Sandrine

S Trapp

L Le Denmat

J.-C Cagnon

Olivier Toussaint

Fruchart

S Da Col

S Jamet

M Staňo

B Trapp

S Le Denmat

L Cagnon

J C Toussaint

O Fruchart

Nucleation, imaging and motion of magnetic domain walls in cylindrical nanowires

published or not. The documents may come

I. INTRODUCTION

The motion of magnetic domain walls (DWs) in onedimensional structures has been a subject of increasing interest over the past two decades [START_REF] Thomas | Micromagnetism[END_REF] . Such structures provide a model playground to investigate DW motion under magnetic field or spin-polarized current. They have been proposed to serve as a basis for logic 2 or memory schemes 3 . So far, fundamental physics and demonstrators made use of flat strips patterned out of thin films, for ease of fabrication and inspection. One may also consider the cylindrical geometry, which we name wire in the following. Wires can be fabricated with bottom-up techniques by electroplating magnetic metals in insulating templates displaying cylindrical pores 4 . Dense arrays of vertical wires would be the natural geometry to implement the proposal of a 3D magnetic race-track memory 3 . For wires, theory and simulations [5][6][START_REF] Thiaville | Spin dynamics in confined magnetic structures iii[END_REF] suggested the existence of two types of DWs: the transverse wall and the Bloch-point wall. The features of their motion under field [5][6][START_REF] Thiaville | Spin dynamics in confined magnetic structures iii[END_REF] or current [START_REF] Thiaville | Nanomagnetism and spintronics[END_REF] were predicted to be even more simple than in strips, mostly precessional in its azimuth in the former case, and purely translational for the second case with speed around 1 km/s, and absence of Walker instabilities.

To search for this physics, there are three requirements: nucleate DWs in a controlled fashion; image them with a simple technique; the material is sufficiently soft so that DWs may be moved under moderate field. These three steps are reported in this manuscript.

II. TECHNIQUES

As regards synthesis, we start from insulating porous templates obtained by anodization of aluminum in oxalic acid 4 . Constant voltage leads to straight pores, while for some wires bursts at higher voltage have been used to create local protrusions, i.e. with larger diameter. Fe 20 Ni 80 * olivier.fruchart@cea.fr wires were then obtained by electroplating at -1.0 V versus saturated calomel electrode in an electrolyte containing 0.5 M Ni 2+ and 0.02 M Fe 2+ with pH = 3. Finally, the template is dissolved in 2 M NaOH , the wires are rinsed several times in water and isopropanol before a drop of solution is left to dry on a supporting surface for further inspection.

Magnetic force microscopy (MFM) was performed with an NT-MDT NTegra Aura instrument. We used Olympus AC240TS cantilevers (stiffness ≈ 2 N/m), customcoated with Co 80 Cr 20 of thickness from 3 to 10 nm. Imaging was performed in air with the ac two-pass technique, monitoring the phase during the lifted pass. The peak-topeak amplitude of the tip is circa 40 nm. The lift height is in the range 20-50 nm. Micromagnetic simulations were performed with the home-made finite-elements micromagnetic code FeeLLGood 9 . MFM contrast was estimated as the map of the second vertical derivative of the vertical component of the simulated stray field.

III. NUCLEATION

Several strategies have been demonstrated to prepare DWs in a controlled manner in patterned strips, such as injection from a large pad 10 or nucleation at the bends of curved wires using a large transverse field 11 . The lesser versatility of design in bottom-up systems makes the preparation of DWs an issue specific to wires.

In a long and narrow wire made of a soft magnetic material, uniform magnetization is the ground state. A nonuniform distribution of magnetization may develop locally at either end such as "C" or curling end domains 5,6 . These eventually lead to nucleation of a DW at a value of magnetic field H n lower than the average transverse demagnetizing field, the latter being close to M s /2. Letting aside thermal activation, the nucleation field is determined by the wire diameter normalized to the dipolar exchange length ∆ d = 2A/(µ 0 M 2 s ), with A the exchange stiffness. The value of H n decreases for increasing diameter 12 . H n may equal or exceed 100 mT for sub-50 nm-diameter wires. Thus, unless the material suffers from very large local pinning (in which case we would disregard it to investigate DW motion) 13 , the nucleation of a DW is immediately followed by its very fast propagation along the wire and annihilation at the other end, leaving again the wire in a single-domain state.

One possibility to nucleate and keep a DW at remanence, is to align magnetization exactly perpendicular to the wire axis using a large external field, before going back to remanence. An oscillatory demagnetization along such a direction may be used as well. One or several DWs may be nucleated at a location away from the ends of the wire, thanks to local imperfections or thermal fluctuations. This has been employed successfully by others, and made possible the first observation of a transverse wall in a wire, using electron holography 14 . However, our experience is that the alignement of the applied field is critical. Indeed, magnetization is already close to parallel to the wire axis under a transverse field back to ≈ 100 mT. Thus, a misalignment of less than one degree is enough to move a created DW towards an end, thus to annihilate it, if the propagation field is of the order of ≈ 1 mT. It is desirable to have more robust procedures for nucleation of DWs. We describe below two such procedures that we implemented successfully.

The first procedure is to modulate the wire diameter along its length. Indeed, as the DW energy increases with the diameter [START_REF] Thiaville | Spin dynamics in confined magnetic structures iii[END_REF][START_REF] Jamet | Magnetic nano-and microwires: Design, synthesis, properties and applications[END_REF] , protrusions and constrictions are expected to act as energy barriers and wells, respectively. The modulation may induce sufficient pinning so that demagnetization with a transverse magnetic field as described previously, is less critical with the alignment. This procedure is illustrated on Fig. 1(a-c), and was used by us to evidence again transverse walls and also for the first time Bloch-point walls by photo-emission electron microscopy 17 . Note however that, as the resulting pinning field is expected to be lower than the longitudinal nucleation field H n , a magnetization process with a magnetic field applied along the wire axis still consists of nucleation-propagation-annihilation and cannot lead to a multi-domain wire [START_REF] Pitzschel | [END_REF] .

The second procedure consists in making use of a curved shape, saturating magnetization across the radius, before coming back to remanence. This procedure is straightforward to implement in strips, designed at will by lithography 11 . As for cylindrical wires, it sometimes happens that a wire is bent during the dispersion procedure. It is possible to drastically increase the yield by first aligning the wires along a given in-plane direction thanks to an applied magnetic field, followed by drying the solution with a blow of air along the in-plane direction transverse to the field. Fig. 1(d,e) shows a wire prepared this way, where head-to-head and tail-to-tail DWs could be nucleated at two opposite bends.

IV. IMAGING

DW motion cannot be tracked with giant magnetoresistance like for strips 18 , as a current-in-plane spin- valve structure is not easy to implement for a wire. Magnetic microscopy is therefore expected to play a key role, with MFM being a tool of choice, as an in-lab technique requiring no special sample preparation. However, MFM has known issues, such as tip-sample interaction and image analysis. We discuss below the specific aspects of these in the case of wires.

Tip-sample interaction must be small enough to avoid dragging DWs during imaging, as low-pinning materials are targeted. The stray field of an MFM tip extends over a length scale similar to the radius of curvature of its apex, which is a few tens of nanometers. Thus, for the wires considered here, the interaction issue is a priori stronger than for flats strips of width several hundreds of nanometers, because the entire DW is under the influence of the stray field. The present MFM images were made using tips with magnetic coating in the range 3-10 nm, which was found to avoid dragging DWs in wires with pinning field as low as 1 mT. This comes at the expense of a lower MFM signal, with a phase shift much smaller than a degree. On the other hand, such low-moment tips avoid most of the mutual contrast expected to scale like the square of the tip moment 19 , thus giving a more faithful map of the stray field emanating from the sample.

As regards contrast analysis, in most instruments the tip oscillates essentially along the normal to the sample, and the tip magnetization is also along the same direction. Thus, it is assumed that MFM reflects a vertical derivative of the vertical component of the stray field, itself related to the neighboring sample magnetic charges.

Thus, DWs are expected to display a monopolar contrast, dark or light depending on their polarity. To first approximation this is the case in Fig. 1(c,e). As already noticed, modulations of diameter [START_REF] Pitzschel | [END_REF] as well as roughness or structural / anisotropy fluctuations 20 also induce a local contrast, with monopolar and dipolar feature along the wire direction, respectively. However, a closer look reveals also a transverse dark/light dipolar contrast perpendicular to the axis at DWs, diameter modulations and wire ends (Fig. 1, and zoom on a DW on Fig. 2a). Rotating the wire by 90 • so that the cantilever is oriented along the wire axis, yields a side contrast again opposite to the contrast on wire, however now symmetric with respect to the wire axis (Fig. 2b). The ground for the contrast is the following. When the magnetic center of mass of the tip is below the mid-height plane of the wire, the vertical component of stray field is opposite to that above the wire (Fig. 2e). The asymmetric contrast occurring for wires horizontal in the images (i.e. transverse to the cantilever direction) stems from the tilt of the tip apex with respect to the sample normal, providing less access to the back side of the wire (Fig. 2e). The angle results from the tilt of the cantilever from the sample plane, combined with the tilt of the tip axis with respect to the normal to the cantilever for the Olympus AC series (Fig. 2d). Note that the former tilt, which induces a non-vertical direction of oscillation, may also play a role in the front-versus back side contrast, weighing spatial derivatives of the stray field along two directions 21,22 .

Fig. 2b is reasonably reproduced by the simulation of MFM contrast of a Bloch-point wall (Fig. 2c), although not taking into account the above-mentioned tilts. Note also that it is not granted that experimentally a Blochpoint wall may be distinguished from a transverse wall, due to the finite spatial resolution. Anyway, it would be wrong to interpret Fig. 2a as a signature for a transverse wall; it is an instrumental feature, which has the strongest signature for large-diameter wires as in Fig. 1c compared to Fig. 1e, and more generally for small thickness of tip coating, small oscillation amplitude and small lift height (Fig. 2f).

V. PROPAGATION

The wires have been subjected to a quasistatic magnetic field during typically 1 s. The MFM tip is parked a few micrometers away from the imaging area during the pulse, so that no magnetic bias from the tip is acting on the wire. Imaging is then performed at remanence.

DWs could be moved in the two afore-mentioned cases: wires with either bends or modulations of diameter. In both cases there exists a distribution of pinning sites along the wire length. The distribution of pinning fields in a one-dimensional system due to statistical disorder, has been one of the earliest concepts to describe magnetization reversal. Its general form is now known as the Becker-Kondorski model 23,24 . It has been detailed re- cently for wires, evaluating the impact of e.g. roughness and fluctuations of magnetocrystalline anisotropy 25 . Examination of several wires reveals a broad distribution of wire-and location-dependent pinning fields, from below 0.1 mT to around 10 mT. No clear correlation was found with roughness, so that it may result from a material issue involving microstructure and/or strain. Note that such distributions are also a common feature for thin strips deposited by physical means such as sputtering, when investigated step by step by magnetic microscopy 26 .

Conclusion

We demonstrated two methods for the controlled nucleation of domain walls (DWs) in cylindrical wires, and highlighted specific features of MFM contrast for such wires. Motion of the DWs is demonstrated with pinning field strength in the range 0.1-10 mT. These values are similar to those in strips with in-plane or out-of-plane magnetization, in which the dynamics of DW motion was

Fig. 1 :

 1 Fig. 1: (a) Scanning electron, (b) AFM (c) MFM of a wire with diameter 80 nm displaying local protrusions with diameter 150 nm, and demagnetized with a large field applied perpendicular to the plane. (d,e) Atomic and magnetic force microscopy of a bent wire with diameter 60 nm, prepared with an in-plane saturating field.

Fig. 2 :

 2 Fig. 2: (a,b) MFM images of a domain wall in a wire with diameter 70 nm, with the cantilever perpendicular and parallel to the wire axis, respectively. (c) Simulated ∂ 2 Hz/∂z 2 , based on a Bloch-point wall. The side sketch shows the 10 nmlift surface where the contrast is calculated, reflecting a conical shape for the tip, rounded with a radius of curvature 10 nm. The width at the base of the wire was set to 105 nm. (d) Schematic shape of the Olympus tips, and their tilt in our microscope. (e) Sketch for the stray field associated with magnetic charges in a wire. The bottom part is the cross-section of the experimental topography associated with (f), with a true aspect ratio (f) Experimental single-line scan close to the end of a 140 nm-diameter wire, while varying the lift height from 20 to 200 nm from left to right (oscillation amplitude kept constant at 50 nm).

Fig. 3 :

 3 Fig. 3: (a-e) Successive steps of remagnetization of the wire shown in Fig. 1c. The images are taken at remanence, each following a magnetic field with an increasing magnitude from top to bottom. Arrows depict the local direction of magnetization. Positive fields point to the right. In (d) and (e) DWs have moved across modulations of diameter.

Acknowledgements

We acknowledge helpful discussions with A. Wartelle (Institut NEEL) and A. Fernandez-Pacheco (Cavendish Laboratory). This project has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n • 309589 (M3d).


Lorenzo Brandolese

CHARACTERIZATION OF SOLUTIONS TO DISSIPATIVE SYSTEMS WITH SHARP ALGEBRAIC DECAY

Keywords: 2000 Mathematics Subject Classification. 42B25, 35K05, 35Q30. Key words and phrases. Heat equation, Navier-Stokes, Decay, Energy, Besov, Diffusion

We characterize the set of functions u0 ∈ L 2 (R n ) such that the solution of the problem ut = Lu in R n × (0, ∞) starting from u0 satisfy upper and lower bounds of the form c(1

Here L is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on u0 for a solution of the Navier-Stokes system to satisfy sharp upper-lower decay estimates as above.

In doing so, we will revisit and improve the theory of decay characters by C. Bjorland, C. Niche, and M.E. Schonbek, by getting advantage of the insight provided by the Littlewood-Paley analysis and the use of Besov spaces.

Introduction

The theory of decay characters was first introduced by C. Bjorland and M.E. Schonbek in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] and further developped by C. Niche and M.E. Schonbek in [START_REF] Niche | Decay characterization to dissipative equations[END_REF], with the motivation of obtaining sharp upper and lower bound estimates for the L 2 -norm of solutions to a large class of linear or semilinear parabolic systems: so far, this theory has been successfully applied, e.g., to the heat equation, the Navier-Stokes equations [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF], the quasi-geostrophic equations [START_REF] Niche | Decay characterization to dissipative equations[END_REF], several compressible approximations of Navier-Stokes [START_REF] Niche | Decay characterization to dissipative equations[END_REF][START_REF]Comparison of Decay of Solutions to Two Compressible Approximations to Navier-Stokes Equation[END_REF] and to the Navier-Stokes-Voigt equation [START_REF] Niche | Decay characterization of solutions to the Navier-Stokes-Voigt equations in terms of the initial datum[END_REF].

The first issue of the present paper is that a slight modification of the original definition of decay character makes the theory more powerful and more widely applicable: our modification allows in particular to get necessary and sufficient conditions on u 0 for the validity of L 2estimates of the form (1.1) c(1

+ t) -γ ≤ e tL u 0 L 2 (R n ) ≤ c ′ (1 + t) -γ ,
(where L is a suitable linear pseudo-differential operator, such as the Laplacian, a fractional Laplacian, etc.), whereas the original approach [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF][START_REF] Niche | Decay characterization to dissipative equations[END_REF] only gave sufficient conditions on u 0 for the validity of (1.1). We also characterize the class of initial data u 0 such that estimates (1.1) hold in terms of suitable subsets of Besov spaces. Applications of our analysis to nonlinear problems include the complete characterization of divergence-free vector fields v 0 ∈ L 2 (R 3 ) 3 such that the corresponding weak solutions of the Navier-Stokes equations satisfy two-side bounds for the energy of the form

(1.2) c(1 + t) -γ ≤ v(t) L 2 (R n ) ≤ c ′ (1 + t) -γ , 0 < γ < 5/4.
One of our main results, Theorem 5.1, will provide three different equivalent conditions on v 0 for the vality of (1.2). Our characterization does not go through when γ = 5/4, because the lower bound of the nonlinear problem in this borderline case is no longer driven by the corresponding lower bound for heat kernel.

The other important results of this paper will be summarized by Eq. (7.2) in the last section.

Notation. We will often use the symbols or in chain of inequalities to avoid the proliferation of different constants. For example, writing f (t) g(t) we mean that f (t) ≤ Cg(t) for some constant C > 0 independent on t. When we have both f (t) g(t) and g(t) f (t) we will often write f (t) ≃ g(t).

2.

Revisiting the theory of decay characters 2.1. Improvement of the basic definitions. The authors in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] introduced the notion of Decay indicator. As the original definition looks somewhat too restrictive, we redefine it in the following way: When P r (u 0 ) -= P r (u 0 ) + , then we can define the decay indicator of u 0 as P r (u 0 ) = P r (u 0 ) -= P r (u 0 ) + .

This definition is interesting only for r ∈ -n 2 , ∞ , as the decay indicator is always zero when r ≤ -n/2. Remark 2.2. Originally, the decay indicator was defined in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] as the limit (2.1) P r (u 0 ) = lim

ρ→0 + ρ -2r-n Bρ | u 0 (ξ)| 2 dξ,
implicitly assuming that the limit does exist. But this is not always the case for u 0 ∈ L 2 (R n ). When the above limit does not exist, the lower and upper decay indicators are convenient substitutes for obtaining some relevant estimates, as we will see below. Of course, P r (u 0 ) -and P r (u 0 ) + are always well defined in [0, +∞]. An example of v 0 ∈ L 2 (R n ) such that P r (v 0 ) is not well defined by the limit (2.1) can be constructed putting fast oscillations for | v 0 (ξ)| near the origin. See the following Example 6.1.

On the other hand, if

u 0 ∈ L 2 (R n ) is such that u 0 (ξ) ∼ |ξ| r as |ξ| → 0, with r ∈ (-n/2, ∞)
, then P r (u 0 ) -= P r (u 0 ) + = r and so P r (u 0 ) is well defined in this case and P r (u 0 ) = r as well.

Definition 2.3. The upper and lower decay characters of u 0 ∈ L 2 (R n ) are respectively defined by

r(u 0 ) + = sup{r ∈ R : P r (u 0 ) + < ∞}, (2.2) r(u 0 ) -= inf{r ∈ R : P r (u 0 ) -> 0}. (2.3)
Remark 2.4. For any u 0 ∈ L 2 (R n ), the upper and lower decay characters are always well defined (adopting the usual convention that inf ∅ = +∞) and satisfy the inequality

(2.4) -n/2 ≤ r(u 0 ) + ≤ r(u 0 ) -≤ ∞.
Indeed, for all r ≤ -n/2, we have P r (u 0 ) + = 0, so the first inequality is immediate. Let us now prove the inequality in the middle of (2.4). If, by contradiction, r(u 0 ) -< r(u 0 ) + , then we could choose two real numbers r < r ′ such that P r (u 0 ) -> 0 and P r ′ (u 0 ) + < ∞. But the last inequality implies P r (u 0 ) + = lim sup ρ→0 ρ 2(r ′ -r) ρ -2r ′ -n Bρ | u 0 (ξ)| 2 dξ = 0. On the other hand, 0 ≤ P r (u 0 ) -≤ P r (u 0 ) + = 0, that contredics the inequality P r (u 0 ) -> 0.

The upper decay character r(u 0 ) + will play a role in obtaining decay estimates from above for e t∆ u 0 , and the lower decay character r(u 0 ) + will be useful for obtaining estimates from below. This is the reason of our terminology.

Depending on u 0 ∈ L 2 (R n ), the sup and the inf in the definition of r(u 0 ) + and r(u 0 ) -can be achieved or not. This observation motivates next definition. Our definition below is an improvement of that of [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF].

Definition 2.5. If u 0 ∈ L 2 (R n ) is such that there exists r * ∈ (-n/2, ∞) such that (2.5) r * (u 0 ) = max{r ∈ R : P r (u 0 ) + < ∞} = min{r ∈ R : P r (u 0 ) -> 0}.
then we call this number r * = r * (u 0 ) the decay character of u 0 . We define also the decay character of u 0 in the two limit situations as follows:

r * (u 0 ) = +∞, if r(u 0 ) + = r(u 0 ) -= +∞, r * (u 0 ) = -n/2, if r(u 0 ) + = r(u 0 ) -= -n/2.
Remark 2.6. For u 0 ∈ L 2 (R n ) the decay character r * (u 0 ) does not always exist. One reason for this is that it can happen that r(u 0 ) + < r(u 0 ) -. See Example 6.3. Another reason is that one could have r(u 0 ) + = r(u 0 ) -, but the supremum or the infimum appearing in the definition of r(u 0 ) + and r(u 0 ) -are nor achieved. See Example 6.2.

In fact, r * (u 0 ) does exist and belongs to (-n/2, ∞) if and only if there exists r * ∈ (-n/2, ∞) such that 0 < P r * (u 0 ) -≤ P r * (u 0 ) + < ∞. The "only if" part of this claim is clear. The "if" part holds because, if r * < r ′ then P r ′ (u 0 ) + = ∞ (otherwise we would get the contradiction P r * (u 0 ) + = 0, as already observed right after (2.4)) and so r * = max{r ∈ R : P r (u 0 ) + < ∞}. In the same way one sees that r * = min{r ∈ R : P r (u 0 ) -> 0}.

Remark 2.7. Let us illustrate the difference between our improved definition of decay character and the original definition in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF]. The main motivation of such an improvement is the validity of the second assertion of Theorem 2.9 below. Our discussion concerns here only the case r * (u 0 ) ∈ (-n/2, ∞), but it could be adapted also to the limit situations r * (u 0 ) = +∞ or -n/2.

Originally, the decay character r * (u 0 ) was defined in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] implicitly assuming that, for all r ∈ R, P r (u 0 ) -= P r (u 0 ) + = P r (u 0 ) and also assuming that, for some r * ∈ (-n/2, ∞), 0 < P r * (u 0 ) < ∞. Under such two conditions, we see that r * = max{r ∈ R : 0 < P r (u 0 ) < ∞} = min{r ∈ R : 0 < P r (u 0 ) < ∞}. For this reason, if u 0 admits a decay character in the sense of [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] then u 0 admits a decay character in the sense of our definition and these are the same. The converse is not true, so the original definition in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] is indeed more restrictive than ours: for example, the function v 0 ∈ L 2 (R n ), constructed in Example 6.1 admits a decay character only in the sense of our definition.

2.2. Applications to upper and lower decay estimates. We start with an application of the notions of upper and lower decay indicators. As in [START_REF] Niche | Decay characterization to dissipative equations[END_REF], we consider the class of (matricial) pseudo-differential operators L with symbol

M(ξ) = P (ξ) -1 D(ξ)P (ξ), for a.e. ξ ∈ R n ,
where D(ξ) and P (ξ) are respectively diagonal and orthogonal matrices of order m, with D(ξ) ij = -c i |ξ| 2α δ i,j , and c i ≥ c > 0, for all i = 1, . . . , m and α > 0. We also assume that P (ξ) ij are homogeneous functions smooth outside ξ = 0. We are interested in establishing L 2 -estimates from above and below for solutions of the linear problem (2.6)

u t = Lu, t > 0, x ∈ R n , u| t=0 = u 0 ,
where u = (u 1 , . . . , u m ) and u 0 = (u 0,1 , . . . , u 0,m ). Basic examples include the heat equation v t = ∆v (in this case P (ξ) = I m and D(ξ) = -|ξ| 2 I m ) or the evolution problem for the fractional Laplacian (P (ξ) = I m and D(ξ) = -|ξ| 2α I m ). Examples of physical interest with P = I m arise, e.g., in fluid mechanics, see [START_REF] Niche | Decay characterization to dissipative equations[END_REF].

We will typically assume u 0 ∈ (L 2 (R n )) m . However, from now on we will not distinguish in our notations between scalar and vector-valued function spaces and write abusively u 0 ∈ L 2 (R n ) also in the vector-valued case. Proposition 2.8. Let u 0 ∈ L 2 (R n ), let L be a pseudo-differential operator as above. Let u be the solution of the linear problem (2.6).

-If P r (u 0 ) -> 0 then there is a constant C 1 > 0 such that, for all t > 0,

C 1 (1 + t) -1 α (r+n/2) ≤ u(t) 2 2 .
.

-If P r (u 0 ) + < ∞, then there is a constant C 2 > 0 such that, for all t > 0,

u(t) 2 2 ≤ C 2 (1 + t) -1 α (r+n/2) .
Proof. The proof below is a minor modification to that of [2, Theorem 5.7] (for the particular case L = ∆) or that of [6, Theorem 2.10] (for the general case).

We have

|e M(ξ)t u 0 (ξ)| = |P (ξ) -1 e D(ξ)t P (ξ) u 0 (ξ)| e -c|ξ| 2α t | u 0 (ξ)|.
Hence, for any function ρ = ρ(t),

u(t) 2 2 ≥ |ξ|≤ρ(t) |e M(ξ)t u 0 (ξ)| 2 dξ |ξ|≤ρ(t) e -2c|ξ| 2α t | u 0 (ξ)| 2 dξ ≥ e -2ctρ(t) 2α |ξ|≤ρ(t) | u 0 (ξ)| 2 dξ. Let Φ r (ρ) = ρ -2r-n |ξ|≤ρ | u 0 (ξ)| 2 dξ.
Under the assumption of the first assertion and by the definition of lim inf, we have 0 < P r (u 0 ) -= lim ǫ→0 + inf ρ∈(0,ǫ] Φ r (ρ). Hence, for some c 0 , ρ 0 > 0 and all 0 < ρ ≤ ρ 0 , we have Φ r (ρ) > c 0 . The choice ρ(t) = ρ 0 (1 + t) -1/(2α) then leads to

u(t) 2 2 e -2ctρ(t) 2α ρ(t) 2r+n Φ r (ρ(t)) ρ(t) 2r+n
and the first claim follows.

Let us prove the second assertion, following again the steps of [START_REF] Niche | Decay characterization to dissipative equations[END_REF]. First of all, the assumption on the symbol M(ξ) allows us to write

1 2 d dt u(t) 2 2 = u, P -1 DP u = -(-D) -1 P u, (-D) -1 P u = -|(-D) -1/2 P u| 2 dξ.
This insures the validity of the energy inequality

d dt u(t) 2 2 ≤ -C |ξ| 2α | u(ξ, t)| 2 dξ.
The classical Fourier splitting idea [START_REF] Schonbek | The Fourier splitting method[END_REF] is then used to deduce the estimate

d dt u 2 2 + ρ(t) 2α u(t) 2 2 ρ(t) 2α |ξ|≤ρ(t) | u(ξ, t)| 2 dξ.
Our assumption now reads P r (u 0

) + = lim ǫ→0 + sup ρ∈(0,ǫ] Φ r (ρ) < ∞. It implies that for some ρ 0 , C > 0 and all 0 < ρ ≤ ρ 0 , we have Φ r (ρ) = ρ -2r-n |ξ|≤ρ(t) | u 0 (ξ)| 2 dξ ≤ C. Hence, ρ(t) 2α |ξ|≤ρ(t) | u(ξ, t)| 2 dξ ρ(t) 2α+2r+n .
Combining the two last estimates, choosing now ρ(t) 2α = M (1 + t) -1 with M > r + n/2, and multiplying by the integrating factor (1 + t) M we arrive at

d dt (1 + t) M u(t) 2 2 (1 + t) M -1-(2r+n)/(2α) .
The upper bound follows by integration.

The following theorem clearly illustrates the importance of the notion of decay character.

Theorem 2.9.

(1) Let u 0 ∈ L 2 (R n ) be such that the decay character r * = r * (u 0 ) ∈ (-n/2, +∞) does exist. Let L as in Proposition 2.8, and u be the solution of the problem (2.6). Then for some C 1 , C 2 > 0, and all positive t,

(2.7) C 1 (1 + t) -1 α (r * +n/2) ≤ u(t) 2 2 ≤ C 2 (1 + t) -1 α (r * +n/2) .
(

) Conversely, if u 0 ∈ L 2 (R n ) is such that the solution u of the problem (2.6) satisfies estimates (2.7) with r * ∈ (-n/2, ∞) then u 0 possess a decay character and r * (u 0 ) = r * . 2
Assertion (1) appears also in [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF][START_REF] Niche | Decay characterization to dissipative equations[END_REF], but with their more restrictive definition of r * (u 0 ). This first assertion is an immediate consequence of the estimates obtained in Proposition 2.8. On the other hand, the validity of assertion ( 2) is made possible by the fact that, compared to [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF][START_REF] Niche | Decay characterization to dissipative equations[END_REF], in our definition we relaxed a little bit the requirements for the existence of the decay character. The proof of the converse part of Theorem 2.9 will be postponed in Remark 4.4, after the characterization of decay characters in terms of subsets of Besov spaces.

Besov space approach to two-side bounds for linear dissipative systems

Let us recall the definition of the homogeneous Besov spaces via the Littlewood-Paley analysis. Let ϕ be a smooth radial function with support contained in the annulus {ξ ∈ R n : 3/4 ≤ |ξ| ≤ 8/3}, such that j∈Z ϕ(ξ/2 -j ) = 1 for all ξ ∈ R n \{0}. Let ∆ j be the usual Littlewood-Paley localization operator around the frequency |ξ| ≃ 2 j , j ∈ Z, namely, ∆ j f = ϕ(•/2 j ) f , see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]. In this paper we will only need to consider the case s < n/p, 1 ≤ p, q ≤ +∞. In this case, the elements of Ḃs,q p can be realized as tempered distributions:

Ḃs p,q = f ∈ S ′ (R n ) : f = j∈Z ∆ j f, the series being convergent in S ′ (R n ), 2 js ∆ j f p j ∈ ℓ q (Z) ,
normed by

f Ḃs p,q = 2 js ∆ j f p ℓ q (Z)
.

The classical characterization of Besov spaces with negative regularity in terms of the heat kernel reads (see, e.g., [1, Theorem 3.4]):

(3.1) f ∈ Ḃ-2σ p,q ⇐⇒ t σ e t∆ f p ∈ L q (R + , dt/t), σ > 0, 1 ≤ p, q ≤ ∞,
with equivalence of the corresponding norms. In particular, for

u 0 ∈ L 2 (R n ), we have u 0 ∈ Ḃ-2σ 2,∞
if and only if e t∆ u 0 2 (1 + t) -σ .

3.1. Two useful subsets of Besov spaces. We now introduce two subsets of Ḃ-σ 2,∞ :

Ȧ-σ 2,∞ =        f ∈ Ḃ-σ 2,∞ : ∃ c, C, M > 0 s.t. ∆ j f 2 ≤ C2 σj ∀j ∈ Z ∆ j f 2 ≥ c2 σj for at least one j ∈ Z in any interval of lenght M        and Ȧ-σ 2,∞ = f ∈ Ḃ-σ 2,∞ : ∃ c, C > 0, ∃(j k ) k∈N ⊂ Z s.t. j k → -∞, (j k -j k+1 ) ∈ ℓ ∞ (N), and ∆ j f 2 ≤ C2 σj ∀j ∈ Z ∆ j k f 2 ≥ c2 σj k ∀k ∈ N .
These sets are not closed under summation and do not have a linear structure. We point out the inclusions

Ȧ-σ 2,∞ ⊂ Ȧ-σ 2,∞ ⊂ Ḃ-σ 2,∞ .
It is worth observing that the intersection L 2 (R n ) ∩ Ȧ-σ 2,∞ is empty, since dyadic blocks of L 2 functions satisfy ∆ j f 2 ≤ f 2 for all j ∈ Z. On the other hand, Ȧ-σ 2,∞ has a nontrivial intersection with L 2 (R n ), which we will characterize below.

3.2. Two-side bounds for the heat kernel. Our first applications of the sets Ȧ-2σ

2,∞ and Ȧ-2σ

2,∞
is the following theorem.

Theorem 3.1. Let σ > 0 and f be a tempered distribution.

(1) Then, f ∈ Ȧ-2σ 2,∞ if and only if

(3.2) ∃ c 1 , c 2 > 0 such that c 1 t -σ ≤ e t∆ f 2 ≤ c 2 t -σ
, for all t > 0.

(2) Moreover, f ∈ Ȧ-2σ 2,∞ if and only if

(3.3) t σ e t∆ f 2 ∈ L ∞ (R + )
, and lim inf t→+∞ t σ e t∆ f 2 > 0.

(

) In addition, f ∈ L 2 (R n ) ∩ Ȧ-2σ 2,∞ if and only if (3.4) ∃ c, c ′ > 0 such that c (1 + t) -σ ≤ e t∆ f 2 ≤ c ′ (1 + t) -σ , for all t > 0. 3
Proof. The property t σ e t∆ f 2 ∈ L ∞ (R + ) and the upper bound in (3.2), are equivalent to f ∈ Ḃ-2σ 2,∞ , as noticed in (3.1). In the same way, the upper bound in (3.4)

is equivalent to the condition f ∈ L 2 (R n ) ∩ Ḃ-2σ
2,∞ . For this reason, in the proof of Theorem 3.1 we need to focus only on the lower bound properties.

Let us start with a simple preliminary computation. For t > 0, let p = p(t) ∈ Z such that 4 p ≤ t < 4 p+1 . Letting d j = ∆ j f 2 2 2 -4jσ we have (the positive constant c is not the same from line to line):

t σ e t∆ f 2 2 ≃ j∈Z t 2σ e -ct2 2j ∆ j f 2 2 ≃ j∈Z t 2σ e -ct4 j 2 4jσ d j ≃ j∈Z 4 2jσ e -c 4 j d j-p , (3.5)
where in the last step was obtained by shifting the summation index. Assume now f ∈ Ȧ-2σ 2,∞ . Then, for some constant M > 0 as in the definition of Ȧ-2σ 2,∞ , and some constant c > 0 independent on p we have (c is not the same from line to line),

t σ e t∆ f 2 2 |j|≤M 4 2jσ e -c 4 j d j-p max |j|≤M d j-p 1,
where in the last inequality we used max |j|≤M 2 -4σ(j-p) ∆ j-p f 2 2 ≥ c, that follows from the definition of Ȧ-2σ 2,∞ . This proves the lower bound in (3.2). If we assume instead the weaker condition f ∈ Ȧ-2σ 2,∞ , then take M = (j kj k+1 ) ℓ ∞ , where j k → -∞ is as in the definition of Ȧ-2σ 2,∞ . Then, there exist an integer p 0 and c > 0 such that for all p ≥ p 0 we have max |j|≤M d j-p = max |j|≤M 2 -4σ(j-p) ∆ j-p f 2 2 ≥ c > 0. As c is independent on p, and hence on t, at least in some interval [t 0 , +∞), we get now

t σ e t∆ f 2 2 1 on [t 0 , ∞).
This in turn gives lim inf t→+∞ t σ e t∆ f 2 > 0.

Conversely, if we assume (3.2), then the lower bound in (3.2) and the computation at the beginning of the proof imply (3.6) j∈Z 4 2jσ e -c 4 j d j-p ≥ c ′ > 0, with 4 p ≃ t and c ′ independent on p. On the other hand, the upper bound (3.2) imply that

u 0 ∈ Ḃ-2σ 2,∞ , hence (d j ) ∈ ℓ ∞ (Z). As (4 2jσ e -c 4 j ) ∈ ℓ 1 (Z), we can find M > 0 such that |j|≥M 4 2jσ e -c 4 j < c ′ /(2 d j ℓ ∞ ).
Combining the two last estimates we get

|j|≤M 4 2jσ e -c4 j d j-p ≥ c ′ /2.
On the other hand, |j|≤M 4 2jσ e -c4 j d j-p ≤ C max |j|≤M d j-p for some C > 0 independent on p. It then follows that, for some c > 0 independent on p,

max |j|≤M d j-p ≥ c.
This insures that f ∈ Ȧ-2σ 2,∞ . If, instead of estimates (3.2) we have the weaker conditions (3.3), then (3.6) holds true only for all p ≥ p 0 , for some p 0 ∈ Z large enough. Applying the above argument to p = p 0 + kM , k ∈ N, we get, for some c > 0 independent on k,

max |j|≤M d j-(p 0 +kM ) ≥ c.
Therefore, we can then construct a sequence of integers (j k ) such that j k → -∞, |j k+1j k | ≤ 3M , and

d j k ≥ c > 0 for all k ∈ N. This implies f ∈ Ȧ-2σ 2,∞ . It only remains to prove the last claim of Theorem 3.1. If f ∈ L 2 (R n ) ∩ Ȧ-2σ 2,∞ then e t∆ f 2 2 ≤ f 2
2 and the upper bound estimate (3.4) immediately follows from the first of (3.3). The lowerbound estimate in (3.4) follows combining the fact that e t∆ f 2 2 ≥ ct -σ , valid in [t 0 , +∞) for some large enough t 0 > 0, by the second of (3.3), with the inequality

e t∆ f 2 2 ≥ e t 0 ∆ f 2 2 , valid in the interval [0, t 0 ].
Conversely, if estimates (3.4) hold then f ∈ Ȧ-2σ 2,∞ by the previous part of the Theorem. Moreover, by Fatou's lemma,

f 2 2 ≤ lim inf t→0 + e -2t|ξ| 2 f (ξ) dξ ≤ (c ′ ) 2 < ∞, so f ∈ L 2 (R n ).

3.3.

A Besov space characterization through e tL . In order establish the natural generalization of Theorem 3.1 to the solutions of the linear problem (2.6), we first need to extend the classical heat kernel characterization of Besov space (3.1) to the setting of the operator e tL . Specifically, we establish the following theorem.

Theorem 3.2. Let σ > 0, 1 ≤ p, q ≤ ∞. Then for any u ∈ S ′ (R n ) such that the Littlewood-Paley series u = j∈Z ∆ j u converges in S ′ (R n ), we have f ∈ Ḃ-2σ p,q (R n ) if and only if t σ/α e tL f p ∈ L q (R + , dt/t). Moreover, the two norms f Ḃ-2σ p,q and t σ/α e tL f p L q (R + , dt/t) are equivalent.

Proof. The proof is an adaptation of that in [1, Theorem 2.34] for the special case

L = ∆. Recall that F(e tL f )(ξ) = P (ξ) -1 e tD(ξ) P (ξ) f (ξ),
where D(ξ) = -|ξ| 2α D, where D = diag(c 1 , . . . , c n ), with α, c j > 0, and P (ξ) is an orthogonal matrix. The entries P (ξ) j,k (j, k = 1, . . . , m) are assumed to be homogeneous functions smooth for ξ = 0.

Lemma 3.3. Let p ∈ [1, ∞],
and f be a tempered distribution with support in the annulus λC, where C = {ξ ∈ R n : 0 < r 1 < |ξ| < r 2 } and λ > 0. Then, for all t > 0.

e tL f p ≤ Ce -ctλ 2α f p ,
for some c, C > 0 independent on λ and t.

Proof. The operator e tL is a convolution operator of the form e tL f = G(•, t) * f . By the homogeneity assumptions on P and D, the symbol G(ξ, t) = P (ξ) -1 e tD(ξ) P (ξ) satisfies the scaling relation G(ξ, t) = G(ξt 1/(2α) , 1), and so G(λx, t) = λ -n G(x, tλ -2α ).

Replacing f with its dilate f (λ•) we can reduce to the case λ = 1. So assume now λ = 1 and consider a smooth cut-off function φ(ξ) identically equal to one on C with compact support bounded away from ξ = 0. As f = φ * f , we have

e tL f p = G(•, t) * φ 1 f p ,
and it only remains to show that G(•, t) * φ 1 ≤ Ce -ct . Indeed, computing the inverse Fourier transform and integrating by parts, G(•, t) * φ(x) = (2π) -n e iξ•x φ(ξ)P (ξ) -1 e tD(ξ) P (ξ) dξ

= (1 + |x| 2 ) -n (I -∆ ξ ) n e iξ•x φ(ξ)P (ξ) -1 e tD(ξ) P (ξ) dξ = (1 + |x| 2 ) -n e iξ•x (I -∆ ξ ) n φ(ξ)P (ξ) -1 e tD(ξ) P (ξ) dξ
But (I -∆ ξ ) n φ(ξ)P (ξ) -1 e tD(ξ) P (ξ) can be bounded by a linear combination of functions of the form |ψ β (ξ)| |∂ β ξ e tD(ξ) |, with ψ β (ξ) smooth with support contained in that of φ(ξ), β ∈ N n and |β| ≤ 2n. Hence, for all ξ, (I -∆ ξ ) n φ(ξ)P (ξ) -1 e -tD(ξ) P (ξ) ≤ C(1 + t) 2d e -ct|ξ| 2α χ(ξ), for some smooth χ with compact support contained in that of φ. It follows,

G(•, t) * φ 1 ≤ Ce -ct
and the assertion of the Lemma follows.

Now assume that u ∈ Ḃ-2σ

p,q . Then, by Lemma 3.3, (3.7) t σ/α e tL f p ≤ j∈Z t σ/α ∆ j e tL f p j∈Z t σ/α 4 jσ e -ct 4 jα η j , with (η j ) ∈ ℓ q and (η j ) ℓ q = f Ḃ-2σ p,q . But, for all σ, c, α > 0, (3.8) sup t>0 j∈Z t σ/α 4 jσ e -ct 4 jα < ∞, as one can easily check splitting the summation in two terms corresponding to j such that 4 j ≤ t -1/α , next 4 j > t -1/α . In the case q = ∞, we readily deduce

t σ/α e tL f p (η j ) ℓ ∞ ≤ f Ḃ-2σ p,∞
, that agrees with (3.14).

In the case 1 ≤ q < ∞, we need to estimate the integral below first applying (3.7), next Hölder inequality to the summation with weight 4 jσ e -ct 4 jα and (3.8), next Fubini's theorem and the definition of the Gamma function:

∞ 0 t σq/α e tL f q p dt t 0 j∈Z t σ/α 4 jσ e -ct 4 jα η j q dt/t t 0 j∈Z t σ/α 4 jσ e -ct 4 jα η q j dt/t Γ(σ/α) j∈Z η q j u q Ḃ-2σ p,q
.

To prove the reverse estimate, let us first establish the following identity:

∞ 0 t σ/α (-D(ξ)) 1+σ/α e tD(ξ) v(ξ) dt = ∞ 0 t 1+σ/α |ξ| 2α(1+σ/α) diag(c 1+σ/α 1 e -c 1 t|ξ| 2α , . . . , c 1+σ/α n e -cnt|ξ| 2α ) v(ξ) dt/t = ∞ 0 t 1+σ/α diag(c 1+σ/α 1 e -c 1 t , . . . , c 1+σ/α n e -cnt ) v(ξ) dt/t = Γ(1 + σ/α) v(ξ)
Applying this to v(ξ) = P (ξ) ∆ j f (ξ) we obtain

∆ j f (ξ) = 1 Γ(1 + σ/α) ∞ 0 t σ/α P (ξ) -1 (-D(ξ)) 1+σ/α e tD(ξ) P (ξ) ∆ j u(ξ) dt = 1 Γ(1 + σ/α) ∞ 0 t σ/α P (ξ) -1 (-D(ξ)) 1+σ/α P (ξ)F(e tL ∆ j u)(ξ) dt
We now make use of the homogeneity properties of P (ξ) and D(ξ) and apply a classical Fourier multiplier theorem (see, e.g. [1, Lemma 2.2]), next the identity e tL = e tL/2 e tL/2 with Lemma 3.3 to get ∆ j f p ∞ 0 t σ/α 4 αj(1+σ/α) e tL ∆ j f p dt ∞ 0 t σ/α 4 αj(1+σ/α) e -ct 4 jα e tL/2 ∆ j f p dt ∞ 0 t σ/α 4 αj(1+σ/α) e -2ct 4 jα e tL f p dt (3.9)

When q = ∞ we just have to observe that

(3.10) ∆ j f p 2 2jσ sup t>0 t σ/α e tL f p to conclude that f Ḃ-2σ p,∞ sup t>0 t σ/α e tL f p .
When 1 ≤ q < ∞, we argue as follow:

f q Ḃ-2σ p,q = j∈Z 2 -2jqσ ∆ j f q p j∈Z 4 -jqσ ∞ 0 t σ/α 4 αj(1+σ/α) e -ct 4 jα e tL f p dt q j∈Z 4 αjq ∞ 0 e -ct 4 jα dt q-1 ∞ 0 t qσ/α e tL f q p e -ct 4 jα dt j∈Z 4 jα ∞ 0 t qσ/α e tL f q p e -ct 4 jα dt ∞ 0 j∈Z
t4 αj e -ct 4 jα t qσ/α e tL f q p dt/t ∞ 0 t qσ/α e tL f q p dt/t

Here we used first (3.9), next Hölder inequality, and in the last inequality a particular case of Eq. (3.8). The proof of Theorem 3.2 is now complete.

We are now in a position of generalizing Theorem 3.1 to the case of the operator L.

Theorem 3.4. Let σ > 0 and f be a tempered distribution. Let also L a pseudo-differential operator as at the beginning of Section 2.2

(1) Then, f ∈ Ȧ-2σ 2,∞ if and only if

(3.11) ∃c 1 , c 2 > 0 such that c 1 t -σ/α ≤ e tL (t) 2 ≤ c 2 t -σ/α
, for all t > 0.

(2) Moreover, f ∈ Ȧ-2σ 2,∞ if and only if

(3.12) t σ/α e tL (t) 2 ∈ L ∞ (R + ), and lim inf t→+∞ t σ/α e tL f 2 > 0, (3) In addition, f ∈ L 2 (R n ) ∩ Ȧ-2σ 2,∞ if and only if (3.13) ∃c, c ′ > 0 such that c (1 + t) -σ/α ≤ e tL (t) 2 ≤ c ′ (1 + t) -σ/α
, for all t > 0.

Proof. Recalling that the multiplication by P (ξ) and P (ξ) -1 conserve the euclidean norm of a vector and that D(ξ) and e tD(ξ) are diagonal matrices, we can write

e tL f 2 2 = |P (ξ) -1 e D(ξ)t P (ξ) f (ξ)| 2 dξ ≃ |e -ct|ξ| 2α f (ξ)| 2 dξ,
By our previous theorem, we have the analogue of the characterization to (3.1) for the operator L, namely,

(3.14) f ∈ Ḃ-2σ p,q ⇐⇒ t σ/α e tL f 2 ∈ L q (R + , dt/t),
with the equivalence of the respective norms.

We now argue as in (3.5): We set as before d j = ∆ j f 2 2 2 -4jσ . For t > 0, let p = p(t) ∈ Z such that 4 pα ≤ t < 4 (p+1)α . Then we have

t σ/α e tL f 2 2 ≃ j∈Z t 2σ/α e -ct2 2jα ∆ j f 2 2 ≃ j∈Z t 2σ/α e -ct4 jα 2 4jσ d j ≃ j∈Z 4 2jσ e -c 4 jα d j-p , (3.15)
The proof can now be carried on making obvious modification to that of Theorem 3.1.

Characterization of L 2 -functions admitting a decay character

For any u 0 ∈ L 2 (R n ), we introduce the quantity σ(u 0 ) ∈ [0, +∞] given by (4.1)

σ(u 0 ) = sup σ ≥ 0 : u 0 ∈ Ḃ-σ 2,∞ .
Notice that, for

u 0 ∈ L 2 (R n ), σ(u 0 ) is always well defined in [0, +∞]. Indeed, by the Plancherel theorem, L 2 (R n ) = Ḃ0 2,2 ⊂ Ḃ0 2,∞ .
The relation between the decay character r * (u 0 ) and σ(u 0 ) is given by the following simple result.

Proposition 4.1. Let u 0 ∈ L 2 (R n ).
When the decay character of u 0 does exist, it is given by the formula (valid also in the limit cases r * (u 0 ) = -n/2 or ∞):

(4.2) r * (u 0 ) + n/2 = σ(u 0 ).
Proof. In the case r * (u 0 ) = -n/2, we have +∞ = P r (u 0 ) -> 0 for all r > -n/2, and so e t∆ u 0 2 2 ≥ c ǫ (1 + t) -ǫ for all ǫ > 0 by the first part of Proposition 2.8. Applying the characterization (3.1) of Besov spaces with negative regularity in terms of the heat kernel we get σ(u 0 ) = 0. In the case r * (u 0 ) = +∞, we have P r (u 0 ) + < ∞ for all r ∈ R and so e t∆ u 0 2 2 ≤ C k (1 + t) -k for all k ≥ 0 by the second part of Proposition 2.8. We apply again (3.1) to conclude that σ(u 0 ) = +∞. In the case -n/2 < r * (u 0 ) < ∞ the conclusion follows immediately combining both parts of Proposition 2.8 with (3.1). Remark 4.2. C. Niche and M.E. Schonbek generalized the notion of decay character by introducing, for s ≥ 0 and u 0 ∈ H s (R n ), the notation

r * s (u 0 ) = r * (Λ s u 0 ), where Λ = √ -∆.
The condition u 0 ∈ H s (R n ) insures that both u 0 and Λ s u 0 belong to L 2 (R n ). They proved in [START_REF] Niche | Decay characterization to dissipative equations[END_REF]Theorem 2.11] that, for u 0 ∈ H s (R n ), r * s (u 0 ) = r * (u 0 ) + s (this formula being valid also in the limit cases). Proposition 4.1 provides a one-sentence proof for this formula to hold: indeed, Λ s is known to be an isomorphism between Ḃ-σ 2,∞ and Ḃ-(σ+s)

2,∞
and therefore σ * (Λ s u 0 ) = σ * (u 0 ) + s.

One can ask if it is possible to give a Besov space characterization of the existence of the decay character of an L 2 -function. Proposition 4.3 below will provide a positive answer.

Proposition 4.3. Let u 0 ∈ L 2 (R n ).
Then the decay character of u 0 does exist and r * = r * (u 0 ) ∈ (-n/2, ∞) if and only if there is σ * > 0 such that u 0 ∈ Ȧ-σ * 2,∞ . In this case, σ * = σ(u 0 ) and formula (4.2) holds.

Proof. Let us start assuming u 0 ∈ Ȧ-σ * 2,∞ , with σ * ∈ (0, ∞). For any ρ > 0, take j ∈ Z such that 2 j-1 < ρ ≤ 2 j . Then,

ρ -2σ * |ξ|≤ρ | u 0 (ξ)| 2 dξ 2 -2jσ * |ξ|≤2 j | u 0 (ξ)| 2 dξ 2 -2jσ * -∞<k≤j ∆ k u 0 2 2 2 -2jσ * -∞<k≤j 2 2kσ * u 0 2 Ḃ-σ * 2,∞ u 0 2 Ḃ-σ * 2,∞ .
Letting r * + n/2 = σ * , such estimates imply P r * (u 0 ) + < +∞.

Moreover, from our assumption we can take (j k ) as in the definition of Ȧ-σ * 2,∞ . Let ρ 0 = 2 j 0 and M = (j kj k+1 ) ℓ ∞ . For any 0 < ρ < ρ 0 , let j k be the largest integer of the sequence (j k ) such that 2 j k ≤ ρ. Then we must have 2 j k +M ≥ ρ, otherwise we would find another integer j h ∈ [j k , j k + M ] such that 2 j h ≤ ρ, thus contredicting the maximality of j k . Then we have

ρ -2σ * |ξ|≤ρ | u 0 (ξ)| 2 dξ 2 -2(j k +M )σ * ∆ j k u 0 2 2 ≥ c > 0, (4.3)
with c independent on ρ ∈ (0, ρ 0 ). This implies, P r * (u 0 ) -> 0. By Remark 2.6 we conclude that the decay character does exist and r * (u 0 ) = r * ∈ (-n/2, +∞).

Conversely, assume that the decay character of u 0 does exist and r * (u 0 ) ∈ (-n/2, +∞). Then we know by the first assertion of Theorem 2.9, that the solution of the heat equation e t∆ u 0 is bounded from below and above by C(1 + t) -1 2 (r * +n/2) . We conclude by the last claim of Theorem 3.1 that u 0 ∈ Ȧ-σ * 2,∞ and that σ * = r * + n/2.

Remark 4.4 (Proof of Theorem 2.9). This first assertion of Theorem 2.9 follows immediately from the estimates obtained in Proposition 2.8. We can now prove the last assertion of Theorem 2.9. If u 0 ∈ L 2 (R n ) satisfies the estimates (2.7), then by Theorem 3.4-(3), we obtain

u 0 ∈ Ȧ-2r * -n 2,∞
and the decay character of u 0 does exist and equals r * by Proposition 4.3. Theorem 2.9 is now completely proved.

Application to nonlinear dissipative systems: The Navier-Stokes equation

As an application of our analysis to nonlinear dissipative systems, let us consider the Navier-Stokes equations in R 3 :

(NS)        u t + u • ∇u + ∇p = ∆u, x ∈ R 3 , t > 0 ∇ • u = 0 u(x, 0) = u 0 (x),
where u = u(x, t) = (u 1 , u 2 , u 3 )(x, t) is the velocity field of an incompressible viscous fluid flow, p = p(x, t) is the pressure, and u 0 = (u 0,1 , u 0,2 , u 0,3 ) is the initial datum. For any u 0 ∈ L 2 (R 3 ), with ∇ • u 0 = 0, we know since J. Leray's classical paper [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] that there is at least one solution

u ∈ C w ([0, ∞), L 2 (R n ))∩L 2 loc (R + , H 1 (R 3
)) solving (NS) in the distributional sense and satisfying the strong energy inequality (5.1)

u(t) 2 2 + 2 t s ∇u(r) 2 2 dr ≤ u(s) 2 2 ,
for s = 0, almost s > 0 and all t ≥ s. The uniqueness and the regularity of such solutions are well known open problems.

The following theorem completely characterize the solutions satisfying sharp two-side decay estimates. It completes well known Wiegner's Theorem by giving several necessary and sufficient conditions for the validity of bounds from below for the decay of the energy. It improves [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF]Theorem 6.5], and completes an earlier study by Skalák [START_REF] Skalák | On the characterization of the Navier-Stokes flows with the power-like energy decay[END_REF] where only the upper bounds were discussed.

Theorem 5.1. Let u 0 ∈ L 2 (R 3 ) and σ > 0. The three following properties are equivalent:

(i) lim inf ρ→0 ρ -2σ |ξ|≤ρ | u 0 (ξ)| 2 dξ > 0 and lim sup ρ→0 ρ -2σ |ξ|≤ρ | u 0 (ξ)| 2 dξ < ∞, (ii) u 0 ∈ Ȧ-2σ 2,∞ , (iii) (1 + t) -σ e t∆ u 0 2 (1 + t) -σ .
If, moreover, u 0 is a divergence-free vector-field, if u is a weak solution of (NS) as above and 0 < σ < 5/4, then the three previous properties are equivalent to

(iv) (1 + t) -σ u(t) 2 (1 + t) -σ .
Proof. The equivalence (i) ⇐⇒ (ii) is just a reformulation of Proposition 4.3. Indeed, as explained in Remark 2.6, the decay character r * (u 0 ) does exist and belongs to (-n/2, ∞) if and only if there exists r * ∈ (-n/2, ∞) such that 0 < P r * (u 0 ) -≤ P r * (u 0 ) + < ∞.

The equivalence (ii) ⇐⇒ (iii) was established in Theorem 3.1.

The implication (iii) ⇒ (iv) relies on classical Wiegner's theorem [START_REF] Wiegner | Decay results for weak solutions of the Navier-Stokes equations on R n[END_REF]: in the 3D case and in the absence of external forces, the Theorem in [START_REF] Wiegner | Decay results for weak solutions of the Navier-Stokes equations on R n[END_REF] can restated as follows: if u 0 ∈ L 2 (R n ) is a divergence-free vector field such that e t∆ u 0 2 (1 + t) -σ , and if u is a weak solution of (NS) as above, then the difference w(t) = u(t)e t∆ u 0 satisfies the decay estimates

w(t) 2        (1 + t) -1/4-σ , if 0 ≤ σ < 1, ln(e + t)(1 + t) -5/4 , if σ = 1, (1 + t) -5/4 , if σ > 1.
Notice in particular that, for 0 < σ < 5/4, one gets t σ w(t) 2 → 0 as t → ∞. Therefore, under the same restrictions on σ and the conditions of item (iii), it easily follows that u = e t∆ u 0 + w satisfies the estimates as in (5.1).

The proof of the implication (iv) ⇒ (iii) relies on the so-called inverse Wiegner's theorem recently established by Z. Skalák [START_REF] Skalák | On the characterization of the Navier-Stokes flows with the power-like energy decay[END_REF]: his results asserts (among other things) that if u is a weak solution of the Navier-Stokes equation as above, satisfying u(t) 2

(1 + t) -σ (with 0 ≤ σ ≤ 5/4) then e t∆ u 0 2

(1 + t) -σ . Hence, Wiegner's theorem applies and decomposing e t∆ u 0 = uw property (iii) follows in the range 0 < σ < 5/4.

In the borderline case σ = 5/4, the implication (iii)⇒(iv) still holds. But the converse implication is no longer true. Indeed, it is possible to construct examples of Navier-Stokes flows such that lim t→+∞ t 5/4 e t∆ u 0 2 = 0 and (1 + t) -5/4 u(t) 2 (1 + t) -5/4 . This can be achieved as follows: one starts with a divergence-free initial datum u 0 ∈ L 2 (R 3 ) such that u 0 vanishes in a neighborhood of the origin. This insures a fast (exponential) L 2 -decay of the solution of the heat equation e t∆ u 0 . Generically (i.e. in the absence of special symmetries), the matrix ∞ 0 (u ⊗ u)(y, s) dy dt will not be a scalar multiple of the identity matrix. By a theorem of T. Miyakawa and M.E. Schonbek [START_REF] Miyakawa | On optimal decay rates for weak solutions to the Navier-Stokes equations in R n[END_REF], one obtains a solution u of the Navier-Stokes equations that satisfies the lower bound estimate u(t) 2

(1 + t) -5/4 , and, by Wiegner's theorem, u(t) 2 (1 + t) -5/4 .

Examples

Example 6.1 (Construction of v 0 ∈ L 2 (R n ) such that the limit P r (v 0 ) is not well defined). Let r ∈ (-n/2, ∞). Decompose the unit ball B 1 = {ξ : |ξ| ≤ 1} in concentric dyadic annuli:

B 1 = Γ 0 ∪ Γ -1 ∪ Γ -2 ∪ . . ., with Γ j = {ξ : 2 j-1 ≤ |ξ| ≤ 2 j }. Then set v 0 (ξ) = |ξ| r on Γ 0 ∪ Γ -2 ∪ .
. . and v 0 (ξ) = 0 elsewhere. A direct computation shows that 0 < P r (v 0 ) -< P r (v 0 ) + < ∞. Indeed, passing to spherical coordinates in the computation of P r (v 0 ) + and P r (v 0 ) -,

P r (v 0 ) + = ω n lim j→-∞ 2 -2j(2r+n) ∞ 0 λ 2r+n-1 1 Γ 2j ∪Γ 2(j-1) ∪... dλ and P r (v 0 ) -= ω n lim j→-∞ 2 -(2j+1)(2r+n) ∞ 0 λ 2r+n-1 1 Γ 2j ∪Γ 2(j-1) ∪... dλ,
where ω n is the surface of the unit sphere of R n and 1 A denotes the indicator function of the set A. Both limits could be easily computed, but in fact it is simpler to observe that Pr(v 0 ) - Pr(v 0 ) + = 2 -(2r+n) < 1 to conclude that P r (v 0 ) is not well defined by formula (2.1). In fact, 0 < P r (u 0 ) -< P r (u 0 ) + < ∞ and by Remark 2.6 v 0 admits a decay character in the sense of Definition 2.5 and r * (v 0 ) = r. Example 6.2. It can happen that r(u 0 ) + = r(u 0 ) -, yet the decay character r * (u 0 ) does not exist. The example is elementary. Let u 0 ∈ L 2 (R n ) be such that u 0 (ξ) = |ξ| r 0 log |ξ| in a neighborhood of ξ = 0, for some r 0 > -n/2. Then by a simple computation r(u 0 ) + = r(u 0 ) -= r 0 . But the decay character r * (u 0 ) is not well defined, because the sup and the inf in the definition of r(u 0 ) + and r(u 0 ) -(see Eq: (2.2)-(2.3)) are not achieved. For this specific initial datum, the corresponding solution of the linear problem (2.6) satisfies estimates similar to those in (2.7), but with a corrective time-dependent logarithmic factor. Notice that for this example there is no r such that 0 < P r (u 0 ) < ∞ (the existence of such an r was needed in the original definition of decay character [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF]). Example 6.3. Let us construct an example of w 0 ∈ L 2 (R n ) verifying r(w 0 ) + < r(w 0 ) -. Fix r ∈ (-n/2, +∞). Let (a k ) and (b k ) two real decreasing sequences such that 0 < b k+1 ≤ a k < b k and b k → 0 for all k ∈ N, and consider two more positive sequences (η k ), (h k ) to be chosen later. We define w 0 through its Fourier transform: For this reason, r(w 0 ) -= r and r * (w 0 ) does not exist. In fact r(w 0 ) -is given by r ′ ∈ (-n/2, ∞), such that the limit lim inf

w 0 (ξ) = ∞ k=0 h k 1 a k ≤|ξ|≤b k . The condition insuring w 0 ∈ L 2 (R n ) is ∞ k=0 h 2 k (b n k -a n k ) < ∞.

Conclusions

Let σ > 0 and f ∈ L 2 (R n ). In [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] C. Bjorland and M.E. Schonbek, proved that:

(7.1a) 0 < lim ρ→0 + ρ -2σ |ξ|≤ρ | f (ξ)| 2 dξ < ∞ =⇒ (1 + t) -σ e t∆ f 2 (1 + t) -σ ,
where e t∆ denotes the heat kernel in R n and the symbol means that the inequality ≤ holds up to a multiplicative constant independent on time. In [START_REF] Niche | Decay characterization to dissipative equations[END_REF] C. Niche and M.E. Schonbek extended this to a large class of pseudo-differential operator L, with homogeneous symbol of degree 2α, proving that (7.1b) 0 < lim

ρ→0 + ρ -2σ |ξ|≤ρ | f (ξ)| 2 dξ < ∞ =⇒ (1 + t) -σ/α e tL f 2 (1 + t) -σ/α .
In the present paper, we showed that the reverse implications in (7.1a)-(7.1b) do not hold, as the limit on the LHS might not exist. Next, suitably relaxing the condition on the LHS, we got the following characterization of the class of L 2 -functions satisfying sharp two-side decay estimates. Namely, we established that:

(7.2)        lim inf ρ→0 + ρ -2σ |ξ|≤ρ | f (ξ)| 2 dξ > 0 lim sup ρ→0 + ρ -2σ |ξ|≤ρ | f (ξ)| 2 dξ < ∞ ⇐⇒ f ∈ Ȧ-2σ 2,∞ , ⇐⇒ (1+t) -σ/α e tL f 2 (1+t) -σ/α ,
where Ȧ-2σ 2,∞ is the subset of the Besov space Ḃ-2σ 2,∞ (R n ) introduced in Section 3.1. Moreover, if f is a divergence-free vector field, then in the special case n = 3, L = ∆ and 0 < σ < 5/4, we proved in Theorem 5.1 that the three previous conditions are in turn equivalent to the two-side energy estimate of weak solutions of the Navier-Stokes equation starting from f : (7.3) (1 + t) -σ u(t) 2 (1 + t) -σ .

When an L 2 function satisfy any of the above equivalent conditions (7.2), the exponent σ ∈ (0, ∞) is uniquely determined. We coin this exponent σ = σ(f ) the Besov character of f . It is related to (a suitable improvement of) Bjorland-Schonbek's [START_REF] Bjorland | Poincaré's inequality and diffusive evolution equations[END_REF] notion of "decay character", denoted r * (f ), by the relation σ(f ) = r * (f ) + n/2.

Definition 2 . 1 .

 21 Let u 0 ∈ L 2 (R n ), B ρ = {ξ ∈ R n : |ξ| ≤ ρ}.The lower and upper decay indicators of u 0 are the two lower and upper limitsP r (u 0 ) -= lim inf ρ→0 + ρ -2r-n Bρ | u 0 (ξ)| 2 dξ and P r (u 0 ) + = lim sup ρ→0 + ρ -2r-n Bρ | u 0 (ξ)| 2 dξ.

  It is then convenient to set ηk = h 2 k (b n k -a n k ), and the first condition is that (η k ) is summable. Let Φ r (ρ) = ρ -2r-n |ξ|≤ρ | w 0 (ξ)| 2 dξ. Then, P r (w 0 ) + = lim sup ρ→0 Φ r (ρ) = lim k→∞ Φ r (b k ) = lim k→∞ ω n b -(2r+n) k ∞ j=k η j , and P r (w 0 ) -= lim inf ρ→0 Φ r (ρ) = lim k→∞ Φ r (a k ) = lim k→∞ ω n a -(2r+n) k ∞ j=k+1 η j ,where ω n is the measure of the unit sphere. Now, we choose,b k = 2 -2 k , a n k = b n kb 2n k , η k = b 2r+n k b 2r+n k+1 . Then, ∞ j=k η j = b 2r+n k. These choices insure that w 0 ∈ L 2 (R n ) as (η k ) is indeed summable and determine (h k ). Moreover, we get 0 < P r (w 0 ) + = ω n < ∞ and we conclude that r(w 0 ) + = r.Let us now study r(w0 ) -. First observe that ∞ j=k+1 η j = b 2r+n k+1 = (b 2r+n k ) 2 . Then, a -(2r+n) k ∞ j=k+1 η j = [b k (1b n k ) -n ] -2r-n (b 2r+n k ) 2 → 0 as k → ∞,so that P r (w 0 ) -= lim inf k→∞ Φ r (a k ) = 0.

ρ→0Φ

  r ′ (ρ) = lim k→∞ Φ r ′ (a k ) = ω n a -(2r ′ +n) k ∞ j=k+1 η j ,is a strictly positive real: we need 2r ′ + n = 2(2r + n). In conclusion, r(w 0 ) -= 2r + n/2, r(w 0 ) + = r.

Supported by the ANR project DYFICOLTI ANR-13-BS01-


Christophe Reutenauer
email: reutenauer.christophe@uqam.ca

THE FOURIER EXPANSION OF ηpzqηp2zqηp3zq{ηp6zq

Keywords: 2010 Mathematics Subject Classification. (Primary) 11F11, 11F20, 14C05, 14G15, 14N10 Dedekind eta function, eta products, Fourier coefficient, punctual Hilbert scheme

We compute the Fourier coefficients of the weight one modular form ηpzqηp2zqηp3zq{ηp6zq in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form ηpzq 4 {ηp2zq 2 .

Introduction

In this note we consider the η-product (1.1) ηpzqηp2zqηp3zq ηp6zq " ź ně1 p1 ´qn q 2 1 ´qn `q2n .

where q " e 2πiz . Recall that ηpzq is Dedekind's eta function ηpzq " e πiz{12 ź ně1 p1 ´qn q.

The η-product ηpzqηp2zqηp3zq{ηp6zq is a modular form of weight 1 and level 6. Since it is invariant under the transformation z Þ Ñ z `1, it has a Fourier expansion of the form (1.2) ηpzqηp2zqηp3zq ηp6zq " ÿ ně0 a 6 pnq q n , where the Fourier coefficients a 6 pnq are integers. For general information on ηproducts, see [START_REF] Köhler | Eta products and theta series identities Springer Monographs in Mathematics[END_REF]Sect. 2.1].

Our first result expresses a 6 pnq in terms of the number rpnq of representations of n as the sum of two squares, i.e the number of elements px, yq P Z 2 such that x 2 `y2 " n. Observe that rpnq is divisible by 4 for all n ě 1 (for n " 0 we have rp0q " 1). The sequence rpnq appears as Sequence A004018 in [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF].

Theorem 1.1. For all non-negative integers m we have a 6 p3mq " p´1q m rp3mq, a 6 p3m `1q " p´1q m`1 rp3m `1q 4 ,

a 6 p3m `2q " p´1q m`1 rp3m `2q 2 .
We next relate ηpzqηp2zqηp3zq{ηp6zq to the weight one modular form ηpzq 4 {ηp2zq 2 and two of its translates. Theorem 1.2. Set j " e 2πi{3 . We have the following linear relation between weight one modular forms:

ηpzqηp2zqηp3zq ηp6zq " 1 4
ηpzq 4 ηp2zq 2 `1 ´j 4 ηpz `1{3q 4 ηp2z `2{3q 2 `1 ´j2 4 ηpz `2{3q 4 ηp2z `1{3q 2 .
Both modular forms ηpzqηp2zqηp3zq{ηp6zq and ηpzq 4 {ηp2zq 2 came up naturally in [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF], where we computed the number C n pqq of ideals of codimension n of the algebra F q rx, y, x ´1, y ´1s of Laurent polynomials in two variables over a finite field F q of cardinality q. Equivalently, C n pqq is the number of F q -points of the Hilbert scheme of n points on a two-dimensional torus. We proved that C n pqq is the value at q of a palindromic one-variable polynomial C n pxq P Zrxs with integer coefficients, which we computed completely (see [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF]Th. 1.3]).

We also showed (see [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF]Cor. 6.2]) that the generating function of the polynomials C n pxq can be expressed as the following infinite product:

(1.3) 1 `ÿ ně1 C n pxq x n q n " ź ně1 p1 ´qn q 2 1 ´px `x´1 qq n `q2n .
It follows from the previous equality that C n p1q " 0. Actually, we proved (see [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF]Th. 1.3 and 1.4]) that there exists a polynomial P n pxq P Zrxs such that C n pxq " px ´1q 2 P n pxq. Moreover, P n pxq is palindromic, has non-negative coefficients and its value at x " 1 is equal to the sum of divisors of n: P n p1q " ř d|n d. When x " e 2iπ{k with k " 2, 3, 4, or 6, then x `x´1 " 2 cosp2π{kq is an integer. For such an integer k, we define the sequence a k pnq by (1.4)

ÿ ně0 a k pnq q n " ź ně1 p1 ´qn q 2 1 ´2 cosp2π{kq q n `q2n .
Since 2 cosp2π{kq is an integer, so is each a k pnq. It follows from (1.3) that these integers are related to the polynomials C n pxq by C n pe 2iπ{k q " a k pnq e 2niπ{k .

In [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF] we computed a 2 pnq, a 3 pnq, and a 4 pnq explicitly in terms of well-known arithmetical functions. In particular, we established the equality

(1.5) a 2 pnq " p´1q n rpnq,
where rpnq is the number of representations of n as the sum of two squares. We also observed in [3, (1.8)] that

(1.6) ÿ ně0 a 2 pnq q n " ηpzq 4 ηp2zq 2 and ÿ ně0 a 6 pnq q n " ηpzqηp2zqηp3zq ηp6zq .
The question of finding an explicit expression for a 6 pnq had been left open in [START_REF] Kassel | On the zeta function of a punctual Hilbert scheme of the two-dimensional torus[END_REF]. This is now solved with Theorem 1.1 of this note. In view of this theorem, of (1.5), and of (1.6), for all m ě 0 we obtain

$ ' ' ' ' ' & ' ' ' ' ' % a 6 p3mq " a 2 p3mq, a 6 p3m `1q " a 2 p3m `1q 4 , a 6 p3m `2q " ´a2 p3m `2q 2 .
(1.7)

We had experimentally observed (see [3, Note 7]) that a 6 pnq " 0 whenever a 2 pnq " 0. As a consequence of (1.7) we can now state that a 6 pnq " 0 if and only if a 2 pnq " 0, i.e. if and only n is not the sum of two squares.

Theorems 1.1 and 1.2 will be proved in the next two sections.

Remarks 1.3. (a) The sequence a 6 pnq is Sequence A258210 in [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF]. The sequence a 6 p3n `1q is probably the opposite of Sequence A258277 in loc. cit. (b) It can be seen from Table 1 that a 6 pnq is not a multiplicative function. Indeed, a 6 p10q ‰ a 6 p2qa 6 p5q or a 6 p18q ‰ a 6 p2qa 6 p9q or a 6 p20q ‰ a 6 p4qa 6 p5q. Next consider the excess function E 1 pn; 4q defined by

E 1 pn; 4q " ÿ d|n , d"1 pmod 4q 1 ´ÿ d|n , d"´1 pmod 4q 1.
It is a multiplicative function, i.e., E 1 pmn; 4q " E 1 pm; 4q E 1 pn; 4q whenever m and n are coprime. It is well known that the excess function can be computed in terms of the prime decomposition of n. Write n " 2 c p a 1 1 p a 2 2 ¨¨¨q b 1 1 q b 2 2 ¨¨¨, where p i , q i 's are distinct prime numbers such that p i " 1 pmod 4q et q i " 3 pmod 4q. Then E 1 pn; 4q " 0 if and only if one of the exponents b i is odd. If all b i 's are even, then (2.3) E 1 pn; 4q " p1 `a1 qp1 `a2 q ¨¨¨.

In the sequel we will need the following result.

ÿ ně0 a k pnq q n " 1 ´4 sinpπ{kq ÿ ně1 ¨ÿ d|n , d odd sin ˆˆ2n d ´d˙π k ˙' q n .
Consider the special case k " 6 of (2.5). Since sinpπ{6q " 1{2, Equality (2.5) becomes

ÿ ně0 a 6 pnq q n " 1 ´2 ÿ ně1 ¨ÿ d|n , d odd sin ˆˆ2n d ´d˙π 6 ˙' q n " 1 `ÿ ně1 ¨ÿ d|n , d odd ξ ˆ2n d ´d˙' q n .
The formula for a 6 pnq follows.

Proof of Theorem 1.1. Let us first mention the following well-known fact (see [ We now distinguish three cases according to the residue of n modulo 3.

(a) We start with the case n " 1 pmod 3q. We have n " 3ℓ `1 for some nonnegative integer ℓ. Since the odd divisors d of n are not divisible by 3, they must satisfy d " 1, 5, 7 or 11 pmod 12q. Such divisors are invertible pmod 12q et we have d 2 " p´1q ℓ`1 2E 1 pn; 4q " p´1q n`ℓ`1 a 2 pnq{2.

Since n and ℓ are of the same parity, we have a 6 pnq " ´a2 pnq{2.

(c) Finally we consider the case when n is divisible by 3. We write n " 3 N m, where m is coprime to 3 and N ě 1. Any odd divisor d of n is of the form d " 3 r s for some odd divisor s of m and 0 ď r ď N. Since m and its divisors s are not divisible by 3 and since s is odd, we again have s " 1, 5, 7 or 11 pmod 12q. Recall that for such s we have s 2 " 1 pmod 12q. Thus, for d " 3 r s, we obtain 2n d ´d " `2 ¨3N´r m ´3r ˘s pmod 12q.

If r " 0, then 2n{d ´d " p6 ¨3N´1 m ´1qs pmod 12q. Therefore,

ξ ˆ2n d ´d˙" ξ `p6 ¨3N´1 m ´1qs ˘" p´1q m ξp´sq " p´1q m´1 ξpsq.
in view of (2.1). If 0 ă r ă N, then 2n{d ´d " p6 ¨3N´r´1 m ´3r qs pmod 12q. Therefore, ξ ˆ2n d ´d˙" ξ `p6 ¨3N´r´1 m ´3r qs ˘" p´1q m ξp´3 r sq " p´1q m´1 ξp3 r sq. Now, 3 r " 3 pmod 12q if r is odd, and 3 r " ´3 if r ą 0 is even. Then by (2.1), ξ ˆ2n d ´d˙" p´1q m´r ξp3sq.

since E 1 p3 N ; 4q " 1 for even N. Finally, m and n being of the same parity, we have a 6 pnq " p´1q m 4 E 1 pm; 4qq " p´1q n 4 E 1 pn; 4qq " a 2 pnq.

Q.e.d.

3. Proof of Theorem 1.2

Set f pqq " ηpzqηp2zqηp3zq{ηp6zq " ř ně0 a 6 pnq q n and gpqq " ηpzq 4 {ηp2zq 2 " ř ně0 a 2 pnq q n ; see (1.6). To prove Theorem 1.2 it suffices to check that f pqq " agpqq `bgp jqq `cgp j 2 qq, where a " 1{4, b " p1 ´jq{4, and c " p1 ´j2 q{4. Now, agpqq `bgp jqq `cgp j 2 qq " a ÿ ně0 a 2 pnq q n `b ÿ ně0 a 2 pnq j n q n `c ÿ ně0 a 2 pnq j 2n q n " pa `b `cq ÿ mě0 a 2 p3mq q 3m

`pa `jb `j2 cq ÿ mě0 a 2 p3m `1q q 3m`1

`pa `j2 b `jcq ÿ mě0 a 2 p3m `2q q 3m`2 .

It follows from (1.7) that agpqq `bgp jqq `cgp j 2 qq " pa `b `cq ÿ mě0 a 6 p3mq q 3m `4pa `jb `j2 cq ÿ mě0 a 6 p3m `1q q 3m`1 ´2pa `j2 b `jcq ÿ mě0 a 6 p3m `2q q 3m`2 .

The right-hand side is equal to f pqq since a `b `c " 1, a `jb `j2 c " 1{4, and a `j2 b `jc " ´1{2. Q.e.d.

Table 1 .

 1 First values of a 6 pnq

	n	1	2	3 4 5 6 7	8	9	10 11 12	13	14 15 16 17 18 19	20
	a 6 pnq	´1 ´2 0 1 4 0 0 ´2 ´4	2	0	0	´2	0	0	1	4	4	0	´4
	We have
					$ '	´1 if m " 1, 5 pmod 12q,
					'
	(2.2)			ξpmq "	' & ' '	´2 if m " 3 pmod 12q, 1 if m " 7, 11 pmod 12q,
					' %	2		if m " 9 pmod 12q.

2. Proof of Theorem 1.1 2.1. For any odd integer m we set ξpmq " ´2 sinpmπ{6q. Because of the wellknown properties of the sine function, ξpmq depends only on the class of m modulo 12 and we have the following equalities for all odd m:

(2.1)

ξp´mq " ´ξpmq and ξpm `6q " ´ξpmq, which is equivalent to ξp´mq " ´ξpmq and ξp6 ´mq " ξpmq.

  Proof. Let d be an odd divisor of n; it is not divisible by 3 since n is not. Therefore, d " 1, 5, 7 or 11 pmod 12q. Observe that d " 1, 5 pmod 12q if and only if d " 1 pmod 4q since d " 3 pmod 12q is excluded. Similarly, d " 7, 11 pmod 12q if and only if d " 3 pmod 4q. Now, ξpdq " ´1 if d " 1, 5 and ξpdq " 1 if d " 7, 11 pmod 12q. Consequently, We now express a 6 pnq in terms of the function ξ introduced above.

	ÿ d|n , d odd	ξpdq "	ÿ d|n , d"3 pmod 4q	1	´ÿ d|n , d"1 pmod 4q	1 " ´E1 pn; 4q.
	Similarly, ξp3dq " ξp3q " ´2 if d " 1, 5 and ξp3dq " ξp9q " 2 if d " 7, 11
	pmod 12q. Therefore,
	ÿ d|n , d odd	ξp3dq "	ÿ d|n , d"3 pmod 4q	2	´ÿ d|n , d"1 pmod 4q	´2 " ´2E 1 pn; 4q.
	2.2. Proposition 2.2. We have
							ˆ2n
	(2.4)			a 6 pnq "	ÿ d|n , d odd	ξ	d	´d˙.

Lemma 2.1. Let n be a positive integer which is not divisible by 3. We have ÿ d|n , d odd ξpdq " ´E1 pn; 4q and ÿ d|n , d odd ξp3dq " ´2E 1 pn; 4q.

Note that 2n{d ´d is an odd integer since d is an odd divisor of n.

Proof. Set u " π{k and ω " d in Formula (9.3) of [2, p. 10]. It becomes

(2.5)

  1, § 51, Th. 65]): the number rpnq of representations of n as a sum of two squares is related to the excess function E 1 pn; 4q by

	(2.6)	rpnq " 4 E 1 pn; 4q
	for all n ě 0. It follows from this fact and from (1.5) that
	(2.7)	a 2 pnq " p´1q n 4 E 1 pn; 4q.

  " 1 pmod 12q. Consequently, Finally observe that n is odd (resp. even) if ℓ is even (resp. odd). Therefore, a 6 pnq " a 2 pnq{4.(b) Now consider the case n " 2 pmod 3q. We have n " 3ℓ `2 for some non-negative integer ℓ. Again the odd divisors d of n must satisfy d " 1, 5, 7, 11 pmod 12q since they are not divisible by 3. Consequently, as above,

					2n d	´d "	2nd 2 d	´d " 2nd ´d pmod 12q.
	Hence,
	ˆ2n
	ξ	d	´d˙"	ξp2nd ´dq " ξp6dℓ `dq " pp´1q d q ℓ ξpdq " p´1q ℓ ξpdq
	in view of (2.1). Therefore, by Proposition 2.2,
					a 6 pnq " p´1q ℓ	ÿ	ξpdq.
					d|n , d odd
	Together with Lemma 2.1 and (2.7), this implies
				a 6 pnq " p´1q ℓ`1 E 1 pn; 4q " p´1q n`ℓ`1 a 2 pnq{4.
				ˆ2n
			ξ	d	´d˙"	ξp2nd ´dq " ξp6dℓ `3dq " p´1q ℓ ξp3dq.
	By Lemma 2.1 and (2.7), we obtain
				a 6 pnq " p´1q ℓ	ÿ	ξp3dq
					d|n , d odd

Now consider the case r " N. If N is odd, then 3 N " 3 pmod 12q and ξ ˆ2n d ´d˙" ξ `p2m ´3N qs ˘" ξpp2m ´3qsq.

Now, if m is odd, then m " 1, 5, 7 or 11 pmod 12q. We have 2m ´3 " 7 or 11 pmod 12q and the multiplication by 7 or by 11 exchanges the sets t1, 5u and t7, 11u. Since by (2.2) ξ takes opposite values on such sets, we have ξpp2m´3qsq " ´ξpsq. Consequently, ξp2n{d ´dq " ´ξpsq when m is odd.

If m is even, then m " 2, 4, 8 or 10 pmod 12q. Then 2m ´3 " 1 or 5 pmod 12q. The multiplication by 1 or by 5 preserves each set t1, 5u and t7, 11u, so that by (2.2) we have ξpp2m ´3qsq " ξpsq. In conclusion, ξp2n{d ´dq " p´1q m ξpsq when r " N is odd.

If r " N is even, then 3 N " ´3 pmod 12q and ξp2n{d ´dq " ξpp2m ´3N qsq " ξpp2m `3qsq. A reasoning as in the odd N case shows that when N is even we have ξp2n{d ´dq " p´1q m´1 ξpsq.

We can now compute a 6 pnq. We start with the case of odd N. Putting the above information together, we obtain

p´1q m´r ¸ξp3sq `p´1q m ξpsq " `p´1q m´1 `p´1q m ˘ÿ s|m , s odd ξpsq " 0.

On the other hand, since the power of 3 in n is odd, then by (2.3) we have a 2 pnq " p´1q n 4 E 1 pn, 4q " 0. Therefore, a 6 pnq " a 2 pnq in this case.

If N is even, then