Matthias Hieber

Sylvie Monniaux

In this paper, we establish maximal L p -L q estimates for non autonomous parabolic equations of the type u ′ (t) + A(t)u(t) = f (t), u(0) = 0 under suitable conditions on the kernels of the semigroups generated by the operators -A(t), t ∈ [0, T ]. We apply this result on semilinear problems of the form u ′ (t) + A(t)u(t) = f (t, u(t)), u(0) = 0.

Heat-kernels and maximal L p -L q -estimates:

the non-autonomous case

Maximal regularity results for linear initial value problems are very effective tools when dealing with nonlinear, in particular quasilinear or fully nonlinear, problems. In fact, it is known that the standard evolution operator approach is in particular not applicable to reaction-diffusion equations where the "diffusion matrices" depend on ∇u. However, involving maximal regularity results and techniques based on the implicit function theorem, one is able to treat problems of the kind described above. For an up-to-date overview of the results and techniques known in this context, we refer to the monographs [START_REF] Amann | Linear and quasilinear parabolic problems[END_REF], [START_REF] Prüss | Evolutionary integral equations and applications[END_REF], [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] and [START_REF] Clément | Global existence for a semilinear parabolic Volterra equation[END_REF].

From the point of view of harmonic analysis, it is natural to replace the property of maximal L p -regularity by the L p -boundedness of certain Banach space valued singular integrals. More precisely, let 1 < p, q < ∞, Ω ⊂ IR N open and -A be the generator of an analytic semigroup on L q (Ω). Then there is maximal L p -L q -regularity for the problem u ′ (t) + Au(t) = f (t), t ∈ [0, T ] u(0) = 0 (1) 1 if and only if the convolution operator R given by (Rf )(t) = t 0 Ae -(t-s)A f (s) ds acts boundedly on L p (0, T ; L q (Ω)). Since Mihlin's theorem for operatorvalued symbols is applicable in this situation if and only if q = 2 one is forced to use other techniques in order to prove L p -boundedness of R. By using the transference principle, Lamberton [START_REF] Lamberton | Équations d'évolutions linéaires associées à des semigroupes de contractions dans les espaces L p[END_REF] proved maximal L p -L q -regularity for (1) provided the semigroup T generated by -A acts as a contraction on L q (Ω) for all q ∈ [1, ∞]. Observe that his approach is necessarily restricted to second order differential operators and does in particular not allow to treat parabolic systems. On the other hand assuming suitable heat-kernel bounds on the semigroup generated by -A, maximal L p -L q -regularity for the solution of (1) was proved recently by Hieber and Prüss in [START_REF] Hieber | Heat kernels and maximal L p -L q estimates for parabolic evolution equations[END_REF]. For generalizations see [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF]). It seems that the idea of using heat-kernel bounds in this context was first used in [START_REF] Stroock | Multidimensional diffusion processes[END_REF] by Strook and Varadhan. The problem of maximal L p -L q -regularity for arbitrary parabolic evolution equations of the form (1) with -A being the generator of an analytic semigroup on some L p -space, as formulated by Brézis (see e.g. [START_REF] Coulhon | Régularité L p pour les équations d'évolution[END_REF]), seems to remain open, in general.

Considering quasilinear problems of the form u ′ (t)+A(t, u(t))u(t) = f (t), u(0) = 0, maximal regularity results for the non-autonomous linear equation

u ′ (t) + A(t)u(t) = f (t), t ∈ [0, T ] u(0) = 0 (2)
are of great interest. If the domains of A(t) are constant, i.e. if D(A(t)) = D for all t ∈ [0, T ] one obtains maximal regularity results for (2) by the one for the autonomous case simply by writing

u ′ (t) + A(0)u(t) = (A(0) -A(t))u(t) + f (t), u(0) = 0
and by using perturbation arguments. Observe, however, that the domains D(A(t)) vary with t for example when A(t) is the L q -realization of a second order differential operator subject to co-normal Neumann boundary conditions.

In this paper we do not only prove maximal L p -L q -regularity results for examples of this kind, but treat the general case where {A(t), t ∈ [0, T ]} satisfies the Acquistapace-Terreni commutator condition and the heat-kernels of the semigroups T t generated by -A(t) satisfy suitable bounds. The method we use is very much inspired by the ones used in [START_REF] Hieber | Heat kernels and maximal L p -L q estimates for parabolic evolution equations[END_REF] and [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF]. Essentially it is based on the technique developed in [START_REF] Duong | Semigroup kernels, Poisson bounds, and holomorphic functional calculus[END_REF] and [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF] allowing to prove L 1 -L 1 w -boundedness of singular integrals with weaker conditions on the kernel than the classical Hörmander almost L 1 -condition.

In order to treat parabolic differential operators acting on Riemannian manifolds we choose spaces of homogeneous type as underlying setting.

This paper is organized as follows. In Section 2, we give the precise assumptions and state our main theorem. The proof of the main result is given in Section 3. In Section 4, we apply of our result to semilinear problems of the form u ′ (t) + A(t)u(t) = f (t, u(t)), u(0) = 0, where the domains of A(t) may vary with t.

Throughout this paper, we denote by L(X, Y ) the space of all bounded linear operators from X to Y , whenever X and Y are Banach spaces. If A is a linear operator in X, we denote its domain by D(A), its resolvent set by ρ(A) and its spectrum by σ(A). Moreover, for any θ ∈ (0, π), we set

Σ θ := {z ∈ l C \ {0}; | arg(z)| < θ}.

Assumptions and the main result

Let (Ω, m, d) be a space of homogeneous type. This means that Ω is a topological space, m is a σ-finite measure on Ω and d is a quasi-metric (i.e. d(x, z) ≤ γ d (d(x, y) + d(y, z)) for all x, y, z ∈ Ω, where γ d ≥ 1) on Ω. We assume the doubling property : there exists a constant C D ≥ 1 such that m(B Ω (x, 2r)) ≤ C D m(B Ω (x, r)) holds for all x ∈ Ω and all r > 0, where B Ω (x, r) := {y ∈ Ω; d(x, y) < r}. We remark that this property implies the strong homogeneity property (H) given as follows :

(H) there exists two constants C H ≥ 1 and ℓ > 0 such that

m(B Ω (x, ar)) ≤ C H a ℓ m(B Ω (x, r))
holds for all x ∈ Ω, a ≥ 1, r > 0.

We consider now M, a measurable subset of Ω, and we let T > 0. Let {A(t), t ∈ [0, T ]} be a family of linear densely defined operators in L 2 (M, m). We assume that (A) there exists ϕ ∈ (0, π 2 ) such that σ(A(t)) ⊂ Σ ϕ , and for all ϑ ∈ (ϕ, π) there exists a constant M ϑ > 0 such that

(λ -A(t)) -1 L(L 2 (M,m)) ≤ M ϑ 1 + |λ| holds for all λ ∈ l C \ Σ ϑ , t ∈ [0, T ].
The condition (A) implies that the operators -A(t) generate uniformly bounded analytic C 0 -semigroups (e -σA(t) ) σ≥0 on L 2 (M, m). We assume that for all t ∈ [0, T ] and for all σ > 0, there exists k t (σ, •, •), bounded and measurable on M × M such that

(e -σA(t) f )(x) = M k t (σ, x, y)f (y)dm(y) m -a.a. x ∈ M
for all f ∈ L 2 (M, m). We assume moreover that the kernels satisfy a uniform estimate of the following type :

(K) there exist a constant n > 0 and a bounded decreasing function g defined on (0, ∞) satisfying lim r→∞ r 2ℓ+γ g(r) = 0 for some γ > 0 such that

|k t (σ, x, y)| ≤ min 1 m(B Ω (x, σ 1 n )) , 1 m(B Ω (y, σ 1 n )) g d(x, y) σ 1 n
holds for all t ∈ [0, T ], σ > 0, and for m -a.a. x, y ∈ M.

It is well-known that condition (K) is satisfied for a large class of differential operators. For details in the context, we refer to [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF], [START_REF] Davies | Heat kernels and spectral theory[END_REF], [START_REF] Fabes | Gaussian upper bounds on fundamental solutions of parabolic equations : the method of Nash[END_REF], [START_REF] Arendt | Gaussian estimates for second order elliptic operators with boundary conditions[END_REF], [START_REF] Daners | Heat kernel estimates for operators with boundary conditions[END_REF]. The condition (K) implies that the semigroups {(e -σA(t) ) σ≥0 , t ∈ [0, T ]} act consistently also on L q (M, m) for 1 ≤ q ≤ ∞. Moreover, they are uniformly bounded and analytic on L q (M, m) for 1 < q < ∞ (see [START_REF] Duong | Semigroup kernels, Poisson bounds, and holomorphic functional calculus[END_REF], Proposition 2.3). Denote their generators by -A q (t). We assume moreover that the operators {A q (t), t ∈ [0, T ]} verify the following commutator conditions on L q (M, m), for all q ∈ (1, ∞).

(C q ) There exists ω q ∈ ϕ, π 2 , α q , β q ∈ [0, 1] with α q < β q and a constant c q > 0 such that

A q (t)(λ -A q (t)) -1 (A q (t) -1 -A q (s) -1 ) L(L q (M,m)) ≤ c q |t -s| βq (1 + |λ|) 1-αq for all λ ∈ l C \ Σ ωq and s, t ∈ [0, T ].
The condition (C 2 ) and the assumption (A) imply in particular that

(c 2 ) (λ -A(t)) -1 -(λ -A(s)) -1 L(L 2 (M,m)) ≤ c 2 (M ω 2 + 1) |t -s| β 2 (1 + |λ|) 1-α 2 for all λ ∈ l C \ Σ ω 2 and s, t ∈ [0, T ].
We assume also the following commutator condition on L 1 (M, m) and L ∞ (M, m).

For q = 1 and q = ∞, it holds (c q ) There exists ω q ∈ ϕ, π 2 , α q , β q ∈ [0, 1] with α q < β q and a constant c q > 0 such that

(λ -A q (t)) -1 -(λ -A q (s)) -1 L(L q (M,m)) ≤ c q |t -s| βq (1 + |λ|) 1-αq for all λ ∈ l C \ Σ ωq and s, t ∈ [0, T ].
Definition 2.1 The family of operators {A(t), t ∈ [0, T ]} belongs to the class MR(p, q), and we say that there is maximal L p -regularity on L q (M, m) if for all function f in L p (0, T ; L q (M, m)), there exists a unique function

u ∈ W 1,p (0, T ; L q (M, m)) with A q (•)u(•) ∈ L p (0, T ; L q (M, m)), verifying (2) in the L p (0, T ; L q (M, m))-sense.
We are now in the position to state the main result of this paper.

Theorem 2.2 Assume that the family {A(t), t ∈ [0, T ]} satisfies the assumptions (A), (K), (C q ) for all q ∈ (1, ∞), (c 1 ) and (c ∞ ). Let 1 < p, q < ∞.

Then the family {A(t), t ∈ [0, T ]} belongs to the class MR(p, q).

Proofs

The results in [START_REF] Acquistapace | A unified approach to abstract linear nonautonomous parabolic equations[END_REF], [START_REF] Monniaux | Semigroup methods to solve non-autonomous evolution equations[END_REF] imply the following facts. If u is a solution of (2) then u fulfills the equation

A(t)u(t) = t 0 A(t) 2 e -(t-s)A(t) (A(t) -1 -A(s) -1 )A(s)u(s) ds + t 0 A(t)e -(t-s)A(t) f (s) ds
for a.a. t ∈ (0, T ). For the time being, let q ∈ (1, ∞) be fixed. If the constant c q in (C q ) is sufficiently small, the operator Q ∈ L(L p (0, T ; L q (M, m))) defined by

(Qg)(t) := t 0 A(t) 2 e -(t-s)A(t) (A(t) -1 -A(s) -1 )g(s) ds, t ∈ (0, T ),
has norm less than 1 2 for all p ∈ (1, ∞). Observe, however, (see also [START_REF] Monniaux | A theorem of the Dore-Venni type for noncommuting operators[END_REF], Remark before Corollary 2), that the family {A(t), t ∈ [0, T ]} belongs to the class MR(p, q) if and only if this holds true for {A(t) + ν, t ∈ [0, T ]}, where ν is an arbitrary constant. Hence, there is no loss of generality in choosing the constants c q in (C q ) as small as we want, by choosing α q slightly larger. It follows that the operator 1 -Q is then invertible in L(L p (0, T ; L q (M, m))). We summarize our observations in the following proposition. Proposition 3.1 The family {A(t), t ∈ [0, T ]} belongs to the class MR(p, q) if and only if the operator S defined by

(Sf )(t) := t 0 A(t)e -(t-s)A(t) f (s) ds t ∈ [0, T ]
acts as a bounded operator on L(L p (0, T ; L q (M, m))).

In [START_REF] Hieber | Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations[END_REF], Theorem 3.2, we proved that assuming (A) and (C 2 ), the family {A(t), t ∈ [0, T ]} belongs to MR(2, 2). Hence the operator

S : L 2 ((0, T ) × M, λ ⊗ m) → L 2 ((0, T ) × M, λ ⊗ m) defined by (Sf )(t, x) = t 0 A(t)e -(t-s)A(t) f (s, •)ds (x) (3) for λ ⊗ m -a.a. (t, x) ∈ (0, T ) × M acts boundedly on L 2 ((0, T ) × M).
Here λ denotes the Lebesgue measure on (0, T ). Moreover, we proved in [START_REF] Hieber | Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations[END_REF], Theorem 3.1, that MR(p, q) is independent of p ∈ (1, ∞). Hence, the Marcinkiewicz interpolation theorem implies the following result. Proposition 3.2 Let S be defined as in [START_REF] Arendt | Gaussian estimates for second order elliptic operators with boundary conditions[END_REF]. Then the assertion of Theorem 2.2 holds true provided S, as well as its adjoint, is of weak type [START_REF] Acquistapace | A unified approach to abstract linear nonautonomous parabolic equations[END_REF][START_REF] Acquistapace | A unified approach to abstract linear nonautonomous parabolic equations[END_REF].

At this point some remarks on our strategy how to prove that S is of weak type (1, 1) are in order. For the time being consider the autonomous case, i.e. A(t) = A for t ∈ [0, T ] and write f ∈ L 1 ((0, T ) × M) in its Calderòn-Zygmund decomposition as f = g + b with b = b i . The strategy used in [START_REF] Hieber | Heat kernels and maximal L p -L q estimates for parabolic evolution equations[END_REF] and [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF] to show that R defined as in Section 1 is of weak type (1, 1) is to decompose Rb i as

RV τ i b i + (R -RV τ i )b i
with an appropriate "smoothing term" V τ i . Observe that in the non-autonomous situation the operator S defined as in (3) seems not to be treatable directly with the semigroup methods developed for the autonomous case. However, semigroup methods do work for S with Sf (t) = t 0 A(s)e -(t-s)A(s) f (s) ds.

In fact, we decompose Sb i in the following as

SU τ i b i + (S τ i -SU τ i )b i + (S -S τ i )b i
with S defined below associated to the kernel -∂ 1 k s (t -s, x, y)1l (0,t) (s). In order to control the above three terms we need assumptions (C q ), (c 1 ) and (c ∞ ).

The rest of this section is devoted to the proof of the fact that S is of weak type (1, 1). To this end, let Q = (0, T ) × M and µ = λ ⊗ m. Let S be the bounded operator on L 2 (Q, µ) defined as above. Denote the kernel of S by p.

Then p : Q × Q → IR is of the form p(t, x; s, y) = -∂ 1 k t (t -s, x, y)1l (0,t) (s) µ -a.a (t, x), (s, y) ∈ Q with the notation ∂ 1 k t (σ, x, y) = ∂k t ∂σ (σ, x, y). This means that Sf (t, x) = - t 0 M ∂ 1 k t (t -s, x, y)f (s, y) ds dm(y)
holds for each continuous function f with compact support in Q, for µ -a.a.

(t, x) / ∈ supp(f ). Next, let p be given by

Q × Q ∋ (t, x; s, y) -→ p(t, x; s, y) = -∂ 1 k s (t -s, x, y)1l (0,t) (s)
for µ-a.a. (t, x), (s, y) ∈ Q. Denote by S the operator in L 2 (Q, µ) associated with p.

Lemma 3.3 Under the conditions (A) and (c 2 ), the operator S is bounded on L 2 (Q, µ).

Proof. Since S is bounded on L 2 (Q, µ) (Proposition 3.1 and [START_REF] Hieber | Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations[END_REF], Theorem 3.2), it suffices to prove that S -S is bounded on L 2 (Q, µ). By assumption (A), we have ( S -S)f (t, •) = t 0 (A(s)e -(t-s)A(s) -A(t)e -(t-s)A(t) )f (s, •) ds

= t 0 1 2iπ Γω 2 λe -(t-s)λ ((λ -A(s)) -1 -(λ -A(t)) -1 )f (s, •) dλ ds
where ω 2 is defined as in (C 2 ) and Γ ω 2 = (∞, 0)e iω 2 ∪ e -iω 2 (0, ∞). Therefore, taking into account the estimates (A) and (c 2 ) it holds for all f ∈ L 2 (Q, µ)

( S -S)f (t, •) L 2 (M,m) ≤ ≤ (M ω 2 + 1)c 2 1 π t 0 ∞ 0 re -(t-s)r cos ω 2 (t -s) β 2 (1 + r) 1-α 2 dr f (s, •) L 2 (M,m) ds ≤ C(M ω 2 , cos ω 2 , c 2 , α 2 , β 2 ) t 0 (t -s) β 2 -α 2 -1 f (s, •) L 2 (M,m) ds,
where

C(M ω 2 , cos ω 2 , c 2 , α 2 , β 2
) is a constant depending only on the listed quantities. Applying Young's inequality, we have

( S -S)f L 2 (Q,µ) ≤ C(M ω 2 , cos ω 2 , c 2 , α 2 , β 2 , T ) f L 2 (Q,µ) .
Consider next for τ > 0, the operator U τ ∈ L(L 2 (Q, µ)) associated to the kernel u τ given by

Q × Q ∋ (t, x; s, y) -→ u τ (t, x; s, y) = 1l ((t-τ ) + ,t) (s) 1 τ k t (τ, x, y).
Moreover, define S τ ∈ L(L 2 (Q, µ)) to be the operator associated to the kernel p τ given by where n was defined in (K). Then (D, µ, δ) is a space of homogeneous type. Define H τ ∈ L(L 2 (D, µ)) associated to the kernel h τ given by

p τ : Q × Q ∋ (t, x; s, y) -→ - 1 τ s+τ s ∂ 1 k t (t -σ + τ, x, y) 1l σ<t dσ.
h τ (t, x; s, y) = e τ e -|t-s| τ min 1 m(B Ω (x, τ 1 n ) , 1 m(B Ω (y, τ 1 n ) g d(x, y)
h τ (ξ, η) ≤ c inf η∈B D (ζ,τ ) h 2 n τ (ξ, η)
holds for all ζ, ξ ∈ D and all τ > 0 ;

(c) there exists a constant C > 0 such that for all τ > 0,

D h τ (η, ξ)v(η) dµ(η) ≤ C sup ρ>0 1 µ(B D (ξ, ρ)) B D (ξ,ρ) |v(η)| dµ(η)
holds for all ξ ∈ D and all v ∈ L 2 (D, µ) ;

(d) for all τ > 0, the operator S -S τ ∈ L(L 2 (Q, µ)) associated to the kernel p -p τ verifies

ν ̺ := sup η∈Q,τ >0 δ(ξ,η)≥̺τ |p(ξ, η) -p τ (ξ, η)|dµ(ξ) < ∞
for all ̺ > 0 ;

(e) for all τ > 0, the operator

S τ -SU τ ∈ L(L 2 (Q, µ)) extends to a bounded operator on L 1 (Q, µ). Moreover, S τ -SU τ L(L 1 (Q,µ)) is independent of τ > 0.
Proof. (a) Taking into account assumption (K), the assertion follows by the choice of u τ and h τ . (b), (c) Assertions (b) and (c) were shown in [START_REF] Hieber | Heat kernels and maximal L p -L q estimates for parabolic evolution equations[END_REF], Lemma 4.2 and Lemma 4.3.

(d) The proof of assertion (d) follows the lines of [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF], Section 3.4. We omit the details.

(e) For all τ > 0 and for all f

∈ L 2 (Q, µ) ∩ L 1 (Q, µ), we have (S τ -SU τ )f (t, •) = = T 0 1 τ s+τ s (A 1 (t)e -(t-σ+τ )A 1 (t) -A 1 (σ)e -(t-σ+τ )A 1 (σ) )1l σ≤t dσ f (s, •) ds = T 0 ds 1 τ s+τ s 1l σ≤t dσ 1 2iπ Γω 1 λe -(t-σ+τ )λ ((λ -A 1 (t)) -1 -(λ -A 1 (σ)) -1 )f (s, •) dλ .
Taking into account the estimate (c 1 ), the argument given in the proof of Lemma 3.3 implies

(A 1 (t)e -(t-σ+τ )A 1 (t) -A 1 (σ)e -(t-σ+τ )A 1 (σ) )f (s, •)1l σ≤t L 1 (M,m) ≤ ≤ c 1 1 π ∞ 0 re -(t-σ+τ )r cos ω 1 (t -σ) β 1 (1 + r) 1-α 1 1l σ≤t dr f (s, •) L 1 (M,m) ≤ C(cos ω 1 , c 1 , α 1 , β 1 ) (t -σ) β 1 (t -σ + τ ) α 1 +1 1l σ≤t f (s, •) L 1 (M,m) .
Therefore, we obtain

(S τ -SU τ )f (s, •) L 1 (M,m) ≤ ≤ C(cos ω 1 , c 1 , α 1 , β 1 ) T 0 1 τ s+τ s (t -σ) β 1 (t -σ + τ ) α 1 +1 1l σ≤t dσ f (s, •) L 1 (M,m) ds ≤ C(cos ω 1 , c 1 , α 1 , β 1 ) t 0 (t -s) β 1 -α 1 -1 f (s, •) L 1 (M,m) ds.
Then, applying Young's inequality, we have

(S τ -SU τ )f L 1 (Q,µ) ≤ C(M ω 1 , cos ω 1 , c 1 , α 1 , β 1 , T ) f L 1 (Q,µ) ,
which implies the assertion. 2

Proposition 3.5 Under the conditions (A), (K), (c 2 ) and (c 1 ), the operator S is of weak type (1, 1) on Q.

Proof. First, remark that the operator S is of weak type (1, 1) on Q if and only if the operator R defined by

Rf := 1l Q S(1l Q f ) is of weak type (1, 1) on D = (0, T ) × Ω.
To prove this last assertion, we define the following bounded operators on L 2 (D, µ) :

Rf := 1l Q S(1l Q f ), R τ f := 1l Q S τ (1l Q f ), V τ f := 1l Q U τ (1l Q f ). (4)
Consider the Calderón-Zygmund decomposition of a function f ∈ L 1 (D, µ) (see [START_REF] Stein | Harmonic Analysis[END_REF], I4, Theorem 2). Then there exist N ∈ IN and κ > 0, depending only on (D, µ, δ), such that for each function f ∈ L 1 (D, µ) and for all r > f 1 µ(D) if µ(D) < ∞ and for all r > 0 otherwise, there exist functions g, b i (i ∈ IN) in L 1 (D, µ) such that we can write f as f = g + i∈IN b i and with the following

properties (i) |g(t, x)| ≤ κr for µ -a.e. (t, x) ∈ D ; (ii) for all i ∈ IN, there exist (t i , x i ) ∈ D and τ i > 0 such that supp(b i ) ⊂ B i := {(t, x) ∈ D; δ((t, x); (t i , x i )) < τ i } and b i 1 ≤ κrµ(B i ); (iii) i∈IN µ(B i ) ≤ κ r f 1 ; (iv) each point of D is contained in at most N balls B i .
In order to prove Proposition 3.5, we have to show that there exists a constant C > 0 such that for all function f ∈ L 2 (Q, µ) ∩ L 1 (Q, µ), and for all r > 0, r µ({(t, x)

∈ D; |Rf (t, x)| > r}) ≤ C f 1 . To this end, let f ∈ L 2 (D, µ) ∩ L 1 (D, µ) be fixed. For 0 < r ≤ f 1 µ(D) , we have r µ({(t, x) ∈ D; |Rf (t, x)| > r}) ≤ rµ(D) ≤ f 1 .
Let r > f 1 µ(D) be fixed. The Calderón-Zygmund decomposition for (f, r) gives then f = g + i∈IN b i where g, b i (i ∈ IN) satisfy (i) -(iv). Remark that the condition (i) of the decomposition implies that g ∈ L ∞ (D, µ) and g ∞ ≤ κr. Moreover, the conditions (ii) and (iii) imply that

i∈IN b i 1 ≤ κ 2 f 1 , and then g 1 ≤ (1 + κ 2 ) f 1 .
The idea of the following proof is to decompose Rb i as

Rb i = RV τ i b i + (R τ i -RV τ i )b i + (R -R τ i )b i
with operators R, R τ and V τ defined as in ( 4) and τ i chosen suitably. We subdivide the proof in four steps.

• Step 1 The function g defined as above belongs to L 1 (D, µ) ∩ L ∞ (D, µ) and

g 2 2 ≤ g 1 g ∞ ≤ r κ(1 + κ 2 ) f 1 .
Since the operator R is bounded on L 2 (D, µ) we have by Chebychev's inequality

µ({(t, x) ∈ D; |Rg(t, x)| > r 4 }) ≤ 16 r 2 Rg 2 2 ≤ 16 r 2 R 2 L(L 2 (D,µ)) g 2 2 ≤ 1 r 16 κ(1 + κ 2 ) R 2 L(L 2 (D,µ)) f 1 . • Step 2 The function i∈IN V τ i b i belongs to L 2 (D, µ)
, where τ i is the radius of the ball B i and the operators V τ were defined in (4). This follows from the estimates (a), (b) and (c) of Lemma 3.4 (see also [START_REF] Hieber | Heat kernels and maximal L p -L q estimates for parabolic evolution equations[END_REF], Section 5, Step II, or [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF], proof of Theorem 2.6). Moreover, there exists a constant cst (depending only on N, κ, the constant C appearing in the assertion (c) of Lemma 3.4 and the norm of the maximal Hardy-Littlewood operator on L 2 (D, µ)) such that i∈IN

V τ i b i 2 2 ≤ cst r f 1 .
Therefore, using the the fact that the operator R is bounded on L 2 (D, µ) (Lemma 3.3) and Chebychev's inequality we have, as in the first step

µ      (t, x) ∈ D; R   i∈IN V τ i b i   (t, x) > r 4      ≤ 16 r 2 R   i∈IN V τ i b i   2 2 ≤ 16 r 2 R 2 L(L 2 (D,µ)) i∈IN V τ i b i 2 2 ≤ 1 r 16 cst R 2 L(L 2 (D,µ)) f 1 .
• Step 3 By Lemma 3.4 (e), there exist a constant K > 0 such that for all i ∈

IN (R τ i -RV τ i )b i 1 ≤ K b i 1 .
Therefore, by Chebychev's inequality, it holds

µ      (t, x) ∈ D; i∈IN (R τ i -RV τ i )b i (t, x) > r 4      ≤ 4 r K i∈IN b i 1 ≤ 1 r 4 K(1 + κ 2 ) f 1 .
• Step 4

We have now to estimate the quantity

r µ      (t, x) ∈ D; i∈IN (R -R τ i )b i (t, x) > r 4      .
For that purpose, set B * i := B D ((t i , x i ), 5γ δ τ i ), where γ δ ≥ 1 is the constant appearing in the triangle inequality for the quasi-metric δ. Choose γ δ = (2γ d ) n if the constant γ d correspond to the quasi-metric d. We then have using the strong homogeneity property (H), µ(B * i ) ≤ C H (5γ δ ) ℓ µ(B i ). Therefore, using Lemma 3.4 (d) with ̺ = 4 and the properties of the Calderón-Zygmund decomposition, it holds

µ      (t, x) ∈ D; i∈IN (R -R τ i )b i (t, x) > r 4      ≤ ≤ i∈IN µ(B * i ) + µ      (t, x) ∈ D \ (∪ i∈IN B * i ); i∈IN (R -R τ i )b i (t, x) > r 4      ≤ C H (5γ δ ) ℓ i∈IN µ(B i ) + i∈IN 4 r D\B * i |(R -R τ i )b i (t, x)| dt dx ≤ C H (5γ δ ) ℓ κ r f 1 + 4 r   sup i∈IN,η∈Q δ(ξ,η)≥4τ i |p(ξ, η) -p τ i (ξ, η)|dµ(ξ)   i∈IN b i 1 ≤ 1 r C H (5γ δ ) ℓ κ + 4ν 4 κ 2 f 1 .
We have proved the existence of a constant C > 0 such that for all f ∈ L 1 (D, µ) ∩ L 2 (D, µ), Rf 1,w ≤ C f 1 , which means that R is of weak type (1, 1) on D. Hence S is of weak type (1, 1) on Q. 2 By interpolation, we know that the operator S acts as a bounded operator on L q (Q, µ) = L q (0, T ; L q (M, m)) , for all q ∈ (1, 2]. In order to prove that this is also the case for q ∈ [2, ∞), we will show that S ′ , the adjoint of S, is also of weak type (1, 1). Proposition 3.6 Under the conditions (A), (K), (c 2 ) and (c ∞ ), the operator S ′ , the adjoint of S, is of weak type (1, 1). as in the proof of Lemma 3. 3, and(c ′ 2 ) and (c ′ 1 ), it follows that the operator S′ defined by

( S′ f ) T (t, •) := t 0 G(t)e -(t-s)G(t) f T (s, •)ds is bounded on L 2 (Q, µ). Hence the operator S ′ -S′ is bounded on L 1 (Q, µ).
It remains to show that the operator S′ is of weak type [START_REF] Acquistapace | A unified approach to abstract linear nonautonomous parabolic equations[END_REF][START_REF] Acquistapace | A unified approach to abstract linear nonautonomous parabolic equations[END_REF]. Since S′ has the same form as S, we may apply Proposition 3.5 and the estimates (c ′ 2 ) and (c ′ 1 )to conclude that S ′ is of weak type (1, 1). 2 Combining Proposition 3.2 with Proposition 3.5 and 3.6 the proof of Theorem 2.2 is complete.

Applications to semilinear problems

Let Ω ⊂ IR N be a bounded domain with smooth boundary and T > 0. We consider the following parabolic initial value problem a i,j (t, x)ν i (x) ∂u ∂x j .

Here (ν 1 (x), ..., ν N (x)) denotes the outer normal vector at a point x ∈ ∂Ω. We assume moreover that (i) a i,j ∈ C([0, T ]; C 1 (Ω)), a ∈ C([0, T ]; C(Ω)), 1 ≤ i, j ≤ N (ii) a i,j = a j,i , 1 ≤ i, j ≤ N (iii) there exists a constant δ > 0 such that N i,j=1

a i,j (t, x)ξ i ξ j ≥ δ|ξ| 2 for all ξ ∈ IR N .

In order to apply our maximal L p -L qregularity result to the semilinear equation ( 5), we can rewrite (5) as an equation in L q (Ω) for 1 < q < ∞. To this end, define D(A q (t)) := {u ∈ W 2,q (Ω); B(t, x, D)u(x) = 0, x ∈ ∂Ω} A q (t)u(x) := A(t, x, D)u(x), for a.e. x ∈ Ω.

Then, for each t ∈ [0, T ], the operator A q (t) generates an analytic semigroup {T t (σ), σ ≥ 0} on L q (Ω), which is bounded provided a is sufficiently large. Assume in addition to (i) that (iv) a i,j ∈ C µ ([0, T ]; L ∞ (Ω)) (1 ≤ i, j ≤ N) and a ∈ C µ ([0, T ]; L N (Ω)) for one µ > 1 2 .

Then it was proved by Yagi [START_REF] Yagi | Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups[END_REF], Theorem 4.1 that (C q ) is satisfied, for all q ∈ (1, ∞) provided properties (i) -(iv) hold. Moreover, it is known (see e.g. [START_REF] Davies | Heat kernels and spectral theory[END_REF], [START_REF] Arendt | Gaussian estimates for second order elliptic operators with boundary conditions[END_REF], [START_REF] Daners | Heat kernel estimates for operators with boundary conditions[END_REF]) that the heat-kernel k t (σ, x, y) of T t (σ) fulfills for suitable constants M, b > 0. Finally, assumptions (c 1 ) and (c ∞ ) hold by [START_REF] Yagi | Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups[END_REF], Theorem 4.2.

Summing up, it follows that {A q (t), t ∈ [0, T ]} fulfills the assumptions of Theorem 2.2. Hence, it follows from this theorem that the family {A(t), t ∈ [0, T ]} belongs to MR(p, q). Now we rewrite (5) as u ′ (t) + A(t)u(t) = F (t, u(t)), t ∈ (0, T ) u(0) = 0 with F ∈ C 1 -,1 -([0, T ] × C 1 (Ω); L q (Ω)) given by F (t, u)(x) := f (t, u(x), ∇u(x)) x ∈ Ω, t ∈ [0, T ].

  Let D := (0, T ) × Ω and define the quasi-metric δ on D by δ((t, x); (s, y)) := |t -s| + d(x, y) n (t, x), (s, y) ∈ D,

τ 1 nLemma 3 . 4

 134 for (t, x), (s, y) ∈ D × D. Finally, set B D (ξ, ρ) = {η ∈ D; δ(η, ξ) < ρ}, the ball in D with center ξ and radius ρ. Under the conditions (A), (K), (c 2 ) and (c 1 ), the following assertions hold (a) |u τ (ξ, η)| ≤ h τ (ξ, η) for µ -a.e. ξ, η ∈ Q and for all τ > 0 ; (b) (Harnack inequality) there exist a constant c ≥ 1 such that sup η∈B D (ζ,τ )

C 1 -

 1 ∂u ∂t + A(t, x, D)u = f (t, u, ∇u) in (0, T ) × Ω B(t, x, D)u = 0 on (0, T ) × ∂Ω u(0, x) ,1 -,1 -([0, T ] × IR × IR N ). (6)The operators A(t, x, D) and B(t, x, D) are given byA(t, x, D)u = -n i,j=1 ∂ ∂x i a i,j (t, x) ∂u ∂x j + a(t, x)u B(t, x, D)u = n i,j=1

2 e -b |x-y| 2 σ

 22 |k t (σ, x, y)| ≤ M σ N , x, y ∈ Ω, σ > 0, t ∈ [0, T ]

Proof. For f ∈ L 2 (Q, µ), we have µ -a.e. S ′ f (t, •) = T t A(s) ′ e -(s-t)A(s) ′ f (s, •)ds where the operators A(s) ′ (s ∈ [0, T ]), acting on L 2 (M, m), are the adjoints of A(s) (s ∈ [0, T ]). We now set G(s) := A(T -s) ′ (s ∈ [0, T ]) and let ϕ be given as in (A). Then the family {G(s), s ∈ [0, T ]} satisfies the following estimates :

(G) σ(G(s)) ⊂ Σ ϕ and for all ϑ ∈ (ϕ, π), there exists a constant M ϑ >

The semigroups generated by G(s) (s ∈ [0, T ]) are associated to kernels k ′ s (σ, x, y) = k s (σ, y, x) m -a.e. satisfying estimate (K). The operators

) verify an assumption analog to the assumption (c 1 ):

With the previous notations, we may express Let τ ∈ (0, T ] and define Z τ := W 1,p (0, τ ; L q (Ω)) ∩ D τ (A p,q ) for 1 < p, q < ∞, where D τ (A p,q ) = {u ∈ L p (0, τ ; L q (Ω)); u(t) ∈ D(A q (t)) for a.e. t ∈ (0, τ ) and A q (•)u(•) ∈ L p (0, τ ; L q (Ω))}. The space Z τ is a Banach space with the norm u Zτ := u W 1,p (0,τ ;L q (Ω)) + A q (•)u(•) L p (0,τ ;L q (Ω))

for u ∈ Z τ . By choosing p and q large enough, we find s

Let now u ∈ Z τ and consider the mapping Φ : u → v, where v is the solution of the linear problem

for all u, u ∈ Z τ and all τ ∈ (0, T ]. The assumption [START_REF] Coulhon | Régularité L p pour les équations d'évolution[END_REF] implies that

Hence we have

for u, u ∈ Z τ . It follows from [START_REF] Coulhon | Maximal regularity and kernel bounds : observations on a theorem by Hieber and Prüss[END_REF] that

By choosing τ small enough, the Banach fixed point theorem implies the following result.

Theorem 4.1 Assume that a i,j (1 ≤ i, j ≤ N), a and f satisfy the assumptions above. Then there exist p 1 , q 1 ∈ (1, ∞), T 1 ∈ (0, T ] such that for all p ∈ [p 1 , ∞) and all q ∈ [q 1 , ∞) there exists a unique u ∈ W 1,p (0, T 1 ; L q (Ω)) with u(t) ∈ D(A q (t)) for a.e. t ∈ (0, T 1 ) and A q (•)u(•) ∈ L p (0, T 1 ; L q (Ω)), satisfying (5) on (0, T 1 ) × Ω.


Contents:

1. A short presentation of Primo a. Ex Libris Group and its product portfolio b. Primo c. Primo 3 d. Primo Central e. URM 2. Primo in the development of the Royal Library Copenhagen (with the Copenhagen University Library)

3. The next step: Primo Central, Summon or …?

The text of this presentation is available in the HAL-system: http://hprints.org/hprints-00531590/en/ Dear colleagues,

Short presentation of Primo

First some facts:

a. Ex Libris Group and its product portfolio

Primo is a product of the Israeli-based, but today international firm, Ex Libris Group, which was founded over 25 years ago. This firm is an important provider of library automation solutions offering a comprehensive product suite for electronic, digital and print materials. Its major competitors are firms like Serials Solutions. Since 2006, Ex Libris Group is owned by Francisco Partners, a private equity firm which focuses exclusively on investments in technology or technology-based services companies. This probably means that Ex Libris has to be quite good at earning money for its owner, but on the other hand the arrangement gives Ex Libris access to extensive financial resources -which it will need to fulfill its ambitious product development strategies. In 2009 its total revenue was actually 84 mio dollars. It has 11 offices around the world, 12 distributors and 4.700 customers. Among the customers are all 10 top universities in the world and 36 of the 50 top European universities, at least according to one of the more prestigious university ranking lists. So, there is no doubt that Ex Libris is a very important player in the international market for library automation systems.

Apart from the discovery and delivery system, Primo, the product portfolio of Ex Libris comprises the two library systems, Aleph and Voyager, as well as SFX, Metalib, Verde, Digitool, Rosetta, and bX Recommender.

b. Primo

Primo was launched in 2007, and the total number of institutions around the world that have chosen Primo now amounts to 738.

It is a user interface providing a one-stop solution for discovery and delivery of local and remote resources, such as books, journal articles, and digital objects.

I shall not describe the Primo system further but move on to its latest avatar, Primo 3.

c. Primo 3

Primo 3 was released in May 2010 and has until now been installed in more than 100 libraries.

It integrates OPAC functionality with the redesigned and streamlined Primo user interface and thus greatly improves the get-it functionality of the system, since users no longer have to pass from a separate discovery tool into a separate delivery tool to get access to the library's literature.

Among the interesting new features of this version are:

 Recommendations: The integration of two recommender systems enables Primo to display the most relevant recommendations generated on the basis of researchers' usage of scholarly materials  Mobile platform: The out-of-the box CSS-based iPhone interface, which can be easily customized by library staff, provides an additional library-branded access point to collections.

d. Primo Central

In sharp competition with Serials Solutions and its Summons, Ex Libris has now launched its Primo Central service.

Primo Central is a mega-aggregate index to scholarly materials. It is hosted by Ex Libris via a cloud computing infrastructure (whatever that is!), and already is claimed to cover hundreds of millions of electronic resources. The index is generated by harvesting metadata from primary and secondary publishers and aggregators and from open access-repositories.

e. URM

The latest development in terms of Ex Libris-based services is the URM-system, of which a first version has now been released to partners, and which has quite recently been taken by seven Australian libraries and library consortia.

Ex Libris describes the system in this way:

"URM integrates back-office processes across all library materials, regardless of type, format, and acquisition method-enabling your library to streamline current processes and to maximize staff efficiency. URM reduces effort and lowers costs associated with metadata management while providing new services for enrichment and enhancement of materials and community engagement with the collections and with each other."

To my mind, the most interesting and the most revolutionary part of the URM concept is the centralization of data and processes which make it possible for individual libraries not to duplicate work and processes done by other libraries.

Instead they are performed centrally and at a much lower cost in terms of staff resources. This applies not only to print resources, but also -and not the least -to digital resources.

The URM will work with a centralized metadata management system (MMS). This system is a joint, international and centralized system maintained by Ex Libris Group, and it contains the basic description of all resources (of all its library customers) in the joint system combined with an inventory of any local information, including institutional holdings.

Dear colleagues, the great players in the game now seem to be moving towards the formula for a distributed world library.

Primo in the development of the Royal Library Copenhagen (with the Copenhagen University Library)

Obviously, the catalogue of the Royal Library and later the OPAC, were always considered a necessary tool for managing print collections and gaining access to them. By 2005/2006 it became a major strategic issue for us -as for libraries in general -to provide a comprehensive discovery tool to our total collections, print and digital. Morever, our discovery tool would have to incorporate relevant developments in information technology and fit into the rapidly evolving communication and information world of our users.

It is no surprise that the major international, commercial providers of automated library systems have responded to these same needs and pressures. One of the first movers in this area was Ex Libris, which had already since 1999 been the supplier of our library system, Rex, based on Aleph.

In the course of 2006-2007 we took the next step in the development of our discovery system and moved from the Aleph OPAC to Primo. Actually we became one of the partner libraries in the development of the new system, which was opened to public use in March 2008.

Deciding to go for Primo came quite naturally, as we were already satisfied with the services of Ex Libris, and Primo appeared to be a natural development of our Aleph system.

By then other systems were being developed. In particular the second largest research library in Denmark, the State Library in Aarhus, had started to develop its own system, Summa, and later entered into a cooperative arrangement with Proquest.

The Royal Library, on the other hand, has for a number of years pursued a policy of avoiding self-development of systems and preferring shelf-ready systems from solid, international suppliers.

With Primo we obtained the following advantages:

* integrated searching of our print resources (located in a great number of individual libraries and collections) * integrated searching of digital periodicals on article level of a significant proportion of our digital periodicals * a smarter and better user interface * simplification of maintenance of the system Overall, Primo has functioned well and it did represent a significant advance in terms of our discovery system. However, it has increasingly been felt as a problem that Primo was "only" a discovery system and that users had to move to a separate delivery system, the old Aleph OPAC, in order to get access to what they had discovered.

Moreover, it is a problem that searching of print resources is not integrated with searching of digital resources.

Finally, we have the problem that our e-content system, DADS, which provides content to our Primo facility for searching digital resources, covers our own portfolio of digital resources inadequately. Ideally, all digital resources should be covered, but that goal may be impossible. Firms which compete with Ex Libris as a producer of discovery and delivery systems and which themselves produce and sell content through their own system, may not be willing to make this content available through Primo.

The first of these problems, i.e. the disconnection between the discovery system and the delivery system, has basically been solved with the implementation of Primo 3, which was opened to our users in the middle of October this year.

But we do need to solve the other problems mentioned, i.e. integrated searching of print and digital resources and much better coverage of our digital resources in the search system. Moreover, we need a much smarter and simpler and cheaper system for managing electronic resources, and we also need a system which integrates getting material from our own collections with getting materials from external systems.

So, where do we go from here?

The next step: Primo Central, Summon or …?

As you are probably aware there are at present at least four major players offering systems which fulfill our requirements as described above.

One is Ex Libris Group with Primo Central. Another is Serials Solutions with Summon. I have the impression that Serials Solutions was a first mover in this area, but Ex Libris Group has now woken up and is working hard to catch up.

A third system is Ebsco's Discovery, which was taken recently by Liverpool's university library, but we are having problems with their proprietary policies, which seem to us to represent a competitive disadvantage.

A fourth system is OCLCs World Cat. It offers a tri-level, international, regional and local solution, which is quite interesting in view of the fact that no single library can today fulfill all its users' needs and has to depend on regional and international networks to fulfill its role properly. This system is now beginning to be taken up by European libraries. WorldCat is rapidly developing its facility for integrated searching of print and digital resources with linkage to full text.

And there are certainly other systems out there, now, and the field is burgeoning with new players and ideas.

What shall non-specialist library directors think?

Personally, I prefer to go for a safe solution offered by a major, international, commercial player. It must of course be smart and very user-friendly, and it must feel comfortable to users attuned to present day information and web-technologies, and it must have a mobile user interface. It must also be competitive in terms of pricing! I consider integrated searching of print and digital to be an essential feature of a 2010-discovery tool, and a major competitive advantage will be the proportion of our digital resources which may be searched on article and chapter level and made available through the system.

Finally, it must provide an Electronic Resources Management System which makes it possible for the library to reduce its local management of electronic resources to a minimum.

Our course has not yet been decided, but I feel that Primo and Summon are two good candidates in the short run. WorldCat may become very attractive in some years, but I would prefer that some other, large European libraries have tried it out successfully before we consider it seriously. In this area I do not want to be in the adventurous avant-garde, but rather just behind!

In the broader picture I see three problems which keep nagging my mind.

The first one is: to what degree is it still relevant for individual researchers and students to use catalogues which only cover the print collections of one library, when they have very easy access to all print collections of -in this case -Danish libraries?

The second one is: researchers and at least graduate students seem to be very familiar with the abstracting and indexing services of their own subject disciplines, like Psych Info in psychology, PubMed in medicine, Historical Abstracts in history and others. Will they really search for digital articles through the library catalogue, or should we rather use our efforts to make them available through relevant subject systems?

The third one is: if our user can find all articles and books through Google etc., why should libraries maintain their own, individual systems?

Whatever the future will bring in the long run there is no doubt that for the present research libraries must still provide discovery tools specific to their own print and digital collections.

And certainly Primo, Summon and similar discovery, delivery and management systems have a key role to play in the next phase of library development. What they will eventually offer remains to be seen, but I have the feeling that a radically new library concept may be developing in front of our eyes: the distributed world library with centralized management of metadata, inventories and acquisition workflows, combined with local service functions and print collections.

How far it will go and how soon, we cannot see now: a veritable battle is going on out there, but when the dust settles I think that we may really have a system which will be much better, though very different … Thank you!


Jiasong Wu
email: jswu@seu.edu.cn

Lu Wang
email: wanglu@seu.edu.cn

Lotfi Senhadji
email: lotfi.senhadji@univ-rennes1.fr

Huazhong Shu

Improved Radix-3 Decimation-in-Frequency Algorithm for the Fast Computation of Forward and Inverse MDCT

Shu et al. recently proposed an efficient radix-3 decimation-in-Frequency (DIF) algorithm for the fast computation of forward and inverse modified discrete cosine transform (MDCT).

Their algorithm decomposes the MDCT of a sequence with length N = 2×3 m and the IMDCT of a sequence with length N = 4×3 m into three N/3-point MDCTs and IMDCTs, respectively. In this paper, we present a refinement of their approach. Comparison with Shu's algorithm shows that the number of arithmetic operations can be reduced from 17.5% to 20.5% for MDCT and from 23.1% to 27.9% for IMDCT when N is varying from 36 to 324.

Introduction

The problem of efficient computation of the modified discrete cosine transform (MDCT) and inverse MDCT (IMDCT) [START_REF] Princen | Subband/transform coding using filter bank design based on time domain aliasing cancellation[END_REF] has attracted attention of many researchers owing to its wide applications in several international standards and commercial products such as MPEG-1 [START_REF]Information technology-Coding of moving pictures and associated audio for digital storage media at up to about 1.5 Mbit/s-Part 3: Audio[END_REF], MPEG-2 [START_REF]Information technology-Generic coding of moving pictures and associated audio-Part 3: Audio[END_REF], and AC-3 [START_REF]Digital audio compression (AC-3) standard[END_REF] in audio coding. In the past decade, many fast algorithms have been reported in the literature (see, e.g., [START_REF] Britanak | Fast computational structures for an efficient implementation of the complete TDAC analysis/ synthesis MDCT/MDST filter banks[END_REF], [START_REF] Britanak | New universal rotation-based fast computational structures for an efficient implementation of the DCT-IV/DST-IV and analysis/synthesis MDCT/MDST filter banks[END_REF] for a survey). It is worth mentioning that the MDCT is equivalent to the modulated lapped transform (MLT) introduced by Malvar [START_REF] Malvar | Signal Processing With Lapped Transforms[END_REF].

Since the layer III specifies two different block sizes: a long block (N = 36) and a short block (N = 12), the MDCT implementation via the complex-valued FFT can not be directly achieved. To solve this problem, Britanak [START_REF] Britanak | An efficient implementation of the forward and inverse MDCT in MPEG audio coding[END_REF][START_REF] Britanak | A new fast algorithm for the unified forward and inverse MDCT/MDST computation[END_REF][START_REF] Britanak | An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice-versa[END_REF][START_REF] Britanak | New fast computational structures for an efficient implementation of the forward/backward MDCT in MP3 audio coding standard[END_REF], Lee [START_REF] Lee | Improved algorithm for efficient computation of the forward and backward MDCT in MPEG audio coder[END_REF], Cheng [START_REF] Cheng | Fast IMDCT and MDCT algorithms⎯a matrix approach[END_REF], and Truong [START_REF] Truong | Fast algorithm for computing the forward and inverse MDCT in MPEG audio coding[END_REF] proposed various approaches based on the computation of lower order discrete cosine transform (DCT). Recently, Shu et al. [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF] developed a fast radix-3 decimation-in-Frequency (DIF) algorithm for computing the MDCT of a sequence with length N = 2×3 m and the IMDCT of a sequence with length N = 4×3 m . Wu et al. [START_REF] Wu | Mixedradix algorithm for the computation of forward and inverse MDCTs[END_REF] proposed a mixed-radix MDCT algorithm based on radix-2 DIF algorithm and radix-3 decimation-in-time (DIT) algorithm.

In this paper, we propose a refinement of the algorithm presented in [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF]. We improve their MDCT algorithm for data sequences with length N = 4×3 m and extend their IMDCT algorithm to support the length N = 2×3 m . We show that the improved algorithm achieves a substantial reduction of the arithmetic operations compared to Shu's algorithm.

Original Shu's MDCT and IMDCT algorithm

The MDCT of an input data sequence x(n), n = 0, 1, 2, …, N -1, is defined as [START_REF] Princen | Subband/transform coding using filter bank design based on time domain aliasing cancellation[END_REF] ,

k = 0, 1, …, N/2 -1,
(1) and its inverse transform is given by ,

n = 0, 1, 2, …, N -1,
(2) where the length N is assumed to be divisible by 6, i.e., N = 2×3 m . In [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF], the following approach to MDCT computation is described.

Computation of

A(k) = X(3k + 1), k = 0, 1, …, N/6 -1. , (3) with a n =x(n), b n =x(N/3+n), c n =x(2N/3+n),
, , , and

for n = 0, 1, …, N/3- 1.

Computation of B

(k) = X(3k) + X(3k + 2), k = 0, 1, …, N/6 -1. (4)
with .

Computation of C

(k) = X(3k) -X(3k + 2), k = 0, 1, …, N/6 -1.
(5) Note that A(k), B(k) and C(k) are all the N/3-length MDCTs. In [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF], a radix-3 algorithm was derived for IMDCT of sequences with length N = 4×3 m which is summarized below.

Computation of

, n = 0, 1, …, N/3 -1.

, [START_REF] Britanak | New universal rotation-based fast computational structures for an efficient implementation of the DCT-IV/DST-IV and analysis/synthesis MDCT/MDST filter banks[END_REF] with

f k =X(k), g k =X(N/6+k), h k =X(N/3+k
), and for k = 0, 1, …, N/6 -1.

Computation of

, n = 0, 1, …, N/3 -1. [START_REF] Malvar | Signal Processing With Lapped Transforms[END_REF] with .

Computation of ,

n = 0, 1, …, N/3 -1. (8)
with and .

Note that the assumption of N being a multiple of 12 instead of a multiple of 6 is necessary to obtain equation (8).

Improved MDCT and IMDCT algorithm

In practical application, the length of sequences, N, is often divisible by 4. For this reason, we present in this section an improved approach for computing the MDCT of sequence length N = 4×3 m . We also propose an improved algorithm for computing the IMDCT in which the sequence length N needs only to be a multiple of 6. More importantly, the proposed IMDCT algorithm is more efficient than that reported in [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF].

Improved MDCT algorithm

In this subsection, we consider the case where the sequence length N is divisible by 4, i.e., N = 4×3 m , m ≥ 1. In this case, A(k) and B(k) can be obtained with ( 3) and ( 4), and C(k) is calculated in the following manner. [START_REF] Britanak | A new fast algorithm for the unified forward and inverse MDCT/MDST computation[END_REF] If N is a multiple of 12, then Eq. ( 9) can be rewritten as Figure 1. Flow graph of a 12-point MDCT [START_REF] Britanak | An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice-versa[END_REF] Eq. [START_REF] Britanak | An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice-versa[END_REF] shows that C(k) is the length-N/3 MDCT. Moreover, the input sequence of ( 10) is similar to that of (4) except for the twiddle factors. Figure 1 shows the flow graph of the realization of 12-point MDCT.

Improved IMDCT algorithm

In this subsection, we show how to calculate the IMDCT in a more efficient way.

Computation of

, n = 0, 1, …, N/3 -1. [START_REF] Britanak | New fast computational structures for an efficient implementation of the forward/backward MDCT in MP3 audio coding standard[END_REF] where ,

,

. ( 14)

For , we have

Since N is a multiple of 6, we obtain , and are all N/6-point IMDCTs. Contrary to the algorithm proposed in [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF], the assumption of N being a multiple of 12 is not required to derive equation (23). Moreover, the input sequence of ( 23) is similar to that

4.

Computational complexity and comparison results

In this section, we will analyze the computational complexity of the improved radix-3 DIF MDCT and IMDCT algorithms and compare the results with the original algorithms [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF]. The comparison of the advantages and disadvantages of these MDCT-based algorithms with the DCT-based algorithms [START_REF] Britanak | Fast computational structures for an efficient implementation of the complete TDAC analysis/ synthesis MDCT/MDST filter banks[END_REF][START_REF] Britanak | New universal rotation-based fast computational structures for an efficient implementation of the DCT-IV/DST-IV and analysis/synthesis MDCT/MDST filter banks[END_REF][START_REF] Malvar | Signal Processing With Lapped Transforms[END_REF][START_REF] Britanak | An efficient implementation of the forward and inverse MDCT in MPEG audio coding[END_REF][START_REF] Britanak | A new fast algorithm for the unified forward and inverse MDCT/MDST computation[END_REF][START_REF] Britanak | An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice-versa[END_REF][START_REF] Britanak | New fast computational structures for an efficient implementation of the forward/backward MDCT in MP3 audio coding standard[END_REF][START_REF] Lee | Improved algorithm for efficient computation of the forward and backward MDCT in MPEG audio coder[END_REF][START_REF] Cheng | Fast IMDCT and MDCT algorithms⎯a matrix approach[END_REF][START_REF] Truong | Fast algorithm for computing the forward and inverse MDCT in MPEG audio coding[END_REF], we refer the readers to [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF] and [START_REF] Wu | Mixedradix algorithm for the computation of forward and inverse MDCTs[END_REF].

Computational complexity of the proposed MDCT algorithm

Since A(k), B(k) and C(k) are all the N/3-length MDCTs, the coefficients X(k) be obtained from A(k), B(k) and C(k), k = 0, 1, …, N/6 -1, with N/3 additions. Hence, we can calculate an N-length MDCT via the computation of three N/3-point MDCTs. Note that the computation of in (4) and in (10) requires four multiplications for each n. However, for n = 4×3 m , when n = (N/4 -1)/2, we have θ n = π/4, . In this case, two multiplications can be saved. Thus, the computational complexity is given by

for N = 4×3 m , m ≥ 1. ( 24
)
with and , where M, A and P denote Multiplications, Additions and the Proposed algorithm, respectively. When applying the algorithm presented in [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF] for N = 4×3 m , neither multiplication nor addition can be saved. The computational complexity of Shu's algorithm is

for N = 4×3 m , m ≥ 1.
(25) where S denotes Shu's algorithm. Table 1 lists the number of arithmetic operations needed by the two approaches for N = 4×3 m , m ≥ 1.

Computational complexity of the proposed IMDCT algorithm

To obtain the input data sequences in (20) and ( 23), four multiplications are required for each k. However, for N = 2×3 m , m ≥ 1, when k = (N/6 -1)/2, we have θ k = π/6, , . In this case, we can save three multiplications and two additions. Meanwhile, we can obtain the sequence from and with 2N/3 additions. The computational complexity of the proposed approach is given by for

N = 2×3 m , m ≥ 1. (26)
with and (See [START_REF] Wu | Mixedradix algorithm for the computation of forward and inverse MDCTs[END_REF]).

For N = 4×3 m , m ≥ 1, when k = (N/4-1)/2, we have θ k =π/4. In this case, two multiplications can be saved.

Moreover, the sequence has the following symmetries [START_REF] Cheng | Fast IMDCT and MDCT algorithms⎯a matrix approach[END_REF] n = 0, from which we can obtain with N/3 additions. The computational complexity is given by for N = 4×3 m , m ≥ 1.

(28) with and . In [START_REF] Shu | Radix-3 algorithm for the fast computation of forward and inverse MDCT[END_REF], the computational complexity is given by: for N = 4×3 m , m ≥ 1.

(29) In Table 2, we summarize the computational complexity of the two algorithms with length N = 4×3 m , m ≥ 1.

Conclusions

We have presented an improvement of the recently published algorithm for computing the MDCT and IMDCT based on the calculation of lower order MDCTs and IMDCTs, respectively. The comparison results that the proposed approach outperforms the original one in terms of the number of arithmetic operations.

.

  18), (20) and (23) show that

Figure 2 .

 2 Figure 2. Flow graph of a 12-point IMDCT of (20) except for the twiddle factors. This permits reducing the number of arithmetic required in the calculation of and . The flow graph corresponding to a 12-point IMDCT computation is shown in Figure 2.

Table 1 .

 1 Comparison of the improved MDCT algorithm with Shu's algorithm in terms of arithmetic complexity for N = 4×3 m , m ≥ 1.

		Shu's algorithm [15]	Improved algorithm	Saved
	N	Mul	Add	Total	Mul	Add	Total	(%)
	12	25	55	80	23	43	66	17.5
	36	123	285	408	115	213	328	19.6
	108	513	1215	1728	487	891	1378	20.3
	324 1971	4725	6696	1891	3429	5320	20.5

Table 2 .

 2 Comparison of the improved IMDCT algorithm with Shu's algorithm in terms of arithmetic complexity for N = 4×3 m , m ≥ 1.

		1, …, N/4 -1,		(27)
	As a result, only	, for n = 0, 1, …, N/4 -1 and n
	= N/2, N/2 + 1, …, 3N/4 -1, need to be calculated.
	Therefore, we can only evaluate	,	and
	for

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60873048, by the National Basic Research Program of China under Grant 2010CB732503, by a Program of Jiangsu Province under Grant SBK200910055 and the Natural Science Foundation of Jiangsu Province under Grant BK2008279. work is also supported by the "Eiffel Doctorate" excellence grant of the French Ministry of Foreign and European Affairs.