Jean-Pierre Demailly

Lawomir Dinew

Vincent Guedj

Pham Hoang Hiep

Lawomir Ko

Ahmed Zeriahi

H ÖLDER CONTINUOUS SOLUTIONS TO MONGE-AMP ÈRE EQUATIONS

Keywords: Mathematics Subject Classification. 32U05, 32U40, 53C55 Monge-Ampère operator, Kähler manifold, pluripotential theory, Hölder continuity. 1

Let (X, ω) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L p right hand side, p > 1. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range MAH(X, ω) of the complex Monge-Ampère operator acting on ωplurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Ko lodziej's result that measures with L p -density belong to MAH(X, ω) and proving that MAH(X, ω) has the "L p -property", p > 1. We also describe accurately the symmetric measures it contains.

introduction

Let (X, ω) be a compact n-dimensional Kähler manifold. Let also Θ = {θ} ∈ H 1,1 (X, R) be a given cohomology class on X. In the note we consider two different cases of interest:

(1) Θ is a Kähler class, i.e. there exists a Kähler form which represents Θ. In this case we assume without loss of generality that ω ∈ Θ;

(2) Θ is a big cohomology class, which means that there exists a (possibly singular) closed

(1, 1) current T representing Θ such that T is strictly positive i.e. T ≥ ε 0 ω for some constant ε 0 > 0.

The study of complex Monge-Ampère equations on compact Kähler manifolds with a Kähler background metric has a long history and many spectacular results have appeared during the years. The big cohomology class setting, on the other hand, was initiated recently in [BEGZ]. This is the most general setting where a meaningful (and nontrivial) theory can be developed. Of course it covers the Kähler class setting as a particular case, but since the latter is more classical and certain technicalities can be avoided we have decided to treat both cases separately.

We deal with the Kähler setting first. We study the range of the (normalized) complex Monge-Ampère operator

MA(u) := 1 V ω (ω + dd c u) n , V ω := V ol ω (X) = X ω n ,
acting on ω-plurisubharmonic (ω-psh for short) Hölder-continuous functions u. Here, as usual d = ∂ + ∂ and d c := 1 2iπ (∂ -∂), and V ω denotes the volume of the cohomology class {ω}, so that MA(u) is a probability measure.

This problem is motivated by the study of canonical metrics on mildly singular varieties: their potentials are solutions to degenerate complex Monge-Ampère equations for which Hölder continuity is the best global regularity one can expect. Furthermore even such weak regularity does imply estimates for the "metric" ω + dd c u which might be relevant for the study of the limiting behavior of the Kähler-Ricci flow. We refer the reader to [START_REF] Song | The Kähler-Ricci flow on surfaces of positive Kodaira dimension[END_REF][START_REF] Song | Canonical measures and Kähler-Ricci flow[END_REF][START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Song | Contracting exceptional divisors by the Kähler-Ricci flow[END_REF][START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF][START_REF] Guedj | Hölder continuous solutions to the complex Monge-Ampère equations equations[END_REF][START_REF] Ko Lodziej | Hölder continuity of solutions to the complex Monge-Ampère equation with the right hand side in L p . The case of compact Kähler manifolds[END_REF][START_REF] Ko | A uniform L ∞ estimate for complex Monge-Ampère equations[END_REF][START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF][START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]To,[START_REF] Tian | On the Kähler-Ricci flow on projective manifolds of general type[END_REF][START_REF] Song | Contracting exceptional divisors by the Kähler-Ricci flow[END_REF] for further geometrical motivations and references.

We let PSH(X, ω) denote the set of ω-psh functions: these are defined as being locally equal to the sum of a plurisubharmonic and a smooth function and any such function u additionally satisfies the inequality ω + dd c u ≥ 0 in the weak sense of currents.

We let Hölder(X, R) denote the set of real valued Hölder-continuous functions on X. Our goal is thus to understand the range MAH(X, ω) := MA(PSH(X, ω) ∩ Hölder(X, R)).

A result of fifth named author [K3] (see [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF] and [Di] for refinements in particular cases) asserts that a probability measure µ = f ω n which is absolutely continuous with respect to the Lebesgue measure belongs to MAH(X, ω) if it has density f ∈ L p for some p > 1. Note that a Monge-Ampère potential u ∈ P SH(X, ω) such that MA(u) = µ is unique, up to an additive constant.

The proof in [K3] does not give any information on the Hölder exponent of the Monge-Ampère potential. We combine here the methods of [K3] and the regularization techniques of the first named author [D1, D4] to establish the following result:

Theorem A. Let µ = f ω n = MA(u) be a probability measure absolutely continuous with respect to Lebesgue measure, with density f ∈ L p , p > 1. Then u is Hölder-continuous with exponent α arbitrarily close to 2/(1 + nq), where q denotes the conjugate exponent of p.

It is a slightly better exponent than the one obtained in some special cases in [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF] and [Di]. It is also asymptotically optimal (see [Pl] and [GKZ] for some local counterexamples which are easily adjustable to the compact setting). The proof uses a subtle regularization result of [D1, D4], as in [Di] and [BD]. The extra tool that allows us to remove symmetry/curvature constraints is the Kiselman minimum principle coupled with Demailly's method of attenuating singularities (the Kiselman-Legendre transform) from [D4].

By keeping track of the Hölder constant together with the exponent one can in fact obtain uniform estimates provided suitable control on the global geometry is assumed. More precisely if we assume uniformly bounded geometry (this notion will be explained in the Preliminaries) the following holds:

Theorem A*. Let (X s , ω s ) be a family of compact Kähler manifolds with uniformly bounded geometry. Consider the Monge-Ampère equations

(ω s + dd c u s ) n = f s ω n s , sup Xs u s = 0.
If f L p (ω n s ) ≤ C are uniformly bounded then the solutions u s are uniformly Hölder continuous for any exponent α < 2/(nq + 1) and the Hölder constant is uniformly controlled by C and the constants from the definition of the uniformly bounded geometry.

We furthermore believe that additional technical improvements of our arguments may lead to analogous statements in the case of classes which are merely semi-positive and big (see [BGZ] for a definition and further developments).

A satisfactory description of MAH(X, ω) is yet to be found. We nevertheless establish a technically involved characterization (Theorem 4.3) that allows us to derive several useful consequences, for example we show:

Theorem B. The set MAH(X, ω) has the L p -property: if µ ∈ MAH(X, ω) and 0 ≤ f ∈ L p (µ) with p > 1 and X f dµ = 1, then f µ ∈ MAH(X, ω).
In particular the set MAH(X, ω) is convex.

It has been recently proved by Dinh-Nguyen-Sibony [DNS] (see also [Ber] for recent developments) that measures in MAH(X, ω) have the following strong integrability property: if

µ ∈ MAH(X, ω), then ( †) exp(-ε PSH(X, ω)) ⊂ L 1 (µ), for some ε > 0.
This is a useful generalization of Skoda's celebrated integrability theorem (see [Sk, Ze]).

It is natural to wonder whether condition ( †) characterizes MAH(X, ω). This is the case when n = 1 (see [DS] and Subsection 4.1). In this note we show that such a characterization still holds in higher dimensions provided the measures under consideration have symmetries:

Theorem C. Let µ be a probability measure with finitely many isolated singularities of radial or toric type. Then µ belongs to MAH(X, ω) if and only if ( †) is satisfied.

Next we turn our attention to the general big cohomology class setting. To this end we choose a smooth (1, 1)-form θ representing Θ. Observe that in general one cannot have θ ≥ 0. Analogously to the Kähler setting we can nevertheless define PSH(X, θ) as the set of functions which are defined again as being locally equal to sum of a plurisubharmonic and a smooth function and any such function ϕ should satisfy θ + dd c ϕ ≥ 0. Observe that by assumption such functions exist, although they all may be singular.

It follows from the regularization theorem of the first author [D4] that we can find a strictly positive closed (1, 1) current T + = θ + dd c ϕ + which represents Θ and has analytic singularities, that is there exists c > 0 such that locally on X we have

ϕ + = c log N j=1 |f j | 2 mod C ∞
where f 1 , ..., f N are local holomorphic functions. Such a current T + is then smooth on a Zariski open subset Ω, and the ample locus Amp(Θ) of Θ is defined as the largest such Zariski open subset (which exists by the Noetherian property of closed analytic subsets). Therefore any θ-psh function ψ with minimal singularities is locally bounded on the ample locus. Here having minimal singularity means that given any other θ-psh function ϕ one has the inequality

ϕ ≤ ψ + O(1).
According to [BEGZ] we can then define the (non-pluripolar) product (θ + dd c ϕ) n , and in case ϕ has minimal singularities, the total mass of this measure is independent of ϕ and equals

X (θ + dd c ϕ) n =: V ol(Θ) > 0.
It is therefore meaningful to study the (normalized) Monge-Ampère equation

MA(ϕ) := 1 V ol(Θ) (θ + dd c ϕ) n = µ,
for a given probability measure µ vanishing on pluripolar sets. When µ = f dV is absolutely continuous with respect to Lebesgue measure with density f ∈ L p (X), p > 1, there is a unique solution modulo additive constant which turns out to have minimal singularities [BEGZ]. The solution is known to be globally continuous on X when the cohomology class Θ is moreover semi-positive ( [START_REF] Eyssidieux | Viscosity solutions to degenerate complex Monge-Ampère equations[END_REF]).

In this context we prove the following analogue of Theorem A:

Theorem D. Let µ be a probability measure absolutely continuous with respect to a fixed smooth volume form, with density f ∈ L p (X), p > 1. Let ϕ be a weak solution of the Monge-Ampère equation MA(ϕ) = µ. Then for any 0 < α < 2/(1 + nq), ϕ is Hölder-continuous of exponent α locally in the ample locus Amp(Θ) of Θ (here q denotes the conjugate exponent of p).

The note is organized as follows. In Section 2 we recall all the basic facts and introduce the necessary definitions. Theorems A and A* are proved in Section 3. After recalling the one dimensional theory in Subsection 4.1, we establish the characterization of MAH(X, ω) in Subsection 4.2. This allows us to prove Theorem B (in Subsection 4.3) and derive further interesting consequences. The case of measures with symmetries is handled in Section 5. Theorem D is proved in Section 6. In the Appendix we briefly explain how bounds on the curvature allow to control the differential of the exponential mapping, a technical information needed in the proof of Theorem A*.

preliminaries

2.1. Curvature and regularization. Let X be a compact Kähler manifold equipped with a fundamental Kähler form ω given in local coordinates by

ω = i 2 k,j g k jdz k ∧ dz j .
Its bisectional curvature tensor in those local coordinates is defined by

R i jk l := - ∂ 2 g k l ∂z i ∂ zj + n p,q=1 g pq ∂g p l ∂ zj ∂g k q ∂z i ,
with g pq denoting the inverse transposed matrix of g pq i.e., n q=1 g pq g sq = δ ps . It is a classical fact that in the Kähler case the bisectional curvature tensor coincides with the Levi-Civita curvature tensor. We say that the bisectional curvature is bounded by A > 0 if for any z ∈ X and any vectors Z, Y ∈ T z X, Z, Y = 0 we have the inequality

n i,j,k,l R i jk l(z)Z i Zj Y k Ȳl ≤ A Z 2 ω Y 2 ω .
Analogously the bisectional curvature is bounded from below (resp. from above) by A if

n i,j,k,l R i jk l(z)Z i Zj Y k Ȳl ≥ A Z 2 ω Y 2 ω , (resp. ≤ )
respectively. It is easy to check that the existence of such bounds is independent of the choice of local coordinates.

Recall that if u is a psh function in a domain in C n then a convolution with a radial smoothing kernel preserves positivity of dd c u. For non-flat metrics, this may not be the case any longer. However, an approximation technique due to the first author allows to control "the negative part" of the smooth form. It is described in detail in [D1] and [D4]. Here we shall briefly highlight its main features.

Consider the exponential mapping from the tangent space of a given point z ∈ X

exp z : T z X ∋ ζ → exp z (ζ) ∈ X,
which is defined by exp z (ζ) = γ(1) with γ being the geodesic starting from z with initial velocity γ ′ (0) = ζ. Given any function u ∈ L 1 (X), we define its δ-regularization ρ δ u to be (2.1)

ρ δ u(z) = 1 δ 2n ζ∈TzX u(exp z (ζ))ρ |ζ| 2 ω δ 2 dV ω (ζ), δ > 0
according to [D1]. Here ρ is a smoothing kernel, |ζ| 2 ω stands for n i,j=1 g i j (z)ζ i ζj , and dV ω (ζ) is the induced measure 1 2 n n! (dd c |ζ| 2 ω ) n . This may be formally extended as a function on X × C by putting U(z, w) := ρ δ u(z) for w ∈ C, |w| = δ. The introduction of the variable w is convenient for an application of Kiselman minimum principle [START_REF] Kiselman | Partial Legendre transformation for plurisubharmonic functions[END_REF][START_REF] Kiselman | Attenuating the singularities of plurisubharmonic functions[END_REF] to that function. It should be noticed that in [D4] the riemannian exponential map "exp" has been replaced by a "holomorphic counterpart" exph, which is defined as the holomorphic part of the Taylor expansion of ζ → exp z (ζ) (the reason is that the calculations then become somewhat simpler, especially in the non Kähler case, but this is not technically necessary; thanks to a well known theorem of E. Borel, such a formal expansion can always be achieved by a smooth function exph : T X → X). The function exph is however not uniquely defined, and this weakens the intrinsic character of the estimates. Therefore, we stick here to the more usual riemannian exp function.

The estimates obtained in [D1] show that all results of [D4] and [BD] are still valid with the unmodified definition of ρ δ u, at least when (X, ω) is Kähler. By Lemma 8.2 of [D1], the exponential function exp :

T X → X, T X ∋ (z, ζ) → exp z (ζ) ∈ X, ζ ∈ T z X
satisfies the following properties:

(1) exp is a C ∞ smooth mapping;

(2) ∀z ∈ X, exp z (0) = z and d ζ exp(0) = Id TzX ;

(3) ∀z ∈ X the map ζ → exp z ζ has a third order Taylor expansion at ζ = 0 of the form

(2.2) exp z (ζ) m -z m -ζ m - 1 2 j,k,l R j kl m zk + 1 3 ζk ζ j ζ l ≤ C(|ζ| 2 (|z| + |ζ|) 2 ), |ζ| < r,
for small enough r > 0. The expansion is valid in holomorphic normal coordinates with respect to the Kähler metric. It is convenient to select a particular smoothing kernel, namely ρ : R + → R + by setting

ρ(t) = η (1-t) 2 exp( 1 t-1 ) if 0 ≤ t ≤ 1, 0 if t > 1 with a suitable constant η, such that (2.3) C n ρ( z 2 ) dV (z) = 1 (dV denotes the Lebesgue measure in C n ).
The crucial estimate of the Hessian of U(z, w) given in [D4], Proposition 3.8 (see also [D1], Proposition 8.5), coupled with Kiselman's theorem provide a lemma stated in this form in [START_REF] Berman | Regularity of plurisubharmonic upper envelopes in big cohomology classes[END_REF]Lemma 1.12]: Lemma 2.1. Fix any bounded ω-psh function u on a compact Kähler manifold (X, ω). Let U(z, w) be its regularization as defined above. Define the Kiselman-Legendre transform with level c by

(2.4) u c,δ := inf 0≤t≤δ U(z, t) + Kt 2 -Kδ 2 -c log t δ .
Then for some positive constant K depending on the curvature, the function U(z, t) + Kt 2 is increasing in t and one has the following estimate for the complex Hessian:

(2.5) ω + dd c u c,δ ≥ -(A min{c, λ(z, δ)} + Kδ 2 ) ω,
where A is a lower bound of the negative part of the bisectional curvature of ω, while

λ(z, t) := ∂ ∂ log t (U(z, t) + Kt 2 ).
2.2. Jensen formula and uniformly bounded geometry. The classical Jensen formula (see, for example [START_REF] Bedford | The Dirichlet problem for the complex Monge-Ampère operator[END_REF]) for a

C 2 function u defined in a ball B(z, 2δ) in C n says that (2.6) (ǔ δ -u)(z) = 2n δ 2n σ 2n-1 δ 0 r 2n-1 r 0 t 1-2n |ζ|≤t ∆u(z + ζ) dV (ζ) dt dr,
where ǔδ is the average of u over B(z, δ) and σ 2n-1 denotes the total surface measure of the unit sphere. Now, if u is defined in a large set, then the integration of the above formula in z provides an estimate of the integral of δ -2 (ǔ δ -u) in terms of the integral of the Laplacian of u. We need such an estimate on compact Kähler manifolds which is uniform as long as the geometry of manifolds is bounded in a certain sense.

Definition 2.2. Consider a family (X s , ω s ) of compact Kähler manifolds. We shall say that it has uniformly bounded geometry if 1) the diameter diam(X s , ω s ) is uniformly bounded, 2) their bisectional curvatures are uniformly bounded, 3) the injectivity radius is uniformly bounded from below.

By well-known estimates [HK], it then follows that the total volumes Vol ωs (X s ) := Xs ω n s are uniformly bounded above and below by constants C and C -1 independent of s.

It turns out that such bounds are enough to ensure various interesting geometric and analytic bounds. Note in particular that they imply lower bounds on the Tian α invariants for the classes of ω s -psh functions which does not depend on s (see [BEGZ]). In potential applications X will usually stay fixed, while the Kähler forms may vary. Note that all conditions are obviously satisfied if the forms ω s are bounded in C ∞ topology and uniformly positive; this can be achieved by selecting appropriate representatives when the cohomology classes [ω s ] are given and contained in a fixed relatively compact region of the Kähler cone of X. Thus an interesting case to treat would be when the classes [ω s ] approach the boundary of this cone. Unfortunately this may in general lead to a blow-up of the curvature and for this reason our argument cannot be applied to study the limiting behavior. On the other hand the method works if the forms ω s approximate a C 1,1 form ω in a fixed cohomology class provided that the curvatures of ω s stay bounded.

We can now state a lemma to be used in the next section.

Lemma 2.3. Assume (X s , ω s ) is a family of compact Kähler manifolds with uniformly bounded geometry. Let u s be continuous ω s -psh functions normalized by min Xs u s = 1, max Xs u s ≤ B for some fixed constant B. If ρ δ u s is the regularization of u s defined as in (2.1) then for δ small enough we have

Xs ρ δ u s -u s δ 2 ω n s ≤ C 0 ,
where C 0 only depends on B and the constants involved in the uniform bounds on the geometry.

Proof. Let us fix s and omit it in the notation for simplicity. By definition

ρ δ u(z) = ζ∈TzX u(exp z ζ)ρ |ζ| 2 ω δ 2 dV ω (ζ) δ 2n = x∈X u(x)ρ | log z x| 2 ω δ 2 dV ω (log z x) δ 2n = x∈X u(x)K δ (z, x) where x → ζ = log z x is the inverse of ζ → x = exp z (ζ).
The map (z, x) → (z, log z x) defines a diffeomorphism from a neighborhood of the diagonal in X × X onto a neighborhood of the zero section of T X by the implicit function theorem. Here

K δ (z, x) = 1 δ 2n ρ | log z x| 2 ω δ 2 dV ω (log z x)
is the semipositive (n, n) form on X × X defined as the pull-back of ρ(|ζ| 2 ω /δ 2 ) dV ω (ζ)/δ 2n by (z, x) → ζ = log z x; it can be viewed as a kernel with compact support in a neighborhood of the diagonal of X × X. By definition, we have x∈X K δ (z, x) = 1 (as is clear by taking u ≡ 1), thus

u(z) = x∈X u(z)K δ (z, x). Therefore X ρ δ u(z) -u(z) dV ω (z) = (x,z)∈X×X u(x) -u(z) K δ (z, x) ∧ dV ω (z) = (x,z)∈X×X u(x) K δ (z, x) ∧ dV ω (z) -K δ (x, z) ∧ dV ω (x)
thanks to a change of variable (z, x) → (x, z). In order to finish the proof we need the following lemma which establishes a pointwise bound for the kernel:

Lemma 2.4. If d ω (z, x) ≤ δ, then | (K δ (z, x) ∧ dV ω (z) -K δ (x, z) ∧ dV ω (x)) | ≤ Cδ 2-2n dV ω (z) ∧ dV ω (x),
for some uniform constant C which only depends on the curvature of ω.

If d ω (z, x) > δ, then K δ (z, x) ∧ dV ω (z) = K δ (x, z) ∧ dV ω (x) = 0. Proof. Given the symmetry of | log z (x)| ω = | log x (z)| ω = d ω (z, x), it is enough to bound the (2n, 2n)-form dV ω (log z x) ∧ dV ω (z) -dV ω (log x z) ∧ dV ω (x). The last assertion follows from the fact that ρ | log z x| 2 ω δ 2 = ρ | log x z| 2 ω δ 2 = 0 if d ω (z, x) > δ.
We now establish the first part of the lemma. Set

ζ = log z x (i.e. x = exp z (ζ)) and y = exp z ( ζ 2 ) = exp z ( 1 2 log z (x)
) (the mid-point of the geodesic joining z and x). Observe that from the expansion (2.2) applied at y (which is identified with zero in this system of normal coordinates) we have

(2.7) ζ m = log z (x) m = x m -z m - 1 2 j,k,l R j kl m zk + 1 3 (x k -z k ) (x j -z j )(x l -z l ) + O( z -x 4 ). Now (2.7) yields dζ m = d(log z x) m = dx m -dz m + O( z -x 2 )(dx, dz),
with an O(...) term depending only on the curvature. By the choice of the center y we have

z j = 1 2 (z j -x j ) + O( z -x 2 )
, where the O(...) term again only depends on the curvature. Thus the expansion

dV ω (ζ) = ω(z) n n! (ζ) = 1 - j,k,l R j kl lz j zk + O( z 3 ) i 2 dζ 1 ∧ dζ 1 ∧ . . . ∧ i 2 dζ n ∧ dζ n at any given point z yields dV ω (log z x) = n j=1 i 2 (dx j -dz j ) ∧ (dx j -dz j ) + O( z -x 2 ).
Thus, by taking the product with dV ω (z), exchanging x and z, and then subtracting and dividing by δ 2n , we obtain the desired bound

dV ω (log z x) ∧ dV ω (z) -dV ω (log x z) ∧ dV ω (x) δ 2n = O( z -x 2 ) δ 2n dV ω (z) ∧ dV ω (x).
The appendix implies that O(...) depends only on global bounds for the geometry.

We can now use Fubini's theorem and the estimates on the kernel to obtain

(x,z)∈X×X u(x) K δ (z, x) ∧ dV ω (z) -K δ (x, z) ∧ dV ω (x) = x∈X z∈B(x,δ) u(x) K δ (z, x) ∧ dV ω (z) -K δ (x, z) ∧ dV ω (x) ≤ x∈X z∈B(x,δ) |u(x)|Cδ 2-2n dV ω (z) ∧ dV ω (x) ≤ x∈X BCδ 2 dV ω (x) ≤ C 0 δ 2 ,
as claimed.

2.3. The H(α) condition and measures uniformly dominated by capacity. A fundamental tool in the study of ω-psh functions is the relative capacity modelled on the Bedford-Taylor relative capacity ( [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]).

Definition 2.5. Let (X, ω) be a compact Kähler manifold. Given a Borel subset K of X, we define its relative capacity with respect to ω by

Cap ω (K) := sup K (ω + dd c ρ) n | ρ ∈ PSH(X, ω), 0 ρ 1 .
The following classes have been considered in [EGZ1]:

Definition 2.6. Let µ be a probability measure on a compact Kähler manifold (X, ω). We say that µ belongs to the class H(α), α > 0 (alternatively, that µ statisfies the H(α) property), if there exists C α > 0 such that for any compact K ⊂ X,

µ(K) ≤ C α Cap ω (K) 1+α
If this holds for any α > 0, we say that µ satisfies H(∞).

It was proved in [K1, K3] that measures of the type µ = f ω n with a density f in L p for some p > 1 do satisfy H(∞) (see also [Ze]). A slightly stronger notion was introduced in [DZ]: Definition 2.7. We say that a probability measure µ is dominated by capacity for L p functions if there exists constants α > 0 and β > 0, such that for any compact K ⊂ X and non-negative f ∈ L p (µ) with p > 1, one has for some constant C independent of K that

µ(K) C • Cap ω (K) 1+α and K f µ C • Cap ω (K) 1+β .
Both notions are variations on the condition (A) introduced by fith named author in [K1]. These conditions, which are actually stronger than condition (A), ensure the existence of bounded solutions u to MA(u) = f µ, as long as X f dµ = 1.

Note that the condition H(∞) is equivalent to domination by capacity for L ∞ functions by a simple application of the Hölder inequality.

2.4. Big cohomology classes. Let X be a compact Kähler manifold of dimension n, and Θ = {θ} ∈ H 1,1 (X, C) ∩ H 2 (X, R) a big cohomology class with a smooth representative θ.

We introduce the extremal function V θ defined by

(2.8) V θ (x) := sup{ϕ(x) | ϕ ∈ P SH(X, θ), sup X ϕ ≤ 0},
where P SH(X, θ) is the set of all θ-plurisubharmonic functions on X. The function V θ is a θ-psh function with minimal singularities.

Similarly to the Kähler case we define the relative capacity:

Definition 2.8. Let X be a compact Kähler manifold. Given a Borel subset K of X, we define its relative capacity with respect to θ by

Cap θ (K) := sup K (θ + dd c ρ) n | ρ ∈ PSH(X, θ), V θ (x) -1 ρ V θ (x) .
Observe that contrary to the Kähler case competitors to maximize the right hand side have minimal singularities but are in general unbounded. The Monge-Ampère measures in the definition are only considered outside the polar locus {x ∈ X | V θ (x) = -∞}. Observe that the latter depends on the cohomology class {θ} but not on the choice of its representative θ.

Most definitions from the Kähler setting have their big counterparts, we refer the readers to [BEGZ] for details and more background regarding big cohomology classes. In particular we can apply the same convolution procedure to any θ-psh function, as well as the Kiselman-Legendre transform.

In order to prove Theorem D we shall need a stability estimate proved in [GZ2]:

Proposition 2.9. Assume that µ is a probability measure absolutely continous with respect to a smooth volume form dV , dµ = f dV , where f ∈ L p (X) with p > 1. Let ϕ, ψ be θplurisubharmonic functions such that MA(ϕ) = µ, -M 0 + V θ ≤ ϕ ≤ V θ and ψ ≤ V θ on X, for some positive constant M 0 > 0. Then for any exponent 0 < γ < 1 nq+1 , there exists a constant

B 0 = B 0 (p, γ, M 0 ) > 0 such that sup X (ψ -ϕ) + ≤ B 0 (ψ -ϕ) + γ L 1 (X) .

Proof of Theorems A and A*

Proof of Theorem A. Fix u ∈ P SH(X, ω) such that MA(u) = µ. Denote by A -1 = A ′ > 0 a bound for the curvature of (X, ω). By [K1] u is continuous, so assume that min X u = 1 and denote by B := max X u the maximum of u. Consider ρ δ u-the regularization of the ω-psh function u defined in (2.1).

Let us set for δ > 0 and α > 0,

(3.1) E(δ, α) := {(ρ δ u -u)(z) > δ α } Let 0 < α 1 < 2 qn+1 . Choose ε > 0, α, α 0 such that α 1 < α < α 0 < 2 -α 0 q(n + ε).
Set θ := e -3AB . Recall (Lemma 2.1) that there exists a constant K which only depends on the curvature such that the functions ρ δ u + Kδ 2 are increasing in δ. Note that for δ small enough

θ α 1 δ α 1 ≥ δ α 0 + Kδ 2 (1 -θ 2 ). Altogether this implies that E(δ, α 0 ) ⊃ E(θδ, α 1 ).
We want to show that E(θδ, α 1 ) is empty. Recall the definition of the Kiselman-Legendre transform at level δ α (see Lemma 2.1)

U δ = inf t∈[0,δ] (ρ t u + Kt 2 -δ α log t δ -Kδ 2 ),
where K is chosen as in the formula (2.4). It follows from [D1] that the same K can be chosen for a family of manifolds with uniformly bounded geometry. In what follows δ 0 and δ j , c j , j = 1, 2, 3; denote constants which are uniform if the geometry is uniformly bounded and f p stays bounded. By Lemma 2.1

ω + dd c U δ ≥ -[(A -1)δ α + Kδ 2 ]ω > -Aδ α ω + 2δ α 0 ω
for 0 < δ < δ 0 , where δ 0 > 0 is small enough. Therefore

u δ := 1 1 + Aδ α U δ
is ω-psh on X and satisfies ω + dd c u δ ≥ δ α 0 ω, provided Aδ α < 1, which we can safely assume. From Lemma 2.3 we have (3.2)

X |ρ δ u -u|ω n ≤ c 1 δ 2 , for 0 < δ < δ 0 . Therefore for E 0 = E(δ, α 0 ) = {(ρ δ u -u)(z) > δ α 0 } we have E 0 ω n ≤ c 1 δ 2-α 0 ,
and, by Hölder inequality,

E 0 f ω n ≤ c 2 δ (2-α 0 )/q .
Let us modify f setting g = 0 on E 0 and g = cf elsewhere, with c such that total integrals of f and g are equal. Solve for continuous ω-psh function v (comp. [K2])

(ω + dd c v) n = gω n , max(u -v) = max(v -u). Observe that f -g L 1 (X) = 2 E 0 f ω n ≤ 2c 2 δ (2-α 0 )/q .
Then by [DZ] there exists c 3 (c 3 depends additionally on ε > 0) such that n+ε) . We claim that there exist small enough constants δ 1 > δ 2 > δ 3 > 0 such that for any 0 < δ < δ 3 there is a set inclusion

(3.3) u -v L ∞ ≤ c 3 δ 2-α 0 q(
(3.4) E(θδ, α 1 ) ⊂ {u δ -v > δ α } ⊂ E(δ, α 0 ).
Indeed, take z in E(θδ, α 1 ). By Lemma 2.1, the function

ρ t u + Kt 2 is increasing in t ∈ [0, δ]. Thus for t ∈ [θδ, δ], ρ t u(z) -u(z) = ρ t u(z) -ρ θδ u + ρ θδ u -u(z) ≥ K(θδ) 2 -Kt 2 + (θδ) α 1 ≥ (θδ) α 1 -Kδ 2 ,
and for t < θδ, since θ = e -3AB , we have

-δ α log(t/δ) ≥ 3ABδ α . Therefore (U δ -u)(z) ≥ min((e -3AB δ) α 1 -Kδ 2 , 3ABδ α ) = 3ABδ α for 0 < δ < δ 1 ,
where δ 1 > 0 is small enough (we can safely assume that δ 1 < δ 0 ). Hence, by (3.3)

(U δ -v)(z) ≥ 3ABδ α -c 3 δ 2-α 0 q(n+ε) > 2ABδ α , for δ < δ 2 , where 0 < δ 2 < δ 1 is small enough. Observe that U δ -u δ ≤ ABδ α . Since AB ≥ 1, it follows that u δ (z) -v(z) > ABδ α > δ α for δ < δ 2 , which proves the first inclusion E(θδ, α 1 ) ⊂ {u δ -v > δ α } in (3.4).
To prove the second inclusion, take z / ∈ E(δ, α 0 ). Since, under our assumptions

u δ < U δ ≤ ρ δ u,
we get, applying (3.3)

(u δ -v)(z) ≤ (ρ δ u -u)(z) + c 3 δ 2-α 0 q(n+ε) ≤ δ α 0 + c 3 δ 2-α 0 q(n+ε) < δ α ,
for 0 < δ < δ 3 , where 0 < δ 3 < δ 2 is small enough. This proves our second inclusion

{u δ -v > δ α } ⊂ E(δ, α 0 )
for 0 < δ < δ 3 and completes the proof of (3.4). Now we want to apply the comparison principle do deduce from (3.4) that the set E(θδ, α 1 ) is empty for δ > 0 small enough. Let us fix 0 < δ < δ 3 and recall that E 0 = E(δ, α 0 ). From (3.4) and the comparison principle [K2], if follows that

{u δ >v+δ α } (dd c u δ + ω) n ≤ {u δ >v+δ α } (dd c v + ω) n ≤ E 0 (dd c v + ω) n = E 0 gω n = 0.
Since u δ is ω-psh and (ω + dd c u δ ) n ≥ δ nα 0 ω n , it follows that the volume of the set {u δ > v + δ α } is zero. Hence it is empty, since u δ and v are ω-psh functions. Therefore from (3.4), it follows that the set E(θδ, α 1 ) is also empty. Setting η = θδ = δe -3AB , we obtain

ρ η u -u ≤ e 3α 1 AB η α 1 , for 0 < η < η 0 = e -3αAB δ 3 .
Note that the above inequality means that locally ["words permutation"] the η-convolution of u is no more than u plus some constant of order η α 1 . Thus repeating the local argument from [GKZ] one obtains that the supremum of u in a coordinate ball of radius η and center z is also controlled by u(z) and a constant of order η α 1 . This proves that u is Hölder continuous with exponent α 1 .

Note that in the proof above we could choose the same δ 1 , δ 2 and δ 3 for uniform α j , c j . Thus, following the lines of this proof, one can obtain an analogous result for families of manifolds with uniformly bounded geometry.

Theorem 3.1 (Theorem A*). Let (X s , ω s ) be a family of n-dimensional compact Kähler manifolds with uniformly bounded geometry. Consider the Monge-Ampère equations

(ω s + dd c u s ) n = f s ω n s , sup Xs u s = 0,
where Xs f s ω n s = Xs ω n s . If f L p (ω n s ) ≤ C are uniformly bounded then the solutions u s are uniformly Hölder continuous for any exponent α < 2/(nq + 1) and the Hölder constant is uniformly controlled by C and the constants from the definition of the uniformly bounded geometry.

As a direct application of Theorem A* one has the following corollary: Corollary 3.2. Suppose X is a compact Kähler manifold and ω is a C 1,1 smooth closed positive form on X. Suppose moreover that ω can be approximated in C 1,1 -norm by smooth closed forms with curvatures bounded by a fixed constant. Let also f be any nonnegative function such that f ∈ L p (ω n ) and X f ω n = X ω n . Then the Monge-Ampère equation

(ω + dd c u) n = f ω n , sup X u = 0
has an α-Hölder continuous solution u for any α < 2/(nq + 1), where q is the conjugate to p.

Finally we remark that in [DZ] the stability result holds not only for measures absolutely continuous with respect to the Lebesgue measure, but also for any measure dominated by capacity for L p functions. Observe that in the proof the sole place where we used the assumption that µ is a measure with density was the application of the Jensen formula in (3.2). Therefore by repeating the above proof one can get the following generalization: Proposition 3.3. Let u ∈ PSH(X, ω) solve the equation MA(u) = µ for µ a probability measure on a compact Kähler manifold (X, ω). Assume that µ satisfies the following additional assumptions:

i) µ satisfies H(∞); ii) ρ δ φ -φ L 1 (µ) = O(δ b ) for some b > 0.
Then u is Hölder continuous with the exponent depending only on n and b.

Examples of such singular measures have been considered in [Hi].

4. Some properties of MAH(X, ω) 4.1. The one dimensional case. In this section we recall for reader's convenience the classical one dimensional theory of Hölder continuous potentials. We refer to [DS] for more details. It is worthwhile to recall that the problem on Riemann surfaces is linear and hence much easier: analogous statements in the case of planar domains are classical in potential theory.

Proposition 4.1. Let (X, ω) be a compact Riemann surface. Let also µ = ω + dd c φ be a probability measure on X, where φ ∈ P SH(X, ω) and B(a, r) be the ball (with respect to the metric induced by ω) centered at point a with radius r. The following properties are equivalent: i) the function φ is Hölder continuous ; ii) there exists constants α, C > 0 such that µ(B(a, r)) ≤ Cr α , for all a ∈ X and 0 < r < 1; iii) there exists ε > 0 such that exp (-εP SH(X, ω)) ⊂ L 1 (µ).

Remark 4.2. As the Laplacian is a linear operator, Proposition 4.1 is actually a local result. It further holds for higher dimensional subharmonic functions. We let the reader check that if u is a subharmonic function in some domain Ω ⊂ R n which contains the origin, and 0 < α < 1, then the following are equivalent:

(1) sup B(δ) u -u(0) ≤ C 1 δ α , for some C 1 > 0 and 0 < δ << 1;

(2)

1 vol(B(δ)) B(δ) u(z) dV (z) -u(0) ≤ C 1 δ α , where C 1 > 0, 0 < δ << 1; (3) B(δ) ∆u ≤ C 3 δ α+n-2 , for some C 3 > 0 and 0 < δ << 1.
It classically follows from this observation that any subharmonic function is α-Hölder continuous (respectively C 1,α ) outside a set of arbitrarily small (n -2 + α)-Hausdorff (respectively (n -1 + α)-Hausdorff ) content.

Characterization of MAH(X, ω).

Let Ω be a bounded domain in C n . Analogously to the formula (2.6) for each u ∈ P SH(Ω) and δ > 0 we set

ǔδ (z) = 1 v 2n δ 2n B δ u(z + w)dV (w) and u δ (z) = sup w∈B δ u(z + w), for z ∈ Ω δ = {z ∈ Ω : d(z, ∂Ω) > δ}. Here B δ = {z ∈ C n : z = (|z 1 | 2 + ... + |z n | 2 ) 1 2 < δ}
and v 2n is the volume of the unit ball B 1 .

Theorem 4.3. Let (X, ω) be a compact Kähler manifold, µ a positive Borel measure on X so that µ(X) = X ω n . The following are equivalent: i) There exists a Hölder continuous ω-psh ϕ such that µ = (ω + dd c ϕ) n .

ii) For every z ∈ X, there exists a neighborhood D of z and a Hölder continuous psh v on D such that µ| D ≤ (dd c v) n .

iii) µ ∈ H(∞) and there exists C, α > 0 such that

K [ǔ δ -u]dµ ≤ C D ∆u δ α , for all u ∈ PSH ∩ L ∞ (Ω), K ⊂⊂ D ⊂⊂ Ω,
where Ω is a local chart.

A positive measure µ thus belongs to MAH(X, ω) if and only if it is locally the Monge-Ampère measure of a Hölder-continuous psh function.

Proof. The implication i) ⇒ ii) is immediate. The implication iii) ⇒ i) was observed to hold in Proposition 3.3.

We now consider the implication ii) ⇒ iii). It is enough to prove the inequality

K [ǔ δ -u](dd v ) n ≤ C D ∆uδ α ,
for all u ∈ PSH ∩L ∞ (Ω), K ⊂⊂ D ⊂⊂ Ω and for any local chart Ω. We can assume without loss of generality that

K = B 1 is the unit ball in C n , D = B 2 and -2 ≤ v ≤ -1, |v(z) -v(w)| ≤ z -w s for all z, w ∈ B 2 . This implies that h(z) := z 2 -4 < v on B 1 , while v < h on B 2 \B r 0 for some 1 < r 0 < 2. Replacing v by max(v, h) we can assume that v = h on B 2 \B r 0 . Fix ρ ∈ C ∞ 0 (C n ) such that ρ ≥ 0, ρ(z) = ρ( z ), suppρ ⊂ B 1 and C n ρ(z) dV (z) = 1. Set vδ (z) = B 1 v(z -δw)ρ(w) dV (w) = 1 δ 2n B(z,δ) v(w)ρ z -w δ dV (w).
Observe that

(1) vδ (z) -v(z) = B 1 [v(z -δw) -v(z)]ρ(w) dV (w) ≤ δ s and (2) ∂ 2 vδ ∂z j ∂z k (z) ≤ C v L ∞ (Ω) δ 2 , (dd c vδ ) n ≤ C dV δ 2n . Choose now φ ∈ C ∞ 0 (C n ) such that 0 ≤ φ ≤ 1, φ = 1 on B r 1 and suppφ ⊂ B r 2 , where r 0 < r 1 < r 2 < 2. Set v δ (z) = B 1 v z -δφ(z)w ρ(w) dV (w). Observe that (3) v δ (z) -v(z) = B 1 [v z -δφ(z)w -v(z)]ρ(w) dV (w) ≤ δ s and (4) v δ (z) = vδ (z) on B r 1 , v δ (z) = v(z) on B 2 \B r 2 .
Fix now any z ∈ B 2 \B r 0 . Since v = h there, we have for any δ < δ 0 ,

∂ 2 v δ ∂z j ∂z k (z) = B 1 [ ∂ 2 h ∂z j ∂z k z -δφ(z)w + δO(1)]ρ(w) dV (w) = B 1 [δ jk + δO(1)]ρ(w) dV (w) = δ jk + δO(1).
Therefore v δ ∈ PSH(B 2 \B r 0 ), ∀δ < δ 0 , hence v δ is actually plurisubharmonic in all of B 2 (if δ is small enough), as follows from (4). Set 3), ( 4) and Stokes fomula we get

T := n-1 j=0 (dd c v) j ∧ (dd c v δ ε ) n-1-j . ¿From (
K [ǔ δ -u](dd c v) n ≤ B 2 [ǔ δ -u](dd c v) n = B 2 [ǔ δ -u][(dd c v) n -(dd c vδ ε ) n ] + B 2 [ǔ δ -u](dd c vδ ε ) n ≤ B 2 [ǔ δ -u]dd c (v -vδ ε ) ∧ T + C δ 2nε B 2 [ǔ δ -u] dV ≤ B 2 [v δ ε -v]dd c (u -ǔδ ) ∧ T + C B 2 ∆uδ 2 δ 2nε ≤ B 2 [v δ ε -v]dd c u ∧ T + C B 2 ∆uδ 2(1-nε) ≤ δ εs Br 2 dd c u ∧ T + Cδ 2(1-nε) B 2 ∆u ≤ C[δ εs v n-1 L ∞ (Ω) B 2 ∆u + δ 2(1-nε) B 2 ∆u] ≤ C B 2 ∆u δ α ,
where ε = 2 s+2n , α = 2s s+2n . 4.3. Proof of Theorem B. Below we derive several simple consequences of this characterization. First, the range of the complex Monge-Ampère operator has the "L p -property": Corollary 4.4. Let ψ ∈ PSH(X, ω) be a Hölder continuous function. Consider a density 0 ≤ f ∈ L p ((ω + dd c ψ) n ) with p > 1 and X f (ω + dd c ψ) n = X ω n . Then there exists a Hölder continuous ω-plurisubharmonic function ϕ such that

(ω + dd c ϕ) n = f (ω + dd c ψ) n .
In particular MAH(X, ω) is a convex set.

Proof. By Hölder inequality we have

K f ω n ψ ≤ f L p (ω n ψ ) [ω n ψ (K)] 1-1 p ,
for any Borel subset K of X. This implies that f ω n ψ ∈ H(∞). On the other hand, by Hölder inequality we have

K [ǔ δ -u]f ω n ψ ≤ f L p (ω n ψ ) [ K [ǔ δ -u]ω n ψ ] 1-1 p ≤ Cδ α ,
for all u ∈ PSH ∩ L ∞ (Ω), K ⊂⊂ D ⊂⊂ Ω and local chart Ω. Therefore using Theorem 2.1 there exists a Hölder continuous ω-psh function ϕ such that

ω n ϕ = f ω n ψ . Fix µ 1 = MA(φ 1 ), µ 2 = MA(φ 2 ) ∈ MAH(X, ω) and set µ = (µ 1 + µ 2 )/2. Observe that ψ := (φ 1 + φ 2 )/2 ∈ PSH(X, ω) ∩ Hölder(X, ω) satisfies (ω + dd c ψ) n ≥ 1 2 n (µ 1 + µ 2 ) hence µ = f (ω + dd c ψ) n with bounded density 0 ≤ f ≤ 2 n-1 .
It therefore follows from the first part of the corollary that µ also belongs to MAH(X, ω), hence the latter is convex.

We also note that the range of the complex Monge-Ampère operator has the product property.

Corollary 4.5. Let (X 1 , ω 1 ), (X 2 , ω 2 ) be two compact Kähler manifolds of dimension n 1 , n 2 , normalized so that

X 1 ω n 1 1 = X 2 ω n 2 2 = 1. Fix µ 1 , µ 2 two probability measures on X 1 , X 2 . The following are equivalent: i) µ 1 ∈ MAH(X 1 , ω 1 ) and µ 2 ∈ MAH(X 2 , ω 2 ). ii) µ = µ 1 × µ 2 ∈ MAH(X 1 × X 2 , ω), where ω = n 1 + n 2 n 1 -1/(n 1 +n 2 ) [ω 1 + ω 2 ].
Here µ = µ 1 × µ 2 denotes the product (probability) measure on X 1 × X 2 , and we still denote by ω 1 , ω 2 the semi-positive forms on X 1 × X 2 obtained by pulling-back ω 1 , ω 2 on each factor.

Proof. i) ⇒ ii) Assume that µ 1 = (ω 1 + dd c u 1 ) n 1 and µ 2 = (ω 2 + dd c u 2 ) n 2
where u 1 , u 2 are Hölder continuous ω i -psh functions on X 1 , X 2 . Pulling back these forms and functions on X = X 1 × X 2 and observing that (ω i + dd c u i ) 1+n i ≡ 0, one obtains

µ = µ 1 × µ 2 = (ω + dd c u) n 1 +n 2 with u = [u 1 + u 2 ] n 1 + n 2 n 1 1/(n 1 +n 2 )
so that µ ∈ MAH(X, ω).

ii) ⇒ i) Since µ satisfies iii) in Theorem 4.3 we infer that µ 1 , µ 2 satisfy the same property. Using Theorem 4.3 again thus yields µ 1 ∈ MAH(X 1 , ω 1 ), µ 2 ∈ MAH(X 2 , ω 2 ).

Measures with symmetries

Generalizing Skoda's celebrated result [Sk], Dinh-Nguyen-Sibony have observed recently [DNS] that if µ is the Monge-Ampère measure of a Hölder-continuous quasi-psh function, then exp(-ε PSH(X, ω)) ⊂ L 1 (µ) for ε > 0 small enough. We show here that the converse holds when µ moreover has radial or toric singularities. The general case is open, see however [Hi] for some partial results. 5.1. Exponential integrability, Lelong numbers and symmetries-basic results. Note for later use that if exp(-ε PSH(X, ω)) ⊂ L 1 (µ), then for all x ∈ X and 0 < r << 1, µ(B(x, r)) ≤ Cr ε and µ(K) ≤ CT (K) ε for all Borel sets K, where T denote the Alexander-Taylor capacity (see [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]). This implies that for all A > 1, there exists

C A > 0 such that µ(K) ≤ C A Cap ω (K) A , for all Borel set K,
where Cap ω denotes the Monge-Ampère capacity. In other words, µ is very well dominated by the Monge-Ampère capacity (it satisfies the condition H(∞)).

Let u be a psh function defined near the origin in C n , with a radial singularity, i.e. such that u(z) = u( z ) for all z. It is then standard that u can be written as u(z) = χ • L(z) where L(z) = log z and χ is a convex increasing function defined in a neighborhood of -∞. Note that -the function u is bounded if and only χ(-∞) > -∞; -the Lelong number ν(u, 0) is non zero if and only if χ(t) ∼ ν(u, 0)t near -∞, which is the maximal growth that χ can have at -∞. Alternatively, ν(u, 0) = 0 if and only if χ ′ (-∞) = 0. The following elementary computation is left to the reader: Lemma 5.1. Let u = χ • L be a radial plurisubharmonic function defined in a ball B ∋ 0. Assume that χ is C 2 smooth. Then u belongs to the domain of definition of the Monge-Ampère operator and

(dd c u) n = ν(u, 0) n δ 0 + c n (χ ′ • L) n-1 χ ′′ • L dV z 2n .
Here δ 0 denotes the Dirac mass at the origin. Note in particular that when ν(u, 0) = 0 then the Monge-Ampère measure (dd c u) n is absolutely continuous with respect to Lebesgue measure.

A similar formula can be derived for Monge-Ampère measures with toric symmetries, but we will not use it: we will handle the toric case by using Theorem 4.3, whereas the radial case will be treated directly, using Lemma 5.1 (the direct method yields better exponents). 5.2. The radial case. We obtain here a complete description of those radial measures which belong to MAH(X, ω).

Proposition 5.2. Let µ be a probability measure on X which is smooth but at finitely many points where it has a radial singularity. The following are equivalent: i) exp(-ε PSH(X, ω)) ⊂ L 1 (µ) for all 0 < ε < ε 0 ; ii) z -a -ε ∈ L 1 (µ) for all 0 < ε < ε 0 and a ∈ X; iii) µ(B(a, r)) ≤ Cr ε for all 0 < ε < ε 0 and a ∈ X; iv) µ = (ω + dd c φ) n , where φ ∈ PSH(X, ω) is Hölder continuous with exponent α arbitrarily close to ε 0 /n.

Proof. The implication i) ⇒ ii) is obvious. The equivalence ii) ⇔ iii) is immediate. The implication iv) ⇒ iii) is classical (successive integration by parts against a cut-off function with support in a corona of radii jr, (j + 1)r) and holds for general (non radial) measures. The implication iv) ⇒ i) was obtained in [DNS], also for general measures. In the sequel we thus focus on the remaining implication ii) ⇒ iv).

Let a ∈ X be one of the finitely many singular points. We fix a local chart near a such that a = 0 is the origin and locally µ = (dd c u) n with u = χ • L, L(z) = log z and χ convex increasing. Observe that u is bounded and χ ′ (-∞) = 0. By Theorem 4.3 it is enough to check that u is Hölder continuous at point a, which is equivalent to showing that 0 ≤ χ(t) -χ(-∞) ≤ C exp(δt) as t → -∞, for some positive constants C, δ > 0.

By assumption there exists ε > 0 such that z -ε ∈ L 1 (µ). We infer from Lemma 5.1 that

0 1 z ε dµ = c 0 (χ ′ • L) n-1 χ ′′ • L dV (z) z 2n+ε = c ′ -∞ (χ ′ (t)) n-1 χ ′′ (t)e -εt dt < +∞.
We now integrate by parts, in finite time, to obtain

ε -A (χ ′ ) n exp(-εt)dt = (χ ′ ) n (-A) exp(+εA) + O(1).
We claim that -∞ (χ ′ ) n exp(-εt)dt is finite. So is the limsup on the right hand side, hence χ ′ (t) ≤ C exp(εt/n), which yields

χ(t) -χ(-∞) ≤ C ′ exp(εt/n).
Therefore u(z) -u(0) ≤ C ′ z ε/n , i.e. u is Hölder continuous.

It remains to prove the claim. If -∞ (χ ′ ) n exp(-εt)dt = +∞, then (χ ′ ) n (-A) exp(+εA) → +∞ as A → +∞. Set

h(t) = (χ ′ ) n (t) exp(-εt) and H(x) = 0 x h(t)dt.
Thus H(x) → +∞ as x → -∞ and εH(x) = H ′ (x) + O(1).

We let the reader check that this implies H(x) = λ exp(-εx) + O(1) for some constant λ ≥ 0. Now χ ′ (t) → 0 as t → -∞ so h(t) = o(exp(-εt)) and H(t) = o(exp(-εt)). This forces λ = 0, hence H(t) = O(1).

5.3. The toric case. We now consider the case of probability measures µ which are smooth but at finitely many points where they have "toric singularities" the origin 0 ∈ C n is called a toric singularity for the measure µ = (dd c u) n , u psh and bounded, if u is (S 1 ) n -invariant, i.e.

u(z 1 , . . . , z n ) = u(|z 1 |, . . . , |z n |), ∀z = (z 1 , . . . , z n ) ∈ ∆ n .
We will call these measures toric measures for short.

Proposition 5.3. Let µ be a toric measure in the unit polydisk ∆ n ⊂ C n . Assume that for all 0 < r < 1 2 and j = 1, ..., n, µ(∆ × ... × ∆ j (r) × .. × ∆) ≤ Cr α , where C, α > 0.

Then

∆n(t) [ǔ δ (z) -u(z)]dµ ≤ C(t)δ β , for all 0 < t < 1 and u ∈ PSH ∩L ∞ (∆ n ) with 0 ≤ u ≤ 1. Proof. Set T u(z) = 1 (2π) n [0,
T ǔδ (z) = 1 (2π) n [0,2π] n ǔδ (e iθ 1 |z 1 |, ..., e iθn |z n |)dθ 1 ...dθ n = 1 (2π) n [0,2π] n 1 c n δ n B δ u(e iθ 1 |z 1 | + w 1 , ..., e iθn |z n | + w n ) dV (w)dθ 1 ...dθ n = 1 (2π) n [0,2π] n 1 c n δ n B δ (|z 1 |,...,|zn|) u(e iθ 1 ξ 1 , ..., e iθn ξ n ) dV (ξ)dθ 1 ...dθ n = 1 c n δ n B δ (|z 1 |,...,|zn|) T u(ξ) dV (ξ) ≤ T u(|z 1 | + δ, ..., |z n | + δ). Since µ is toric, ∆n(1/2) [ǔ δ (z) -u(z)]dµ = ∆n(1/2) [T ǔδ (z) -T u(z)]dµ, thus ∆n(1/2) [ǔ δ (z) -u(z)]dµ ≤ C n j=1 ∆n(1/2) log 1 + δ |z j | dµ ≤ nC 1/2 0 δt α t 2 + δt dt ≤ C ′ δ β
with β = α/(α + 2), as can be checked by cutting the integral into two pieces

δ γ 0 + 1/2
δ γ , where γ = 1/(α + 2).

Corollary 5.4. A toric probability measure µ belongs to MAH(X, ω) if and only if exp(-ε PSH(X, ω)) ⊂ L 1 (µ) for some ε > 0.

Proof. If µ belongs to MAH(X, ω), then exp(-ε PSH(X, ω)) ⊂ L 1 (µ) for some ε > 0, as follows from [DNS]. Assume now that exp(-ε PSH(X, ω)) ⊂ L 1 (µ) for some ε > 0. As explained earlier, this implies that µ is very well dominated by the Monge-Ampère capacity, in particular µ ∈ H(∞). The previous proposition shows that item (iii) of Theorem 4.3 applies, hence µ ∈ MAH(X, ω).

In view of the above proofs, one may wonder whether all probability measures satisfying condition H(∞) belong to MAH(X, ω). The following example shows this is far from being the case.

Example 5.5. We assume here (X, ω) = (P 1 , ω F S ) is the Riemann sphere equipped with the Fubini-Study form. We let φ ∈ PSH(X, ω) be a function that is smooth in P 1 but at one point which we choose as the origin 0 in some affine chart C and so that

φ(z) = exp --log |z| - 1 2 log[1 + |z| 2 ]
near the origin. The reader will easily check, following the arguments in Example 4.2 in [BGZ], that µ = ω + dd c φ is very well dominated by the logarithmic capacity , in particular satisfies H(∞), although φ is not Hölder continuous.

The case of big cohomology classes

Proof of Theorem D. In order to deal with the general case of big cohomology classes, we use again the regularization techniques of the first author, coupled now with Proposition 2.9. We let ϕ be a θ-psh function solution of (θ + dd c ϕ) n = µ, where the density f ≥ 0 of µ with respect to a smooth volume form belongs to L p for some p > 1. The solution is unique up to an additive constant, it is θ-psh with minimal singularities (see [BEGZ]). We can thus assume, without loss of generality, that -C 0 + V θ ≤ ϕ ≤ V θ . We let ϕ → ρ δ ϕ again denote the regularization operator defined in (2.1). As in the Kähler case t → ρ t ϕ + Kt 2 is increasing for 0 ≤ t ≤ δ 0 and some constant K.

We consider the Kiselman-Legendre transform,

ψ c,δ (z) := inf t∈]0,δ] ρ t ϕ(z) + Kt 2 -c log(t/δ) ,
where 0 ≤ δ ≤ δ 0 and c > 0 will be carefully chosen below. Observe that

ϕ ≤ ψ c,δ ≤ ρ δ ϕ + Kδ 2 .
The fundamental curvature estimate is now

θ + dd c ψ c,δ ≥ -(Ac + Kδ 2 )ω
for some constant A > 0. Since the coholomogy class Θ = {θ} is big, there exists a θ-psh function ψ 0 on X such that θ + dd c ψ 0 ≥ ε 0 ω, for some small constant ε 0 > 0. Subtracting a large constant, we can always assume that ψ 0 ≤ 0 hence ψ 0 ≤ V θ . It follows that the function

ϕ c,δ := Ac + Kδ 2 ε 0 ψ 0 + 1 - Ac + Kδ 2 ε 0 ψ c,δ
is θ-plurisubharmonic on X. Fix 0 < δ < δ 0 and choose c > 0 such that Ac + Kδ 2 = ε 0 δ α , where α := 2γ, and observe that c = ε 0 A -1 δ α -KA -1 δ 2 = O(δ α ). In the sequel we set

ϕ δ := ϕ c,δ .
Since ψ 0 ≤ V θ ≤ ϕ + C 0 , we see from the definition that on the ample locus,

ϕ δ -ϕ = δ α (ψ 0 -ϕ) + (1 -δ α )(ψ c,δ -ϕ) ≤ C 0 δ α + (1 -δ α )(ρ δ ϕ -ϕ + Kδ 2 ). Furthermore, since ϕ ≤ V θ ≤ 0, we get ̺ δ ϕ ≤ 0, thus ψ c,δ ≤ Kδ 2 ≤ C 0 δ α if δ ≤ δ 0 small enough, and so ϕ δ ≤ C 0 δ α . This implies ψ := ϕ δ -C 0 δ α ≤ V θ . By Proposition 2.9, it follows that sup X (ϕ δ -ϕ) ≤ B 0 max(ϕ δ -ϕ -C 0 δ α , 0) γ L 1 (X) + C 0 δ α ≤ B 0 ρ δ ϕ + Kδ 2 -ϕ γ L 1 (X)
+ C 0 δ α for some constant B 0 > 0 which depends only on γ and the uniform norm of ϕ -V θ .

Applying Lemma 2.3, the last estimate yields sup

X (ϕ δ -ϕ) ≤ C 1 δ α ,
where C 1 := B 0 C ω + K γ + C 0 and C ω is the constant in Lemma 2.3. This inequality ϕ δ ≤ ϕ + C 1 δ α yields a uniform lower bound on the parameter t = t(z) which realizes the infimum in the definition of ϕ δ (z) for a fixed z ∈ Ω. Namely the last inequality gives

ϕ δ (z) -ϕ(z) = δ α (ψ 0 (z)-ϕ(z)) + (1-δ α ) (ρ t ϕ(z)+Kt 2 -ϕ(z)-c log(t/δ)) ≤ C 1 δ α .
Since V θ -ϕ ≥ 0 and ρ t ϕ(z) + Kt 2 -ϕ(z) ≥ 0, it follows that c(1 -δ α ) log[t(z)/δ] ≥ δ α (ψ 0 (z) -V θ (z) -C 1 ).

Since c = ε 0 A -1 δ α -KA -1 δ 2 , the choice δ ≤ δ 1 := min{δ 0 , (ε 0 /2K) 1/(2-α) } yields c ≥ 1 2 ε 0 A -1 δ α and therefore t(z) ≥ δκ(z), where (6.1) κ(z) := exp (C 2 (ψ 0 (z) -V θ (z) -C 1 ) , (6.2)

C 2 := 2A ε 0 (1 -δ α 0 )
.

We are now in position to conclude. Fix z ∈ Amp(Θ). Since t(z) ≥ κ(z)δ and t → ρ t ϕ + Kt 2 is increasing, we get ρ κ(z)δ ϕ(z) -ϕ(z) ≤ ρ t(z) ϕ(z) + Kt(z) 2 -ϕ(z) = ψ c,δ (z) -ϕ(z) = 1 1 -δ α (ϕ δ (z) -δ α ψ 0 (z)), and by the above and the assumption ϕ ≤ V θ ≤ 0 we find ϕ δ -δ α ψ 0 ≤ ϕ + C 1 δ α -δ α ψ 0 ≤ C 1 δ α + δ α (V θ -ψ 0 ), ρ κ(z)δ ϕ(z) -ϕ(z) ≤ (1 -δ α 0 ) -1 )δ α (C 1 + V θ (z) -ψ 0 (z)). Replacing δ by κ(z) -1 δ and using (6.1), we obtain for δ ≤ δ 0 κ(z), ρ δ ϕ(z) -ϕ(z) ≤ (1 -δ α 0 ) -1 δ α (C 1 + V θ (z) -ψ 0 (z)) • exp (αC 2 (C 1 + V θ (z) -ψ 0 (z))) ≤ C 3 exp (2αC 2 (C 1 + V θ (z) -ψ 0 (z))) , (6.3) where (6.4)

C 3 := (αC 2 ) -1 (1 -δ α 0 ) -1 . This finishes the proof of Theorem D, since ψ 0 (z) -V θ (z) is locally bounded from below on Amp(Θ) as well as κ(z) given by (6.1).

Appendix

We briefly explain below how bounds on the curvature may be used to control the differential of the exponential mapping. This is essentially a variation on the theme of Jacobi vector fields.

Estimates for the differential of the exponential. For accurate computations with the exponential we need to control its differential in terms of the curvature. To this end we determine the Jacobi equations which calculate the variation of geodesics.

Let namely u → u + v be a small perturbation of the geodesic t → u(t) with initial velocity ζ. Its linearization satisfies In particular the formula holds at ζ := u ′ (0). Thus if the curvature is bounded by the constant R 2 0 (the square being taken for the ease of notation), then |(D v(t) D v(t) )| ≤ 2R 2 0 |ζ| 2 |v|. This is a vector analogue of the scalar equation y ′′ = 2py. By Gronwall's lemma the solution to the corresponding Cauchy problem with data v(0) = v 0 , Dv(0) = v 1 is estimated by

|v(t)| ≤ |v 0 | cosh( √ 2R 0 |ζ|t) + |v 1 | √ 2R 0 |ζ| sinh( √ 2R 0 |ζ|t).
Let us denote by τ z,ζ (t) : T Z X → T exp z (tζ) X the parallel translation along the geodesic. Let also ṽ(t) := τ z,ζ (t) -1 v(t) ∈ T z X. Then ṽ satisfies the analogous equation with curvature transported back to T z X. Thus (6.8)

|ṽ(t) -v 0 -v 1 t| ≤ |v 0 | cosh( √ 2R 0 |ζ|t) + |v 1 | √ 2R 0 |ζ| sinh( √ 2R 0 |ζ|t) -|v 0 | -|v 1 |t.
The differential of the ordinary exponential mapping evaluated at (h, η) ∈ T (T X) (z,ξ) ≃ T z X ⊗ T z X is precisely v(1) for the solution of the Cauchy problem v(0) = h, Dv(0) = η. Thus (6.8) gives us the bound

|τ z,ζ (1) -1 d exp z (ζ)(h, η) -(h + η)| ≤ h cosh( √ 2R 0 |ζ|) + η √ 2R 0 |ζ| sinh( √ 2R 0 |ζ|) -h -η.
If |ζ| is small (|ζ| ≤ ε 2R 0 , say), then elementary Taylor expansion gives us the bound

|τ z,ζ (1) -1 d exp z (ζ)(h, η) -(h + η)| ≤ (1 + O(ε))(c 1 ε 2 |h| + c 2 ε 2 |η|).
Thus there exists some uniform ε 0 such that in the balls |ζ| ≤ ε √ 2R 0

for any ε ≤ ε 0 the differential is a diffeomorphism and is even O(ε 2 ) close to the identity. Remark 6.1. Similar estimates can be obtained in the Hermitian case either, geodesics being defined by the Chern connection rather than the Levi-Civita connection. One then has to assume additionally a uniform bound on |∂ω| ω and |D(∂ω)| ω to accommodate the presence of torsion. However, replacing exp by exph as was done in [D4] and [BD] would be a challenge, because we would then need an "effective" version of E. Borel's theorem to show that exph can be chosen to satisfy the same estimates as exp, and this is certainly non trivial.

  2π] n u(e iθ 1 |z 1 |, ..., e iθn |z n |)dθ 1 ...dθ n . Note that T u(z) = T u(|z 1 |, ..., |z n |) is increasing and logarithmically convex. This implies that T u(|z 1 | + δ 1 , ..., |z n | + δ n ) -T u(|z 1 |, ..., |z n |)| ≤ C n j=1 log 1 + δ j |z j | , for all z ∈ ∆ n (1/2). It follows from Fubini theorem that

  Moreover if D denotes the Levi-Civita connection with respect to ω then along the geodesic u(t) O(|u(t)|)ζ. Let us now put ζ = v. Then the Jacobi equation takes the intrinsic form

Acknowledgements. This project was initiated during the BIRS conference "Complex Monge-Ampère equation" held in 2009. The authors would like to acknowledge the perfect working conditions provided by the organizers. Part of the research was done while the second and fifth authors were visiting the Erwin Schrödinger institute in Vienna in 2009. They would like to express their gratitude for the hospitality. The second author was supported by Polish ministerial grant "Iuventus Plus" and Kuratowski fellowship granted by the Polish Mathematical Society (PTM) and Polish Academy of Science (PAN). The research was done while the fourth author was a postdoctoral research fellow at Centro Internazionale per la Ricerca Matematica, Trento, Italy. He would like to thank the members of the Institute for their kind hospitality. The second and fifth authors were partially supported by NCN grant 2011/01/B/ST1/00879.


José Manuel Marinho

Vincent Nélis

Stefan M Petters

Isabelle Puaut
email: isabelle.puaut@irisa.fr

Preemption Delay Analysis for Floating Non-Preemptive Region Scheduling

In real-time systems, there are two distinct trends for scheduling task sets on unicore systems: non-preemptive and preemptive scheduling. Non-preemptive scheduling is obviously not subject to any preemption delay but its schedulability may be quite poor, whereas fully preemptive scheduling is subject to preemption delay, but benefits from a higher flexibility in the scheduling decisions. The time-delay involved by task preemptions is a major source of pessimism in the analysis of the task Worst-Case Execution Time (WCET) in real-time systems. Preemptive scheduling policies including non-preemptive regions are a hybrid solution between non-preemptive and fully preemptive scheduling paradigms, which enables to conjugate both world's benefits. In this paper, we exploit the connection between the progression of a task in its operations, and the knowledge of the preemption delays as a function of its progression. The pessimism in the preemption delay estimation is then reduced in comparison to state of the art methods, due to the increase in information available in the analysis.

I. INTRODUCTION

Nowadays processors are composed of several subsystems (such as caches, pipelines, transfer lookaside buffers, etc.) which display, at any time-instant, an associated "state". In the context of this work, what we understand by "state of a subsystem" is the history information enclosed in the subsystem, as well as its logical configuration, at a particular time-instant. For example, the state of a cache at a given time-instant can be seen as a snapshot of all the information stored in that cache at that instant. The objective of this state is to accelerate average-case execution times. All these processor subsystems quasi-continuously face state changes at run-time and we are concerned with state changes which affect the temporal behavior of the executing tasks. In particular, it is the case for task preemptions: when a task resumes its execution (after being preempted), for example, the cache(s) will display a state which is different from its state at the time the task got interrupted. If its state has been substantially altered during the time the task was pending, it is likely that it might be needed to reconstruct at least partially its working set after the the task resumes execution.

Reconstructing the subsystems' states is attached to time penalties in real processors, which may increase the execution time of a task by non-negligible amounts of time. In general purpose computing this effect is balanced by the usually huge performance gains by deploying such state carrying sub-systems and hence is in most cases ignored. In embedded real-time systems, where timeliness is an essential property of the system and is hence thoroughly analysed, these penalties need to be carefully evaluated to ensure that all deadlines are met.

In this work we will mainly focus on the cache-related preemption delay (CRPD), because this delay has the most important impact on the variation of the execution time of a preempted task [START_REF] Petters | Scheduling analysis with respect to hardware related preemption delay[END_REF]. Knowledge of preemption delays is crucial for the assessment of the timing behavior of task-sets when scheduled by a given scheduling policy.

Real-time scheduling policies may be broken into three broad categories, with respect to how preemptions are handled: (a) Nonpreemptive scheduling, where task preemptions are not allowed, (b) Fully-preemptive scheduling, where the highest priority active task always gets hold of the processor as soon as it arrives in the system (by interrupting the current executing task if needed), and (c) Limited preemptive scheduling, a hybrid solution between non-and fully-preemptive scheduling. This latter category can be itself divided into two subcategories: fixed non-preemptive region scheduling, where preemption points are hard-coded in the task's code and preemptions are allowed only when the execution of a task reaches one of these preemption points, and floating nonpreemptive region scheduling. In the latter one, whenever a higher priority task is released, the currently running task starts to execute in a non-preemptive region. The length of this non-preemptive region is constant and defined in static time for each task. When the duration of the non-preemptive region elapses, the regular priority relationship between tasks is applied and the task with the highest priority is dispatched onto the processor potentially collating several preemptions into a single point.

On the one hand the floating non-preemptive regions model is more flexible than the fixed one and does not require modifications in the applications. On the other hand it restricts the time-locations at which the preemptions may take place, which makes it more predictable than fully-preemptive scheduling policies. These policies thus provide the system designer with more information about how the system will behave and decrease the pessimism involved in the analysis. It is important to state that the schedulability of these restricted preemption policies dominate over the fully preemptive ones [START_REF] Bertogna | Limited preemption edf scheduling of sporadic task systems[END_REF]. The theory devised onwards assumes the scheduling using floating non-preemptive regions and proposes a new approach to safely but more tightly bound the preemption delay suffered by a task when compared to the state-of-the-art.

II. RELATED WORK AND CONTRIBUTION

CRPD estimation has been a subject of wide study. Several methods have been proposed that provide an off-line estimation based on static analysis, for this inter-task interference value. Even though the work was later refined we will only discuss initial work. Lee et al. presented one of the earliest works on CRPD estimation for instruction-caches [START_REF] Lee | Analysis of cache-related preemption delay in fixed-priority preemptive scheduling[END_REF] where the concept of useful cache blocks was introduced.

Computation of the CRPD in data caches has been proposed by Ramaprasad and Mueller [START_REF] Ramaprasad | Bounding preemption delay within data cache reference patterns for real-time tasks[END_REF]. Since the assumption used in [START_REF] Lee | Analysis of cache-related preemption delay in fixed-priority preemptive scheduling[END_REF], that the value of CRPD throughout a control flow graph's basic block would remain constant, no longer holds for data caches a different approach had to be devised.

Preemption delay estimation is of little value without its integration into the schedulability test of the systems. Since preemption delay is affected by all elements of the task-set several approaches exist to handle this situation. Scheduling analysis by Lee [START_REF] Lee | Analysis of cache-related preemption delay in fixed-priority preemptive scheduling[END_REF] is based on response-time analysis (RTA) by using the k highest values of preemption delay and incorporating that quantity into the response time of the task. Lee uses integer linear programming (ILP) to compute the preemption delay suffered by each task.

Busquets et al. also used RTA [START_REF] Busquets-Mataix | Adding instruction cache effect to schedulability analysis of preemptive real-time systems[END_REF], but considered the maximum effect the preempted task may suffer by multiplying the number of preemptions with the maximum CRPD. While this is more pessimistic than Lee's approach, it removes the complex analysis of intersecting cache sets which for realistically sized programs suffers from heavy state explosion.

Also a less complex algorithm in comparison to Lee's resorting to RTA was presented by Petters and Färber [START_REF] Petters | Scheduling analysis with respect to hardware related preemption delay[END_REF]. Opposed to Busquets' approach Petters uses the knowledge of the maximum damage each preempting task may cause instead of only considering the worst-case preemption delay. The ILP problem is addressed by using an iterative algorithm.

Altmeyer et al. presents a summary of all of the literature so far relative to preemption delay on fully preemptive fixed task priority [START_REF] Altmeyer | Pre-emption cost aware response time analysis for fixed priority pre-emptive systems[END_REF]. The authors also presented an enhancement to the available work by merging the approaches of Petters and Busquets in a safe way and considering jitter and the preemption delay suffered by the shared resource execution. A demand-bound function based procedure has been proposed by Ju et al. [START_REF] Ju | Accounting for cache-related preemption delay in dynamic priority schedulability analysis[END_REF], while the general approach of computing the CRPD is similar to Lee's approach.

All of the presented preemption delay-aware schedulability tests are specific to fully preemptive scheduling and are much more pessimistic than the one presented in this work since they do not consider the evolution of the preemption delay with the program progression of the preempted task. Our approach differs from past work in the sense that it ties the preemption delay with program-execution progression, thus enabling less pessimism in the preemption delay estimation.

Restricting preemption points presents a viable way to address the problem of preemption delay. The mechanism of preemption deferral was first proposed by Burns et al. [START_REF] Burns | Preemptive priority-based scheduling: an appropriate engineering approach[END_REF]. It has a number of advantages as has been pointed out in several works e.g. [START_REF] Wang | Scheduling fixed-priority tasks with preemption threshold[END_REF], [START_REF] Yao | Comparative evaluation of limited preemptive methods[END_REF]. In particular, Gang Yao et al. provide a comparison of all the available methods described so far in literature [START_REF] Yao | Comparative evaluation of limited preemptive methods[END_REF] regarding restricted preemptive scheduling using fixed task priority.

Bertogna and Baruah have devised a method to compute the size of the non-preemptive regions, for earliest deadline first (EDF) scheduling policy, using a demand-bound function based technique [START_REF] Bertogna | Limited preemption edf scheduling of sporadic task systems[END_REF]. In this work the slack in the schedule depending on the length of the interval, assuming synchronous release of all the tasks, is computed. The method fits both the fixed non-preemptive region model and the floating one.

Several methods addressing the same issue in fixed task priority exist [START_REF]Feasibility analysis under fixed priority scheduling with limited preemptions[END_REF], [START_REF] Marinho | Job phasing aware preemption deferral[END_REF]. A fixed priority scheduling method has been devised by Gang Yao et al. [START_REF]Feasibility analysis under fixed priority scheduling with limited preemptions[END_REF], where a maximum bound on the length of fixed non-preemptive regions is provided. In this situation the computed length of the fixed non-preemptive regions are generally larger than in previous work, as the last chunk of a task's execution is not subject to further preemptions. This enables a further reduction on the number of preemptions.

Marinho and Petters presented a method to increase, at run-time, the length of the preemption triggered floating nonpreemptive regions for fixed task priority [START_REF] Marinho | Job phasing aware preemption deferral[END_REF]. This method is taking advantage of off-line knowledge and on-line task release information to increase the length of the non-preemptive regions. This leads generally to a steep decrease on the preemptions suffered. Similar to previous work the preemption delay problem was not addressed in their work. Reducing the number of preemptions helps decreasing the pessimism added to the schedulability test.

The preemption delay estimation problem using fixed nonpreemptive region scheduling was presented by Bertogna et al. [START_REF] Bertogna | Optimal selection of preemption points to minimize preemption overhead[END_REF]. In order to reduce CRPD, the usage of fixed non-preemptive areas of code is proposed. The preemption points are thus reduced to a small number of well defined points. In this way the maximum CRPD is decreased and overall system's response time is enhanced. This work has the limitation that it requires manipulation of the code of tasks and thus is not very amenable to system developers. In particular, it is not straightforward to take into account tasks with complex control flow graphs [START_REF] Bertogna | Optimal selection of preemption points to minimize preemption overhead[END_REF]. Additionally it can not be easily applied in situations where the task-sets are subject to run-time change, since the maximum allowed distance between preemption points is defined by the higher priority workload.

Our work addresses the computation of the preemption delay in systems using preemption triggered floating non-preemptive regions which was previously not covered in the literature.

III. SYSTEM MODEL

The system consists of a task set τ = {τ 1 , . . . , τ n } scheduled to run on a single core processor. Each task τ i may generate a potentially infinite sequence of jobs, which are the entities that contend for the processor usage. For each task, we assume that we have an estimation of its worst-case execution time (WCET) denoted by C i .

There is an inherent priority relation between the jobs which governs the contention for the processor. This contention will be treated in a limited preemption model, which means that preemptions are allowed but are subject to some restrictions. This work supports both fixed task priority [START_REF]Feasibility analysis under fixed priority scheduling with limited preemptions[END_REF] and EDF [START_REF] Bertogna | Limited preemption edf scheduling of sporadic task systems[END_REF] with floating non-preemptive region scheduling policies.

A floating non-preemptive region starts when the highest priority job is executing on the processor and a higher priority job is released. We denote by Q i the length of the non-preemptive regions of task τ i . This means that once a floating non-preemptive region has started, it will last for Q i time units unless the currently running job completes before. Therefore, the preemption points which lead to the worst-case cumulative preemption delay are subject to the constraint of being distanced by at least Q i time units apart. The first preemption can only happen after the task τ i has completed Q i units of execution. In this situation a higher priority release occurred at the same exact moment at which τ i started execution. It is likely that the first preemption will occur after τ i has progressed further than Q i . The Q i value is a characteristic of each task. If the currently running job has not yet finished after the Q i time units elapse then the highest priority job in the ready queue preempts it. The determination of Q i can be performed by following the approaches determined in Bertogna et al. [START_REF] Bertogna | Limited preemption edf scheduling of sporadic task systems[END_REF] or Marinho and Petters [START_REF] Marinho | Job phasing aware preemption deferral[END_REF] and is assumed given within this work.

When a preempted task (say τ i ) resumes its execution, its remaining execution time will eventually increase, in comparison to the situation in which it was not preempted. This effect is due to the loss of working set in the hardware state. Within this work we focus on the largest contributor which is the CRPD. We call this increase in the remaining execution time the preemption delay that the task τ i has to account for. This delay is as high as the amount of information, useful for the remaining execution of τ i , evicted during the preemption. The preemption delay varies during the execution of the job. Let illustrate that with a simple example. Suppose that a task starts its execution by loading from the memory an important amount of data. Then the task processes all these data in a short period of time and finally, it performs a long-time computation using only a small subset of the data. In this case, the maximum preemption delay during the beginning of the task will be high, since in the worst-case scenario all the loaded data might be evicted during a preemption, hence forcing the task to reload them at the return from preemption. Then, once the data have been processed, the maximum preemption delay falls drastically, since a preemption during the long-time computation can only force the task to reload the few data that it needs when resuming its execution.

Each task is then characterised by a task-specific preemption delay pattern. As jobs execute their preemption delay varies with their progression through their execution. We model this varying cost of every task τ i using a preemption delay function f i . As such, it displays, for any time-instant t where the function is defined, an upper bound on the preemption cost that the task would incur if it was preempted at time t. This function is only valid for the first preemption since it does not take into account the preemption delay that has to be paid in the post-preemption execution.

IV. COUPLING PREEMPTION DELAY COST WITH EXECUTION

POINTS This section focuses on determining the initial preemption delay function f i of each task τ i . For that purpose, one first needs to obtain for every task represented by its control-flow graph (see Figure 1 ]. This method can be extended in a straightforward manner to programs with natural loops. The algorithm presented above can be applied to every loop individually, starting with the innermost. A loop can then be considered as a single node with known earliest and latest start offsets when analyzing the outer loop of the whole program. Similarly, tasks containing function calls can be analyzed provided that their call graph is acyclic by first analyzing the leaves in the call graph.

(s min x + e min x ) (2)
s max b def = max x∈pred(b) (s max x + e max x ) (3)
Knowing the possible execution interval [s min b , e max b ] of every basic block b, the set of basic blocks that might execute at a given time instant t, noted BB(t) is known. For each basic block b in this set, state of the art methods like the one described in [START_REF] Lee | Analysis of cache-related preemption delay in fixed-priority preemptive scheduling[END_REF] is used to compute the maximum CRPD when preempting the task in basic block b, noted CRPD b . More formally, function f i can be defined as follows:

f i (t) def = max ∀b∈BB(t) (CRPD b )

V. DETERMINATION OF PREEMPTION DELAY UPPER-BOUNDS

As stated in Section III, a task will always execute nonpreemptively for at least Q i time units before a preemption occurs, unless it completes before the end of the non-preemptive region.

A naive thought to upper-bound the cumulative preemption delay over a task's execution (say τ i ) might be to select from f i the maximum number of points p k (each distanced from every other by at least Q i time units) such that the sum ∀p k f i (p k ) is maximum. However, the simple example depicted in Figure 2 shows that this solution is not correct. As we can see, on the top plot where f i is depicted, there are at most two points that may be selected (since no three points could be distanced by at least Q i time units in time). The bottom plot presents an hypothetical run of task τ i , where the run-time preemption delay cost is presented. At run-time, since time is spent paying preemption delay after each preemption, more points can be selected (see the bottom plot), hence providing a higher cumulative preemption delay. A pessimistic, but correct, solution to upper-bound the execution time C i of a task τ i while taking into account all the possible preemption delays that τ i might undergo during its execution, is simply to multiply the maximum number of preemptions that can occur during τ i 's execution (i.e.,

p 1 f (t) PD hypothetical run p 1 p 2 p 2 p 3 Q 2Q 3Q 4Q

Ci

Qi

, this is discussed in more detail in [START_REF] Marinho | Job phasing aware preemption deferral[END_REF]) by the maximum delay of one preemption (i.e., max t∈[0,Ci] f i (t)). Given the increase in the WCET due to this cumulative overhead, the maximum number of preemptions that can occur eventually increases as well. Therefore, this computation has to be performed iteratively, in the style of the well-know task response-time computation, i.e., C

(0) i = C i and C (k) i = C i + C (k-1) i Q i × max t∈[0,Ci] f i (t) (4)
The pessimism of this computation comes directly from the fact that it considers a constant cost for every possible preemption, and this constant cost is assumed to be the maximum possible cost. That is, this approach is not sensitive to the preemption cost pattern of the task. As it was claimed in the abstract, using this additional information (the tasks preemption cost pattern) enables us to derive a much more accurate upper-bound. This second technique is described in Algorithm 1, and a detailed explanation is provided below. Description of Algorithm 1. Initially we will explain the intuition of the approach on Figure 3 before presenting the actual algorithm. In Figure 3 the gray curve is the f i function. Suppose that prog is the current progression in the task execution. Considering the next preemption point, the approach is looking for the lower bound on the progression which will be achieved within the next Q i time units in any preemption scenario. For this, function f i is investigated from the current prog to prog +Q i . On the ordinate also at length Q i a line D(x, t) is drawn to prog +Q i . The point p ∩ where f first crosses D(x, t) limits the range of values which need to be considered. A preemption past this value would lead to a situation where this point would again be considered in a subsequent iteration, since then prog would not pass this point in the current iteration. Within the interval, delay max is determined. That means in an interval Q i under any preemption scenario at least Q idelay max progress in program execution will be achieved. It is a conservative bound as a later preemption means that also the non preemptible region will only start then. This point prog +Q idelay max will serve as new starting point.

Returning to the Algorithm 1: Lines 1-4 initialise the variables. The variable prog memorizes the current progression in the task's operations while total delay records the cumulative preemption delay accounted for up to the current progression. As the task τ i executes, it accounts for progressing in its execution (and the variable prog is increased) and for the preemption delay (which updates the variable total delay). The algorithm is iterative, and at each iteration the variables delay max and p next (lines 3 and 4) are the preemption delay taking place only in the current iteration and the next progression point in τ i 's execution at which the next iteration will start, respectively. Lines 1-4 can be seen as the first iteration of the algorithm. delay max is set to 0 and p next to

Q i , prog + Q i p max prog D(x, t) p ∩ p next delay max Q i -delay max delay max Q i Fig. 3. Algorithm iteration sketch
because no preemption can occur during the first Q i time units of τ i 's execution.

The algorithm starts iterating at line 5, and it iterates as long as the next computed progression point p next does not fall beyond τ i 's execution boundary. Line 6 shifts the current progression point of τ i to the computed value p next . Then, lines 12 and 13 compute the next progression point p next and the maximum delay that τ i could suffer while progressing in its operations from its current progression point to p next . Finally, line 14 adds this maximum delay to the current cumulative delay accounted so far.

In the following Theorem 1, we prove that the value returned by Algorithm 1 is an upper-bound on the cumulative preemption delay that the given task τ i might suffer during its execution. This implies that the WCET of τ i (while taking into account all the possible preemption delays that τ i might suffer during its execution) is given by

C i def = C i + total delay (5)
where total delay is the value returned by Algorithm 1.

Theorem 1. Algorithm 1 returns an upper-bound on the preemption delay that a given task τ i can suffer during the execution of any of its jobs.

Proof: Algorithm 1 computes the maximum cumulative preemption delay iteratively, by progressing step by step through the execution of the task τ i . Hereafter, we use the notation prog (k) to denote the progression through τ i 's execution at the beginning of the k th iteration of the algorithm. Similarly, total delay (k) will be used to denote the cumulative preemption delay that τ i has suffered until it reached a progression of prog (k) . In proof, we show that at each iteration k > 0, total delay (k) actually provides an upper-bound on the cumulative preemption delay that τ i might suffer before reaching a progression of prog (k) in its execution. The proof is made by induction.

Basic step. Algorithm 1 first considers that τ i progresses by Q i time units in its execution without suffering any preemption delay (since it cannot get preempted during these first Q i time units). We consider this first step as the first iteration of the algorithm. That is, straightforwardly, total delay (1) = 0 is an upper (and even exact) bound on the cumulative preemption delay that τ i may suffer before reaching a progression of Q i time units in its execution.

Induction step. We assume (by induction) that total delay (k) , k > 1, is an upper-bound on the cumulative preemption delay that τ i might suffer before reaching a progression of prog (k) time units in its execution.

During the k th iteration, Algorithm 1 computes prog (k+1) and total delay (k+1) as follows:

prog (k+1) = prog (k) +Q i -delay max (6)
total delay (k+1) = total delay (k) + delay max [START_REF] Ju | Accounting for cache-related preemption delay in dynamic priority schedulability analysis[END_REF] where

delay max = f i (p max ) (8)
p max = arg max px∈[prog (k) ,p∩] {f i (p x )} (9)
p ∩ = min{p x } such that (10) ) , prog (k) +Q i ] and f i (p x ) = -p x + prog (k) +Q i Equations 6 and 7 can be interpreted as follows. During the k th iteration, Algorithm 1 assumes that τ i executes for Q i time units during which τ i progresses by Q idelay max units of time in its execution and suffers from a delay of delay max ; The algorithm assumes that τ i gets preempted when its progression reaches p max given by Equation 9. Below we show that choosing any other preemption point p other = p max would ultimately 1 result in a cumulative preemption delay lower than the one returned by Algorithm 1, thus showing that the value returned by Algorithm 1 is an upper-bound. Two cases may arise: p other > p next or p other ≤ p next .

p x ∈ [prog (k
Case 1: p other > p next . This means that τ i progresses in its execution until it reaches p next without being preempted, i.e., from a progression of prog (k) , τ i reaches a progression of p next by being executed only for (p nextprog (k) ) time units, and with an unchanged cumulative preemption delay of total delay (k) . On the other hand, in the execution scenario built by Algorithm 1, τ i 's execution reaches a progression of prog (k+1) = p next by being executed for Q i time units, and with a cumulative preemption delay of total delay (k+1) = total delay (k) + delay max ≥ total delay (k) . In other words, Algorithm 1 manages to progress slower in τ i 's execution while suffering from a greater preemption delay. Furthermore, p other is still a candidate preemption point for a further iteration of Algorithm 1.

Case 2. p other ≤ p next . After executing τ i for Q i time units, we have that 1) the delay of the preemption that occurs when τ i 's progression reaches p other has been totally accounted for (since p other < p next ≤ p ∩ ). 2) the progression of τ i in this scenario becomes

prog other = prog (k) +Q i -f i (p other ) ≥ prog (k) +Q i -f i (p max ) ≥ prog (k+1) (11)
3) the cumulative preemption delay becomes total delay other = total delay (k) +f i (p other )

≤ total delay (k) +f i (p max ) ≤ total delay (k+1) (12)
Thus, after executing τ i for Q i time units Algorithm 1 progressed less in the execution of τ i (Inequality 11) while suffering from a higher preemption delay (Inequality 12). As a consequence of Cases 1 and 2, it holds at each iteration of Algorithm 1 that choosing to preempt the task when it reaches a progression of p max ultimately leads to an upper-bound on its cumulative preemption delay.

1 when τ i 's execution will be completed

VI. EVALUATION AND DISCUSSION

Three synthetic f i functions have been created in order to compare the performance of the proposed preemption delay estimation with the state-of-the-art method using the procedure described by Equation 4. The three functions used are two bell shaped functions, the first one with σ 2 = 300 and µ = 2000 and a vertical offset of 10 units and the second one with a bigger variance, σ 2 = 3000, the same average and no offset. Finally a function with two local maxima separated in time is used. All functions have maximum value of 10 units, and have C = 4000. We vary the Q i value so having a fixed C still paints a generic picture of how the methods behave.

The synthetic functions used represent distinct generic memory usage patterns from tasks. The sole purpose of the set of functions provided is to validate the method proposed in this paper. Having generic patterns, rather than function f i extracted from a set of real benchmarks (which would present more complex patterns), enables for a more clear evaluation of Algorithm 1 performance. These functions are portrayed in Figure 4.

The proposed algorithm is shown empirically, in Figure 5, to provide a considerably less pessimistic upper-bound on the preemption delay value for a task, specially for smaller values of Q i . Since the state of the art method purely relies on Q i , C i and the maximum preemption delay of f i then its preemption delay estimate is strictly the same for all the benchmark functions, since they all have the same C and maximum value. The preemption delay axis in Figure 5 is in logarithmic scale so that the differences between the state of the art and the proposed algorithm are more easily observed across the entire Q i spectrum.

There are fluctuations in the results which are analysis artifacts and imply that the analysis is pessimistic. In some cases increasing the Q i results in bigger preemption delay. This is caused as the analysis checks for the preemption delay in the window of prog and t A , but conservatively considers the actual preemption to occur at prog. An actual preemption at p max in physical terms would initiate a new window of length Q i to start only at that p max instead of prog , thus the method, as is proven in Theorem 1, provides an upper bound for the preemption delay in any conceivable real task execution scenario.

VII. CONCLUSION AND FUTURE WORK

In this work we have proposed a new algorithm to compute an upper-bound on the preemption delay suffered by a task which executes in a system scheduled with preemption triggered floating non-preemptive regions. This algorithm has been shown to dominate over the state-of-the-art method. The method is easy to implement with small overhead and builds up on existing staticanalysis methods.

As of future work we intend to tighten our result by (i) discarding less information during the computation of function f i (t) and (ii) reducing the number of preemptions (i.e., the number of iterations) considered in Algorithm 1 -it is indeed impossible for a task to get preempted every Q i time units as assumed by Algorithm 1 unless the periods of the other tasks enable such a preemption scenario.

  .a), the interval of time [s min b , e max b ] during which every basic block b might execute, considering the execution of τ i in isolation. Computing execution intervals on loop-free code requires to know for every basic block b its earliest and latest start offsets s min b and s max b. This can be done by a breadth-first traversal of the CFG, applying to every traversed basic block b the following formulas:

Fig. 1 .

 1 Fig. 1. Example of CFG for loop-free code. The CFG is composed of several basic blocks (0..10) connected by directed edges that represent jumps in the code. Each basic block is a set of sequential instructions delimited by a jump. In the left part, intervals [c min b , c max b ] represent the minimum and maximum execution times of basic block b. In the right part, intervals [s min b , s max b ] represent the earliest and latest start time of every basic block b.

Fig. 2 .

 2 Fig. 2. Comparison Between Function f i and the Run-time Preemption Delay Development

  Fig. 4. Synthetic Benchmark Functions

Algorithm 1 :

 1 Upper-Bound the Preemption DelayInput : f i (): preemption delay function of task τ i Q i : length of the non-preemptive region Output: total delay: cumulative preemption delay suffered by τ i1 prog ← 0 ; 2 total delay ← 0; 3 delay max ← 0 ; 4 pnext ← Q i ;/ * While the next progression is not beyond C i * /

	5 while pnext < C i do
		/ * Update time, progression and delay	* /
	6	prog ← pnext ; / * Compute the next progression step and the next delay to account for * /
	7	p∩ ← min{px} such that

8

px ∈ [prog

(k)

, prog

(k)

+Q i ] 9 and f i (px) = -px + prog (k) +Q i } ; 10 if p∩ = null then p∩ ← prog +Q i ; 11 pmax ← arg max px∈[prog,p∩] {f i (px)}; 12 delay max ← f i (pmax); 13 pnext ← prog +Q idelay max ; 14 total delay ← total delay + delay max ; 15 return total delay ;