Souan Abja
email: soufian.abja@im.uj.edu.pl

Guillaume Olive

Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation

Keywords: degenerate elliptic equations, local regularity, Hessian equations AMS Subject Classications. 35J70, 32W20

In this paper, we establish a local regularity result for W 2,p loc solutions to complex de- generate nonlinear elliptic equations F (D 2 C u) = f when they are comparable to the Monge-Ampère equation. Notably, we apply our result to the so-called k-Monge-Ampère equation.

Introduction and main results

In this work, we are interested in the local regularity theory for nonlinear complex degenerate elliptic equations of the form

F (D 2 C u) = f, (1)
where D 2 C u denotes the complex Hessian of u.

Such equations have been studied extensively in the literature, going back to the celebrated work [START_REF] Caarelli | The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian[END_REF] on the Dirichlet problem in the real case and its counterpart [START_REF] Li | On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian[END_REF] for the complex setting. We recall that an important feature of degenerate equations is that, unlike uniformly elliptic equations, the C 2,α -regularity may fail. Perhaps one of the most important equation of the form (1) is the complex Monge-Ampère equation:

det(D 2 C u) = f > 0.
(2)

The regularity of the solution of this equation is studied in the literature by many authors and by dierent tools. It was rst proved in pioneering work [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF] that the solution u to the Dirichlet problem in a ball B belongs to C 1,1 (B) ∩ C 0 (B) provided that the right-hand side f ∈ C 2 (B) and the boundary data is C 2 (∂B). Another important regularity result concerning the Dirichlet problem associated with (2) in a strictly pseudoconvex domain, established in [START_REF] Caarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF], is the smooth regularity up to the boundary of its solution when the right-hand side, the boundary data and the domain are all smooth (the strict positivity of the right-hand side is also essential).

The local regularity of (2) (no boundary data) was also studied. A sharp result was obtained in [START_REF] Bªocki | A local regularity of the complex Monge-Ampère equation[END_REF] by developing some methods in [START_REF] Trudinger | Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations[END_REF]: solutions u ∈ W 2,p loc of (2) with f ∈ C ∞ are necessarily C ∞ whenever p > n(n -1), and no smaller exponent p can be expected in general.

The local regularity of other equations of the form (1) were also studied. Notably, in [START_REF] Dinew | A priori estimates for complex Hessian equations[END_REF], a counterpart of [START_REF] Bªocki | A local regularity of the complex Monge-Ampère equation[END_REF] was proved for the so-called complex k-Hessian equation under the assumption that u belongs to W 2,p loc with p > n(k -1).

The goal of the present paper is to extend the approach of [START_REF] Bªocki | A local regularity of the complex Monge-Ampère equation[END_REF] and [START_REF] Dinew | A priori estimates for complex Hessian equations[END_REF] to more general nonlinear complex degenerate elliptic equations. We will introduce simple conditions on the nonlinearity F to obtain general local regularity results, thus considerably broadening the eld of application of this method.

In particular, we shall see that our results apply to the so-called complex k-Monge-Ampère equation or MA k -equation, k ∈ {1, . . . , n}, that has recently received much attention ([HL09, Sad18, Din20b, Din20a]):

1≤i1<...<i k ≤n λ i1 (D 2 C u) + . . . + λ i k (D 2 C u) = f,
where λ 1 (D 2 C u), . . . , λ n (D 2 C u) denote the eigenvalues of D 2 C u. For k = 1 we have the Monge-Ampère equation det(D 2 C u) = f and for k = n this is the Poisson equation ∆u = f .

Interior estimates for this equation in the real setting have been studied recently in [START_REF] Dinew | Interior estimates for p-plurisubharmonic functions[END_REF].

In the complex setting this operator was discussed in [START_REF] Sadullaev | Further developments of the pluripotential theory (survey), Algebra, complex analysis, and pluripotential theory[END_REF] and, in the special case k = n -1, also in [START_REF] Tosatti | The Monge-Ampère equation for (n -1)plurisubharmonic functions on a compact Kähler manifold[END_REF]. It has been shown in [START_REF]m-subharmonic and m-plurisubharmonic functions on two problems of Sadullaev[END_REF] that this operator does not satisfy an integral comparison principle, which makes the associated potential theory much harder to be developed.

Finally, let us also mention that the Dirichlet problem associated to this operator was studied in [START_REF] Zhou | Representation and regularity for the dirichlet problem for real and complex degenerate hessian equations[END_REF] using a probabilistic approach.

The outline of the paper is as follows. After recalling some basic notations, we precise in • Let H n be the set of n × n Hermitian matrices. The (i, j)-th entry of a matrix A ∈ H n will be denoted by a i j . We recall that H n is a vector space over the eld R of dimension n 2 . The inner product on H n is the standard Frobenius inner product:

A, B H n = Trace (AB) ∈ R, A, B ∈ H n .
The n × n identity matrix will be denoted by Id.

We will use the classical cones

C n = {A ∈ H n | A > 0} , C 1 = {A ∈ H n | Trace (A) > 0} . • The Fréchet derivative at A ∈ C of a function F ∈ C 1 (C, R),
DF (A)B = 1≤i,j≤n ∂F ∂a i j (A)b i j , B ∈ H n .
• In this work we use the following convention for ellipticity. We say that a function F :

C -→ R is: (degenerate) elliptic in C if A ≤ B =⇒ F (A) ≤ F (B),
for every A, B ∈ C, where we recall that A ≤ B (resp. A < B) means that the Hermitian matrix B -A is non-negative-denite (resp. positive-denite).

uniformly elliptic in C if there exist 0 < m ≤ M such that

A ≤ B =⇒ m B -A H n ≤ F (B) -F (A) ≤ M B -A H n , for every A, B ∈ C.
For instance with these conventions the equation ∆u = f is (uniformly) elliptic.

• The complex Hessian is denoted by D 2 C u = u i j 1≤i,j≤n , where we use the standard notations u j and uj to denote, respectively, ∂u ∂zj = 1 2 ∂u ∂xj -i ∂u ∂yj and ∂u

∂ zj = 1 2 ∂u ∂xj + i ∂u ∂yj .
The Laplacian is denoted by ∆u = n j=1 u j j = Trace (D 2 C u) ∈ R. Note that it is the complex Laplacian, so it is one fourth the real Laplacian.

• Finally, we shall denote by C(n), C(n, Ω), etc. a positive number that may change from line to line but that depends only on the quantities indicated between the brackets.

Main results

Throughout this work we will always make the following assumptions: (e) Concavity:

F 1/d is concave in C.
(f ) Comparison with determinant: there exists C > 0 such that

F (P ) 1/d ≥ C(det(P )) 1/n , ∀P ∈ C n . (3)
Some examples will be presented in Section 3 below.

The key assumption here is the assumption (f ), that will help us to compare our general nonlinear equation F (D 2 C u) = f with the complex Monge-Ampère equation and thus use the recent uniform estimates obtained in [START_REF] Abja | Uniform estimates for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation[END_REF].

The assumption (a) is quite standard (see e.g. [START_REF] Caarelli | The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian[END_REF]). Note that this excludes the case of the whole space C = H n . It is important to point out that we cannot only consider homogeneous functions of degree 1 by normalizing F into F 1/d since this normalization is not in general regular up to the boundary (e.g.

F (A) = det(A) 1/n in C = C n ).
Finally, by concavity and homogeneity, we can check that F satises (3) for every A ∈ C if, and only if,

F (A + P ) 1/d ≥ F (A) 1/d + C (det P ) 1/n ,
for every A ∈ C and P ∈ C n . In particular, F is necessarily elliptic in C (but, in general, not uniformly elliptic). We refer for instance to [ADO20, Remark 1.3] for more details.

The rst result of the present paper is the following:

Theorem 1.1. Let C and F satisfy the above standing assumptions (a), (b), (c), (d), (e) and

(f ). Let f > 0 in Ω with f 1/d ∈ C 1,1 (Ω). Let u ∈ W 2,p loc (Ω) with (D 2 C u)(z) ∈ C for a.e. z ∈ Ω satisfy (almost everywhere) F (D 2 C u) = f in Ω, and assume that p > n max {d -1, 1} . (4)
Then, for every non empty open subset ω ⊂⊂ Ω, we have ∆u ∈ L ∞ (ω) with

sup ω ∆u ≤ R, for some R > 0 depending only on n, p, d, dist (ω, ∂Ω), D 2 C u L p (Ωε) , ∆(f 1/d ) L ∞ (Ωε) and inf Ωε f if d > 1 (resp. sup Ωε f if d < 1),
where Ω ε = {z ∈ Ω | dist (z, ∂Ω) > ε} and ε > 0 is some number depending only on dist (ω, ∂Ω).

Theorem 1.1 generalizes [BD11, Theorem] and [DK14, Theorem 4.1], where the particular cases of the complex Monge-Ampère equation and the complex k-Hessian equation were considered.

An important consequence of Theorem 1.1 and of the classical theory for uniformly elliptic real equations from [START_REF] Caarelli | Fully nonlinear elliptic equations[END_REF] will be the following local regularity result: Corollary 1.2. Under the framework of Theorem 1.1, assume in addition that F satises the following property:

(g) Sets of uniform ellipticity: there exists R 0 > 0 such that, for every R > R 0 , F is uniformly elliptic in C R , where

C R = A ∈ C Trace (A) < R, 1 R < F (A) . (5)
Then, we have the property

F ∈ C ∞ (C), f ∈ C ∞ (Ω) =⇒ u ∈ C ∞ (Ω). (6)
Remark 1.3. Note that C R is convex because F is concave. However, it is never a cone and it does not contain C n . Note as well that lim R→+∞ C R = C.

We also point out that we do not need to assume that C R is compact. For instance, C R is not bounded for the simplest example F (A) = Trace (A) on A ∈ C 1 (n ≥ 2). However, this is a rather pathological case and we will see in Proposition 3.1 below that C R is always bounded whenever ∂C ∩ ∂C 1 = {0} .

Remark 1.4. Whether the condition (4) yields the smallest value of p to induce the C ∞regularity (6) may depend on the equation itself. This condition is for instance sharp in the case of the Monge-Ampère equation, see [START_REF] Bªocki | A local regularity of the complex Monge-Ampère equation[END_REF]p. 412], but it is still an open problem for other equations such as the complex k-Hessian equation, the k-Monge-Ampère equation (see Section 4.3 below), etc.

The rest of the article is organized as follows. In the next section we prove our main results.

In Section 3 we present some examples of Hessian equations that are covered by our framework.

Finally, in Section 4, we discuss a bit more in detail the case of the k-Monge-Ampère equation.

2

Proofs of the main results

The proof of Theorem 1.1 is inspired from the proof of [Tru80, Theorem 1] and the ideas of [START_REF] Bªocki | A local regularity of the complex Monge-Ampère equation[END_REF] (see also [START_REF] Dinew | A priori estimates for complex Hessian equations[END_REF]Section 4]).

We introduce the normalization

G(A) = (F (A)) 1/d , A ∈ C. Clearly, G is elliptic in C. The equation F (D 2 C u) = f then becomes G(D 2 C u) = f 1/d .
Throughout Section 2 the notation L u is exclusively saved for the linearization of

G about A = (D 2 C u)(z): (L u w)(z) = 1≤i,j≤n ∂G ∂a i j ((D 2 C u)(z))w i j (z). (7)
Note that L u is (degenerate) elliptic and that its coecients are Lebesgue measurable (as composition of a continuous function with a Lebesgue measurable function).

Preliminaries

Geometric conguration. Let ω ⊂⊂ Ω and set δ = dist (ω, ∂Ω) > 0. For z 0 ∈ ω, let

ω = ω ∩ B δ/4 (z 0 ),
where B R (z 0 ) ⊂ C n denotes the open ball of center z 0 and radius R > 0. Up to the transformation z → (z -z 0 )/(δ/2), we can always assume that B δ/2 (z 0 ) is the open unit ball, that will simply be denoted by B 1 in the sequel. Thus, B δ (z 0 ) becomes B 2 (0), that will simply be denoted by B 2 . Therefore, from now on, we are in the following geometric conguration:

ω ⊂⊂ B 1 ⊂⊂ B 2 ⊂⊂ Ω, dist (ω, ∂B 1 ) ≥ 1 4 dist (ω, ∂Ω). (8)
Approximation of the Laplacian. In order not to consume too much regularity, we will need a suitable approximation of the Laplacian. This is done as in [BD11, p. 415] and [DK14, Lemma 4.2] (see also [BT76, Proposition 6.3]).

Lemma 2.1. For every

0 < ε < 1, let T ε : L p (B 2 ) -→ L p (B 1 ) (1 ≤ p ≤ ∞)
be the linear operator dened by

(T ε u)(z) = n + 1 ε 2 (u ε (z) -u(z)) , with u ε (z) = 1 µ n (B ε (z)) Bε(z) u dµ n ,
where here and in what follows µ n denotes the Lebesgue measure in C n . Then, we have the following properties:

(i) Positivity: If u is subharmonic in B 2 , then (T ε u)(z) ≥ 0 for every z ∈ B 1 . (ii) Regularity: If u ∈ C 0 (B 2 ), then T ε u ∈ C 0 (B 1 ). If u ∈ W 2,p (B 2 ) and p < ∞, then u ε ∈ W 2,p (B 1 ) with D 2 C u ε = (D 2 C u) ε (with a slight but obvious abuse of notation). (iii) Convergence: If u ∈ W 2,p (B 2 ) and p < ∞, then T ε u ∆u weakly in L p (B 1 ) as ε → 0. (iv) Uniform bound: If u ∈ W 2,p (B 2 )
then, for every 0 < ε < 1, we have

T ε u L p (B1) ≤ C(n, p) ∆u L p (B2) .
The A uniform estimate for strong supersolutions. Recently, it was obtained in [ADO20, Theorem 1.4] some new type of Alexandrov-Bakelman-Pucci estimate for various nonlinear complex degenerate equations. Thanks to the assumptions of the present article, it applies in particular to our linearized operator L u dened in (7) for which it results in the following:

Theorem 2.2. Let r, q > n. For every g ∈ L q (B 1 ) and every

w ∈ W 2,r loc (B 1 ) ∩ C 0 (B 1 ) such that L u (-w) ≤ g in B 1 , w ≤ 0 on ∂B 1 ,
we have sup

B1

w ≤ C(n, r, q) g + L q (B1) ,

where g + = max(g, 0) denotes the positive part of g.

Estimate of the operator norm. We will need the following estimate:

Lemma 2.3. There exists C > 0 such that, for every A ∈ C,

DG(A) L(H n ,R) ≤ C(G(A)) 1-d A d-1 H n . (9)
Proof. By continuity of DF on the compact A ∈ C A H n = 1 and the fact that DF is homogeneous of degree d -1, we have, for some C > 0,

DF (A) L(H n ,R) ≤ C A d-1 H n , ∀A ∈ C.
The desired estimate (9) then follows from the computation

DG(A)B = 1 d (F (A)) (1/d)-1 DF (A)B, A ∈ C, B ∈ H n .
We point out that it is only to show (9) that we use that F is C 1 up to the boundary.

Despite not needed, we mention that the inequality (9) also shows that the coecients of the operator L u belong to L ∞ if d = 1, and to L p/(d-1) if d > 1, which are then in L n by the assumption (4).

Jensen's inequality in convex subsets of

1 µ n (Ω) Ω H(z) dµ n ∈ C,
and

G 1 µ n (Ω) Ω H(z) dµ n ≥ 1 µ n (Ω) Ω G (H(z)) dµ n ,
(the right-hand side possibly being equal to -∞).

Proof of Theorem 1.1

We are now ready to prove our rst main result.

1) Strategy of the proof.

Let 0 < ε < 1. For α, β ∈ [2, +∞) (to be determined later) let us introduce an auxiliary function w given by w

(z) = η(z)(T ε u(z)) α , z ∈ B 1 , where η(z) = dist (z, ∂B 1 ) 2β = 1 -|z| 2 β
.

Note that:

• w is unambiguously dened since T ε u ≥ 0 by item (i) of Lemma 2.1 and subharmonicity of u, which follows from the assumptions that D 2 C u ∈ C ⊂ C 1 (see e.g. [Kli91, Theorem 2.5.8]).

• w ∈ C 0 (B 1 ) by item (ii) of Lemma 2.1 since u ∈ C 0 (Ω) by the Sobolev embedding W 2,p loc (Ω) ⊂ C 0 (Ω) as p > n by assumption (4). • w ∈ W 2,p loc (B 1 ) thanks to the regularity of u and Sobolev embeddings (using that p ≥ n).

The goal will be to bound sup B1 w from above by a positive number depending only on the quantities indicated in the statement of Theorem 1.1. This will also provide an upper bound for sup ω T ε u since η(z) ≥ (dist (ω, ∂Ω)/4) 2β > 0 for z ∈ ω (recall (8)). This will in turn imply that ∆u ∈ L ∞ (ω) with the same bound as for T ε u. Indeed, denoting by C a bound for T ε u we will have shown that T ε u ∈ S, where

S = g ∈ L ∞ (ω), g L ∞ (ω) ≤ C .
We can check that this set is closed in L p (ω) (using for instance the partial converse of the Lebesgue dominated convergence theorem [Rud87, Theorem 3.12]). Since it is also clearly convex, it is then weakly closed in L p (ω) (see e.g. [Rud91, Theorem 3.12]). As T ε u ∆u weakly in L p (ω) (item (iii) of Lemma 2.1) and T ε u ∈ S, it follows that ∆u ∈ S as well, which is exactly what we want. Now, in order to bound sup B1 w we are going to use the condition (4) on p to show that (L u (-w)) + ∈ L q (B 1 ) for some q > n, with estimate

(L u (-w)) + L q (B1) ≤ C sup B1 w 1-2/β + 1 , (10)
for some C > 0 depending only on the quantities indicated in the statement of Theorem 1.1. The conclusion will then follow from the uniform estimate of Theorem 2.2 with g = (L u (-w)) + . Note that w = 0 on ∂B 1 (in fact, this is the (only) obstruction to simply take η = 1).

2) Computation of L u (-w).

We have

w i = η i (T ε u) α + ηα(T ε u) α-1 (T ε u) i ,
and

w i j = η i j (T ε u) α + η i α(T ε u) α-1 (T ε u)j + ηjα(T ε u) α-1 (T ε u) i + ηα(α -1)(T ε u) α-2 (T ε u)j(T ε u) i + ηα(T ε u) α-1 (T ε u) i j .
Consequently,

L u (-w) = 1≤i,j≤n - ∂G ∂a i j (D 2 C u)w i j = I + II,
where (using also the identity v i = vī for any real-valued function v)

I = 1≤i,j≤n - ∂G ∂a i j (D 2 C u)η i j (T ε u) α + 2α(T ε u) α-1 Re   1≤i,j≤n - ∂G ∂a i j (D 2 C u)η i (T ε u)j   + 1≤i,j≤n - ∂G ∂a i j (D 2 C u)ηα(α -1)(T ε u) α-2 (T ε u)j(T ε u) i ,
and

II = 1≤i,j≤n - ∂G ∂a i j (D 2 C u)ηα(T ε u) α-1 (T ε u) i j .
3) Estimate of the term I.

To estimate this term we will make use of the ellipticity and of the estimate (9). Let us introduce the following sesquilinear form on C n :

ϕ(a, b) = 1≤i,j≤n - ∂G ∂a i j (D 2 C u)a i b j , a, b ∈ C n .
This form is nonpositive by ellipticity of G. Therefore, we can use the basic inequality 2Re

ϕ(a, b) ≤ -ϕ(a, a) -ϕ(b, b) with a i = 1 √ t η i and b j = √ t(T ε u) j (t > 0 to be chosen below), to obtain 2Re   1≤i,j≤n - ∂G ∂a i j (D 2 C u)η i (T ε u)j   ≤ - 1 t 1≤i,j≤n - ∂G ∂a i j (D 2 C u)η i ηj -t 1≤i,j≤n - ∂G ∂a i j (D 2 C u)(T ε u) i (T ε u)j.
Taking now t = (α -1)η/T ε u > 0, this removes the third term in I to give

I ≤ (T ε u) α 1≤i,j≤n - ∂G ∂a i j (D 2 C u) η i j - α (α -1)η η i ηj . Let B = (b i j ) 1≤i,j≤n , b i j = -η i j + α (α -1)η η i ηj.
Direct computations show that

η i = -β zi η 1-1/β , η i j = -βδ ij η 1-1/β + β(β -1)z i z j η 1-2/β ,
where δ ij denotes the Kronecker delta. Since η ≤ 1, we obtain

B H n ≤ C(α, β, n)η 1-2/β = C(α, β, n)w 1-2/β 1 (T ε u) α(1-2/β) .
Consequently, using (9), we have

I ≤ C(α, β, n)C (f, d) D 2 C u d-1 H n (T ε u) 2α/β w 1-2/β , where C (f, d) = (inf B2 f ) (1/d)-1 if d ≥ 1 and C (f, d) = (sup B2 f ) (1/d)-1 if d < 1.
Note that we cannot use the better estimate (3) since B ∈ C n .

4) Estimate of the term II.

For this term, we will use the concavity of F to show that

II ≤ ηα T ε (f 1/d ) (T ε u) α-1 .
To prove such an estimate, we would like to use the concavity inequality

G(B) ≤ G(A) + DG(A)(B -A), A, B ∈ C, (11) with A = (D 2 C u)(z), B = (D 2 C u ε )(z). But rst observe that B = (D 2 C u) ε (z) (see Lemma 2.1
) and apply Jensen's inequality (Lemma 2.4) to see that indeed B ∈ C for a.e. z ∈ B 1 with, in addition,

G((D 2 C u) ε ) ≥ G(D 2 C u) ε .
We can now use (11) to obtain the desired estimate:

1≤i,j≤n

- ∂G ∂a i j (D 2 C u)(T ε u) i j ≤ n + 1 ε 2 G(D 2 C u) -G(D 2 C u) ε = -T ε (f 1/d ).
5) Choices of α and β.

In summary, we have obtain the inequality

(L u (-w)) + ≤ C(α, β, n)C (f, d) D 2 C u d-1 H n (T ε u) 2α/β w 1-2/β + ηα T ε (f 1/d ) (T ε u) α-1 .
Thanks to the condition (4) on p, we can nd q > n such that p/q ≥ 1 and (p/q)-(d-1) > 0.

Consequently, taking

α = 1 + p q , β = 2α p q -(d -1)
,

we have α, β ≥ 2 and (L u (-w)) + ∈ L q (B 1 ), with the following estimate (by Hölder's inequality and (iv) of Lemma 2.1):

(L u (-w)) + L q (B1) ≤ C(α, β, n)C (f, d)C(n, p) D 2 C u d-1 L p (B1) ∆u 2α/β L p (B2) sup B1 w 1-2/β + ηαC(n, p) ∆(f 1/d ) L ∞ (B2) ∆u p/q L p (B2) .
This yields the desired estimate (10).

C ∞ regularity

In this section, we show how to deduce Corollary 1.2 from Theorem 1.1 and the classical theory for uniformly elliptic real equations from [START_REF] Caarelli | Fully nonlinear elliptic equations[END_REF].

We follow the presentations of [START_REF] Wang | On the C 2,α -regularity of the complex Monge-Ampère equation[END_REF][START_REF] Tosatti | α estimates for nonlinear elliptic equations in complex and almost complex geometry[END_REF]. Let S 2n be the space of 2n × 2n real symmetric matrices. The inner product on S 2n is the standard Frobenius inner product.

In this section, a function u of n complex variables (z 1 , . . . , z n ) will also be a considered as a function of 2n real variables (x 1 , . . . , x n , y 1 , . . . , x n ), where z j = x j + iy j . The real Hessian of u is then denoted by

D 2 R u =     u xixj u xiyj u yixj u yiyj     ∈ S 2n .
The open ball in R 2n of center 0 radius r > 0 will be denoted by B r .

Corollary 1.2 is essentially a consequence of the following fundamental general result and Schauder's estimates:

Theorem 2.5. Let F : S 2n -→ R be concave and uniformly elliptic, and let f ∈ C 0,β (B 1 )

(0 < β ≤ 1). If u ∈ W 2,p loc (B 1 ) is a strong solution to F (D 2 R u) = f in B 1
, and p ≥ 2n, then there exist α ∈ (0, 1) and ε ∈ (0, 1) such that

u ∈ C 2,α (B ε ).
We recall that strong solution simply means that the equation holds almost everywhere. This theorem is a consequence of [CC95, Theorem 8.1] (see e.g. [Wan12, Section 2] for details).

In this reference the result deals in fact with C-viscosity solutions, but we recall that a strong solution u ∈ W 2,p loc with p ≥ 2n is a C-viscosity solution, see [Lio83, Section III].

In order to apply this result we need two preliminary observations: rstly, we have to associate to our equation F (D 2 C u) = f an equation for the real Hessian F (D 2 R u) = f and, secondly, this F has to be dened over the whole space S 2n . This is done in a standard way (see e.g.

[Bªo99, Wan12, TWWY15], etc.):

• We identify n × n Hermitian matrices with the subspace of S 2n given by matrices invariant by the canonical complex structure:

ι(H n ) = A ∈ S 2n AJ -JA = 0 , J = 0 -Id Id 0 ,
where the map ι : H n -→ S 2n is given by

ι(A + iB) = A -B B A .
Let us also introduce the projection π : S 2n -→ ι(H n ) given by π(S) = S + J Tr SJ 2 .

The complex and real Hessians are then related by the identity

ι(2D 2 C u) = π(D 2 R u). As a result, if u solves F (D 2 C u) = f , then it solves as well F (D 2 R u) = f,
where F : E R -→ R is given by

F (A) = F 1 2 ι -1 (π(A))
,

and E R ⊂ S 2n is dened by E R = π -1 (ι (2C R )) .
We recall that C R is dened in (5) and that it is not a cone, so that the factor 2 cannot be removed.

It is clear that E R is an open convex set and that F is concave in E R . Since F is uniformly elliptic in C R by assumption of Corollary 1.2, it is also clear that F is uniformly elliptic in E R .

• Let us now extend F to S 2n . This is done by considering F : S 2n -→ R dened by

F (A) = inf L(A) L : S 2n -→ R ane linear, mId ≤ DL ≤ M Id, L ≥ F in E R ,
where 0 < m ≤ M denote the ellipticity constants of F . We can check that F is concave and uniformly elliptic in S 2n , and that we have F = F in E R (see e.g. [TWWY15, Lemma

4.1]).

Corollary 1.2 is now an easy consequence of the previous results.

Proof of Corollary 1.2. After translation and dilation of the coordinates if necessary, it is sucient to show that u ∈ C ∞ (B ε ) for some ε ∈ (0, 1) when B 1 ⊂⊂ B 2 ⊂⊂ Ω.

From Theorem 1.1 we know that, for some R 1 > 0, ∆u < R 1 in B 2 .

On the other hand, by our assumptions on f , there exists R 2 > 0 such that

F (D 2 C u) = f > 1 R 2 .
Consequently, for any R > max {R 0 , R 1 , R 2 } we have

D 2 C u ∈ C R .
It now follows from the previous discussion that u is a strong solution to the concave and uniformly elliptic real equation

F (D 2 R u) = f in B 1 . Besides, u ∈ W 2,p (B 1 ) for every 1 ≤ p < ∞ by Calderón-Zygmund estimates since u, ∆u ∈ L ∞ (B 2 ).
We can then apply Theorem 2.5 and obtain that u ∈ C 2,α (B ε ) for some α ∈ (0, 1) and ε ∈ (0, 1).

As a result, u

∈ C 2,α (B ε ) solves F (D 2 R u) = f in B ε . Since F ∈ C ∞ (C) =⇒ F ∈ C ∞ (E R ),
it follows that u ∈ C ∞ (B ε ) from the classical Schauder's estimates, see e.g. [CC95, Proposition 9.1] (the proof in this reference is carried out for E R = S 2n but all the arguments go through after noticing that tD 2 u(x

+ he k ) + (1 -t)D 2 u(x) t ∈ [0, 1], x ∈ H, 0 ≤ h ≤ dist (H, ∂Ω)/2 is a compact set of S 2n which is included in E R ).

Application to Hessian equations

Let us now present some examples covered by our framework. We emphasize that our results

Theorem 1.1 and Corollary 1.2 do not require that our equations are Hessian, but all the examples of application that we present here will be Hessian equations.

Let us rst recall some notations.

• For A ∈ H n , its eigenvalues (which are real) will always be sorted as follows:

λ 1 (A) ≤ λ 2 (A) ≤ • • • ≤ λ n (A).
We then introduce λ : H n -→ R n dened by λ(A) = (λ 1 (A), . . . , λ n (A)).

• A function F : C -→ R is said to be a Hessian operator if there exist a set Γ ⊂ R n and a function F : Γ -→ R such that

C = λ -1 (Γ), F (A) = F (λ(A)), ∀A ∈ C.
The notation Γ will always be saved for sets of R n whereas the notation C will always be saved for sets of H n .

• For k ∈ {1, . . . , n}, we introduce the classical cones

Γ k = {λ ∈ R n | σ (λ 1 , . . . , λ n ) > 0, ∀ ∈ {1, . . . , k}} ,
where σ k is the k-th elementary symmetric polynomial:

σ k (λ) = (i1,...,i k )∈E k n λ i1 • • • λ i k ,
where, here and in the rest of this article, we use the notation

E k n = (i 1 , . . . , i k ) ∈ {1, . . . , n} k i 1 < . . . < i k .
We recall that card

E k n = C k n = n!/(k!(n -k)!).
• We recall that a set S ⊂ R n is symmetric if (λ τ (1) , . . . , λ τ (n) ) ∈ S for every (λ 1 , . . . , λ n ) ∈ S and every permutation τ . A function F : S -→ R is symmetric if F (λ τ (1) , . . . , λ τ (n) ) = F (λ 1 , . . . , λ n ) for every permutation τ .

Let us now provide practical conditions on Γ and F to guarantee that C = λ -1 (Γ) and F = F • λ satisfy the assumptions of our main results.

Proposition 3.1. Let F ∈ C ∞ (R n , R) be a symmetric polynomial, homogeneous of degree d ∈ N * , satisfying, for some C > 0, F (λ) 1/d ≥ C n i=1 λ i 1/n , ∀λ ∈ Γ n , (12)
and let Γ ⊂ R n be a nonempty open symmetric convex cone such that

Γ n ⊂ Γ ⊂ Γ 1 , ∂Γ ∩ ∂Γ 1 = {0} , F > 0 in Γ, F = 0 on ∂Γ, F 1/d is concave in Γ.
Then, C = λ -1 (Γ) satises (a) and F = F • λ satises (b), (c), (d), (e), (f ) and (g).

We emphasize that the map λ has few nice properties, so this result is far from being trivial. The proof of Proposition 3.1 is postponed to the end of this section for the sake of the presentation.

Remark 3.2. The concavity assumption is satised by a large class of polynomials, namely, the hyperbolic polynomials. We recall that a polynomial F homogeneous of degree d ∈ N * is called hyperbolic with respect to e ∈ R n if F (e) = 0 and if, for every λ ∈ R n , the one-variable polynomial t ∈ C -→ F (λ -te) has only real roots. In this case, F 1/d is concave in the open convex cone (called hyperbolicity cone)

Γ = λ ∈ R n F (λ -te) = 0, ∀t ≤ 0 .
We refer to [CNS85, Section 1] and [START_REF] Gårding | An inequality for hyperbolic polynomials[END_REF] for more details.

Let us now nally present some concrete examples.

• The complex Monge-Ampère equation:

F (λ) = n i=1 λ i , Γ = Γ n .
The degree of homogeneity is obviously d = n and the comparison property ( 12) is trivial. This polynomial is hyperbolic with respect to e = (1, 1, . . . , 1). We can check that the associated hyperbolicity cone is indeed Γ n . It clearly satises all the desired assumptions.

• The complex k-Hessian equation: for k ∈ {1, . . . , n},

F = σ k , Γ = Γ k .
The degree of homogeneity is obviously d = k and the comparison property (12) follows from Maclaurin's inequality: for any k ≥ 2,

Γ k-1 ⊂ Γ k , σ k C k n 1/k ≤ σ k-1 C k-1 n 1/(k-1) in Γ k-1 .
This polynomial is hyperbolic with respect to e = (1, 1, . . . , 1). We can check that the associated hyperbolicity cone is indeed Γ k . It clearly satises all the desired assumptions (k > 1).

• For s ∈ [0, 1), let F (λ) = (1 -s) 2 λ 1 λ 2 + s(λ 1 + λ 2 ) 2 = (λ 1 + sλ 2 )(sλ 1 + λ 2 ), Γ = Γ 2-s = λ ∈ R 2 λ 1 + sλ 2 > 0, sλ 1 + λ 2 > 0 .
The cones Γ 2-s interpolate between Γ 2 and Γ 1 . The degree of homogeneity is obviously d = 2 and the comparison property ( 12) is trivial since F (λ) ≥ (1 -s) 2 λ 1 λ 2 . The cone clearly satises all the desired assumptions ( F is also a hyperbolic polynomial with respect to e = (1, 1) but the concavity of F 1/2 is easily checked by a direct computation here).

• The complex k-Monge-Ampère equation: for k ∈ {1, . . . , n},

F = MA k , Γ = Γ k .
where

MA k (λ) = (i1,...,i k )∈E k n (λ i1 + . . . + λ i k ) , Γ k = λ ∈ R n λ i1 + . . . + λ i k > 0, ∀(i 1 , . . . , i k ) ∈ E k n .
The degree of homogeneity is d = C k n . The comparison property ( 12) is a consequence of the following inequality: for any k ≥ 2,

Γ k-1 ⊂ Γ k , 1 k (MA k ) 1/C k n ≥ 1 k -1 (MA k-1 ) 1/C k-1 n in Γ k-1 . ( 13
)
This property will be detailed in Section 4.2 below. This polynomial is again hyperbolic with respect to e = (1, 1, . . . , 1). We can check that the associated hyperbolicity cone is indeed Γ k . It clearly satises all the desired assumptions (k < n).

We conclude this section with the proof Proposition 3.1.

Proof of Proposition 3.1. We only establish the non obvious properties.

1) Regularity of F . Since F is a symmetric polynomial, by Newton's fundamental theorem of symmetric polynomials we can write

F (λ) = p(σ 1 (λ), . . . , σ n (λ)),
for some polynomial p. Since all the functions A ∈

H n -→ σ k (λ(A)) are C ∞ (H n , R) (recall that they are equal to (-1) k (∂ n-k h/∂t n-k )(0, A) where h ∈ C ∞ (R × H n , R) is the function h(t, A) = det(tId -A)), we deduce that F ∈ C ∞ (H n , R) by composition.
2) Convexity of C and concavity of F 1/d . We point out that the concavity of F 1/d is also shown in [CNS85, Section 3] but we would like to present a dierent proof here. Let

A, B ∈ C and t ∈ [0, 1] be xed. We have to show that

λ(tA + (1 -t)B) ∈ Γ, F (λ(tA + (1 -t)B)) 1/d ≥ tF (λ(A)) 1/d + (1 -t)F (λ(B)) 1/d .
To this end, it is convenient to make use of the theory of majorization, for which we refer to [START_REF] Marshall | Inequalities: theory of majorization and its applications[END_REF]. The starting point is that, for every k ∈ {1, . . . , n}, the function

A ∈ H n -→ λ 1 (A) + . . . + λ k (A) is concave.
From this property, we have that tλ(A)

+ (1 -t)λ(B) majorizes λ(tA + (1 -t)B), that is              k j=1 λ j (tA + (1 -t)B) ≥ k j=1 (tλ j (A) + (1 -t)λ j (B)) , ∀k ∈ {1, . . . , n -1} , n j=1 λ j (tA + (1 -t)B) = Trace (tA + (1 -t)B) = n j=1 (tλ j (A) + (1 -t)λ j (B)) . (14)
It follows that (see e.g. [MOA11, Chapter 1, B.1. Lemma])

λ(tA + (1 -t)B) Tr = T N T N -1 • • • T 1 (tλ(A) + (1 -t)λ(B)) Tr ,
for some matrices T 1 , . . . , T N ∈ R n×n of the form

T i = t i Id + (1 -t i )P i ,
for some t i ∈ [0, 1] and some elementary permutation matrix P i ∈ R n×n (i.e. switching only two components). Since Γ is convex and symmetric, it is clear that such transformations map Γ into itself. As a result, we have λ(tA

+ (1 -t)B) ∈ Γ.
On the other hand, since F 1/d is concave and symmetric, it is Schur-concave (see e.g.

[MOA11, Chapter 3, C.2. Proposition]), meaning that the majorization (14) implies that

F (tλ(A) + (1 -t)λ(B)) 1/d ≤ F (λ(tA + (1 -t)B)) 1/d .
Using again the concavity of F 1/d , we obtain the concavity of F 1/d .

3

) Uniform ellipticity of F in C R . Let R > 0 be xed. Let Γ R = λ ∈ Γ n i=1 λ i < R, 1 R < F (λ) .
We rst show that the assumption ∂Γ ∩ ∂Γ 1 = {0} implies that Γ R is bounded. To this end, it is sucient to prove that there exists ε > 0 such that

Γ ⊂ λ ∈ Γ ε |λ| < n i=1 λ i .
To show this property, we argue by contradiction and assume that there exists a sequence

(λ ε ) ε>0 ⊂ Γ such that ε |λ ε | ≥ n i=1 λ ε i . (15)
Since Γ ⊂ Γ 1 , we have in particular that λ ε = 0 and we can normalize the sequence by considering λε = λ ε / |λ ε |. We have λε ∈ Γ since Γ is a cone. We can extract a subsequence, still denoted by ( λε ) ε>0 , that converges to some λ ∈ Γ. Besides, λ = 0 since λ = 1.

Passing to the limit ε → 0 in (15), we obtain that

0 ≥ n i=1
λi .

Since Γ ⊂ Γ 1 , it follows that λ ∈ ∂Γ ∩ ∂Γ 1 and thus λ = 0 by assumption, a contradiction.

As a result, Γ R is compact and there exist real numbers m ≤ M such that, for every i ∈ {1, . . . , n},

m ≤ ∂ F ∂λ i (λ) ≤ M, ∀λ ∈ Γ R . (16)
Let us now prove that m > 0. To this end, it is sucient to show that

0 < ∂ F ∂λ i (λ), ∀λ ∈ Γ, ∀i ∈ {1, . . . , n} , Γ R ⊂ Γ.
The inclusion Γ R ⊂ Γ immediately follows from the assumption that F = 0 on ∂Γ.

Let us now prove the inequality. We use some ideas of the proof of [CNS85, Corollary, pp.

269-270]. Let i ∈ {1, . . . , n} be xed. Let e i denote the i-th canonical vector of R n . We have e i ∈ Γ n ⊂ Γ. From the concavity and homogeneity of Ĝ = F 1/d , which belongs to

C 1 (Γ) ∩ C 0 (Γ) since F > 0 in Γ, we have 0 ≤ Ĝ(e i ) ≤ Ĝ(λ) + D Ĝ(λ)(e i -λ) = D Ĝ(λ)e i = ∂ Ĝ ∂λ i (λ).
Let us now prove that this inequality is strict. Assume not, and let then λ * ∈ Γ be such that (∂ Ĝ/∂λ i )(λ * ) = 0. Since Γ is open, there exists ε > 0 such that λ * + te i ∈ Γ for every t ∈ [0, ε]. By concavity, we then have

D Ĝ(λ * + te i ) -D Ĝ(λ * ) (te i ) ≤ 0, ∀t ∈ [0, ε]. As a result, ∂ Ĝ ∂λ i (λ * + te i ) = 0, ∀t ∈ [0, ε]. Since F > 0 in Γ, this is equivalent to ∂ F ∂λ i (λ * + te i ) = 0, ∀t ∈ [0, ε].
By analyticity, we deduce that this identity holds for any t ∈ R, in particular for negative ones. Let then

T (λ * ) = inf {t ∈ R | λ * + te i ∈ Γ} . Since Γ ⊂ Γ 1 , we necessarily have T (λ * ) > -∞. Integrating the identity (∂ F /∂λ i )(λ * + te i ) = 0 over [T (λ * ), 0] then yields F (λ * ) = F (λ * + T (λ * )e i ).
This is a contradiction since F > 0 in Γ λ * , whereas F = 0 on ∂Γ λ * + T (λ * )e i .

Therefore, (16) holds with m > 0. To conclude, it remains to show that this implies the uniform ellipticity of

F = F • λ in C R . Let then A, B ∈ C R with A ≤ B.
m B -A H n ≤ F (B) -F (A) ≤ M B -A H n ,
for some 0 < m ≤ M (that do not depend on A, B). We only prove the rst inequality, the other one being proved similarly. Since B -A ≥ 0, the rst inequality is equivalent to

m Trace (B -A) ≤ F (B) -F (A),
for some 0 < m . Since Trace (B -A) = Trace (B) -Trace (A), this is equivalent to the following property for F :

m n i=1 (µ i -λ i ) ≤ F (µ) -F (λ). (17)
Let us then prove this inequality. By denition, we have λ, µ ∈ Γ R . Assume rst that

(µ 1 , . . . , µ n-1 , λ n ) ∈ Γ R , (µ 1 , . . . , µ n-2 , λ n-1 , λ n ) ∈ Γ R , . . . (µ 1 , λ 2 , . . . , λ n ) ∈ Γ R . (18)
Then, taking t(µ 1 , . . . , 16), iterating this process and summing all the obtained inequalities, we eventually obtain (17) with m = m.

µ n-1 , µ n ) + (1 -t)(µ 1 , . . . , µ n-1 , λ n ) ∈ Γ R in (16) and integrating over t ∈ [0, 1], we have m(µ n -λ n ) ≤ F (µ 1 , . . . , µ n-1 , µ n ) -F (µ 1 , . . . , µ n-1 , λ n ). Taking then t(µ 1 , . . . , µ n-1 , λ n )+(1-t)(µ 1 , . . . , µ n-2 , λ n-1 , λ n ) ∈ Γ R in (
The general case can be deduced from the case (18) by an approximation argument (using the compactness of Γ R ).

for λ ∈ Γ k-1 . This is equivalent to show that

(i1,...,i k )∈E k n (λ i1 + . . . + λ i k ) ≥ k k -1 C k n    (j1,...,j k-1 )∈E k-1 I k λ j1 + . . . + λ j k-1    C k n /C k-1 n .
Using (19) and the inequality of arithmetic and geometric means (note that card E k-1

I k = k), we have λ i1 + . . . + λ i k = k k -1 (j1,...,j k-1 )∈E k-1 I k λ j1 + . . . + λ j k-1 k ≥ k k -1    (j1,...,j k-1 )∈E k-1 I k λ j1 + . . . + λ j k-1    1/k .
To conclude, it remains to observe the identity

(i1,...,i k )∈E k n    (j1,...,j k-1 )∈E k-1 I k λ j1 + . . . + λ j k-1    =   (j1,...,j k-1 )∈E k-1 n λ j1 + . . . + λ j k-1   kC k n /C k-1 n ,
which can be established as before by counting the terms λ j1 + . . . + λ j k-1 that appear in the left-hand side (note that kC k n /C k-1 n = n -k + 1).

A lower bound for p

Let us conclude Section 4 by mentioning that, even if the condition p > n(C k n -1) is not known to be sharp, it is however necessary for the conclusion of Theorem 4.1 to be true that p is not too small. More precisely, we will show that it cannot be smaller than

p * = (n -k)C k n .
Note that p * is only equal to n(C k n -1) for k = 1 (and k = n but we have excluded this value).

This follows from the following type of Pogorelov's example, which is built on the one of [START_REF] Bªocki | On the regularity of the complex Monge-Ampère operator[END_REF].

Proposition 4.5. Let 1 ≤ m < n and β > 0. The function

u(z , z ) = 1 + |z | 2 |z | 2β , where (z , z ) ∈ C m × C n-m , satises: (i) u ∈ C ∞ (C n \N ), where N = {z ∈ C n | z = 0}, and (D 2 C u)(z) > 0 for every z ∈ C n \N . (ii) u ∈ W 2,p loc (C n ) (p ≥ 1) if, and only if, β ≥ 1 or β < 1 and p < n -m 1 -β . ( 20
) (iii) u ∈ C 1,α loc (C n ) (α ∈ [0, 1]) if, and only if, α ≤ 2β -1. (iv) For β = 1 -C k m /C k n (C k m = 0 if m < k), we have MA k (λ(D 2 C u)) ∈ C ∞ (C n ) with MA k (λ(D 2 C u)) > 0 in C n . For β = 1 -1/C k
n , the largest p that we can obtain from the condition (20) is when m = k and it gives p < (n -k)C k n . Consequently, we have constructed an example that satises

u ∈ W 2,p loc (C n ), ∀p ∈ 1, (n -k)C k n , D 2 C u > 0 in C n \ {z ∈ C n | z = 0} , MA k (λ(D 2 C u)) ∈ C ∞ (C n ), MA k (λ(D 2 C u)) > 0 in C n ,                        but u ∈ C 1,α loc (C n ), ∀α > 1 - 2 C k n .
Remark 4.6. The same function can be used to obtain lower bounds for various degenerate elliptic equations (Monge-Ampère, k-Hessian equation, etc.), although not always optimal.

Proof of (i).

(D 2 C u)(z) =     δ ij |z | 2β β zi z j |z | 2β-2 β zi z j |z | 2β-2 β 1 + |z | 2 (β -1)z i z j |z | 2β-4 + δ ij |z | 2β-2     , (21)
where δ ij denotes the Kronecker delta and 1 ≤ i, j ≤ m for the left upper block, 1 ≤ i ≤ m and m + 1 ≤ j ≤ n for the right upper block, etc.

Let us now nd all the eigenvalues of (D 2 C u)(z). From this expression we see that:

-m) = m -1. • Similarly, β 1 + |z | 2 |z | 2β-2
is an eigenvalue of multiplicity at least n -m -1. Besides, in case of equality with the rst type of eigenvalue, it is then of multiplicity n -2.

Therefore, in every case we know n -2 eigenvalues and to nd the two remaining ones, we are going to nd their sum S and product P . The computation of S is easy. Indeed, the sum of the eigenvalues is

Trace (D 2 C u) = S + (m -1) |z | 2β + (n -m -1)β 1 + |z | 2 |z | 2β-2 .
On the other hand, from (21), we have

Trace (D 2 C u) = m |z | 2β + β 1 + |z | 2 (n -m + β -1) |z | 2β-2 . Therefore, S = |z | 2β + β 2 1 + |z | 2 |z | 2β-2 .
In order to nd the product P , we will use the relation with the determinant of D 2 C u. As product of the eigenvalues, we have

det(D 2 C u) = P |z | 2β m-1 β 1 + |z | 2 |z | 2β-2 n-m-1 = P β n-m-1 1 + |z | 2 n-m-1 |z | 2β(n-2)-2(n-m-1) . (22)
Let us now compute det(D 2 C u) directly from (21). Doing the line substitutions To compute det(A), we note that it is the sum of a rank one matrix with an invertible matrix, for which we have the formula det(J + v 1 v Tr 2 ) = det(J)(1 + v Tr 2 J -1 v 1 ),

L i ← L i -
for any invertible J and vectors v 1 , v 2 . Using this formula we obtain Comparing with (22), we nally obtain that

P = β 2 |z | 4β-2 .
From the knowledge of the sum S and product P , the two remaining eigenvalues are given by

S + √ S 2 -4P 2 = φ(z) |z | 2β-2 and 2P S + √ S 2 -4P = β 2 φ(z) |z | 2β ,
where φ is the smooth positive function dened by

φ(z) = |z | 2 + β 2 1 + |z | 2 + |z | 2 + β 2 1 + |z | 2 2 -4β 2 |z | 2 2 .
Finally, note that all the eigenvalues are positive outside N .

Proof of (ii). We only investigate the integrability of the second order derivatives since we can check afterwards that the rst order derivatives yield a less restrictive condition on the exponent of integrability p. We recall that the only problem for the integrability of u i j is near N , see the expression (21). From this expression, we also see that the worst term is then

β 1 + |z | 2 (β -1)z i z j |z | 2β-4 ,
(unless β = 1 but in such a case we clearly have u i j ∈ C 0 (C n ) for every i, j). r 2(n-m)-1+2p+(2β-4)p dr.

It is well-known that this integral is nite if, and only if, the power of r is larger than -1, which gives the desired condition (20).

Proof of (iv). Let us now compute MA k (λ(D 2 C u)). We recall that MA k (λ) =

•

  Section 1.2 what kind of nonlinear operators F are considered in this article and we present our main results. Their proofs are the purpose of Section 2. Some examples of Hessian equations covered by such a framework are then given in Section 3. Finally, in Section 4 we detail the case study of the MA k -equation. All along this work, Ω ⊂ C n (n ≥ 1) is a nonempty open bounded connected subset.

  (a) Domain: C ⊂ H n is a nonempty open convex cone such that C n ⊂ C ⊂ C 1 . (b) Regularity: F ∈ C 1 (C, R).(c) Positivity: F > 0 in C.(d) Homogeneity: F is positively homogeneous of degree d > 0 (i.e. F (αA) = α d F (A) for ever α > 0 and A ∈ C).

  of multiplicity at least m -1. Indeed, since the right upper block denes a matrix of rank at most 1, the rank of D 2 C u-|z | 2β Id is thus at most 1+n-m. By the rank-nullity theorem, |z | 2β is an eigenvalue of multiplicity at least n -(1 + n

β

  zi z r |z | -2 L r , for each i ∈ {m + 1, . . . , n}, we obtain det(D 2 C u) a ij = β β -1 + |z | 2 zi z j |z | 2β-4 + βδ ij 1 + |z | 2 |z | 2β-2 . Denoting by A = (a ij ) m+1≤i,j≤n we have det(D 2 C u) = |z | 2β m det(A).

  2 C u) = β n-m+1 1 + |z | 2 n-m-1 |z | 2βn-2(n-m) .

(

  Let us then consider the ball B R × B ε , with R, ε > 0, where B r (resp. B r ) denotes the open ball of C m (resp. C n-m ) with center 0 and radius r > 0. We recall that µ d denotes the Lebesgue measure in C d . We have1 + |z | 2 zi z j |z | 2β-4 p L p (B R ×B ε ) |z i | |z j |) p |z | (2β-4)p dµ n-m .The rst integral in the right-hand side is clearly nite. For the second one, using standard formula (see e.g. [Kli91, p. 23]) we have B ε (|z i | |z j |) p |z | (2β-4)p dµ n-m = C(n -m, ε, p) ε 0

(

  i1,...,i k )∈E k n (λ i1 + . . . + λ i k ) .

From 2 ..

 2 what precedes we know that there are m eigenvalues of the form |z | 2β and n -m eigenvalues of the form |z | 2β-2(by of the form we mean up to the multiplication by a smooth positive function). Now, observe that:• The sum λ i1 + . . . + λ i k is of the form |z | 2β if so is each element of the sum. There are C k m possibilities for this situation to happen (C k m = 0 if m < k).• In all the other cases, the sum λ i1 + . . .+ λ i k is of the form |z | 2β-Since there are C k n = card E k n products overall in MA k , this means that such sums appear C k n -C k m times.In summary, we haveMA k (λ(D 2 C u)) = ψ(z) |z | C k m (2β)+(C k n -C k m )(2β-2) ,for some smooth positive function ψ.Consequently, MA k (λ(D 2 C u)) ∈ C ∞ (C n ) whenthe power of |z | is exactly equal to zero, which gives the desired condition on β, namely β For this value of β we also have MA k (λ(D 2 C u)) = ψ > 0 in C n .

  It can be identied with the matrix

	∂F ∂a i j (A)	1≤i,j≤n	∈ H n through the formula

dened on a non empty open subset C ⊂ H n , will be denoted by DF (A) ∈ L(H n , R).

  We denote the eigenvalues of A (resp. B) by (λ 1 , . . . , λ n ) (resp. (µ 1 , . . . , µ n )). By denition, we have to show that

  It is clear that D 2 C u is smooth o N . Let z ∈ C n \N . A directcomputation shows that we have the block decomposition

Acknowledgements

Both authors were supported by the Polish National Science Centre Grant 2017/26/E/ST1/00955. The authors would like to thank Sªawomir Dinew for numerous discussions and fruitful suggestions.

4

Additional properties for the MA k -equation

In this section, we detail the case of the k-Monge-Ampère equation, which is important for applications.

Let us rst state explicitly the local regularity result that we have obtained for this equation.

We recall once again that, unless k = 1 (the Monge-Ampère equation), it is not known if this condition on p is sharp. However, we will provide in Section 4.3 below an example that shows that a threshold does exist for such a result to be valid.

k-plurisubharmonic functions

The importance in the study of the operator MA k lies in its connection with the notion of k-plurisubharmonic function.

Denition 4.2. A function u : Ω -→ R is said to be k-plurisubharmonic in Ω if it is upper semi-continuous and if it is subharmonic whenever it is restricted to any ane complex plane of dimension k. We denote by k-plurisubharmonic the vector space of such functions.

For instance, 1-plurisubharmonic functions are the plurisubharmonic functions, and n-plurisubharmonic functions are the subharmonic functions.

To make clearer the link between k-plurisubharmonic functions and the operator MA k , we rst introduce some notations.

By choosing the canonical basis of the complex space

D A can be identied with a matrix of size C k n × C k n , that will still be denoted by D A .

Example 4.4. Let A = u i j 1≤i,j≤n .

• For n = 3 and k = 2, we have

• For n = 4 and k = 3, we have

• For n = 4 and k = 2, we have

.

We can check that D A is Hermitian if so is A and that the spectrum of D A is exactly

and

This cone is linked to the notion of k-plurisubharmonic function as follows: when u ∈ C 2 (Ω), we have the characterization

For more information about k-plurisubharmonic functions we refer for instance the reader to [START_REF]Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds[END_REF][START_REF]p-convexity, p-plurisubharmonicity and the Levi problem[END_REF].

4.2

Comparison between MA k and MA k-1

In this section, we show that the crucial comparison property (12) holds for MA k . In fact, we prove the more precise property (13) that allows to compare MA k and MA k-1 . Let k ∈ {2, . . . , n} be xed. Let us rst show the inclusion

Let then λ ∈ Γ k-1 and let us show that λ i1 + . . . + λ i k > 0. For any (i 1 , . . . , i k ) ∈ E k n , we denote by

Then, the claim follows from the identity

To show this identity, we have to count how many λ i1 , λ i2 , . . . are in the sum of the left-hand side. For λ i1 the only possibility is to have j 1 = i 1 . Therefore, there are C k-1 k-2 = k -1 such terms. For λ i2 there are two and only two possibilities, namely j 1 = i 2 or j 2 = i 2 (in this second case, necessarily j 1 = i 1 ). This gives C k-1 k-3 + C k-2 k-3 terms, which is equal to C k-1 k-2 by Pascal's rule. Repeating this reasoning leads to (19).

Let us now show the inequality MA k (λ) ≤ MA k-1 (λ),


A ma grande soeur, ma Caro. Tu as été une seconde maman pour nous et les années ne font que nous rapprocher. Il me tarde de passer encore beaucoup de moments avec toi. Merci de toujours veiller sur moi et de m'avoir soutenue tout au long de ces années. Merci pour ton aide indispensable et ton soutien sans faille dans cette dernière ligne droite. Tes conseils resteront toujours précieux et importants pour moi.

A mon grand petit frère, Loulou. Bien que je te devance uniquement de 5 minutes, pour moi tu es mon petit frère quand même (même si tu me dépasse largement en hauteur !). Merci d'avoir été mon compagnon d'enfance avec ces jeux, ces chamailleries et notre compréhension mutuelle. Amandine, merci d'être là pour ce petit frère. Tu fais ressortir de ce Loulou ses meilleurs côtés. Merci de m'accueillir de temps en temps chez vous, c'est toujours avec plaisir de venir vous voir dans ma première ville étudiante.

A mes grands-parents, Coco et René. Même si vous n'êtes plus là, il y a une partie de vous qui restera toujours avec moi. Merci de m'avoir transmis votre passion pour la science et la nature. Mamie, je pense que je réalise ton rêve d'avant. Tu as été la première au courant de ma réussite d'entrée dans les études médicales et, à chaque avancée je pense à toi. Papi, tu m'as transmis le goût de l'aventure et de la voile que je conserverai toujours. A Annie et Jean-Pierre, merci d'avoir été une présence importante durant mon enfance et de m'avoir suivie tout le long de mon parcours. Malgré les bêtises d'enfants que nous avons pu faire, nous avons passés de belles vacances entre cousins chez vous. J'en garde pleins de souvenirs heureux et amusants.

Aux membres de mon jury,

A monsieur le Professeur Gilles Cambonie, merci de me faire l'honneur de présider le jury de cette thèse. Merci pour le partage de vos connaissances, et de votre enseignement. Merci de vos conseils précieux sur mes choix, de m'accueillir prochainement dans votre service et de la confiance que vous me m'accordez. J'espère en être à la hauteur.

A Monsieur le Professeur Gérald Chanques, merci d'avoir accepté de juger cette thèse. Vos travaux ont inspiré ce travail. Je vous remercie de l'intérêt que vous portez sur ce sujet, qu'est la musicothérapie.

A Monsieur le Professeur Nicolas Sirvent, merci d'avoir accepté de juger cette thèse. Merci pour votre gentillesse, bienveillance et disponibilité durant ces années d'internat. Merci pour votre accueil lors de mon semestre en oncologie-hématologie pédiatrique. Vous avez toujours été à l'écoute de chacun de nous.

A Monsieur le Docteur Christophe Milési, merci de m'avoir proposé et fait confiance pour ce beau sujet qu'est la musicothérapie. Ton engagement auprès des patients et de leur confort est un exemple pour moi. Merci de ton accompagnement indispensable et du temps que tu y as consacré. Merci pour ta gentillesse lors des semaines de travail en service et pendant la rédaction de cette thèse. Ton aide et ton accompagnement m'ont été indispensables pour aboutir à ce travail et ce résultat.

A mes amis, ceux de mon enfance, Alexia, je pense la plus vieille de mes amies. De la primaire, au collège puis lycée on ne s'est pas lâchée. Merci pour ton accueil au Vietnam. Pleins de bons souvenirs et péripéties ont été racontées depuis ! Tu as été la seule à pouvoir me porter en scooter, et ce n'est pas rien ! Charlotte, on s'est suivie plus longtemps sur nos études depuis la primaire. Tu as été d'une aide indispensable pendant cette fameuse P1. Puis s'en est suivi de bons souvenirs de jeunes carabins.

Mathilde, Fabrice et bien sûr la petite Sophia, vous êtes arrivés plus tard, mais comme on dit, mieux vaut tard que jamais ! Merci d'avoir toujours été présents tout au long de ces années. Vous êtes ma bouffée d'air frais.

Honorine et Quentin, merci d'avoir aussi été là. On s'est rencontré sur le tard, mais les liens d'amitiés se sont vites construits. Je vous souhaite plein de bonheur dans cette nouvelle aventure qui arrive.

Mathilde, tu es une belle rencontre de ces années lycée. Pleins de souvenirs d'adolescence restent depuis. Nous en avons bien profité. Je te souhaite de poursuivre à fond tes envies et j'espère que l'on pourra se voir un peu plus souvent.

A tous ceux rencontrés durant ces années d'apprentissages,

Aux Grenoblois, Sophie, Aude, Claire, Elsa, Béré, Charlotte, Corentin, Amélie, Florian, merci pour ces bons moments partagés tout au long de l'externat. L'internat nous a bien dispatché dans toute la France. Passez quand vous voulez dans le Sud ! A Charline, ma première rencontre à l'internat de Perpi, qui a signé le début d'une belle amitié. Un beau voyage partagé malgré les aléas, avec de sacrées rencontres et bien sur le Schnaps maison. Rien de mieux pour se réchauffer ! Merci pour ces bons moments partagés. Je reconnaitrai sans aucun doute le bruit du cerf la prochaine fois ! A Marion, on s'est rencontré un peu plus tard durant l'internat. Tu m'as supportée pendant 1 an au boulot, j'espère que ça t'a suffi ! Merci pour ces sessions escalade-sushis (les meilleures). Tu es la plus pédiatre des pédiatres, et surtout reste le ! A Thien-An, merci pour tous ces bons repas préparés puis dévorés. Tu es un ami de confiance, d'écoute et discussion.

A l'équipe de Pédiatrie du CH de Perpignan, et mes co-internes, merci d'avoir été les premiers à nous voir en temps qu'internes. Ce fut un hiver intense en virus, gastros et grippes mais nous y avons survécu. Merci à cette équipe paramédicale et médicale incroyable. Merci pour cette ambiance folle de premier semestre d'internat ! A l'équipe de Néonatologie du CHU de Nîmes, merci de nous avoir accueilli avec beaucoup de gentillesse et pédagogie. Anaïs et Mathilde on s'est retrouvé pour un nouveau semestre, moins virulent cependant ! Merci d'avoir été mes premières co-internes pendant 1 an. Marie Emmanuelle, tu as été ma première chef de néonat, merci de m'avoir transmis tes connaissances et ta précision dans cette discipline.

A l'équipe des Urgences Pédiatriques du CHU de Nîmes, vous avez été formidables ! Je cite très souvent ce semestre aux urgences comme exemple. De bons moments passés ensemble malgré les temps épidémiques difficiles, des CRPites extrêmes chez les internes et un hiver fiévreux. Merci Philippe de ton encadrement, ta confiance et de tes diagnostics, tu es un exemple. Merci à toute l'équipe médicale et paramédicale, ça a été un très grand plaisir de travailler avec vous tous.

A l'équipe d'Oncologie-Hématologie Pédiatrique, merci pour votre accueil remplit de bienveillance. Ça n'a pas toujours été facile, mais votre humanité a rendu ce semestre inoubliable. Merci Josiane du partage de ton bureau et de ton soutien infaillible. Merci Isabelle pour ta force tranquille et ton expérience, tu as pris soin de nous. Merci Anne-Charlotte pour ta gentillesse sans limite, à Maïdou pour ton humour atypique mais ta rigueur sans faille. Merci à mes co-internes Marion et Sita pour ce semestre riche en émotion. Houria merci pour ton sourire à toute épreuve. A l'équipe de Réanimation Néonatale, même si on se connaissait des gardes ce fut un grand plaisir de travailler de ce côté cette fois-ci. Merci pour cet apprentissage et cette collaboration étroite médicale-paramédicale. Merci à Béné et Vincent (encore !) pour ces trois mois courts mais intenses en ce contexte si particulier ! J'ai hâte de revenir travailler avec vous. Clem', alias Mida merci pour ces blagues pleines de spontanéité, j'ai bien musclé mes abdos ! Edouard, merci pour ton sérieux inébranlable et tes tiramisus incroyables, c'était un plaisir de travailler à tes côtés ! A très vite ! A toute l'équipe de co-interne, nos petites Manon et Marion, Chloé, Sita, Anaïs, et Carole, on s'en souviendra de ce semestre. Camille merci pour tes conseils et tes encouragements ! Toujours au top ! Merci à toute l'équipe médicale, Maliha, Sabine, les deux Odile, Florence, Pépé, Steph, Héléna, Renaud, Damien, Laurène, Flora.

A l'équipe Nantaise, merci pour votre accueil si chaleureux. On se sent vite bien chez vous (malgré une météo parfois capricieuse). Merci Cyril pour ta gentillesse dès les premiers mails échangés et ton accueil chaleureux, ce fut un grand plaisir de travailler avec toi et toute l'équipe. Merci aux médecins de Réa Ped, Manon, Bénédicte, Isabelle, Jean-Michel, Alexis, Nicolas, JEP, Brendan, Pierre pour votre bonne humeur, partage et gentillesse. Merci aux médecins de Néonat pour cette gentillesse et douceur partagées : Louis, le roi de la blague et des couvre chefs tous plus atypiques les uns des autres, Pauline ta bonne humeur et douceur, JB pour tes connaissances inépuisables, et ainsi que Marion, Camille, Arnaud, Louis-Marie, Magali, Laure et Anne. A mes co-internes, le trio infernal : M♡m♡ (oui j'ai osé), merci pour ta bonne humeur, tes blagues constantes et inépuisables ! Juju, merci de m'avoir fait rire aux larmes : c'était sur la plus pédiatres des urgentistes que ça devait tomber ! Qu'un mot à dire : Vitamines. En tous cas merci pour ce semestre un peu écourté, riche en rire. Merci de m'avoir soutenue dans mes derniers moments d'interne et mes folies passagères ! Vous avez rendu ce semestre inoubliable. Merci à tous, Géraldine, Elise, Sandra, Mathilde, Mélanie, Antoine, Damien et Marin, je garde vos p'tites bouilles en souvenir. Pour conclure ce chapitre, il y a juste un mot à dire deux fois : Bisous ! Bisous ! A toutes les équipes paramédicales, vous avez été formidables. Vous êtes indispensables et de bons conseils. Merci pour ces rires, pleurs, partages, p'tits déj' et apéros ! Le travail en équipe est ce que j'apprécie le plus. Pain has been defined by the International Association for the Study of Pain as "an unpleasant sensory and emotional experience associated with actual or potential tissue damage or described in terms of such damage"(1). Pain is a subjective complex multidimensional experience with emotional, physiological, sensory, behavioral, affective, cognitive, sociocultural and environmental components [START_REF] Bond | Pain : its nature and treatment[END_REF]. Comfort is a state of a physical ease and a lack of pain with pleasant feelings, state of well-being, and a relief of sorrow or distress. It includes physical, psychospiritual, sociocultural and environmental components. It is more than the absence of pain (3).

A Manon

Pain is a common adverse event in children hospitalized, due to illness, injury or medical procedures (4). In Pediatric Intensive Care Unit (PICU), children are more exposed to painful and stressful events of moderate to severe intensity because of necessary invasive management [START_REF] Groenewald | Prevalence of moderate-severe pain in hospitalized children: Pain prevalence in hospitalized children[END_REF][START_REF] Bosch-Alcaraz | A literature review of comfort in the paediatric critical care patient[END_REF].

A simple care, such as a cleaning care, may become painful or uncomfortable. Moreover, insufficient pain control in children can have immediate and longterm consequences, including negative effects on acceptance of later care procedures, reduced pain thresholds and negative emotional outcomes [START_REF] Young | Pediatric procedural pain[END_REF][START_REF] Von Baeyer | Children's memory for pain: overview and implications for practice[END_REF][START_REF] Wright | Complementary, Holistic, and Integrative Medicine: Music for Procedural Pain[END_REF].

Pharmacotherapy with administration of analgesics or sedative agents, is a fundamental part of the pain treatment and is effective in reducing it. However, it can cause severe side effects including over sedation, prolonged length of stay, longer length of ventilation, drug tolerance, iatrogenic withdrawal syndrome, and delirium [START_REF] Harris | Clinical recommendations for pain, sedation, withdrawal and delirium assessment in critically ill infants and children: an ESPNIC position statement for healthcare professionals[END_REF][START_REF] Anand | Tolerance and Withdrawal From Prolonged Opioid Use in Critically Ill Children[END_REF][START_REF] Ista | Withdrawal symptoms in children after long-term administration of sedatives and/or analgesics: a literature review. "Assessment remains troublesome[END_REF].

According to the pain and comfort definitions, their management should include a personalized and holistic approach (4). The combined use of pharmacological and nonpharmacological treatments can alleviate patient pain and provide comfort during procedural interventions [START_REF] Czarnecki | Procedural Pain Management: A Position Statement with Clinical Practice Recommendations[END_REF], and reduce pharmacological requirements [START_REF] Cepeda | Music for pain relief[END_REF][START_REF] Kim | Music therapy inhibits morphine-seeking behavior via GABA receptor and attenuates anxiety-like behavior induced by extinction from chronic morphine use[END_REF][START_REF] Dijkstra | The effects of music on physiological responses and sedation scores in sedated, mechanically ventilated patients[END_REF]. Holistic comfort measures are superior to strictly pharmacological pain management in the treatment of pediatric procedural pain [START_REF] Bice | Exploring Holistic Comfort in Children Who Experience a Clinical Venipuncture Procedure[END_REF][START_REF] Bice | Holistic Comfort Interventions for Pediatric Nursing Procedures: A Systematic Review[END_REF].

Music therapy is one of the non-pharmacological interventions [START_REF] Wright | Complementary, Holistic, and Integrative Medicine: Music for Procedural Pain[END_REF][START_REF] Klassen | Music for Pain and Anxiety in Children Undergoing Medical Procedures: A Systematic Review of Randomized Controlled Trials[END_REF]. It has been used to decrease pain and discomfort throughout human history [START_REF] Kemper | Music as therapy[END_REF]. It employs specific musical elements with sound, rhythm, melody, harmony, dynamic and tempo to facilitate positive interactions, improve emotional and cognitive state. Music therapy has been defined by the World Federation of Music Therapy as "the professional use of music and its elements as an intervention in medical, educational, and everyday environments with individuals, groups, families, or communities who seek to optimize their quality of life and improve their physical, social, communicative, emotional, intellectual, and spiritual health and wellbeing" [START_REF]World Federation of Music Therapy[END_REF]. Music therapy can alleviate pain and anxiety by distracting children during procedures. It blocks pain pathways by closing the neural "gate" in the spinal cord, and alters pain perception inhibiting the amount of pain transmitted to the brain (21-24) (APPENDIX 1). It also modulates activity into the mesolimbic structures, which are involved in reward processing and induces feelings of wellbeing [START_REF] Mavridis | Music and the nucleus accumbens[END_REF][START_REF] Menon | The rewards of music listening: Response and physiological connectivity of the mesolimbic system[END_REF].

Previous studies have reported that music intervention can have a positive effect on pain level in adult unit [START_REF] Tang | The Use of Music Intervention in Healthcare Research: A Narrative Review of the Literature[END_REF][START_REF] Shabanloei | Effects of Music Therapy on Pain and Anxiety in Patients Undergoing Bone Marrow Biopsy and Aspiration[END_REF], including intensive care units (ICU) [START_REF] Jaber | Effets de la musicothérapie en réanimation hors sédation chez des patients en cours de sevrage ventilatoire versus des patients non ventilés[END_REF]. Among children population, it was also proved to be effective in reduction of pain or anxiety for burn injury [START_REF] Hanson | Nonpharmacological Interventions for Acute Wound Care Distress in Pediatric Patients With Burn Injury: A Systematic Review[END_REF][START_REF] Li | The effects of music intervention on burn patients during treatment procedures: a systematic review and meta-analysis of randomized controlled trials[END_REF][START_REF] Whitehead-Pleaux | Exploring the Effects of Music Therapy on Pediatric Pain: Phase 1[END_REF], during postoperative period [START_REF] Nelson | Relaxation Training and Postoperative Music Therapy for Adolescents Undergoing Spinal Fusion Surgery[END_REF], venipuncture [START_REF] Hartling | Music to Reduce Pain and Distress in the Pediatric Emergency Department: A Randomized Clinical Trial[END_REF][START_REF] Caprilli | Interactive Music as a Treatment for Pain and Stress in Children During Venipuncture: A Randomized Prospective Study[END_REF], lumbar puncture [START_REF] Nhan Nguyen | Music Therapy to Reduce Pain and Anxiety in Children With Cancer Undergoing Lumbar Puncture: A Randomized Clinical Trial[END_REF], in neonatology [START_REF] Shah | Trial of Music, Sucrose, and Combination Therapy for Pain Relief during Heel Prick Procedures in Neonates[END_REF] and emergency departments [START_REF] Van Der Heijden | Children Listening to Music or Watching Cartoons During ER Procedures: A RCT[END_REF]. Therefore, pain and discomfort can be reduced or avoided by providing comfort interventions such as music therapy. In many children's hospitals, music therapy is becoming an important part of clinical cares [START_REF] Wright | Complementary, Holistic, and Integrative Medicine: Music for Procedural Pain[END_REF]. Nevertheless, more research needs to be carried out in order to establish the effectiveness of music therapy in PICUs. A previous pilot study suggested that it was possible to implement music therapy in our PICU [START_REF] Roux | La Musicothérapie Outil D'amélioration Du Confort Des Patients Lors De La Toilette En Réanimation Pédiatrique[END_REF]. This work also highlighted that the most common procedure in our PICU was "cleaning care", which caused frequent discomfort for children.

To our knowledge, the effect of music therapy during procedural cares in pediatric intensive care unit has not been investigated. Our objective was to evaluate this technique in a population of critically ill children during cleaning care. We planned to compare the rise of discomfort during this procedure with or without music therapy. Our hypothesis was that the music therapy was able to attenuate this rise. with music therapy. Children, parents or caregivers could choose the type of music from the "Music Care©" selection.

During the sequence with music therapy, cleaning care and videotape recording were launched after 10 minutes of music listening. Music continued until the end of the care. The cleaning care was performed at least one hour after the bolus administration of an analgesic or an anxiolytic.

Outcomes

The primary outcome was "the discomfort variation": difference between the Face Legs Activity Cry Consolability (FLACC) scale's score before and during the cleaning. The first FLACC evaluation was performed right before the beginning of the cleaning care. The second evaluation was the highest score during the care. The FLACC scores were assessed on video at least 8 days following the last recording, by two independent investigators. The averages between ratings of the two reviewers were recorded. If there was a variation of more than 1 point between the two reviewers' assessments, a third reviewer was then appointed to give a third evaluation.

Then, a collegial value was attributed after discussion between the 3 reviewers.

The secondary outcomes were physiological variations such as heart rate, mean arterial blood pressure, respiratory rate before and during the procedure. They were continuously measured with a cardiorespiratory monitor (IntelliVue MP70, Philips Medical Systems). The first value was selected just before the cleaning, and the second was the highest value during the procedure. The use of pharmacological drugs has been compared between the two groups.

FLACC scale's scores

The Face, Legs, Activity, Cry, Consolability (FLACC) Scale's score is a validated pain assessment tool in children from 0 to 18 years old to evaluate procedural pain [START_REF] Manworren | Pediatric Pain Measurement, Assessment, and Evaluation[END_REF][START_REF] Babl | The Use of the Faces, Legs, Activity, Cry and Consolability Scale to Assess Procedural Pain and Distress in Young Children[END_REF]. This scale includes five domains (face, legs, activity, cry, and consolability) with scores of 0, 1, and 2 for each domain and a total score ranging from 0 to 10 (46) (APPENDIX 4). Therapeutic intervention threshold had not been established to date. However, a pain score greater or equal to 4, is generally considered as indicator of significant pain (47).

Primary outcome

The increase of discomfort assessed by FLACC score variation before and during cleaning cares was attenuated with music therapy (1.58 (1.55) vs 2.14 (1.87); p = 0.02) (Figure 2). The comfort assessed by the FLACC score during the cleaning was better with music therapy (2.21 (1.92) vs 3.46 (2.68); p<0.001) (Table 2). There was an upward trend of the number of patients with a FLACC's score greater than 3 during cleaning cares without music therapy (20 (44%) versus 12 (27%), p = 0.12) (APPENDIX 6, Figure 4). The order of cleaning cares with or without music therapy did not affect pain scores' variation between the two groups (APPENDIX 6, Table 4). Age did not influence the effect of music therapy on pain.

Secondary outcomes

The increase of heart rate was attenuated with music therapy (8 (14) vs 17 [START_REF] Czarnecki | Procedural Pain Management: A Position Statement with Clinical Practice Recommendations[END_REF], p = 0.002).

The variation of respiratory rate and mean arterial blood pressure did not differ during cleaning cares with or without music therapy (Figure 3). The use of sedative/analgesic drugs and duration of the cleaning cares were not different. (Table 2). * DISCUSSION This is one of the first randomized controlled crossover study comparing the effect of music on pain and comfort in children hospitalized in Pediatric Intensive Care Unit. We found that passive music therapy using a "U" sequence during cleaning cares improved the comfort, attenuated the rise of discomfort and heart rate in critically ill children.

Pain and comfort

Benefits of music therapy to reduce pain in our study are similar to other pediatric studies [START_REF] Klassen | Music for Pain and Anxiety in Children Undergoing Medical Procedures: A Systematic Review of Randomized Controlled Trials[END_REF][START_REF] Whitehead-Pleaux | Exploring the Effects of Music Therapy on Pediatric Pain: Phase 1[END_REF][START_REF] Nelson | Relaxation Training and Postoperative Music Therapy for Adolescents Undergoing Spinal Fusion Surgery[END_REF][START_REF] Nhan Nguyen | Music Therapy to Reduce Pain and Anxiety in Children With Cancer Undergoing Lumbar Puncture: A Randomized Clinical Trial[END_REF]. The results of this study showed that patients who had music therapy intervention during cleaning cares had an improvement in their pain level and a reduction of discomfort. But we also noticed a downward trend in pain scores before cleaning cares with music therapy compared without music therapy. It is probably related to the beginning of the music 10 minutes before the start of cleaning cares with the first pain score evaluation at this moment.

FLACC scale's score

We chose the FLACC Scale's score because it is a validated pain assessment tool to evaluate children's procedural pain [START_REF] Caprilli | Interactive Music as a Treatment for Pain and Stress in Children During Venipuncture: A Randomized Prospective Study[END_REF][START_REF] Nhan Nguyen | Music Therapy to Reduce Pain and Anxiety in Children With Cancer Undergoing Lumbar Puncture: A Randomized Clinical Trial[END_REF]. It is recommended for assessing procedural pain and distress among pediatric patients, including the very young children. [START_REF] Von Baeyer | Systematic review of observational (behavioral) measures of pain for children and adolescents aged 3 to 18 years[END_REF][START_REF] Shen | Evaluation of nurse accuracy in rating procedural pain among pediatric burn patients using the Face, Legs, Activity, Cry, Consolability (FLACC) Scale. Burns[END_REF] It has the advantage to be easy to use, with a high level of inter-rater reliability and validity with other pediatric pain scales. [START_REF] Shen | Evaluation of nurse accuracy in rating procedural pain among pediatric burn patients using the Face, Legs, Activity, Cry, Consolability (FLACC) Scale. Burns[END_REF][START_REF] Beltramini | Pain Assessment in Newborns, Infants, and Children[END_REF][START_REF] Voepel-Lewis | Reliability and Validity of the Face, Legs, Activity, Cry, Consolability Behavioral Tool in Assessing Acute Pain in Critically Ill Patients[END_REF]. This scale is commonly used in our PICU and can be compared to other pain scales using 0-to-10 rating scores, and could allow comparison of our results with further studies [START_REF] Voepel-Lewis | Reliability and Validity of the Face, Legs, Activity, Cry, Consolability Behavioral Tool in Assessing Acute Pain in Critically Ill Patients[END_REF].

Music intervention

This passive music intervention with "U" sequence can have multiple effects on patients with physiological and psychological effects. Psychological effect is a result of promotion of a "listening" relationship between patients and caregivers [START_REF] Kemper | Music as therapy[END_REF]. Results of these actions are reduction of pain, anxiety and a significant decrease of drug consumption [START_REF] Guétin | Évaluation et standardisation d'une nouvelle technique de musicothérapie dans la prise en charge de la douleur : le montage en « U »[END_REF][START_REF]MusicCare[END_REF][START_REF] Rennick | A Pilot Randomized Controlled Trial of an Intervention to Promote Psychological Well-Being in Critically Ill Children: Soothing Through Touch, Reading, and Music*[END_REF]. This technique of passive music therapy has advantages such as being low cost and easy to use at every time, without requiring the intervention of a professional music therapist.

Various types of instrumental music were selected based on the patient's preferences.

Musical choices can be related to cultural background with differences in music preferences of various populations [START_REF] Good | Cultural Differences in Music Chosen for Pain Relief[END_REF]. During our study, children were able to choose the type of music. When the children could not choose, parents chose the music, and otherwise as a third option, caregivers did. When possible, it is essential that children are involved in choosing and deciding music therapy intervention and the choice of the type of music [START_REF]World Federation of Music Therapy[END_REF].

No adverse event occurred during our study. Some studies reported some adverse events such as feelings of isolation, loss control during the use of headphones [START_REF] Wright | Complementary, Holistic, and Integrative Medicine: Music for Procedural Pain[END_REF]. During our study, we only used speakers without causing any adverse event.

Other passive forms of distraction include listening to a story, viewing television or movies.

Active forms of distraction include active music therapy, interactive toys or electronic games, virtual reality, controlled breathing, guided imagery and relaxation [START_REF] Koller | Distraction Techniques for Children Undergoing Procedures: A Critical Review of Pediatric Research[END_REF]. These active distractions demand the participation of an additional therapist, who is not necessarily available. In addition, passive intervention requires only attention to the stimuli [START_REF] Koller | Distraction Techniques for Children Undergoing Procedures: A Critical Review of Pediatric Research[END_REF], and maybe more adapted to critically ill children. It has been showed that passive music therapy was as effective as active music therapy [START_REF] Klassen | Music for Pain and Anxiety in Children Undergoing Medical Procedures: A Systematic Review of Randomized Controlled Trials[END_REF][START_REF] Colwell | Impact of Music Therapy Interventions (Listening, Composition, Orff-Based) on the Physiological and Psychosocial Behaviors of Hospitalized Children: A Feasibility Study[END_REF]. Perhaps, the choice of passive or active form of distraction from a list of possibilities, should be decided by the children when possible [START_REF] Koller | Distraction Techniques for Children Undergoing Procedures: A Critical Review of Pediatric Research[END_REF].

Physiological responses and drugs consumption

Music can affect a person physiologically by reducing heart rate, respiratory rate and arterial blood pressure [START_REF] Cepeda | Music for pain relief[END_REF][START_REF] Dunn | Music and the reduction of post-operative pain[END_REF][START_REF] Jaber | Effets de la musicothérapie en réanimation hors sédation chez des patients en cours de sevrage ventilatoire versus des patients non ventilés[END_REF][START_REF] Li | The effects of music intervention on burn patients during treatment procedures: a systematic review and meta-analysis of randomized controlled trials[END_REF][START_REF] Nhan Nguyen | Music Therapy to Reduce Pain and Anxiety in Children With Cancer Undergoing Lumbar Puncture: A Randomized Clinical Trial[END_REF]. In our study we showed a decrease in heart rates with music therapy in line with other studies. However, no differences were demonstrated in respiratory rates or mean arterial blood pressure. The use of pharmacological drugs before or during procedure did not differ during cleaning cares. It could be explained by a low number of children in need of additional analgesic therapies during cleaning cares. In other studies, sedation or antalgic requirements were reduced during procedural interventions in order to make procedures safer [START_REF] Kulkarni | Music during interventional radiological procedures, effect on sedation, pain and anxiety: a randomised controlled trial[END_REF].

Limitations

A limitation of this study includes the impossibility to blind the children and care givers because of music intervention. Caregivers could also benefit from psychological and physiological effects during procedures with music therapy. As the result, decrease in the variation of the FLACC score could be partly related in the manner of how the care was provided with or without music therapy. This study respected the crossover randomized and two separate blinded evaluations of pain scores, enabling an accurate assessment of the FLACC scores during cleaning cares.

We regret the absence of FLACC evaluation before the beginning of the music therapy during cleaning cares with therapy. Indeed, it seems to show a possible effect of music therapy during the first 10 minutes of listening on pain and comfort before the beginning of the procedure.

As the FLACC score were lower in the music care before the intervention, the efficacy of music (difference of FLACC during and before the care) may have been underestimated in the music group even if the difference is still significant.

Our results concern only cleaning care procedures for children hospitalized in PICU. It could be relevant to carry out further multicentric studies, aiming to assess the effect of music therapy during other procedures, such as central venous line placement, invasive and non-invasive ventilation, sleep quality, pharmacologic consumption (morphinic and sedative), improvement of neurological outcomes, and chronic pain.

CONCLUSION:

This study demonstrates the efficacy of music therapy in decreasing pain and discomfort during cleaning cares in children hospitalized in our Pediatric Intensive Care Unit. The current study suggests that music therapy can be used to reduce pain during procedure in children hospitalized in PICU and should be considered as a part of pain treatment.

  ABBREVIATIONS .....................................................................................................................

Figure 2 .

 2 Figure 2. Variations of FLACC scale's scores during cleaning cares with and without music therapy. Data were analyzed using the mixed models (mean difference of 0,56 point, β= 0.62 [0.1 ; 1.14] p = 0.02). The limits of the box plots denote lower and upper quartiles.

Figure 3 .

 3 Figure 3.Variations of heart rate, respiratory rate and mean arterial blood pressure with and without music therapy (mean difference of -10 points, β 8.9[3,3;14,4], p=0,002). Data were analyzed using the mixed models. The limits of the box plots denote lower and upper quartiles. HR: heart rate in beats per minute; RR: Respiratory rate in frequency per minute; MABP: mean arterial bloop pressure in mmHg.

APPENDIX 6 :Figure 4 .

 64 Figure 4. Pain scores during cleaning cares showing an upward trend of the number of patients with a FLACC's score greater than or equal to 4 during cleaning cares without music therapy (20(44%) versus 12 (27%), p = 0.12). Data were analyzed using the Chi-square with Yates' correction.

  MUSIDORÉ : Prise en charge de la douleur avec de la musicothérapie en réanimation pédiatrique. NCT03916835.Introduction : La douleur est un effet indésirable courant chez les enfants hospitalisés dans les unités de réanimations pédiatriques. La musicothérapie est l'un des traitements non pharmacologiques pouvant soulager la douleur et l'inconfort des enfants pendant les soins. Le but de cette étude est d'évaluer l'efficacité de la musicothérapie passive sur le confort des enfants lors de toilettes en réanimation pédiatrique.Méthodes : Nous avons réalisé un essai clinique randomisé en cross over, prospectif, avec une évaluation aveugle du critère de jugement principal. Les patients inclus étaient des enfants entre 6 mois et 15 ans, admis dans l'unité de réanimation pédiatrique du CHU de Montpellier en France.Nous avons utilisé un programme de musicothérapie spécifique, "Music Care©", basé sur une séquence « U ».-Critère de jugement principal : Différence entre la variation du score de douleur FLACC (Face Legs Activity Cry Consolability) avant et pendant les toilettes, avec et sans musicothérapie.-Les critères de jugement secondaires : Variation des paramètres physiologiques tels que la fréquence cardiaque, la pression artérielle moyenne, la fréquence respiratoire et l'utilisation des médicaments antalgiques.Résultats: 50 enfants ont été inclus de Mai 2019 à Mai 2020 avec un âge moyen (DS) de 4,7 (4) ans. La variation de score de douleur avant et pendant les toilettes était plus faible avec la musicothérapie 1,58 (1,55) point contre 2,14 (1,87) points sans musicothérapie (p = 0,02). Le confort évalué par le score de FLACC durant les toilettes était amélioré avec la musicothérapie (2.21 (1.92) vs 3.46 (2.68) ; p<0.001). Conclusion : Les résultats de cette étude démontrent l'efficacité de la musicothérapie pour diminuer la douleur durant les toilettes chez les enfants hospitalisés dans notre unité de réanimation pédiatriques. La musicothérapie peut être employée pour réduire la douleur pendant les soins chez les enfants hospitalisés en réanimation pédiatrique et devrait être systématiquement proposée en association avec les traitements pharmacologiques. Mots clés : Musicothérapie, Réanimation pédiatrique, Douleur, FLACC

équipe de Réanimation Pédiatrique, merci

  de nous avoir accueilli dans ce stage un peu redouté. Ce fut une découverte et une révélation si on peut dire. Merci à l'équipe paramédicale pour son efficacité à toute épreuve. Merci d'avoir été patientes et patients et de m'avoir aidé dans la réalisation du travail sur la musicothérapie, qui sans vous n'aurait pas de sens. Séverine et Christiane, vous avez été les piliers et l'origine de ce travail, je vous en remercie. Merci à Julien pour ton savoir transmis et ta patience lors de mes premières voies centrales. Un grand merci à Maud et Vincent, au futur Berdeau & Co. C'est grâce à vous qu'on se revoit très prochainement pour travailler ensemble. Merci d'avoir cru en moi et fait confiance, je n'aurai pas pu rêver mieux pour ce semestre. A mes co-internes,

Merci Anne, Stéphanie et Laure pour votre disponibilité durant ce stage.

A l'

Floflo, Chlochlo et Benji, merci

	de m'avoir fait rire à n'en
	plus pouvoir. Aux co-internes d'à côté,

Anaïs, Julie, Marion, Éléonore et Maxime, la belle équipe, on se revoit très vite ! A l'équipe de Cardiologie et Pneumologie Pédiatrique, un

  semestre inattendu si je devais le résumé. Merci Pascal pour ta gentillesse et ton exigence qui nous rendent meilleurs. Oscar et Arthur merci pour votre disponibilité, et vos cours d'échographie (avec beaucoup de patience… ou pas !). Johan, le Zouk c'est le futur pour les visites ! Riyadh merci pour ta gentillesse et confiance, Marika pour tes visites longues mais remplies de connaissances, Stéphane pour ta patience lors des épreuves d'efforts et ta bonne humeur constante, Sophie et ton incroyable savoir, Grégoire et ton déhanché d'anthologie. Anne et Sabine les reines des explos et du pédalage ! Hamouda notre référent étude (merci de voler à notre secours à toutes nos questions). Et bien sûr une équipe de co-internes de choc, Chlochlo et la malchance en hospit' mais toujours prête pour un p'tit Zouk, Paupau (alias Paulette en Breton) notre maman cardioped' (merci de m'avoir sauvé la vie sur le GR34), Nico, sa bouteille en verre et ponctualité à toute épreuve !

A l'équipe d'Anesthésie Pédiatrique, merci

  d'avoir accueilli pendant 3 mois la pédiatre que je suis. Vous avez été patients, accueillants et pédagogues. Cela m'a permis de me sentir bien dans cet environnement inconnu. Ce partage a été essentiel pour moi et continuera à être utile lors de nos prochaines collaborations.

Table 1

 1 Baseline characteristics on admission in PICU

	n = 45

Table 2 .

 2 Outcomes

		Music therapy	Without music therapy		p
		n = 45	n = 45
		Before	During	Before	During
	FLACC score	0.66 (1.31)		1.40 (2.16)			0.05
			2.21 (1.92)		3.46 (2.68)		<0.001
	HR, per min (SD)	132 (28)	139 (29)	127 (25)	144 (29)		0.002
	MABP, mmHg (SD)	80 (19)	85 (20)	80 (18)	81 (19)		0.74
	RR, per min (SD)	33 (12)	37 (14)	33 (14)	40 (14)		0.1
	Use of		4 (9)		6 (13)	OR 0.6373	0.74
	pharmacological					95% CI
	drugs, n (%)					0.12 ;2.92
	Duration of cleaning	9,1 (4,4)	9,6 (4,2)		0.50
	cares, min (SD)

Values are numbers (%) or means (SD). FLACC: Face, Legs, Activity, Cry, Consolability; HR: Heart rate; MABP: mean arterial blood pressure; RR: respiratory rate.

Table 3 .

 3 Primary outcome in the study groups FLACC scale's score variation Values are means with Standard Deviation.

		Group A	Group B	p
	With Music Therapy	1,44 (1,62)	1,62 (1,46)	0,69
	Without Music Therapy	2 (1,72)	2,11 (1,96)	0,8

ACKNOWLEDGEMENTS :

A ma famille, D'abord mes Parents, qui m'ont soutenue depuis le début de ces longues études. Sans vous, je n'aurais pu réussir ce parcours. Merci de m'avoir offert une enfance heureuse, pleine d'aventure et découverte. Merci est un si petit mot pour désigner la chance d'avoir grandi dans notre famille.

METHODS

Patients

This two-arm prospective crossover clinical study with blind assessment of the primary outcome has been carried out from May 2019 to May 2020 in the department of Pediatric Intensive Care Unit of University Hospital in Montpellier, France.

The inclusion criteria were: (1) Conscious children. [START_REF] Bond | Pain : its nature and treatment[END_REF] Age between 6 months and 15 years old. (3) Expected length of stay > 2 days. (4) Absence of hearing impairments. [START_REF] Groenewald | Prevalence of moderate-severe pain in hospitalized children: Pain prevalence in hospitalized children[END_REF] Signed consent from both parents and authorization for videotape recording.

The exclusion criteria were: [START_REF]Pain terms: a list with definitions and notes on usage. Recommended by the IASP Subcommittee on Taxonomy[END_REF] Patient discharged or dead before the second cleaning. [START_REF] Bond | Pain : its nature and treatment[END_REF] Withdrawal of parental consent.

Protocol and Treatment allocation

Once the children have been included in the protocol, two consecutive cleaning cares were performed with videotape recording without sound during the care: one with music therapy and the other without it. Order of music therapy intervention was randomly allocated during the inclusion stage. The randomization was centralized (Department of Medical Information, University Hospital of Montpellier), with a computer-generated randomization process. Children in arm "A" received music therapy during the first cleaning care and not during the second cleaning care. Children in arm "B" received music therapy during the second cleaning care and not during the first cleaning care. Each patient's acted as their own control. Several health care providers or parents performed cleaning cares.

Music Therapy Intervention and videotaping

We used a specific program, "Music Care©" (MUSIC CARE © Paris, France). It is based on a "U" sequence, using a rhythm variation to relax patients, with slow and flowing music with a tempo of 60 to 80 beats per minute [START_REF] Guétin | Musicothérapie et algologie[END_REF][START_REF] Guétin | Évaluation et standardisation d'une nouvelle technique de musicothérapie dans la prise en charge de la douleur : le montage en « U »[END_REF]. Tempo of the music is an important factor as it reduces pain and provides relaxation to patients [START_REF] Cepeda | Music for pain relief[END_REF][START_REF] Nilsson | The Anxiety-and Pain-Reducing Effects of Music Interventions: A Systematic Review[END_REF]. Music was non-lyrical, with maximum volume level at 60 dB and a minimum duration of 20 minutes (42) (APPENDIX 3). It uses the principles of hypno-analgesia. [START_REF]MusicCare[END_REF] The preparation of each cleaning care strictly followed the same equipment installation process, in order to keep comparability and blind methodological options. Speakers were placed around the child's head during the two sequences. They were activated only during the sequence

Statistics

Sample size was calculated based on our previous pilot study results that suggested an increase of FLACC score of 4 (+/-1.5) points of children undergoing cleaning care, provided variance data for FLACC scale's scores [START_REF] Roux | La Musicothérapie Outil D'amélioration Du Confort Des Patients Lors De La Toilette En Réanimation Pédiatrique[END_REF]. It was predicted that music therapy would reduce pain levels by 1 point. In order to detect a clinically and statistically significant difference with an alpha error of 0,05 and statistical power of 90%, the sample size of 50 patients was calculated as being appropriate.

For the primary outcome, FLACC scale's score was calculated for each child and each procedure. The variation was compared between procedures, with and without music therapy using the mixed models. Our primary analysis was based on an intention to treat approach where all children who were randomly assigned to a study group were included. We used the mixed models, χ 2 test, Fisher's exact test and Student's t-tests to compare the baseline characteristics, primary and secondary outcomes. All statistical tests were performed at a significance level of 0,05 using SAS V.9.2. The distribution of the variables has been verified graphically.

Ethics

Written authorization from parents and children old enough was obtained for inclusion in the study and for being videotaped. It was also possible to withdraw from the study. The study was approved by the Île-de-France IX Ethics Committee. The trial was registered prior to patient recruitment in the National Library of Medicine registry (ClinicalTrials.gov ID: NCT03916835).

RESULTS

Population

Between May 2019 and May 2020, we enrolled 50 consecutive patients hospitalized in the Pediatric Intensive Care Unit of University Hospital in Montpellier. Twenty-five patients were randomized into the arm "A" and 25 into arm "B". Four patients randomized into the Arm "A" were not included in the analysis because three of them were discharged from PICU after the first cleaning care, and one patient's parents withdrew consent after randomization. One patient randomized into the arm "B" was not included in the analysis because he passed away before intervention (figure 1). Patient demographic and clinical characteristics are presented in Table 1.

Descriptive statistics showed no significant differences between the Arms "A" and "B" (APPENDIX 5).

APPENDICES: APENDIX 1:

The Gate control Theory [START_REF] Dunn | Music and the reduction of post-operative pain[END_REF][START_REF] Mann | Pain: Creative Approaches to Effective Management[END_REF] APPENDIX 2: Glossary of musical terms

Term Definitions Dynamics

The loudness or softness of a musical piece Harmony Consonant combination of notes sounded simultaneously to produce chords or to accompany tunes

Melody

The dominant tune of composition

Rhythm

The organizational pattern of sound in time or the timing of musical sound Tempo

The speed at which music is played Volume

The loudness level of music (in dB)

Orchestra Formation

The group of instruments organized to perform ensemble music APPENDIX 3: The ''U'' sequence, S. Guétin et al [START_REF] Guétin | Intérêt de la musicothérapie dans la prise en charge de la lombalgie chronique en milieu hospitalier (Étude contrôlée, randomisée sur 65 patients)[END_REF] Musical sessions last about 20 to 40 minutes. There are multiple phases in order to bring the patient to relaxation. The effect acts by reducing the musical rhythm, orchestra formation, frequency and volume (descending phase). A re-dynamizing phase (ascending branch of the "U"), after a phase of maximum relaxation (lower part of the "U") follows.

APPENDIX 5: Baseline characteristics on admission in PICU

Arm A (n=21) Arm B (n=24) p Age, years (SD) 5,2 (

SERMENT

• En présence des Maîtres de cette école, de mes chers condisciples et devant l'effigie d'Hippocrate, je promets et je jure, au nom de l'Etre suprême, d'être fidèle aux lois de l'honneur et de la probité dans l'exercice de la médecine.

• Je donnerai mes soins gratuits à l'indigent et n'exigerai jamais un salaire au-dessus de mon travail.

• Admise dans l'intérieur des maisons, mes yeux ne verront pas ce qui s'y passe, ma langue taira les secrets qui me seront confiés, et mon état ne servira pas à corrompre les moeurs, ni à favoriser le crime.

• Respectueuse et reconnaissante envers mes Maîtres, je rendrai à leurs enfants l'instruction que j'ai reçue de leurs pères.

• Que les hommes m'accordent leur estime si je suis fidèle à mes promesses. Que je sois couverte d'opprobre et méprisée de mes confrères si j'y manque.