Anton Baranov
email: anton.d.baranov@gmail.com

Rachid Zarouf
email: rzarouf@cmi.univ-mrs.fr

A BERNSTEIN-TYPE INEQUALITY FOR RATIONAL FUNCTIONS IN WEIGHTED BERGMAN SPACES

Keywords: 2000 Mathematics Subject Classification. Primary 32A36, 26A33; Secondary 26C15, 41A10 Rational function, Bernstein-type inequality, weighted Bergman norm

Given n ≥ 1 and r ∈ [0, 1), we consider the set Rn, r of rational functions having at most n poles all outside of 1 r D, were D is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in Rn, r in weighted Bergman spaces with "polynomially" decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with "super-polynomially" decreasing weights.

Introduction

Estimates of the norms of derivatives for polynomials and rational functions (in different functional spaces) is a classical topic of complex analysis (see surveys by A.A. Gonchar [START_REF] Gonchar | Degree of approximation by rational fractions and properties of functions[END_REF], V.N. Rusak [START_REF] Rusak | Rational Functions as Approximation Apparatus, Izdat. Beloruss[END_REF], and P. Borwein and T. Erdélyi [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF]Chapter 7]). Such inequalities have applications in many domains of analysis; to mention just some of them: 1) matrix analysis and in operator theory (see "Kreiss Matrix Theorem" [START_REF] Leveque | On the resolvent condition in the Kreiss matrix Theorem[END_REF][START_REF] Spijker | On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem[END_REF] or [START_REF] Zarouf | Analogs of the Kreiss resolvent condition for power bounded matrices[END_REF][START_REF] Zarouf | Sharpening a result by E.B[END_REF] for resolvent estimates of power bounded matrices), 2) inverse theorems of rational approximation (see [START_REF] Danchenko | An integral estimate for the derivative of a rational function[END_REF][START_REF] Peller | Hankel operators of class and their applications (rational approximations, Gaussian processes, the problem of majorizing operators)[END_REF][START_REF] Pekarskii | Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation[END_REF]), 3) effective Nevanlinna-Pick interpolation problems (see [START_REF] Zarouf | Effective ∞ interpolation[END_REF][START_REF] Zarouf | Effective ∞ interpolation constrained by Hardy and Bergman weighted norms[END_REF]).

Here, we present Bernstein-type inequalities for rational functions f of degree n with poles in {z : |z| > 1}, involving Hardy norms and weighted Bergman norms. Let P n be the complex space of polynomials of degree less or equal to n ≥ 1. Let D = {z ∈ C : |z| < 1} be the unit disc of the complex plane and D = {z ∈ C : |z| ≤ 1} its closure. Given r ∈ [0, 1), we define R n, r = p q : p, q ∈ P n , d • p < d • q, q(ζ) = 0 |ζ| < 1 r , (where d • p denotes the degree of p ∈ P n ), the set of all rational functions in D of degree less or equal than n ≥ 1, having at most n poles all outside of 1 r D. Notice that for r = 0, we get R n, 0 = P n-1 .

1.1. Definitions of Hardy spaces and radial weighted Bergman spaces. We denote by Hol (D) the space of all holomorphic functions on D. From now on, if f ∈ Hol (D) then for every ρ ∈ (0, 1) we define

f ρ : ξ → f (ρξ) , ξ ∈ 1 ρ D.
We consider the two following scales of Banach spaces X ⊂ Hol (D) :

a. The Hardy spaces H p = H p (D), 1 ≤ p ≤ ∞ :

H p = f ∈ Hol (D) : f p H p = sup 0≤ρ<1 T |f ρ (ξ)| p dm(ξ) < ∞ ,
where m stands for the normalized Lebesgue measure on T = {z ∈ C : |z| = 1}. As usual,we denote by H ∞ the space of all bounded analytic functions in D. b. The radial weighted Bergman spaces L p a (w), 1 ≤ p < ∞ (where "a" means analytic),

L p a (w) = f ∈ Hol (D) : f p L p a (w) = 1 0 ρw (ρ) T |f ρ (ζ)| p dm(ζ)dρ < ∞ ,
where the weight w satisfies w ≥ 0 and

1 0 w(ρ)dρ < ∞. For the classical power weights w(ρ) = w β (ρ) = (1 -ρ) β , β > -1, we have L p a (w β ) = L p a (1 -|z|) β dA(z) , A being the normalized area measure on D.
For general properties of these spaces we refer to [START_REF] Hedenmalm | Theory of Bergman spaces[END_REF][START_REF] Zhu | Operator theory in function spaces[END_REF]. From now on, for two positive functions a and b, we say that a is dominated by b, denoted by a b, if there is a constant c > 0 such that a ≤ cb; and we say that a and b are comparable, denoted by a ≍ b, if both a b and b a.

Statement of the problem and known results

. By Bernstein-type inequalities for rational functions one usually understands the inequalities of the form

(1.1) f ′ X ≤ φ X, Y (n) f Y , f ∈ R n ,
where R n is the set of all proper rational functions of degree at most n with the poles in {|z| > 1}, X and Y are some normed spaces of functions analytic in the unit disc, and φ is some increasing (often polynomially growing) function. Thus, for a given pair of the function spaces X and Y , the question is to determine the dependence on n for the norm of the differentiation operator (R n , • X ) to Y . Bernstein-type inequalities of E.P. Dolzhenko [START_REF] Dolzhenko | Rational approximations and boundary properties of analytic functions[END_REF] and A.A. Pekarskii [START_REF] Pekarskii | Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation[END_REF] are of this form; e.g., it is shown in [START_REF] Dolzhenko | Rational approximations and boundary properties of analytic functions[END_REF] that

f H 1 1 ≤ c 1 n f ∞ , f B 1/2 2,2 ≤ c 2 n 1/2 f ∞ , f ∈ R n ,
where H 1 1 is the Hardy-Sobolev space, and B 1/2 2,2 is the Besov (or Dirichlet) space, see the definition in Section 3. Let us also mention that this problem is a part of a more general one given by G. Lorentz in a letter sent to T. Erdélyi in 1988 (see [START_REF] Erdélyi | George Lorentz and inequalities in approximation[END_REF]).

Looking at (1.1), we notice that for some choices of X and Y , we have φ X, Y (n) = +∞ for every n = 1, 2, . . . . Indeed, it may happen for instance when the poles of our function f are allowed to be arbitrary close to the torus T : we can observe this phenomenon for example in the special case X = Y = H p , 1 ≤ p ≤ +∞ but also when X = Y = L p a (w), 1 ≤ p ≤ +∞. This observation leads us to come back on the problem in (1.1) and to state it more generally : that is replacing R n by R n, r (for any fixed r ∈ [0, 1)) and φ X, Y (n) by φ X, Y (n, r) so that to focus on this phenomenon of "natural dependence on the parameter r". For most of the classical cases already studied by others (for instance E. P. Dolzhenko [START_REF] Dolzhenko | Rational approximations and boundary properties of analytic functions[END_REF], A. A. Pekarskii [START_REF] Pekarskii | Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation[END_REF], V.V. Peller [START_REF] Peller | Hankel operators of class and their applications (rational approximations, Gaussian processes, the problem of majorizing operators)[END_REF]) the spaces X and Y are such that sup r∈(0, 1) φ X, Y (n, r) < +∞: in this case we can set φ X, Y (n) = sup r∈(0, 1) φ X, Y (n, r). As a consequence, if sup r∈(0, 1) φ X, Y (n, r) = +∞, it may be of interest to search (as a continuation of the investigations of the second author [START_REF] Zarouf | Asymptotic sharpness of a Bernstein-type inequality for rational functions in H 2[END_REF][START_REF] Zarouf | Application of a Bernstein type inequality to rational interpolation in the Dirichlet space[END_REF]) for the "best possible" φ X, Y (n, r) in an asymptotically sense, that is to say as n → ∞ and r → 1 -. This question has already been answered for the case X = Y = H p , 1 ≤ p ≤ +∞ by K. M. Dyakonov [6] see (1.2) below. In this paper, we answer the same question for the case X = Y = L p a (w), 1 ≤ p ≤ +∞. Let us give a general formulation of our problem for the special case X = Y for which we set C n, r (X) = φ X, Y (n, r) : given a Banach space X of holomorphic functions in D, we are searching for the best possible constant C n, r (X) such that

f ′ X ≤ C n, r (X) f X , f ∈ R n, r .
For the case where X = H p is a Hardy space, an estimate which gives a correct order of growth for C n, r (X) was obtained by K.M. Dyakonov [START_REF] Dyakonov | Kernels of Toeplitz operators, smooth functions, and Bernstein-type inequalities[END_REF] (as a very special case of more general results): for any p ∈ [1, ∞] there exist positive constants A p and B p such that (1.2)

A p n 1 -r ≤ C n, r (H p ) ≤ B p n 1 -
r for all n ≥ 1 and r ∈ [0, 1). More precisely, the upper estimate for p ∈ (1, +∞) is treated in [6, Theorem 1], the case p = 1, in [6, Corollary 1], and the case p = +∞ (known much earlier) is given in [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF]Theorem 7.1.7]. The below estimate follows trivially when applying the differentiation operator to the test function

f (z) = (1 -rz) -n .
For the case p = 2 an asymptotically sharp result was obtained later in [START_REF] Zarouf | Asymptotic sharpness of a Bernstein-type inequality for rational functions in H 2[END_REF]: for any r ∈ (0, 1) there exists the limit

lim n→∞ C n, r H 2 n = 1 + r 1 -r .
Related results about Bernstein-type inequalities in a more general setting of the so-called model or star invariant subspaces may be found in [START_REF] Dyakonov | Smooth functions in the range of a Hankel operator[END_REF]Theorems 10,[START_REF] Hedenmalm | Theory of Bergman spaces[END_REF], [START_REF] Dyakonov | Differentiation in star-invariant subspaces I. Boundedness and compactness[END_REF]Theorem 1], and [START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF][START_REF] Baranov | Embeddings of model subspaces of the Hardy space: compactness and Schatten-von Neumann ideals[END_REF]. 1.3. Main results. We obtain estimates for the derivatives of rational functions with respect to weighted Bergman norms. It turns out that there is an essential difference between slowly (polynomially) decreasing weights and fast (superpolynomially) decreasing weights. In the first case we have a two-sided estimate analogous to (1.2), while in the second case only the above estimate remains true. Let us give the precise definitions. Recall that w is always an integrable nonnegative function on (0, 1).

Definition 1.1. (Polynomially decreasing weights)

The weight w is said to be γ-polynomially decreasing if there exists γ > 0 such that

ρ → (1 -ρ) -γ w(ρ),
is increasing on [r 0 , 1) for some 0 ≤ r 0 < 1. We say that w is polynomially decreasing if it is γ-polynomially decreasing for some γ > 0.

Definition 1.2. (Super-polynomially decreasing weights) The weight w is said to be super-polynomially decreasing if for any γ > 0 there exists r(γ) ∈ (0, 1) such that the function ρ → (1 -ρ) -γ w(ρ), decreases on the interval [r(γ), 1).

Typical example of the weights from the first class are given by w

(r) = (1 -r) β , β > -1, or w(r) = (1 -r) β (| log(1 -r)| + 1) γ , β > -1, γ ∈ R. The weights w(r) = exp -c(1 -r) -γ , c > 0, γ > 0 are super-polynomially decreasing.
Our first result may be considered as an analogue of Dyakonov's theorem for the radial weighted Bergman spaces.

Theorem 1.3. Let 1 ≤ p < ∞ and let w be an integrable nonnegative function on [0, 1). Then there exists a positive constant K depending only on p (but not on the weight w) such that

(1.3) C n, r (L p a (w)) ≤ K n 1 -r
for all r ∈ [0, 1) and n ≥ 1. Moreover, if we fix r ∈ (0, 1) and let n tend to infinity, then we have

(1.4) Kr 1 -r ≤ lim inf n→∞ C n,r (L p a (w)) n ≤ lim sup n→∞ C n,r (L p a (w)) n ≤ K 1 -r ,
where K is, as K, a positive constant depending only on p.

The next theorem shows that for the polynomially decreasing weights the quantity C n, r (L p a (w)) admits a below estimate of the same form. Theorem 1.4. If w is γ-polynomially decreasing, then there exists a positive constant K ′ depending only on w and p such that

(1.5) K ′ n 1 -r ≤ C n, r (L p a (w)) ≤ K n 1 -r ,
where K is defined in (1.3) and where the left-hand side inequality of (1.5) holds for all r ∈ [0, 1) and n ≥ γ+3 p + 1. In particular, (1.5) holds for the classical weights

w(ρ) = w β (ρ) = (1 -ρ) β ρ, β > -1.
The polynomial decrease is essential and provides a sharp bound for the validity of the uniform estimate (1.5) for all possible values of n and r. Namely, if the weight is super-polynomially decreasing, then (1.5) will fail along some sequence of radii.

Theorem 1.5. Suppose that w is super-polynomially decreasing. Then there exists a sequence r n → 1such that for any p,

C rn,n (L p a (w)) n = o 1 1 -r n , n → ∞.
Acknowledgements. The authors are deeply grateful to Nikolai Nikolski, Alexander Borichev, and Evgueny Doubtsov for many helpful discussions, constructive comments and precious remarks which definitively helped to improve the manuscript.

Proofs of Theorems 1.3 and 1.4

Proof of Theorem 1.3. First, we notice that for any 0 ≤ α < 1, (2.1)

f p L p a (w) ≍ u∈Cα ρ |f (ρξ)| p w(ρ)dm(ζ)dρ
for all f ∈ L p a (w), where C α = {z : α < |z| < 1}. Let f ∈ R n, r with r ∈ [0, 1) and n ≥ 1. Using (2.1) with α = 1 2 we get

f ′ p L p a (w) ≍ ρξ∈C 1/2 |f ′ (ρξ)| p w (ρ) dm(ζ)dρ = 1 1 2 ρ w(ρ) 1 ρ p (f ρ ) ′ p H p dρ.
Now using the fact that f ρ ∈ R n, ρr ⊂ R n, r for every ρ ∈ (0, 1), we get

1 1 2 ρw(ρ) 1 ρ p (f ρ ) ′ p H p dρ ≤ (2C n, r (H p )) p 1 1 2 ρw (ρ) f ρ H p p dρ ≍ (C n, r (H p )) p f p L p
a (w) . In particular, using the right-hand side inequality of (1.2), we get

C n, r (L p a (w)) ≤ K p n 1 -r ,
for all p ∈ [1, ∞), and β ∈ (-1, ∞), where K p is a constant depending on p only. Now, let us prove (1.4). Let

f n (z) = 1 (1 -rz) n ∈ R n,r , and D = {z ∈ D : |1 -rz| ≤ 2|1 -r|}. We claim that f n p L p a (w) ∼ D |f n (z)| p w(z)dA(z), n → ∞,
and, analogously,

f ′ n p L p a (w) ∼ D |f ′ n (z)| p w(z)dA(z), n → ∞.
Indeed, by a very rough estimate

D\D |f n (z)| p w(z)dA(z) ≤ C 1 2 pn (1 -r) pn ,
where C 1 > 0 depends only on w. On the other hand, if we put D = {z ∈ D :

|1 -rz| ≤ 3 2 |1 -r|}, then D |f n (z)| p w(z)dA(z) ≥ 1 (3/2) pn (1 -r) pn D w(z)dA(z).
Since r (thus D and D) are fixed we see that

1 2 pn (1 -r) pn = o 1 (3/2) pn (1 -r) pn D w(z)dA(z) , n → ∞. Thus, f ′ n p L p a (w) f n p L p a (w) ∼ D |f ′ n (z)| p w(z)dA(z) D |f n (z)| p w(z)dA(z).
Obviously,

D |f ′ n (z)| p w(z)dA(z) = D n p r p |1 -rz| pn+p w(z)dA(z) ≥ n p r p 2 p (1 -r) p D 1 |1 -rz| pn w(z)dA(z) = n p r p 2 p (1 -r) p D |f n (z)| p w(z)dA(z).
Thus,

lim inf n→∞ f ′ n L p a (w) n f n L p a (w) ≥ r 2(1 -r)
.

For the proof of Theorem 1.4 we will need two lemmas.

Lemma 2.1. Let r ∈ [0, 1) and t ≥ 0. We set

I(t, r) = T |1 -rξ| -t dm(ξ) and ϕ r (t) = T |1 + rξ| t dm(ξ).
Then,

I(t, r) = 1 (1 -r 2 ) t-1 ϕ r (t -2)
for every t ≥ 2, and t → ϕ r (t) is an increasing function on [0, +∞) for every r ∈ [0, 1). Moreover, both r → ϕ r (t -2) and r → I(t, r), are increasing on [0, 1), for all t ≥ 0.

Proof. Indeed, supposing that t ≥ 2, we can write

I(t, r) = 1 1 -r 2 T |b ′ r (ξ)| 1 |1 -rξ| t-2 dm(ξ),
where b r (z) = r-z 1-rz ). Using the fact that b r • b r (z) = z and changing the variable in the above integral we get for every t ≥ 0, r ∈ [0, 1). Thus, ϕ r is a convex fonction on [0, ∞) and ϕ ′ r is increasing on [0, ∞) for all r ∈ [0, 1). Moreover, ), and so ϕ r is increasing on [0, ∞). The fact that r → I(t, r), is increasing on [0, 1) for all t ≥ 0 is obvious since

I(t, r) = 1 1 -r 2 T |b ′ r (ξ)| 1 |1 -rb r • b r (ξ)| t-2 dm(ξ) = 1 1 -r 2 T 1 |1 -rb r (ξ)| t-2 dm(ξ) = 1 (1 -r 2 ) t-1 ϕ r (t -2), since 1 -rb r (z) = 1-rz-r(r-z) 1-rz = 1-r 2 1-rz . Now, ϕ r (t) = 2π 0 exp t 2 ln 1 + r 2 -
ϕ ′ r (0) = 1 4 2π 0 ln 1 + r 2 -2r cos s ds = 0. Thus, ϕ ′ r (t) ≥ ϕ ′ r (0) = 0, ∀t ∈ [0, ∞), r ∈ [0, 1
I(t, r) = 1 (1 -rz) t/2 2 H 2 = k≥0 a 2 k (t)r 2k ,
where a k (t) is the kth Taylor coefficient of (1 -z) -t/2 . The same reasoning gives that r → ϕ r (t) is increasing on [0, 1).

Lemma 2.2. If for some r 0 ∈ [0, 1) and γ > 0 the function w(ρ)

(1-ρ 2 ) γ is increasing on [r 0 , 1), then 1 r ρw(ρ)I(t, rρ)dρ ≍ 1 r0 ρw(ρ)I(t, rρ)dρ,
for all t such that t ≥ γ + 3 and for all r ≥ r 0 , with constants independent on t.

Proof. Clearly,

(ρ) (1 -ρ 2 ) γ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 ϕ rρ (t)dρ ≤ w(r) (1 -r 2 ) γ r r0 ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 ϕ rρ (t)dρ ≤ w(r) (1 -r 2 ) γ ϕ r 2 (t) 1 r0 ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ, because u → ϕ u (t)
is increasing for all t > 0. For the same reason,

1 r ρw(ρ) 1 (1 -(rρ) 2 ) t-1 ϕ rρ (t)dρ = 1 r w(ρ) (1 -ρ 2 ) γ ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 ϕ rρ (t)dρ ≥ w(r) (1 -r 2 ) γ ϕ r 2 (t) 1 r ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ. Now note that r r0 ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ 1 r ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ, r ∈ [r 0 , 1),
with constants independent on t ≥ γ + 3. Indeed, this estimate holds for t = γ + 3, and, hence, by monotonicity of the function ρ → (1 -(ρr) 2 ) -1 , for all t ≥ γ + 3. Thus, using Lemma 2.1 and the fact that the function (1-ρ) -γ w(ρ) is increasing on [r 0 , 1), we obtain

r r0 ρw(ρ)I(t, rρ)dρ ≤ w(r) (1 -r 2 ) γ ϕ r 2 (t -2) r r0 ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ ≤ κ 1 w(r) (1 -r 2 ) γ ϕ r 2 (t -2) 1 r ρ 1 -ρ 2 γ (1 -(rρ) 2 ) t-1 dρ ≤ κ 2 1 r ρw(ρ) 1 (1 -(rρ) 2 ) t-1 ϕ rρ (t)dρ,
(where κ 1 , κ 2 are positive constants which do not depend on t), which completes the proof.

Proof of Theorem 1.4. We need to prove only the lower bound, the upper bound is already proved in Theorem 1. By Lemma 2.1, this is equivalent to the estimate

1 r ρw(ρ) ϕ rρ (pn + p -2) (1 -(rρ) 2 ) pn+p-1 dρ ≥ C (1 -r) p 1 r ρw(ρ) ϕ rρ (pn -2) (1 -(rρ) 2 ) pn-1 dρ.
The last statement is obvious since

1 r ρw(ρ) ϕ rρ (pn + p -2) (1 -(rρ) 2 ) pn+p-1 dρ ≥ 1 (1 -r 2 ) p 1 r ρw(ρ) ϕ rρ (pn + p -2) (1 -(rρ) 2 ) pn-1 dρ ≥ 1 (1 -r 2 ) p 1 r ρw(ρ) ϕ rρ (pn -2) (1 -(rρ) 2 ) pn-1 dρ,
where the last inequality is due to the fact that t → ϕ u (t) is increasing for all 0 ≤ u < 1.

3. The case of super-polynomially decreasing weights. Proof of Theorem 1.5:

For the proof of Theorem 1.5 we will need a definition from the theory of model subspaces of the Hardy space. For a finite subset σ of D with card σ = n, consider the finite Blaschke product

B σ = λ∈σ b λ , where b λ (z) = λ-z 1-λz , λ ∈ D. Define the model space K Bσ by K Bσ = B σ H 2 ⊥ = H 2 ⊖ B σ H 2 .
Consider the family (e k ) 1≤k≤n in K Bσ (known as Malmquist basis, see [13, p. 117]),

e 1 (z) = (1 -|λ 1 |) 1/2 1 -λ 1 z and e k (z) = k-1 j=1 b λj (z) (1 -|λ k |) 1/2 1 -λ k z , k ∈ [2, n],
The family (e k ) 1≤k≤n associated with σ is an orthonormal basis of the n-dimensional space K Bσ .

In what follows we denote by L p a (w, sD) and by H p (sD), s > 0, the weighted Bergman space and the Hardy space in the disc sD = {z : |z| < s}, respectively. If w ≡ 1, we write simply L p a (sD) and we write

L p a if s = 1. Lemma 3.1. Let n ≥ 1, r, s ∈ [0, 1) and p ∈ [1, +∞]. We set M p, s (n, r) = sup |f (ξ)| : ξ ∈ D, f ∈ R n, r , f L p a (sD) ≤ 1 . Then (3.1) M p, 2 3 (n, r) ≤ d c n (1 -r) n+b ,
where d > 0, b > 0, c > 1 are some absolute positive constants (may be, depending on p). Remark 3.2. Lemma 3.1 is valid not only for s = 2 3 , but for every s ∈ (0, 1), with constants d > 0, b > 0, c > 1 depending both on s and p.

Proof. For every f ∈ R n, r and ξ ∈ D, we have

f 1 2 ξ = f 2 3 3 4 ξ = D f 2 3 (u) k 3 4 ξ (u) 2 dA(u) ,
where k λ (z) = 1 1-λz is the standard Cauchy kernel associated with λ ∈ D, and A is the normalized area measure on D. Applying Hölder's inequality we obtain

f 1 2 ξ ≤ f 2 3 L p a k 3 4 ξ 2 L p ′ a = 3 2 1/p f L p a( 2 3 D) k 3 4 ξ 2 L p ′ a , ξ ∈ T,
where p ′ is such that 1 p + 1 p ′ = 1. Now, note that

k 3 4 ξ 2 L p ′ a ≤ k 3 4 ξ 2 H ∞ = 1 1 -3 4 2 = 16. Finally, supposing f L p a( 2 3 D) ≤ 1, we obtain f L p a( 1 2 D) ≤ f H ∞ ( 1 2 D) ≤ 16
(3.3) M 2, 1 2 (n, r) ≤ 2 √ n 2 1 -r n+ 1 2 . For every f ∈ R n, r , we have f 1 2 ∈ R n, 1 2 r ⊂ R n, r . If {1/λ 1 , . . . , 1/λ n }
f (ξ) = n k=1 α k   k-1 j=1 λj 2 -2ξ 1 -λ j ξ   1 -1 4 |λ k | 1 -λ j ξ , ξ ∈ D,
and by the Cauchy-Schwarz inequality,

(3.5) |f (ξ)| ≤ n k=1 |α k | 2 1/2 n k=1 k-1 j=1 λj 2 -2ξ 1 -λ j ξ 1 -1 4 |λ k | 2 1/2 1 -λ j ξ 2 1/2 . for any ξ ∈ D. Now, if λ ∈ rD and ξ ∈ D, λ 2 -2ξ 1 -λξ = 2 λ 4 -ξ 1 -λ 4 ξ 1 -λ 4 ξ 1 -λξ = 2b λ 4 (ξ) 1 + 3λ 4(1 -λξ) , which gives λ 2 -2ξ 1 -λξ ≤ 2 1 + 3r 4 1 1 -r = 4 -r 2(1 -r) ≤ 2 1 -r .
We get

(3.6) n k=1 k-1 j=1 λj 2 -2ξ 1 -λ j ξ 1 -1 4 |λ k | 2 1/2 1 -λ j ξ 2 ≤ 1 (1 -r) 2 n k=1 2 2(k-1) 1 1 -r 2(k-1) ≤ 1 4 2 1 -r 2n+1 . Now we first notice that n k=1 |α k | 2 1/2 = f 1 2 H 2 .
For any function ϕ

(z) = k≥0 ϕ(k)z k in H 2 , one has ϕ 2 H 2 = k≥0 | ϕ(k)| √ k + 1 √ k + 1 | ϕ(k)| ≤ ϕ L 2 a ϕ B 1/2 2, 2
,

We now use the upper bound of [21, Theorem A, (4)]: for ϕ ∈ R n, ρ one has

ϕ 2 B 1/2 2, 2 = ϕ ′ 2 L 2 a + ϕ 2 H 2 ≤ (2 + r) n 1 -r ϕ 2 H 2 + ϕ 2 H 2 ≤ 4n 1 -r ϕ 2 H 2 ,
which gives

ϕ H 2 ≤ 2 n 1 -ρ ϕ L 2 a . In particular, with ϕ = f 1 2 we get ϕ ∈ R n, 1 2 r and (3.7) f 1 2 H 2 ≤ 2 n 1 -r/2 f 1 2 L 2 a ≤ 2 √ 2n f 1 2 L 2 a .
We conclude from (3.5), (3.6) and (3.7) that for any ξ ∈ D,

|f (ξ)| ≤ f 1 2 H 2 1 4 2 2n+1 (1 -r) 2n+1 1 2 ≤ 1 2 2 1 -r n+ 1 2 2 √ 2n f 1 2 L 2 a , that is, |f (ξ)| ≤ √ 2n 2 1 -r n+ 1 2 f L 2 a ( 1 2 D) , ξ ∈ D.
Taking the supremum over ξ ∈ D and f ∈ R n, r we obtain (3. Here and everywhere below in this proof, C i , i = 1, . . . , 5, are positive constants, depending, may be, only on p and w (but not on n and r). By (1.2), we have for the first integral

I 1 ≤ C 1 n 1 -R p R 0 f ρ p p w(ρ)dρ ≤ C 2 n 1 -R p f p L p a (w) .
Note that f ρ ∈ R n, ρr ⊂ R n, r , and, thus, f ρ ∞ ≤ M p, 2 3 (n, r) f ρ L p a ( 2 3 D) . Applying (1.2) once again together with an obvious inequality f ρ p ≤ f ρ ∞ , we get n such that the function w(r)(1 -r) -γn decreases on [r • n , 1). Now for a fixed n take r, R so that r • n < R < r < 1 and 1 -R = (1 -r) 1/2 , 1 -r • n = (1 -r) 1/4 . We have

I 2 ≤ C 3 n 1 -r p 1 R f ρ p ∞ w(ρ)dρ ≤ C 3 f p L p a( 2
w(R) ≤ w(r • n ) (1 -R) γn (1 -r • n ) γn = w(r • n )(1 -r) γn/4 .
Hence, using the fact that w is bounded on [r • 1 , 1), we obtain

I 2 ≤ C 4 n 1 -r p f p L p a (w) • c pn (1 -r) γn/4
(1 -r) pn+pb .

Let us show that for sufficiently large n, c pn (1 -r) γn/4

(1 -r) pn+pb → 0, r → 1 -.

Indeed, choosing r so that c < (1 -r) -1 , we get c pn (1 -r) γn/4

(1 -r) pn+pb ≤ (1 -r) The corresponding estimate for I 1 is obvious since 1 -R n = (1 -r n ) 1/2 .

1 r0 1 r

 11 ρw(ρ)I(t, rρ)dρ ≥ ρw(ρ)I(t, rρ)dρ, r ∈ [r 0 ,

3 . 1 ( 1 - 1 r

 3111 Let us prove the minoration with the test function f (z) = rz) n . Using (2.1) with α = r 0 , we need to show that )I(pn, rρ)dρ = C (1 -r) p f p L p a (w) . Since r ∈ [r 0 , 1) and n ≥ γ+3 p , by Lemma 2.2 applied with t = pn + p and t = pn this means that 1 r ρw(ρ)I(pn + p, rρ)dρ ≥ C (1 -r) p ρw(ρ)I(pn, rρ)dρ.

and f 1 2 ∈α

 2 is the set of the poles of f (thus, |λ j | ≤ r, j = 1, . . . , n), then f ∈ K Bσ with σ = {λ 1 , . . . , λ n } ⊂ rD), whereas the set {2/λ 1 , . . . , 2/λ n } is the set of the poles of the functionf 1 2 K B σ ′ with σ ′ = 1 2 λ 1 , . . . ,1 2 λ n ⊂ r 2 D. Hence, there exist α 1 , . . . , α n ∈ C k e k , on D, where (e k ) n k=1 is the Malmquist basis associated with the set σ ′ . Since both f 1 2 and n k=1 α k e k are meromorphic in C the equality (3.4) is in fact valid everywhere in C. Thus,

1 + I 2 , 1 R

 121 3). Combining (3.2) and (3.3) and choosing d = 48, b = 1 2 and c > 2 such that 2 n √ n ≤ c n for any n ≥ 1, we complete the proof and obtain (3.1). Proof of Theorem 1.5. Take r ∈ (0, 1) and R ∈ (0, r) and let us represent the norm f ′ p L p a (w) of a function f ∈ R n, r as I (f ρ ) ′ p p w(ρ)dρ.

  r) pn+pb w(R), where the last inequality follows from Lemma 3.1. Note that r) pn+pb w(R) f p L p a (w) . Now, choose a positive increasing sequence (γ n ) n∈N such that n = o (γ n ), as n → +∞. For any n we fix r •

γn 4 -

 4 2pn-pb → 0, r → 1-, since n = o(γ n ), n → ∞.Hence, there exists a sequence (r n ), r n → 1-, such that I

  2r cos s ds,

	ϕ ′ r (t) =	1 4	0	2π	ln 1 + r 2 + 2r cos s exp	t 2	ln 1 + r 2 + 2r cos s ds,
	and
	ϕ ′′ r (t) =	1 4	0	2π		ln 1 + r 2 -2r cos s	2 exp	t 2	ln 1 + r 2 -2r cos s ds ≥ 0,

1/2

The first author is supported by the Chebyshev Laboratory (St.Petersburg State University) under RF Government grant 11.G34.31.0026 and by RFBR grant 12-01-00434.


Audrey Hohmann

Gonéri Le Cozannet

Emmanuel Kouokam

Maria Ilaria

Pannaccione Apa

Amélie Vagner

Pierre Thierry

Applying a structured multicriteria risk mapping method in Mount Cameroon

come   L'archive ouverte pluridisciplinaire

Applying a structured multicriteria risk mapping method in Mount Cameroon

A. Hohmann 1 , G. Le Cozannet 1 , E. Kouokam 2 , M. Ilaria Pannacione Apa 3 , A.Vagner 1 , P.Thierry 1 1 : BRGM 2 : MINIMIDT 3 : INGV This study aims at mapping the volcanic, induced landslides and earthquake risk in Mount Cameroon in an integrated way by applying a structured multicriteria method. The proposed methodology is an adaptation of Saaty"s Analytic Hierarchy Process (AHP, Saaty 1977) to risk mapping. AHP is a decision approach designed to solve complex multiple criteria problems (Saaty, 1990). After having determined the geographic entities involved in the analysis (i.e. in our study area, the villages in the Fako division of Mount Cameroon), the AHP method requires the identification of each risk component to be taken into account and a data collection. One of the advantages of AHP is to include both qualitative and quantitative criteria in the evaluation. In a typical hierarchy process, the top level reflects the overall objective (focus) of the decision problem, which is the risk in our case. The main criteria affecting the decision are represented in intermediate levels. The lowest level comprises the sub-criteria to assess the main criteria. Once a hierarchy is set up, the decision-maker can start a prioritization procedure to determine the relative importance of each criterion in each level of the hierarchy. All identified criteria are compared against each other in a pairwise comparison matrix which is a measure to express the relative preference among the criteria. For this, numerical values expressing a judgment of the relative importance (or preference) of one criterion against another have to be assigned to each criterion. Then the entity profiles are evaluated with respect to all sub-criteria, criteria and finally the risk to create a synthesis table finally.

Such a methodology has already been applied at the Merapi volcano by CVGHM/BPPTK. The same definition and components of risk have been kept for our study and so the same hierarchical scheme. The main criteria that are considered in risk assessment are the hazard itself, exposure aspect to the threat of disaster, vulnerability aspects and capacity to face disaster. Among these main criteria, we considered that each component of the risk has an equal importance.

The use of GIS tool helps spatial analysis to address multidimensional problems such as risk assessment. A matrix was created and organized according to a specific hierarchical model as defined in the previous step, and then potential datasets were selected for each sub-criterion. This matrix included 4 main criteria and 24 sub-criteria. Hazards and Exposure criteria are based on the "Gestion des Risques Naturels et Protection Civile (GRINP)" project, and the Vulnerability and Capacity criteria are associated with the report on Mount Cameroon "Socioeconomic vulnerability and resilience" written by MINIMIDT/INGV for MIAVITA project.

In the GRINP project, a zoning map of risks associated with geological phenomena in the vicinity of Mount Cameroon was produced by crossing the hazards and elements at stake in a GIS system (Thierry and al., 2007). The elements at stake which were considered as most important were identified and mapped by the GRINP team. Therefore, in our matrix, multihazard levels correspond to sub-criteria to assess the Hazard main criterion. In the same way, the result of crossing each GIS layer of stakes with multi-hazard levels correspond to subcriteria to assess the Exposure main criterion. In our study, we defined the Exposure criteria with data on the exposure of resident population, agricultural area, and number of drinking water supply stations that would be cut off if a disaster occurs.

In MIAVITA an anthropological-based perceived exposure study was carried out in Fako division of the Mount Cameroon Region by MINIMIDT and INGV. This study enabled to collect demographic and socio-economic data in several villages. Methods used in this work include a questionnaire-based survey, in-depth interviews, focus group discussions, field observations and check list to evaluate processes. Two sampling techniques were used on a geographical and population basis. For an overview of vulnerability and capacity to face disaster in this region, we took into account only the responses to questionnaires. Finally, the responses of about 1000 permanent residents were selected for analysis. As the geographical coordinates of sampled interviewees were taken using GPS and a question relative to their place of residence was asked in the questionnaire. Therefore we were able to link each person to a point in a village in our GIS project. The second step of this work was to identify indicators among the 430 questions collected in the questionnaire in order to complete the matrix for the Vulnerability and Capacity main criteria. Technically, for each village and subcriteria, the answers of residents have been analyzed, divided into 3 classes, and then class values were estimated. This way we have identified in the Vulnerability main criterion all things that are susceptible to increase the vulnerability of an area to disaster like age of respondents, level of education, poverty level, community participation etc. And as well for the Capacity main criterion, with all things which are able to reduce the impact of disaster caused by Mount Cameroon volcano eruptions like land use planning, public awareness programs, emergency response drills, disaster management plan, insurance, public adaptation, etc. This study shows an example where GIS analysis techniques combined with AHP can support the definition and calculation of spatial indicators allowing an integrated risk mapping at the scale of the entities (villages).

The MIAVITA project is financed by the European Commission under the 7th Framework Programme for Research and Technological Development, Area "Environment", Activity 6.1 "Climate Change, Pollution and Risks". Saaty, T.L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15, 234-281 Saaty, T. L. (1990). Multicriteria decision making: The analytic hierarchy process. Pittsburgh, PA: RWS Publications Thierry and al. (2007), Projet GRINP -Composante 1 Réalisation d'une carte de zonage des risques du Mont Cameroun Rapport final -Rapport BRGM 333p