 (X, m) → [0, ∞]
be a convex and l.s.c. functional. In all the paper we assume that E is not the constant +∞. Let (T t ) t 0 be the semigroup of nonlinear operators generated by -∂E, where ∂ denotes the subdifferential operator, via the differential equation

∂ t T t f ∈ -∂E(T t f ), ∀t ∈ (0, ∞), ∀u ∈ L 2 (X, m), T 0 f = f, ∀f ∈ L 2 (X, m). (2)
Equation ( 2) is well-posed for all f ∈ L 2 (X, m). Its solution is usually called the gradient flow of E starting at f . See [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] and refer to Section 1.1 for more background.

We say that E is a non-bilinear Dirichlet form if, for all t 0, the operator

T t : L 2 (X, m) → L 2 (X, m) verifies 1.
order preservation: T t f T t g for all f, g ∈ L 2 (X, m) such that f g (for the pointwise order up to a negligible set);

2. L ∞ -contraction: T t f -T t g ∞ f -g ∞ for all f, g ∈ L 2 (X, m).
This class of forms was introduced by Cipriani and Grillo [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF] and we will provide an equivalent "static" definition in Theorem 1.3 without reference to the underlying semigroup (also see Theorem 2.1).

Our main goal is to verify the normal contraction property for non-bilinear Dirichlet forms. A normal contraction is a 1-Lipschitz function φ : R → R, such that φ(0) = 0. We denote by Φ the set of all normal contractions. We say that a functional E over L 2 (X, m) has the normal contraction property if

E(φ(f )) E(f ), ∀φ ∈ Φ, ∀f ∈ L 2 (X, m). (3)
We provide the following sharp solution to this problem.

Theorem 1.2. Let E be a non-bilinear Dirichlet form. Then E has the normal contraction property Eq. (3) if and only if

E(-f ) E(f ) ∀f ∈ L 2 (X, m). (4)
This theorem goes in the same direction as the well-established one for the bilinear case, [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF][START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF][START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF]. We merely prove that a form will operate on all normal contractions, once it operates on the simplest one. Henceforth, we say that E is symmetric if Eq. ( 4) holds and, equivalently, E(-f ) = E(f ) for all f ∈ L 2 (X, m). As witnessed by Example 1.1, the necessary assumption Eq. ( 4) needs to be made, since this non-bilinear Dirichlet form does not have the normal contraction property.

Let us highlight that Theorem 1.2 may be viewed as a strengthening of the result of Claus [18, Corollary 2.40], whose proof follows the far more conventional approach of [START_REF] Bénilan | Completely accretive operators[END_REF][START_REF] Bénilan | Quelques aspects non linéaires du principe du maximum[END_REF]. The class of normal contractions we consider is richer and it controls, for example, the absolute value of the argument of the non-bilinear Dirichlet form, which can be very useful (see e.g. [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]), as well as more complicated contractions.

In order to prove Theorem 1.2, we establish two results, both of which may be of independent interest. Firstly, we provide the following equivalent characterisation of non-bilinear Dirichlet forms, which turns out to be more wieldy for our purposes than the other equivalent static characterisation of [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces[END_REF]Theorem 3.8], recalled in Theorem 2.1.

f (x) H α (f, g)(x) g(x) 0 2α Figure 1: Graph of the function H α (f, g)(x) for fixed g(x).
Theorem 1.3. Let E be a non-negative, l.s.c. functional over L 2 (X, m). For all f, g ∈ L 2 (X, m), and α ∈ [0, ∞) we denote by f ∨ g and f ∧ g denote the pointwise maximum and minimum and set (see Fig. 1)

H α (f, g)(x) =      g(x) -α f (x) -g(x) < -α, f (x) f (x) -g(x) ∈ [-α, α], g(x) + α f (x) -g(x) > α. (5) If, for all f, g ∈ L 2 (X, m), and α ∈ [0, ∞), E verifies E(f ∨ g) + E(f ∧ g) E(f ) + E(g), (6)
E(H α (f, g)) + E(H α (g, f )) E(f ) + E(g), ( 7
)
then E is a non-bilinear Dirichlet form and, in particular, E is convex. Conversely, all non-bilinear Dirichlet forms verify Eqs. (6) and [START_REF] Barthélemy | Invariance d'un convexe fermé par un semi-groupe associé à une forme non-linéaire[END_REF].

The advantage of Theorem 1.3 as compared to Theorem 2.1 is that conditions Eqs. ( 6) and ( 7) are easier to verify and useful to develop other functional inequalities such as Eq. ( 3).

The second important step towards Theorem 1.2 is a reduction.

Lemma 1.4. Let G be the set of all normal contractions φ ∈ Φ such that |φ ′ | = 1 and φ ′ has at most two points of discontinuity. Let G be the collection of all finite compositions of elements in G. Then, G is dense in Φ for the pointwise convergence on R.

We observe that the elements of G are irreducible with respect to composition, so that G is minimal in this sense. While the space G is quite simple, proving Eq. ( 3) for φ ∈ G by hand from Eqs. ( 4), ( 6) and ( 7) is still delicate, albeit elementary.

Plan of the paper

The remainder of the paper is structured as follows. In Section 2, we establish Theorem 1.3. In Section 3, we prove Theorem 1.2, relying on Theorem 1.3. This is the heart of our work. Finally, we discuss future directions of research in Section 4.

Efficient equivalent characterisation of nonbilinear Dirichlet forms

The goal of the present section is to prove Theorem 1.3.

Preliminaries

We introduce the subsets

C 1 and C 2,α , for α ∈ [0, ∞), of L 2 (X, m; R 2 ) : C 1 = (f, g) ∈ L 2 (X, m; R 2 ) : f g , (8)
C 2,α = (f, g) ∈ L 2 (X, m; R 2 ) : |f -g| α . (9)
We notice that for all α, the sets C 1 and C 2,α are convex and closed in the L 2 -topology. For any closed and convex subset C, the 1-Lipschitz projection operator P C : L 2 (X, m; R 2 ) → C is defined by

P C (f, g) = arg min (w,z)∈C f -w 2 2 + g -z 2 2 .
The projection map sends any point (f, g) to the closest point P C (f, g) in C. We denote by P 1 C and P 2 C the two components of the projection operator in L 2 (X, m). More properties of projection maps are studied in [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF]. If one considers the sets C 1 and C 2,α , we have an explicit expression for the projections, thanks to [17, Lemma 3.3]:

P 1 (f, g) = f - 1 2 ((f -g) ∨ 0), g + 1 2 ((f -g) ∨ 0) , (10)
P 2,α (f, g) = g + 1 2 ϕ α • (f -g), f - 1 2 ϕ α • (f -g) , (11)
where ϕ α : R → R is given by

ϕ α (z) = ((z + α) ∨ 0) -((z -α) ∧ 0). (12)
We further recall [17, Definition 3.1, Remark 3.2, Theorem 3.6].

Theorem 2.1. Let E : L 2 (X, m) → [0, ∞] be a l.s.c. functional. If

E P 1 1 (f, g) + E P 2 1 (f, g) E(f ) + E(g), (13)
E P 1 2,α (f, g) + E P 2 2,α (f, g) E(f ) + E(g), (14)
for all f, g ∈ L 2 (X, m) and α ∈ [0, ∞), then E is a non-bilinear Dirichlet form and, in particular, E is convex. Conversely, all non-bilinear Dirichlet forms fulfil Eqs. ( 13) and ( 14).

The key argument is the well-known fact from [START_REF] Barthélemy | Invariance d'un convexe fermé par un semi-groupe associé à une forme non-linéaire[END_REF][START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] stating that

E P 1 C (f, g) + E P 2 C (f, g) E(f ) + E(g)
for all f, g ∈ L 2 (X, m) if and only if the semigroup T t from Eq. ( 2) preserves C :

T t C ⊂ C, ∀t 0,
where C can be any convex and closed set. Thus, Eqs. ( 13) and ( 14) correspond to the order-preservation and the L ∞ -contraction properties for (T t ) t , respectively. In [17, Theorem 3.8] one more step is made.

Theorem 2.2. Let E : L 2 (X, m) → [0, ∞] be a l.s.c. functional. Then, E satisfies Eq. ( 6) if and only if E is convex and satisfies Eq. (13).

Indeed, the last statement is a consequence of the more general [7, Proposition 2.5].

Theorem 2.3. Let C be a closed convex subset of L 2 (X, m; R 2 ), let P C = (P 1 C , P 2 C ) be the associated orthogonal projection. Let E : L 2 (X, m) → [0, ∞] be a l.s.c. functional. Let h, k : L 2 (X, m; R 2 ) → L 2 (X, m) be two continuous mappings such that, for all u, v ∈ L 2 (X, m) and t, s ∈ [0, 1] it holds that

h(u t , v s ) = u 1-s , k(u t , v s ) = v 1-t , (15)
where

u t = (1 -t)u + th(u, v), v s = (1 -s)v + sk(u, v).
Moreover, assume

P C (u, v) = (u 1/2 , v 1/2 ). (16)
Then, we have that for all u, v ∈ L 2 (X, m)

E P 1 C (u, v) + E P 2 C (u, v) E(u) + E(v),
if and only if E is convex and for all u, v ∈ L 2 (X, m)

E(h(u, v)) + E(k(u, v)) E(u) + E(v).

Proof of Theorem 1.3

To conclude the section, we show that the convex sets C 2,α verify the hypotheses of Theorem 2.3.

Proof of Theorem 1.3. Fix α > 0. Recalling the explicit expression of ϕ α from Eq. ( 12), for any u, v ∈ L 2 (X, m) we have

ϕ α • (u -v)(x) =      u(x) -v(x) -α u(x) -v(x) -α, 2u(x) -2v(x) |u(x) -v(x)| α, u(x) -v(x) + α u(x) -v(x) α.
Further recalling the expression of P 2,α from Eq. ( 11), in order to satisfy Eq. ( 16), we now choose h, k

: L 2 (X, m; R 2 ) → L 2 (X, m) such that v + 1 2 ϕ α • (u -v) = u + h(u, v) 2 , u - 1 2 ϕ α • (u -v) = v + k(u, v) 2 .
Therefore, the expressions for h, k are the following

h(u, v)(x) =      v(x) -α u(x) -v(x) -α, u(x) |u(x) -v(x)| α, v(x) + α u(x) -v(x) α, k(u, v)(x) =      u(x) + α u(x) -v(x) -α, v(x) |u(x) -v(x)| α, u(x) -α u(x) -v(x) α,
and we notice that h(u, v) = H α (u, v) and k(u, v) = H α (v, u).
It remains to verify the twist condition Eq. [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF]. Fix s, t, u, v as in the hypothesis. Since the values of H α is defined pointwise, we also fix x ∈ X and drop this parameter for compactness of notation. Suppose that |u -v| α, then

H(u, v) = u, H(v, u) = v, so u t = u 1-s = u, v s = v. The case u-v < -α
is analogous to that with u -v > α, since the role of u and v is symmetric. Hence, we will discuss only the former. Here we have

u t = (1 -t)u + t(v -α), v s = (1 -s)v + s(u + α).
We need not discuss more subcases for the expression of H α (u t , v s ), since

u t -v s = (1 -t)u + t(v -α) -(1 -s)v -s(u + α) = (1 -t -s)(u -v) -(t + s)α < -α.
x 1 0 Hence,

x 2 x 3
H α (u t , v s ) = v s -α = (1 -s)v + su + (s -1)α = u 1-s ,
The second condition in Eq. ( 15) follows similarly, so we omit it. Thus, applying Theorem 2.3, Eq. ( 14) is equivalent to E being convex and Eq. ( 7). Yet, Theorem 2.2 gives that the convexity and Eq. ( 13) are equivalent to Eq. ( 6), so Theorem 1.3 reduces to Theorem 2.1.

The normal contraction property

Throughout this section we fix a measure space (X, m) and a functional on L 2 (X, m) satisfying Eqs. ( 4), ( 6) and ( 7) for all f, g ∈ L 2 (X, m) and α ∈ [0, ∞). We will prove Eq. (3) progressively, starting from simple functions φ. More specifically, for k ∈ {0, 1, . . . }, x 1 , . . . ,

x k ∈ R such that -∞ = x 0 < x 1 < • • • < x k < x k+1 = ∞,
we consider the continuous function φ x 1 ,...,x k : R → R (see Fig. 2) defined by φ x 1 ,...,x k (0) = 0 and

φ ′ x 1 ,...,x k (x) = (-1) i (17)
for x ∈ (x i , x i+1 ). Let us denote F k = {φ x 1 ,...,x k : x 1 < • • • < x k ∈ R}, so that F 0 = {id}. We further set Φ x 1 ,...,x k = E • φ x 1 ,...,x k . 3.1 Basic contractions Proposition 3.1. For any x ∈ R and f ∈ L 2 (X, m) we have Φ x (f ) E(f ).
Proof. Fix x 0 (the case x < 0 is treated identically) and f . By Eq. ( 6)

Φ x (f ) + E(0 ∨ f ) E(f ) + E(σ • f ) (18)
(see Fig. 3), where 19). Thus, it suffices to show that E(0 ∨ f ) E(σ • f ).

σ(y) =      0 y 0, y y ∈ (0, x), 2x -y y x. x 0 id σ x 0 0 ∨ id φ x
But Eqs. ( 4) and ( 7) with α = 2x (see Fig. 4) give

2E(σ • f ) E(σ • f ) + E(-σ • f ) E(0 ∨ f ) + E(-(0 ∨ f )) 2E(0 ∨ f ), (19)
concluding the proof.

Proposition 3.2. For any 0

x 1 < x 2 or x 1 < x 2 0 and f ∈ L 2 (X, m) it holds that Φ x 1 ,x 2 (f ) E(f ).
Proof. Fix 0 x 1 < x 2 and f , the case x 1 < x 2 0 being analogous. Let

σ(x) =      0 x x 1 , x 1 -x x ∈ (x 1 , x 2 ), x + x 1 -2x 2 x x 2 , ψ(x) = 0 x 0, φ x 1 ,x 2 (x) x > 0. x 1 0 x 2 0 ∨ id σ x 1 x 1 0 x 2 0 ∨ (id -x 1 )
ψ Figure 5: Illustration of Eq. ( 20). Then Eq. ( 7) with α = x 1 (see Fig. 5) gives

x 1 0 0 ∨ (id -x 1 ) 0 ∧ (x 1 -id) 2(x 2 -x 1 ) x 1 0 x 2 -σ σ
E(ψ • f ) + E(0 ∨ (f -x 1 )) E(0 ∨ f ) + E(σ • f ). (20)
Moreover, by Eqs. ( 4) and ( 7) for α = 2(x 2 -x 1 ) (see Fig. 6) we get

2E(0 ∨ (f -x 1 )) E(0 ∨ (f -x 1 )) + E(0 ∧ (x 1 -f )) E(σ • f ) + E(-σ • f ) 2E(σ • f ), (21)
so that E(ψ • f ) E(0 ∨ f ). Furthermore, Eq. ( 6) gives

Φ x 1 ,x 2 (f ) + E(0 ∨ f ) E(ψ • f ) + E(f ) (22)
(see Fig. 7), yielding the desired conclusion.

Proposition 3.3. For any x 1 < 0 < x 2 and f ∈ L 2 (X, m) it holds that Φ x 1 ,x 2 (f ) E(f ). 22). Proof. Without loss of generality assume that x 2 > -x 1 and fix f . Consider

x 1 0 x 2 id ψ x 1 0 x 2 0 ∨ id φ x 1 ,x 2
x 1 0 id ψ 2x 2 x 1 x 2 0 id ∧ x 2 φ x 1 ,x 2
ψ(x) =      x -2x 1 x < x 1 , -x x 1 x x 2 , -x 2 x > x 2 .
Then Eq. ( 7) with α = 2x 2 (see Fig. 8) gives

Φ x 1 ,x 2 (f ) + E(f ∧ x 2 ) E(f ) + E(ψ • f ). (23)
Yet,

ψ = φ x 1 • (id ∧ x 2 ), so by Proposition 3.1 we have E(ψ • f ) E(f ∧ x 2 ).
Combining this with Eq. ( 23) yields the desired conclusion.

Reduction to basic contractions

As we will see, the next proposition is essentially Lemma 1.4.

Proposition 3.4. Any φ ∈ F k with k 0 can be written as φ

∈ F k for k 3, with -∞ = x 0 < x 1 < • • • < x k < x k+1 = ∞. Consider i ∈ {1, . . . , k -1} such that x i+1 -x i <
x j+1 -x j for all j = i (we may assume that the inequality is strict by perturbing the x i and taking a limit if necessary). We consider the following cases.

• If x i+1 0, then set

x ′ j = x j + 2(x i+1 -x i ) 1 j < i, x j+2 i j k -2.
• If x i 0, then set

x ′ j = x j 1 j < i, x j+2 -2(x i+1 -x i ) i j k -2. • If x i < 0 < x i+1 , then set x ′ j = x j -x i 1 j < i, x j+2 -x i+1 i j k -2.
Then it suffices to prove that

φ = φ x ′ 1 ,...,x ′ k-2 • φ x i ,x i+1 .
To do this, we verify Eq. ( 17) in each case. We will only treat the case x i 0, the others two being analogous. We have that

φ ′ x ′ 1 ,...,x ′ k-2 (φ x i ,x i+1 (x)) × φ ′ x i ,x i+1 (x) (24)
changes sign at x i and x i+1 due to the second factor. Moreover, φ x i ,x i+1 takes the values in I = R \ [2x i -x i+1 , x i ] exactly once and

φ x i ,x i+1 (x j ) = x ′ j 1 j < i, x ′ j-2 i + 2 j k.
But our choice of i implies I ⊃ {x ′ 1 , . . . , x ′ k-2 }, so the first factor in Eq. ( 24) changes sign precisely at x 1 , . . . , x i-1 , x i+2 , . . . , x k , concluding the proof. With Proposition 3.4 it is immediate to deduce Lemma 1.4.

Proof of Lemma

1.4. Observe that G = {id, -id} • (F 0 ∪ F 1 ∪ F 2 ). Thus, G ⊃ {id, -id} • F 2 • (F 0 ∪ F 1 ) ⊃ {id, -id} • ∞ k=0 F k ⊃ G , (25)
where the first and third inclusions follow by definition, while the second one is Proposition 3.4. Thus, G = {id, -id} • ∞ k=0 F k is indeed dense in Φ, concluding the proof.

We are ready to assemble the proof of Theorem 1.2.

Proof of Theorem 1.2. By Theorem 1.3, any non-bilinear Dirichlet form E satisfies Eqs. ( 6) and ( 7) and is l.s.c. Since Eq. ( 4) is a hypothesis of Theorem 1.2, Propositions 3.1 to 3.3 and Eq. ( 4) yield that for any φ ∈ G (recall Lemma 1.4) and f ∈ L 2 (X, m) it holds that E(φ • f ) E(f ). Indeed, F 0 is trivial, Proposition 3.1 deals with F 1 , Propositions 3.2 and 3.3 give F 2 and then Eq. ( 4) allows us to take opposites. Therefore, Eq. ( 3) also holds for all φ ∈ G .

Fix f ∈ L 2 (X, m) and an arbitrary normal contraction φ ∈ Φ. By Lemma 1.4, there exists a sequence φ n ∈ G such that φ n (x) → φ(x) for all x ∈ R, as n → ∞, and

E(φ n • f ) E(f ) for all n. We have that φ n (f ) → φ(f ) pointwise in X, but |φ n • f | 2 |f | 2 ∈ L 1 (X, m),
as all functions φ n are normal contractions. Then, by Lebesgue's dominated convergence theorem

φ n • f → φ • f in L 2 (X, m
). Thus, we obtain the desired inequality via the l.s.c. of E.

Locality

Let us conclude this section with a concept of locality allowing a much more direct proof of Theorem 1.2 under this hypothesis. We say that a nonbilinear Dirichlet form E is local if for all c ∈ R and u, v ∈ L 2 (X, m) such that u(x)(v(x) -c) = 0 for all x ∈ X, we have

E(u + v) = E(u) + E(v).
Proof of Theorem 1.2 in the local case. Fix a symmetric local non-bilinear Dirichlet form E. As in the proof of Theorem 1.2 it suffices to establish Eq. ( 3) for all φ ∈ ∞ k=1 F k (this part of the proof does not rely on Theorem 1.3 and Propositions 3.1 to 3.4). Fix φ = φ x 1 ,...,x k for some x 1 < • • • < x k . Observe that φ(x) = ((x -x 1 ) ∧ 0) + k i=1 (-1) i ((0 ∨ (x -x i )) ∧ (x i+1 -x i )).

Since all summands satisfy the locality condition, we get

E(φ • u) = E((u -x 1 ) ∧ 0)) + k i=1 E (-1) i ((0 ∨ (u -x i )) ∧ (x i+1 -x i )) = E((u -x 1 ) ∧ 0)) + k i=1 E((0 ∨ (u -x i )) ∧ (x i+1 -x i )) = E(u),
using symmetry and locality for the second and third equalities.

Future directions

Two challenges which are still open are the following. Firstly, we are not aware of any attempt to obtain a structural decomposition analogous to the one of [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]in the non-bilinear setting. Secondly, the theory of [START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF][START_REF] Sturm | On the geometry of metric measure spaces. I[END_REF][START_REF]On the geometry of metric measure spaces[END_REF] covers even the case where Cheeger's energy of the metric measure space is a non-bilinear form, while an analogue of [START_REF]Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds[END_REF] for the non-bilinear case is missing. It is our opinion that the subject of metric measure spaces would profit from a study in this direction.

These two problems are strong motivations behind our paper, as we foresee that the normal contraction property would be crucial in developing such theories. One difficulty we anticipate is the generalisation of the computations in [START_REF] Bakry | Diffusions hypercontractives[END_REF], which looks complicated even in the case of Finsler manifolds. Finally, establishing the normal contraction property adds one structural argument in favour of the choice made by Cipriani and Grillo of the generalisation of bilinear Dirichlet forms to the non-bilinear setting.

Figure 2 :

 2 Figure 2: Graph of the function φ x 1 ,x 2 ,x 3 .

Figure 3 :Figure 4 :

 34 Figure 3: Illustration of Eq. (18).

Figure 6 :

 6 Figure 6: Illustration of Eq. (21).

Figure 7 :

 7 Figure 7: Illustration of Eq. (22).

Figure 8 :

 8 Figure 8: Illustration of Eq. (23).

  •• • ••φ ⌊k/2⌋ •ψ with φ i ∈ F 2 for all i ∈ {1, . . . , ⌊k/2⌋} and ψ ∈ F k-2⌊k/2⌋ .Proof. We proceed by induction on k. The statement is trivial for k ∈ {0, 1, 2}. Assume that φ = φ x 1 ,...,x k

Acknowledgements

The first author has been funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 754362. Partial support has been obtained from the EFI ANR-17-CE40-0030 Project of the French National Research Agency. The second author was supported by ERC Starting Grant 680275 "MALIG." We are grateful to Giuseppe Savaré for numerous stimulating and helpful comments. We thank the organisers of the CEREMADE Young Researcher Winter School 2022, where part of the work was done.