A Walther

C Marcoux

B Desloges

R Grechishkin

D Givord

N M Dempsey

Micro-patterning of NdFeB and SmCo magnet films for integration into Micro-Electro-Mechanical-Systems

The integration of high performance RE-TM (NdFeB and SmCo) hard magnetic films into Micro-Electro-Mechanical-Systems (MEMS) requires their patterning at the micron scale.

In this paper we report on the applicability of standard micro-fabrication steps (film deposition onto topographically patterned substrates, wet etching and planarization) to the patterning of 5 µm thick RE-TM films. While NdFeB comprehensively fills micron scaled trenches in patterned substrates, SmCo deposits are characterized by poor filling of the trench corners, which poses a problem for further processing by planarization. The magnetic hysteresis loops of both the NdFeB and SmCo patterned films are comparable to those of non-patterned films prepared under the same deposition/annealing conditions. A micron-scaled multipole magnetic field pattern is directly produced by the unidirectional magnetization of the patterned films. NdFeB and SmCo show similar behavior when wet etched in an amorphous state: etch rates of approximately 1.25µm/minute and vertical side walls which may be attributed to a large lateral over-etch of typically 20 µm. Chemical-Mechanical Planarization (CMP) produced material removal rates of 0.5-3µm/min for amorphous NdFeB. Ar ion etching of such films followed by the deposition of a Ta layer prior to film crystallization prevented degradation in magnetic properties compared to nonpatterned films.

Introduction

There are many potential applications for high performance micro-magnets in magnetic MEMS. 1 However, the emergence of magnetic MEMS has been hindered by the challenge of integrating high quality magnets using techniques compatible with today's MEMS technologies. [START_REF] Gutfleisch | Magnetic Nanostructures in Modern Technology[END_REF] One of the most promising reports to date, from the viewpoint of integration, concerns the fabrication of Co 80 Pt 20 magnets {cylinders of height 2 µm and diameter 5 or 10 µm, µ 0 H c = 0.5 T, (BH) max = 52 kJ/m 3 ) by electro-deposition at room temperature. [START_REF] Zana | [END_REF] The successful integration of RE-TM magnets into devices, for which much higher coercivities and energy densities may be expected, should provide a significant boost to the field of magnetic MEMS. To achieve this, two challenges are faced. The first of these is the preparation of thick film magnets (1-100 µm) at reasonable deposition rates over relatively large surface areas. High rate triode sputtering has been shown to be suitable for the preparation of thick films of both SmCo 4 and NdFeB 5 alloys and we have recently used this technique to prepare high energy product 5 µm thick films of these materials on Si substrates (SmCo 7 -140 kJ/m 3 , NdFeB -400 kJ/m 3 ). 6,7 The second challenge to be solved is the lateral structuring of the films at the micron-scale.

The use of Si substrates greatly facilitates lateral structuring using standard microtechnology processes. However, the use of these standard processes for the structuring of rare earth based alloys, in particular steps based on chemical attack, is challenging because of the poor resistance of these alloys to oxidation. Features of size 5x5 µm up to 100x100 µm were wet-etched into 1 µm thick NdFeB films sputtered onto AL 2 O 3 substrates using salpetric acid. 8 Reports exist on both the dry and wet etching of SmCo films deposited on Al 2 O 3 substrates. 9,10 Wang et al. compared the etching rates achieved using different Ar based plasmas and found that addition of Cl 2 to an Ar plasma led to a 10-fold increase in etching rate (a maximum value of 0.7 µm/min. was reported) compared with a pure Ar plasma. 9 Budde et al. reported an increase in etching rate of SmCo from 0.02 µm/min for dry etching to 12.5µm/min for wet etching. 10 A major concern is whether coercivity is altered during structuring. Budde et al. referred to the magnetic properties of the nonpatterned films but not to those of the wet etched films, 10 while Wang et al. only gave the magnetic properties of films etched at very low rates. 9 Lemke et al. reported that no significant change was observed in the coercivity of their patterned NdFeB films, which however, had relatively low coercivities to begin with (0.2 T). 8 In this paper we report on a study of three different processes ((i) film deposition onto patterned substrates, (ii) wet etching and (iii) planarization) for the micro-patterning of RE-TM films. The influence of the processing steps on the magnetic properties of the hard magnetic material is presented.

Experimental details

Patterned Si/SiO 2 substrates containing trench motifs with individual trenches of depth 6 µm, length 500 µm and trench/wall widths on the scale of 5-100 µm were prepared by standard micro-fabrication processes: deposition of 6µm of SiO 2 on Si followed by photolithography through a mask containing the trench motif and finally dry etching of the SiO 2 .

The magnetic (NdFeB, SmCo) and buffer/capping (Ta,Cr) layers were deposited using DC triode sputtering at rates of the order of 18 µm/hour, 6,7 onto patterned Si/SiO 2 wafers and standard Si wafers. The NdFeB films were deposited at temperatures ≤ 400°C, and were thus amorphous. 7 The SmCo films were deposited in both the amorphous (no-substrate heating) and crystallised state (T sub =400°C). 6 Post-deposition annealing of amorphous films was carried out at 750°C for 10 minutes under secondary vacuum (10 -6 mbar) either in-situ (full wafer) or ex-situ on samples of typical surface area 5 x 5 mm. Film thicknesses were measured using a Dektak profilometer. Magnetic measurements were made at 300 K on a Vibrating Sample Magnetometre (VSM) with a maximum magnetic field of 8T.

Observations were made with an FEG-SEM (Gemini 1530). heating the substrate to 400°C during deposition, 6 did not lead to an improvement of the trench corners. Nor did a decrease in the sputtering rate from 18 to 10µm/h, achieved by decreasing the target potential from 900 to 300V. The difference in trench filling behavior of the NdFeB and SmCo alloys is not presently understood.

Results

The amorphous as-deposited films were annealed at a temperature of 750°C for 10 minutes to crystallize the hard magnetic Nd 2 Fe 14 B and SmCo 7 phases. 6,7 Post-deposition annealing of the entire 100 mm patterned wafers with NdFeB layers did not lead to film fracture or peel-off. This is in contrast with the case of continuous planar films deposited onto 100 mm Si wafers, for which peel-off occurs in the central region of the film. 7 In contrast, annealing of the amorphous SmCo films deposited on the patterned wafers does lead to film peel-off, as was already observed with continuous planar films deposited on Si substrates. 6 Peel-off occurs via fracture below the SmCo/substrate interface (i.e. some SiO 2 remains adhered to the SmCo film). When deposited directly in the crystallized state (T sub = 400°C), peel-off from patterned SmCo films does not occur in the trenched regions, but only in the continuous planar regions between different motifs.

Film peel-off is attributed to the build up of strain due to the difference in thermal expansion coefficients of the magnetic layer and the Si wafer. While Si has a thermal expansion coefficient of 2.6 x10 -6 /K, values in the range of 6 x10 -6 /K to 12 x10 -6 /K have been measured for RE-TM alloys at 750°C. 11 The fact that NdFeB films have distinctly less tendency to peel-off than SmCo 7 films may be attributed to two particular features. The first is that Nd 2 Fe 14 B has a lower thermal expansion anisotropy than SmCo 5 (SmCo 7 is closely related to the SmCo 5 phase). 11 The second is the fact that Nd 2 Fe 14 B is an Invar system. A large positive magnetovolume anomaly develops in the ferromagnetic state, below 592 K, which reduces very significantly the material's thermal expansion. Note finally that a study of the mechanical properties of these films as a function of the film dimensions is underway.

II -Wet etching of RE-TM films

Wet etching trials on non-capped 5µm thick RE-TM films were carried out on continuous planar films in the amorphous state. This is because it is expected that amorphous films should be less prone to oxidation than crystalline films, due to the absence of grain boundaries which can act as paths for oxygen diffusion. 2). Budde had also reported vertical flanges for wet-etched SmCo films, which he attributed to large lateral over-etching. 9 Note that patterns as small as a few tens of µm in size could in principle also be defined this way, the actual size used here was chosen to suit the overall process flow being studied (see below).

III -Planarization of NdFeB films

Chemical-Mechanical-Planarization can be used to eliminate topographical relief. This step was developed for NdFeB only, as the poor quality of the trench corners of SmCo deposited on patterned substrates and stress issues render them unsuitable for planarization.

5µm thick NdFeB films were deposited onto non-heated planar Si substrates. Wet etching was used to topographically pattern the magnetic films. These patterned films were planarized to a final thickness of 1 µm using chemical-mechanical planarization (CMP).

Film removal rates of 0.5 to 3µm/min were achieved by varying the composition of the polishing slurry.

2) Magnetic properties of micro-magnets

The extrinsic properties of a permanent magnet (coercivity, remanence, loop squareness…) depend on both the material's intrinsic magnetic properties and the magnet's microstructure (grain size, grain boundary quality, oxygen content, crystallographic texture…). As the latter may be influenced by how the magnet is prepared, the different steps involved in the micro-patterning of magnets may be expected to influence the micro-magnet's extrinsic properties. The magnetic properties of micro-magnets prepared using the various steps described above were thus evaluated.

I -Magnetic properties following deposition onto patterned substrates

VSM measurements of RE-TM films deposited onto patterned substrates are compared with those of films deposited onto non-patterned substrates and processed under similar conditions (Figure 3). More specifically, the NdFeB films were deposited amorphous at 400°C and crystallised during a post-deposition anneal (750°C/10 minutes), while the SmCo films were deposited in the crystallised state (T sub = 400°C). These deposition temperatures were chosen to induce crystallographic texture and thus maximize the remanent magnetization and energy product of the hard magnetic material. 6,7 It can be seen that, for a given material, the hysteresis loops of the patterned and planar films are very similar.

The patterned NdFeB films, which have an out-of-plane texture, were magnetized by the application of an out-of-plane magnetic field of 7T. A polar Kerr image of a uniaxial Magneto-Optic-Indicator-Film (MOIF) [START_REF] Grechishkin | Magnetic Nanostructures in Modern Technology[END_REF] placed on top of the magnetized patterned NdFeB film is shown in Figure 4. The black/white contrast corresponds to the up/down direction of magnetization in the MOIF. This image reveals that the patterned sections of the film produce micron scaled multi-polar magnetic field patterns at the position of the MOIF with a pole pitch approximately equal to the pitch of the trench walls (note that the nonpatterned sections of film between motifs do not produce enough stray field to saturate the MOIF). This is only a qualitative image and quantitative imaging using a planar MOIF is underway. The relevance of this result is that following a simple fabrication procedure (deposition + annealing of the magnetic layer), the trivial application of a uniaxial magnetic field can be used to produce a micron-scaled multi-polar field pattern. These micro-patterned films have been used to levitate diamagnetic beads, [START_REF] Chetouani | Solid-State Sensors, Actuators and Microsystems Conference[END_REF] and have potential applications in lab-on-chip experiments.

II -Magnetic properties following etching and planarization

Rare earth elements have a high affinity for oxygen and it is well known that surface oxidation of RE-TM magnets leads to degradation in their magnetic properties. [START_REF] Katter | Proc. of the 19 th Int. Workshop on Rare Earth Magnets and their Appl[END_REF] Thus fabrication steps involving chemicals could be critical for the processing of RE-TM films.

To assess the impact of the etching/planarization steps, the 1 µm thick amorphous NdFeB patterns produced were further processed under two different conditions : (#1) simple annealing and (#2) annealing following Ar ion etching during 5 min (to remove any damaged surface layer produced during the chemical attack) and deposition of a Ta capping layer. The annealing treatments were carried out at 750°C for 10 min. in a rapid heating furnace. [START_REF] Walther | [END_REF] In-plane VSM measurements of these samples are compared in figure 5. It can be seen that good magnetic properties (single phase behavior, high coercivity (µ 0 H C = 1.5 T), good squareness for an isotropic sample) are achieved for the sample #2 which was cleaned and capped before annealing. Sample #1 shows approximately the same coercivity value but it has poor loop squareness indicative of a wide distribution in coercivity values of individual grains. Note that samples which were capped with Ta but not etched prior to annealing had a small amount of a soft magnetic phase (data not shown). It can be concluded that light etching serves to remove some damaged material and that the Ta capping layer protects the magnetic film from oxidation during the annealing step.

Conclusions

The applicability of some standard micro-fabrication steps has been tested for 5 µm thick The first fully integrated magnetic micro-switch was made with electrodeposited hexagonal close packed CoPtP magnets with relatively low values of coercivity and remanence (µ 0 H c = 0.3 T, µ 0 M r = 0.3 T). [START_REF] Dieppedale | Proceedings of IEEE Sensors[END_REF] The integration of high performance NdFeB into a modified version of this micro-switch, by the consecutive application of the processing steps developed in this work (figure 6), is presently being studied. [START_REF]ANR (French National Research Agency) "Nanomag2" project, which is a collaborative action between[END_REF] Note that in this process flow the wet etching step is not used to define the final size of the magnet but as a coarse patterning step prior to CMP. Thus the large lateral over-etch observed does not pose a problem. The applicability of these micro-fabrication steps to thicker RE-TM films needs to be established. Finally, it should be noted that the high processing temperatures needed for producing NdFeB (550-750°C) films dictate that these films need to be integrated onto a given wafer before other temperature sensitive components. On the other hand, SmCo alloys, which are produced at relatively low temperatures (350-400°C), may be processed "above IC" (integrated circuit).

Figure captions

and discussion 1 )

 1 Micro-patterning of RE-TM films I -Deposition of RE-TM films onto patterned substrates )/NdFeB(8µm)/Ta(100nm)} and {Cr(100nm)/SmCo(8µm)/Cr(100nm)} films, deposited onto non-heated, patterned Si/SiO 2 substrates, are shown in figures 1a,b. It was found that NdFeB fills comprehensively and continuously the patterns even for trenches with an aspect ratio of 1.2 (6µm deep, 5µm wide). Though SmCo fills the trenches, the film structure is different at the trench corners, appearing granular and less dense than the rest of the film (fig 1c). Note that because the substrates were not heated during deposition, the asdeposited films are amorphous. Deposition of SmCo in the crystallized state, achieved by

  rare earth transition metal (NdFeB and SmCo) films. It has been found that the deposition of films onto pre-patterned Si/SiO 2 substrates with trench motifs with individual trenches of depth 6 µm and trench/wall widths on the scale of 5-100 µm works better for NdFeB than for SmCo. While NdFeB fills comprehensively and continuously the trenches, the SmCo deposits are characterized by poor filling of the trench corners (i.e. full density is not achieved in these areas). The use of patterned substrates was found to improve the mechanical behavior of the RE-TM films. In contrast to the case of films deposited on nonpatterned substrates, film fracture did not occur during the post-deposition high temperature (750°C) annealing treatment of amorphous NdFeB films deposited on patterned wafers. However fracture still occurs upon annealing of amorphous SmCo films deposited on patterned wafers, which should thus be directly deposited in the crystallized state. The magnetic properties (hysteresis loops) of patterned NdFeB and SmCo films are comparable to those of non-patterned films prepared under the same deposition/annealing conditions. The unidirectional magnetization of such a topographically patterned film directly produces a micron-scaled multi-pole magnetic field pattern.Wet etching of NdFeB and SmCo gave very similar results: etch rates of approximately 1.25µm/min., large lateral over-etch of typically 20 µm and vertical side walls. Chemical-Mechanical Planarization was studied for NdFeB films. Material removal rates of 0.5 to 3µm/min were achieved. Degradation in magnetic properties of the thus patterned NdFeB films, relative to continuous planar films, was prevented by the application of Ar ion etching of the film surface followed by the deposition of a Ta layer prior to film annealing.

Figure 1 :

 1 Figure 1: SEM cross sectional images of RE-TM films deposited onto patterned wafers

Figure 2 :

 2 Figure 2: SEM cross sectional images of a fracture surface of wet etched RE-TM films.

Figure 3 :

 3 Figure 3: Comparison of the hysteresis loops of non-patterned continuous (cont.) and

Figure 4 :Figure 5 :

 45 Figure 4: Polar Kerr optical image of a Magneto-Optic-Indicator-Film (MOIF) placed on

Figure 6 :

 6 Figure 6: Process flow diagram for the micro-patterning of magnetic films.

  Optical lithography was used to define mm sized patterns in a resist layer deposited on the RE-TM film. Removal rates of approximately 1.25µm/min. were achieved for both NdFeB and SmCo wet etched at room temperature in an acidic bath containing (NH 4 ) 2 S 2 O 8 , H 2 O and H 2 SO

4

. A large lateral overetch of typically 20 µm occurred in both materials. The side walls of the etched films are relatively vertical (Figure

Acknowledgments

This work was carried out in the framework of the ANR "Nanomag2" project. R.G. is grateful for a CNRS visiting research fellowship at Institut Néel.


Manuel Del Pino

Jean Dolbeault

AND Stathis Filippas

Achiles Tertikas

Manuel Del Pino

Achilles Tertikas

A logarithmic Hardy inequality

Keywords: December 2, 2009 Hardy inequality, Sobolev inequality, interpolation, logarithmic Sobolev inequality, Hardy-Sobolev inequalities, Caffarelli-Kohn-Nirenberg inequalities, scale invariance, Emden-Fowler transformation, radial symmetry, symmetry breaking Mathematics Subject Classification (2000). 26D10, 46E35, 58E35

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved among radial functions, in some range of the parameters.

Introduction and main results

The classical Hardy inequality in R d , d ≥ 3, states that for any smooth, compactly supported function u ∈ D(R d ), the following inequality holds:

(1)

R d |u| 2 |x| 2 dx ≤ 4 (d -2) 2 R d |∇u| 2 dx .
The constant 4/(d -2) 2 is the best possible one. Many studies have been devoted to extensions and improvements of Hardy's inequality in bounded domains containing zero. In this direction, the first result is due to Brezis and Vázquez; see [START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF]. In [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF], nonlinear improvements have been established, whereas in [41,42,[START_REF] Adimurthi | An improved Hardy-Sobolev inequality and its application[END_REF] linear and Sobolev type improvements are given. In [START_REF] Adimurthi | On the best constant of Hardy-Sobolev inequalities[END_REF], the best constant in the correction term of Sobolev type is computed. We also refer to [START_REF] Cianchi | Hardy inequalities with non-standard remainder terms[END_REF] for improvements involving nonstandard correction terms. A recent trend seems to be oriented towards weights involving a distance to a manifold rather than a distance to a point singularity; see for instance [START_REF] Badiale | A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics[END_REF][START_REF] Dávila | Hardy-type inequalities[END_REF][START_REF] Alvino | On the best constant in a Hardy-Sobolev inequality[END_REF][START_REF] Tertikas | On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality[END_REF]. In particular, when taking distance to the boundary, the dependence of the correction term on the geometry of the domain has been established in [START_REF] Hoffmann-Ostenhof | A geometrical version of Hardy's inequality[END_REF][START_REF] Filippas | On a question of Brezis and Marcus[END_REF][START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF]. In the special case of the half-space in three space dimensions, the best constant of the Sobolev term in the improvement of Hardy's inequality has been found in [START_REF] Benguria | The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space[END_REF] and it turns out to be the best Sobolev constant.

On the other hand a subject of particular interest has been the analysis of the link between Hardy's inequality (1) and Sobolev's inequality. A family of inequalities that interpolate between Hardy and Sobolev inequalities is given by the Hardy-Sobolev inequality,

(2)

R d |u| p |x| d-d-2 2 p dx 2 p ≤ C HS (p) R d |∇u| 2 dx
for any u ∈ D(R d ), where 2 ≤ p ≤ 2 d/(d -2), d ≥ 3, for a certain C HS (p) > 0. Extremals for (2) are radially symmetric and the best constant C HS (p) can be explicitly computed: see [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF][START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF]. We shall recover the expression of C HS (p) at the end of Section 3.1. Extensions and improvements of the Hardy-Sobolev inequalities, and more generally of the Caffarelli-Kohn-Nirenberg inequalities established in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF], have been the object of many papers. We refer the reader for instance to [START_REF] Badiale | A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics[END_REF][START_REF] Wang | Caffarelli-Kohn-Nirenberg inequalities with remainder terms[END_REF][START_REF] Abdellaoui | Some improved Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Alvino | On the best constant in a Hardy-Sobolev inequality[END_REF][START_REF] Tertikas | On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality[END_REF][START_REF] Adimurthi | On the best constant of Hardy-Sobolev inequalities[END_REF] for various contributions to this topic.

The purpose of this paper is to investigate the connection between (1) and another classical Sobolev type inequality: the optimal logarithmic Sobolev inequality in R d established in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] which, expressed in a scale invariant form due to Weissler in [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF], reads

(3) R d |u| 2 log |u| 2 dx ≤ d 2 log 2 π d e R d |∇u| 2 dx
for any u ∈ H 1 (R d ) such that R d u 2 dx = 1. We point out a parallel between these inequalities: just like ( 1) is an endpoint of the family [START_REF] Adimurthi | An improved Hardy-Sobolev inequality and its application[END_REF], that connects with Sobolev's inequality, Inequality (3) can be viewed as an endpoint of a family of optimal Gagliardo-Nirenberg inequalities that also connects to Sobolev's inequality; see [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF]The optimal Euclidean L p -Sobolev logarithmic inequality[END_REF] for more details.

We emphasize that Hardy's inequality [START_REF] Abdellaoui | Some improved Caffarelli-Kohn-Nirenberg inequalities[END_REF] in R d cannot be improved in the usual sense, that is, there is no nontrivial potential V ≥ 0 and no exponent q > 0 such that, for any function u,

C R d V (x) |u| q dx 2/q ≤ 4 (d -2) 2 R d |∇u| 2 dx - R d |u| 2 |x| 2 dx
for some positive constant C, as one can easily see by testing the above inequality with u ǫ (x) = |x| -d-2

2 +ǫ , |x| ≤ 1, and u ǫ (x) = |x| -d-2

2 -ǫ , |x| > 1, and sending ǫ to zero. Instead of improving on the potential, we study here the possibility of improving on the control of u. The weight is fixed to be 1/|x| 2 and we try to get a control on |u| 2 log |u| 2 instead of a control on |u| 2 only, as can sometimes be done for inequalities which appear as endpoints of a family, like [START_REF] Adimurthi | On the best constant of Hardy-Sobolev inequalities[END_REF]. As a result, we obtain inequalities of logarithmic Sobolev type, with weight 1/|x| 2 in the term involving the logarithm. Such an inequality is somewhat unusual, because in most of the cases, logarithmic Sobolev inequalities involve bounded positive measures. The euclidean case with Lebesgue's measure is an exception and can actually be reinterpreted in terms of the gaussian measure, see for instance [START_REF] Carlen | Logarithmic Sobolev inequalities and spectral gaps[END_REF][START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF] for some recent contributions in this direction. In the case of bounded measures, there is a huge literature: one can refer to [START_REF] Muckenhoupt | Hardy's inequality with weights[END_REF][START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF][START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF] for a few key contributions.

The logarithmic Sobolev and Hardy inequalities play an important role in a number of instances. The first one is a very natural tool for obtaining intermediate asymptotics for the heat equation, see [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF][START_REF] Toscani | Sur l'inégalité logarithmique de Sobolev[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Dolbeault | A new class of transport distances between measures[END_REF][START_REF] Bartier | Improved intermediate asymptotics for the heat equation[END_REF] with natural extensions to nonlinear diffusions (see for instance [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] and references therein). These inequalities are also useful in obtaining heat kernel estimates (see for instance [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF][START_REF] Davies | Heat kernels and spectral theory[END_REF]). A related logarithmic Sobolev inequality recently appeared in [START_REF] Filippas | Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains[END_REF][START_REF]Improving L 2 estimates to Harnack inequalities[END_REF], where it was used for obtaining upper bounds for the heat kernel of a degenerate equation.

We shall denote by

D 1,2 (R d ) the completion of D(R d ) under the L 2 (R d ) norm of the gradient of u. Let S = 1 π d (d -2) Γ (d) Γ d 2 2 d = C HS 2 d d -2
be the optimal constant in Sobolev's inequality, according to [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF]. Our first result states the validity of the following logarithmic Hardy inequality.

Theorem A. Let d ≥ 3. There exists a constant C LH ∈ (0, S] such that, for all u ∈ D 1,2 (R d ) with R d |u| 2 |x| 2 dx = 1, we have (4) R d |u| 2 |x| 2 log |x| d-2 |u| 2 dx ≤ d 2 log C LH R d |∇u| 2 dx .
Inequality (4) can be viewed as an infinitesimal form of the Hardy-Sobolev inequality at p = 2: we observe that its left hand side is nothing but the derivative in p at p = 2 of the left hand side of (2), up to a factor 2. Compared to an entropy term with respect to the measure |x| -2 dx, there is however a log(|x| d ) term. Such a term is easily recovered by scaling considerations and compensates for the presence of a superquadratic nonlinearity |u| 2 log |u| 2 . The quantities involved in (4) give a precise account of the fact that, to exert control by the Dirichlet integral of a power larger than two of u, the singularity has to be at the same time milder.

It is natural to search for the optimal constant and extremals for Inequality (4). Our second result answers this question in the class of radially symmetric functions, depending only on |x|, x ∈ R d .

Theorem B. Let d ≥ 3. If u = u(|x|) ∈ D 1,2 (R d ) is radially symmetric, and R d |u| 2 |x| 2 dx = 1, then R d |u| 2 |x| 2 log |x| d-2 |u| 2 dx ≤ d 2 log C * LH R d |∇u| 2 dx .
where

C * LH := 4 d Γ d 2 2 d π (8 π e) 1 d d -1 (d -2) 2 1-1 d .
Equality in the above inequality is achieved by the function

u = ũ R d |ũ| 2 |x| 2 dx where ũ(x) = |x| -d-2 2 exp -(d-2) 2 4 (d-1) log |x| 2 .
For d ≥ 2 and a < (d -2)/2, by starting from a more general weighted Hardy inequality,

R d |u| 2 |x| 2 (a+1) dx ≤ 4 (d -2 -2 a) 2 R d |∇u| 2 |x| 2 a dx , (5)
we prove the validity of a whole class of weighted logarithmic Hardy inequalities. If we denote by D 1,2 a (R d ) the completion with respect to the norm defined by the right hand side of (5

) of D(R d \ {0}) if d ≥ 2 and of {u ∈ D(R) : u ′ (0) = 0} if d = 1, our result reads: Theorem A'. Let d ≥ 1. Suppose that a < (d -2)/2, γ ≥ d/4 and γ > 1/2 if d = 2. Then there exists a positive constant C GLH such that, for any u ∈ D 1,2 a (R d ) normalized by R d |u| 2 |x| 2 (a+1) dx = 1, we have (6)
R d |u| 2 |x| 2 (a+1) log |x| d-2-2 a |u| 2 dx ≤ 2 γ log C GLH R d |∇u| 2 |x| 2 a dx .
On the other hand, in the radial case, we have a more general family of sharp inequalities:

Theorem B'. Let d ≥ 1, a < (d -2)/2 and γ ≥ 1/4. If u = u(|x|) ∈ D 1,2 a (R d
) is radially symmetric, and

R d |u| 2 |x| 2 (a+1) dx = 1, then (7)
R d |u| 2 |x| 2 (a+1) log |x| d-2-2 a |u| 2 dx ≤ 2 γ log C * GLH R d |∇u| 2 |x| 2 a dx ,
where

(8) C * GLH = 1 γ Γ d 2 1 2 γ (8 π d+1 e) 1 4 γ 4 γ -1 (d -2 -2 a) 2 4 γ-1 4 γ if γ > 1 4 and C * GLH = 4 Γ d 2 2 8 π d+1 e if γ = 1 4 .
If γ > 1 4 , equality in (7) is achieved by the function

u = ũ R d |ũ| 2 |x| 2 dx where ũ(x) = |x| -d-2-2 a 2 exp -(d-2-2 a) 2 4 (4 γ-1) log |x| 2 .
Theorems A and B are special cases of Theorems A' and B' corresponding to a = 0, γ = d/4, d ≥ 3. The family of inequalities of Theorem B' imply on the one hand the logarithmic Sobolev inequality, and on the other hand the Hardy inequality, with optimal constants, as we shall see in Section 4. In dimension d = 1, radial symmetry simply means that functions are even.

We notice that Inequalities (6) and ( 7) are both homogeneous and scale invariant. Actually, all integrals are individually scale invariant, in the sense that their values are unchanged if we replace u(x) by u λ (x) = λ (d-2-2 a)/2 u(λ x). This is of course consistent with the fact that the inequalities behave well under the Emden-Fowler transformation [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] u

(x) = |x| -d-2-2 a 2 w(y) with y = (s, ω) := -log |x|, x |x| ∈ C := R × S d-1
and have an equivalent formulation on the cylinder C, which goes as follows.

C

|w| 2 log |w| 2 dy ≤ 2 γ log C * GLH C |∇w| 2 dy + 1 4 (d -2 -2 a) 2 .
The value of the optimal constant C * GLH is given by [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]. If γ > 1 4 , equality in (11) is achieved by the function

w(s) = w(s) C w2 dy where w(s) = exp - (d -2 -2 a) 2 4 (4 γ -1) s 2 . If d = 1, C is equal to R.
For any d ≥ 1, one may suspect that the optimal constant for (6) (resp. ( 10)) is achieved in the class of radially symmetric functions (resp. functions depending only on s ∈ R) and therefore C GLH = C * GLH . Using the method developed in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF], it turns out that there is a range of the parameters a and γ for which this is not the case.

Theorem C. Let d ≥ 2 and a < -1/2. Assume that γ > 1/2 if d = 2. If, in addition, d 4 ≤ γ < 1 4 + (d -2 a -2) 2 4 (d -1) ,
then the optimal constant C GLH in inequality (6) is not achieved by a radial function and C GLH > C * GLH . This paper is organized as follows. In Section 2, we derive Theorems A, A' and A" as a consequence of a Caffarelli-Kohn-Nirenberg interpolation inequality. In Section 3, we present a complete study of the radial case and in particular we prove Theorems B, B' and B". This study is based on a sharp one-dimensional interpolation inequality. In Section 4, we show that Theorem B' implies both the logarithmic Sobolev inequality (3) and the Hardy inequality (1). In the final section, we study the symmetry breaking of the interpolation inequalities as well as of the logarithmic Hardy inequality, thus establishing Theorem C.

Interpolation inequalities. Proof of Theorems A, A' and A"

In this section, we will give the proofs of Theorems A, A' and A" with the help of a general inequality of Caffarelli-Kohn-Nirenberg type and a differentiation procedure with respect to some of the parameters of the inequality. Our starting point is the following inequality, which has been established in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]:

(12) R d |u| p |x| b p dx 2 p ≤ C CKN (p, a) R d |∇u| 2 |x| 2 a dx ∀ u ∈ D(R d ) .
Restrictions on the exponents are given by the conditions: b See for instance [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] for a review of various known results like existence of optimal functions. In the limit case b = a+1, p = 2, ( 12) is equivalent to [START_REF] Alvino | On the best constant in a Hardy-Sobolev inequality[END_REF] and the optimal constant is then

∈ (a + 1/2, a + 1] in case d = 1, b ∈ (a, a + 1] when d =
C CKN (2, a) = 4/(d-2-2 a) 2 .
The range a > (d -2)/2 can also be covered with functions in the space D(R d \ {0}). Inequalities are not restricted to spaces of smooth functions and can be extended to the space D a,b (R d ) obtained by completion of D(R d \ {0}) with respect to the norm defined by

u 2 = |x| -b u 2 L p (R d ) + |x| -a ∇u 2 L 2 (R d ) ,
but some care is required. For instance, if a > (d -2)/2, it turns out that, for any

u ∈ D a,b (R d ), lim r→0+ r -d u L 2 (B(0,r)) = 0 .
See [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF] for more details.

A key issue for ( 12) is to determine whether equality is achieved among radial solutions when C CKN is the optimal constant, or, alternatively, if symmetry breaking occurs. See [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF][START_REF] Lin | Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF][START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] for conditions for which the answer is known. Here are some cases for which radial symmetry holds:

(i) The dimension is d = 1. (ii) If d ≥ 3,
we assume either a ≥ 0 or, for any p ∈ (2, 2 * ), a < 0 and |a| is small enough, or for any a < 0, p -2 > 0 is small enough. (iii) If d = 2, we assume either a < 0 with |a| small enough and |a| p < 2, or, for any a < 0, p -2 > 0 is small enough.

In such cases, optimal functions are known and

C CKN (p, a) is explicit (see Section 3.3). Alternatively, it is known that for d ≥ 2, if (14) a < b < 1 + a - d 2 1 - d -2 -2 a (d -2 -2 a) 2 + 4 (d -1) ,
minimizers are not radially symmetric. In such a case the explicit expression of C CKN is not known. More details will be given in Section 5.

Let 2 * = ∞ if d = 1 or d = 2, 2 * = 2d/(d -2) if d ≥ 3 and define ϑ(p, d) := d (p -2) 2 p .
We have a slightly more general family of interpolation inequalities than [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF], which has also been established in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and goes as follows.

Theorem 1 (According to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]). Let d ≥ 1. For any θ ∈ [ϑ(p, d), 1], there exists a positive constant C(θ, p, a) such that

(15) R d |u| p |x| b p dx 2 p ≤ C(θ, p, a) R d |∇u| 2 |x| 2 a dx θ R d |u| 2 |x| 2 (a+1) dx 1-θ ∀ u ∈ D 1,2 a (R d ) .
Inequality [START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF] coincides with [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF] if θ = 1. We will establish the expression of C(θ, p, a) when minimizers are radially symmetric and extend the symetry breaking results of Felli and Schneider to the case θ < 1 in Sections 3 and 5 respectively. Before, we give an elementary proof of ( 15), whose purpose is to give a bound on C(θ, p, a) in terms of the best constant in [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF], and to justify the limiting case that is obtained by passing to the limit θ → 0 + and p → 2 + simultaneously.

Proposition 2. Let b ∈ (a + 1/2, a + 1] when d = 1, b ∈ (a, a + 1] when d = 2 and b ∈ [a, a + 1] when d ≥ 3.
In addition, for any d ≥ 1, we assume that (13) holds. Then we have

(i) Let K := {k ∈ (0, 2) : k ≤ d -(d -2) p/2 if d ≥ 3}. For any θ ∈ [ϑ(p, d), 1] ∩ (1 -2/p, 1], we have C(θ, p, a) ≤ inf k∈K C CKN 2 (p -k) 2 -k , a 1-k p 2 d -2 -2 a 2( k p +θ-1) (ii) Let d ≥ 2. Suppose that a < (d -2)/2, γ ≥ d/4 and γ > 1/2 if d = 2.
We have

C GLH ≤ C CKN 4 γ 2 γ-1 , a .
Proof. If d ≥ 3 and p = 2 * , that is for b = a, then ϑ(p, d) = 1 = θ and ( 15) is reduced to a special case of [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF]. Assume that p < 2 * . Let u ∈ D(R d ). For any k ∈ (0, 2), we have: ( 16)

R d |u| p |x| b p dx = R d |u| |x| 1+a k |u| p-k |x| b p-k (1+a) dx ≤ R d |u| 2 |x| 2 (a+1) dx k 2 R d |u| 2 p-k 2-k |x| 2 b p-k (1+a) 2-k dx 2-k 2 .
We observe that ( 12) holds for some a, b and p if, due to the scaling invariance, these parameters are related by the relation

b = a + 1 + d 1 p -1 2 = a + 1 -ϑ(p, 1)
. For any k ∈ (0, 2), we also have the relation

B = A + 1 + d 1 P -1 2 if A = a , P = 2 (p -k) 2 -k and B P = 2 (b p -k (1 + a)) 2 -k .
Hence, using [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF], we have that ( 17)

R d |u| 2 p-k 2-k |x| 2 b p-k (1+a) 2-k dx 2-k p-k ≤ C CKN 2 p -k 2 -k , a R d |∇u| 2 |x| 2 a dx provided that 2 < 2 (p -k)/(2 -k) ≤ 2 * if d ≥ 3, which is equivalent to k ≤ d -(d -2) p/2 using the fact that k < 2.
On the other hand, we may estimate the first integral in the right hand side of ( 16) by ( 5) and get ( 18) 16), ( 17) and ( 18) we get

R d |u| 2 |x| 2 (a+1) dx ≤ 4 (d -2 -2 a) 2 R d |∇u| 2 |x| 2 a dx 1-α R d |u| 2 |x| 2 (a+1) dx α for any α ∈ [0, 1]. Combining (
R d |u| p |x| b p dx 2 p ≤ C(θ, p, a) R d |∇u| 2 |x| 2 a dx θ R d |u| 2 |x| 2 (a+1) dx 1-θ with θ = 1 -αk/p ∈ [ϑ(p, d), 1]
and this proves [START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF]. Notice that for d ≥ 3, the restriction θ ≥ ϑ(p, d) [START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF] still holds true. For this case, we refer to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]. If d = 1, one knows that the Inequality (15) holds true under the condition θ > 1 -2/p, but the inequality still holds under the weaker condition θ ≥ ϑ(p, 1) = 1/2 -1/p. See Section 3.1 for further details in the one-dimensional case.

comes from the condition k ≤ d -(d -2) p/2. If d = 2, θ > ϑ(p, 2) = 1 -2/p is due to the restriction k < 2. However, if θ = ϑ(p, 2), Inequality
Let P ∈ (2, 2 * ] if d ≥ 3, P > 2 if d = 1 or 2. For any p ∈ [2, P ) we choose k = 2 P -p P -2 ∈ (0, 2], which also satisfies k ≤ d-(d-2) p/2 so that P = 2 p-k 2-k ≤ 2 * , if d ≥ 3 and k < 2.
We also set B := a+ 1 -

d 1 P -1 2 , so that B P = 2 b p-k (1+a) 2-k
. Then ( 16) can be written as

(19) R d |u| p |x| b p dx ≤ R d |u| 2 |x| 2 (a+1) dx P -p P -2 R d |u| P |x| BP dx p-2 P -2 .
Here we assume that b 19) is valid for any 2 < p < P ≤ 2 * , and it is an equality for p = 2 and any P ∈ (2, 2 * ]. By differentiating [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF] with respect to p at p = 2, we get that for any P ∈ (2, 2 * ],

= a + 1 + d 1 p -1 2 . If d ≥ 3, Estimate (
R d |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R d |u| 2 |x| 2 (a+1) dx   dx ≤ P P -2 R d |u| 2 |x| 2 (a+1) dx log    R d |u| P |x| BP dx 2 P R d |u| 2 |x| 2 (a+1) dx    . For d ≥ 3, let 2 γ := P P -2 ∈ [ d 2 , ∞). For d = 1, 2, let 2 γ := P P -2 ∈ (1, ∞). Then, for any γ ≥ d/4 if d ≥ 3 and any γ > 1/2 if d = 1,
2 and any a < (d -2)/2, using once more [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF], we have shown (ii).

Proof of Theorems A, A' and A":

The existence of C GLH is a straightforward consequence of Proposition 2 if d ≥ 2. This proves Theorem A', except for the case d = 1 which will be considered in Section 3.4. Theorem A follows with γ = d/4, d ≥ 3 and a = 0. In particular we get an upper bound for the optimal constant:

C LH ≤ C CKN (2 * , 0) = S.
By the Emden-Fowler transformation [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF], the inequalities of Proposition 2, on R d , are transformed into equivalent ones on the cylinder C = R × S d-1 . More precisely [START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF] can be reformulated as (20)

C |w| p dy 2 p ≤ C(θ, p, a) C |∇w| 2 dy + 1 4 (d -2 -2 a) 2 C |w| 2 dy θ C |w| 2 dy 1-θ .
Notice that by standard arguments, the sharp constant in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] is achieved in H 1 (C) when the parameters are in the range corresponding to the assumptions of Proposition 2, provided p > 2 and θ > ϑ(p, d).

By [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF], the logarithmic Hardy inequality (6) of Theorem A' takes the form [START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF] of Theorem A" if γ ≥ d/4, d ≥ 3 and a < (d -2)/2, or γ > 1/2, d = 2 and a < 0. The one-dimensional case, for which C = R, will be directly investigated in the next section.

3. The one-dimensional and the radial cases. Proof of Theorems B, B' and B"

In this section we will study the Caffarelli-Kohn-Nirenberg interpolation inequality [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] as well as the logarithmic Hardy inequality [START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF] in the one-dimensional case and under the restriction to the set of radial functions. As a consequence, we shall also establish Theorems B, B' and B". 21) is however of interest by itself and can be considered as depending on the parameters θ, p and σ, independently of a and d.

R |w| p ds 2 p ≤ K(θ, p, σ) R |w ′ | 2 ds + σ 2 R |w| 2 ds θ R |w| 2 ds 1-θ , for any w ∈ H 1 (R), with σ = (d -2 -2 a)/2, provided C(θ, p, a) |S d-1 | 1-2/p = K(θ, p, σ). Inequality (
If θ > ϑ(p, 1), the proof of the existence of an optimal function is standard. After optimizing the inequality under scalings, ( 21) reduces to a Gagliardo-Nirenberg inequality, whose optimal function is defined up to a scaling and a multiplication by a constant. Let us give some details.

If we optimize Inequality [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF] under scalings, we find that it is equivalent to the one-dimensional Gagliardo-Nirenberg inequality. Let

Q[w] := R |w ′ | 2 ds + σ 2 R |w| 2 ds θ R |w| 2 ds 1-θ
be the functional which appears in the right hand side of Inequality [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF] and consider w λ (s) = λ 1/p w(λ s), λ > 0. This scalings leaves the left hand side of Inequality (21) invariant, while

Q[w λ ] = λ 2-b R |w ′ | 2 ds + σ 2 λ -b R |w| 2 ds θ R |w| 2 ds 1-θ with b = (p -2)/(p θ). We observe that a = 2 -b is positive if and only if θ > (p -2)/(2 p) = ϑ(p, 1). Hence we find that (i) if θ < ϑ(p, 1), then inf λ>0 Q[w λ ] = lim λ→∞ Q[w λ ] = 0, and Inequality (21) does not hold. (ii) if θ = ϑ(p, 1), then inf λ>0 Q[w λ ] = lim λ→∞ Q[w λ ] = R |w ′ | 2 ds ϑ(p,1) R |w| 2 ds 1-ϑ(p,1)
, so that ( 21) is equivalent to the one-dimensional Gagliardo-Nirenberg inequality [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] w

L p (R) ≤ C GN w ′ ϑ(p,1) L 2 (R) w 1-ϑ(p,1) L 2 (R) ∀ w ∈ H 1 (R) .
Hence K(ϑ(p, 1), p, σ) = C 2 GN is independent of σ > 0, and Inequality (21) admits no optimal function if θ = ϑ(p, 1), σ > 0. It degenerates into [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] in the limit σ → 0 + , for which an optimal function exists.

(iii) if θ > ϑ(p, 1), then inf λ>0 Q[w λ ] is achieved for λ 2 = b a σ 2 R |w| 2 ds R |w ′ | 2 ds , that is inf λ>0 Q[w λ ] = κ R |w ′ | 2 ds ϑ(p,1) R |w| 2 ds 1-ϑ(p,1) with κ = a b σ 2 p-2 2 p a+b a σ 2 θ , i.e. 1 κ = (p -2) σ 2 2 + (2θ -1) p p-2 2 p 2 + (2θ -1) p 2 p θ σ 2 θ .
As a consequence, K(θ, p, σ) = κ -1 C 2 GN and optimality is achieved in Inequality [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF], since ( 22) admits an optimal function; see for instance [START_REF] Agueh | Gagliardo-Nirenberg inequalities involving the gradient L 2 -norm[END_REF][START_REF] Petersson | Best constants for Gagliardo-Nirenberg inequalities on the real line[END_REF].

The above Gagliardo-Nirenberg inequality ( 22) is equivalent to the Sobolev inequality corresponding to the embedding H 1 0 (0, 1) ֒→ L q (0, 1) for some q = q(p) > 2; see [START_REF] Benguria | Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane[END_REF][START_REF] Veling | Lower bounds for the infimum of the spectrum of the Schrödinger operator in R N and the Sobolev inequalities[END_REF][START_REF]Lower bounds for the infimum of the spectrum of the Schrödinger operator in R N and the Sobolev inequalities[END_REF] for more details. Also notice that, using the radial symmetry of the minimizers of the optimal functions of the Hardy-Sobolev inequality (2), we recover the expression of C HS (p) = |S d-1 | -(p-2)/p K(1, p, (d -2)/2) that can be found in [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF][START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF],

using |S d-1 | = 2 π d/2 / Γ (d/2), √ π Γ(d) = 2 d-1 Γ (d/2) Γ ((d + 1)/2
) and Lemma 3 below. Similarly, in all cases for which optimal functions are known to be radially symmetric in Inequality [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF] (see Section 2), we have

C CKN (p, a) = |S d-1 | -(p-2)/p K(1, p, (d -2 -2 a)/2
). The constant K(θ, p, σ) can be computed as follows.

Lemma 3. Let σ > 0, p > 2 and θ ∈ [ϑ(p, 1), 1]. Then the best constant in Inequality [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF] is given by:

(23) K(θ, p, σ) = (p -2) 2 σ 2 2 + (2θ -1) p p-2 2 p 2 + (2θ -1) p 2 p θ σ 2 θ 4 p + 2 6-p 2 p   Γ 2 p-2 + 1 2 √ π Γ 2 p-2   p-2 p
.

If θ > ϑ(p, 1), the best constant is achieved by an optimal function w(s), which is unique up to multiplication by constants and shifts and is given by

w(s) = cosh(λ s) -2 p-2 with λ = 1 2 (p -2) σ p+2 2+(2θ-1) p 1 2 .
Proof. Using the Emden-Fowler transformation, the value of K(θ, p, σ) can be computed using the equation [START_REF] Byeon | Symmetry breaking of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] (p -2) 2 w ′′ -4 w + 2 p |w| p-2 w = 0 such that w ′ (0) = 0 and lim |s|→∞ w(s) = 0. A minimizer for ( 21) is indeed defined up to a translation (a scaling in the original variables) and a multiplication by a constant, which can be adjusted to fix one of the coefficients in the Euler-Lagrange equation as desired. An optimal function can therefore be written as w(λ s) for some λ > 0, on which we can optimize. The solution w of ( 24) is unique if we further assume that it is positive with a maximum at s = 0. This can be seen as follows. Multiply (24) by w and integrate from s to +∞. Since lim |s|→∞ w ′ (s) = 0, the function s → 1 2 (p -2) 2 w ′ (s) 2 -2 w(s) 2 + 2 w(s) p is constant and therefore equal to 0. This determines w(0) = 1, so that the solution is unique and given by

w(s) = (cosh s) -2 p-2 ∀ s ∈ R . Hence K(θ, p, σ) = max λ>0 λ -1 I p 2 p (λ J 2 + σ 2 λ -1 I 2 ) θ (λ -1 I 2 ) 1-θ where I q := R |w(s)| q ds and J 2 := R |w ′ (s)| 2 ds . With λ = µ θ , let g(µ) := µ 2 p +2θ-1 J2 I2 + σ 2 µ 2 p -1 and observe that K(θ, p, σ) = I 2 p p min µ>0 g(µ) θ I 2 .
If θ > ϑ(p, 1), the minimum of g(µ) is achieved at

λ 2 = µ 2θ = p-2 2+(2θ-1) p σ 2 I2 J2 and min µ>0 g(µ) θ = h(p, θ, σ) J 2 I 2 1 2 -1 p where h(p, θ, σ) := 2 + (2θ -1) p (p -2) σ 2 p-2 2 p 2 p θ σ 2 2 + (2θ -1) p θ . If θ = ϑ(p, 1), then inf µ>0 g(µ) = J2
I2 and we set h(p, θ, σ) := 1. For any θ ∈ [ϑ(p, 1), 1], we thus obtain

K(θ, p, σ) = I 2 p p h(p, θ, σ) J 1 2 -1 p 2 I 1 2 + 1 p 2 .
Using the formula

R ds (cosh s) q = √ π Γ q 2 Γ q+1 2 =: f (q) ,
we can compute

I 2 = f 4 p -2 , I p = f 2 p p -2 = f 4 p -2 + 2 ,
and get

I 2 = √ π Γ 2 p-2 Γ p+2 2 (p-2) , I p = 4 I 2 p + 2 , J 2 := 4 (p -2) 2 (I 2 -I p ) = 4 I 2 (p + 2)(p -2)
.

Hence

K(θ, p, σ) = (p -2) 2 σ 2 2 + (2θ -1) p p-2 2 p 2 + (2θ -1) p 2p θ σ 2 θ 4 p + 2 6-p 2 p 1 I 2 p-2 p
, which proves (23). Substituting θ with γ (p -2) in the expression of K(θ, p, σ) given by ( 23) and taking the logarithm, we get

log K(γ (p -2), p, σ) = p -2 2 p log 2σ 2 2 γ p -1 + γ (p -2) log 2 γ p -1 2 p γ σ 2 + 6 -p 2 p log 4 p + 2 + p -2 p log   Γ 2 p-2 + 1 2 √ π 2 p-2 Γ 2 p-2   .
Using Stirling's formula, it is easy to see that lim t→∞

Γ(t+ 1
2 )

√ t Γ(t) = 1, so that lim p→2 K(γ (p -2), p, σ) = 1. Hence, for any γ > 1/(2 p), let

K(γ, σ) := -2 d dp K(γ (p -2), p, σ) |p=2
and consider the limit as p → 2 in Inequality [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF]. We observe that 1/4 > 1/(2 p) for any p > 2 so that γ > 1/4 guarantees γ > 1/(2 p) uniformly in the limit p → 2 + . The case γ = 1/4 is achieved as a limit case.

Lemma 4. Let σ > 0 and γ ≥ 1/4. Then for any w ∈ H 1 (R) the following inequality holds true

(25) R |w| 2 log |w| 2 R |w| 2 ds ds + K(γ, σ) R |w| 2 ds ≤ 2 γ R |w| 2 ds log R |w ′ | 2 ds R |w| 2 ds + σ 2 ,
with sharp constant given by

(26) K(γ, σ) = 2 γ log γ - 4 γ -1 2 log 4 γ -1 4 σ 2 + 1 2 log (2 π e)
if γ > 1/4, and equality holds in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]

for w(s) = exp -σ 2 4 γ-1 s 2 . If γ = 1/4, then K(γ, σ) = 2 γ log γ + 1 2 log (2 π e).
Proof. The result follows by taking the logarithm of ( 21) and differentiating at p = 2 for γ > 1/4. The equality case in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] can be checked by a direct computation. Similarly, Inequality ( 25) is equivalent to [START_REF] Badiale | A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics[END_REF] with K(γ, σ)

+ log |S d-1 | = -2 γ log C * GLH .
This proves Theorem B". Theorem B' follows by the Emden-Fowler change of coordinates [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. Theorem B corresponds to the special case a = 0, d ≥ 3. Notice that, with σ = (d -2 -2 a)/2, Inequality [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] written in terms of a function f on R + such that u(x) = f (|x|) takes the form

∞ 0 r d-3-2 a |f | 2 log r d-2-2 a |f | 2 dr + K (γ, σ) ≤ 2 γ log ∞ 0 r d-1-2 a |f ′ | 2 dr , under the normalization condition ∞ 0 r d-3-2 a |f | 2 dr = 1.
3.4. The sharp interpolation inequality in the case of the one-dimensional real line. Recall that inequalities written on the euclidean space R d are equivalent to one-dimensional inequalities on C by the Emden-Fowler transformation (9) only in case of radial functions and, for d = 1, only for even functions. However, in this case, we may notice that the restriction a < (d -2)/2 = -1/2 means that the weight |x| -2a corresponds to a positive power, so that we may consider the problem on R + and R -as two independent problems when dealing with a smooth function u such that u ′ (0) = 0.

Using X + log X + + X -log X -≤ (X + + X -) log(X + + X -) with X ± = R ± |u| 2 |x| 2 (a+1) dx, we get R |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R |u| 2 |x| 2 (a+1) dx   dx = R |u| 2 |x| 2 (a+1) log |x| d-2-2 a |u| 2 dx-(X + +X -) log(X + +X -) ≤ R - |u| 2 |x| 2 (a+1) log |x| d-2-2 a |u| 2 dx -X -log X -+ R + |u| 2 |x| 2 (a+1) log |x| d-2-2 a |u| 2 dx -X + log X + = R - |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R - |u| 2 |x| 2 (a+1) dx   dx + R + |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R + |u| 2 |x| 2 (a+1) dx   dx
By the Emden-Fowler transformation, we know that

R ± |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R ± |u| 2 |x| 2 (a+1) dx   dx ≤ 2 γ X ± log C GLH Y ± X ± with Y ± = R ± |∇u| 2 |x| 2 a dx. Using X -log Y- X-+ X + log Y+ X+ ≤ (X + + X -) log Y++Y- X++X-, we end up with the inequality R |u| 2 |x| 2 (a+1) log   |x| d-2-2 a |u| 2 R |u| 2 |x| 2 (a+1) dx   dx ≤ 2 γ R |u| 2 |x| 2 (a+1) dx log   C GLH R |∇u| 2 |x| 2 a dx R |u| 2 |x| 2 (a+1) dx   ,
which completes the proof of Theorem A' in the one-dimensional case.

Connection with logarithmic Sobolev and Hardy inequalities

In this section, we study the connection of the Logarithmic Hardy inequality (4) and its generalized form [START_REF] Alvino | On the best constant in a Hardy-Sobolev inequality[END_REF] with the Euclidean Logarithmic Sobolev inequality (see below), the Hardy inequality (1) and its generalized form [START_REF] Adimurthi | An improved Hardy-Sobolev inequality and its application[END_REF], and the Logarithmic Sobolev inequality on C (see below).

As we have seen in the previous section, the weighted logarithmic Hardy inequality of Theorem B' (radial case) is equivalent to the one-dimensional inequality [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] with sharp constant given by [START_REF] Carlen | Logarithmic Sobolev inequalities and spectral gaps[END_REF]. With the choice 2σ = 4 γ -1, we observe that lim γ→1/4 K(γ, 4 γ -1) = 1 2 log π e 2 and recover the one-dimensional logarithmic Sobolev inequality written in the scale invariant form (see [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF]) with optimal constant, namely

(28) R |w| 2 log |w| 2 R |w| 2 ds ds ≤ 1 2 R |w| 2 ds log 2 π e R |w ′ | 2 ds R |w| 2 ds .
Actually, Inequality ( 25) can be written in a simpler form in terms of a function v ∈ H 1 (R), such that

w(s) = v (d -2 -2 a) s/ √ 4 γ -1 as follows. For any γ ≥ 1/4 and any v ∈ H 1 (R), R |v| 2 log |v| 2 R |v| 2 ds ds + 2 γ log γ + 1 2 log(2 π e) R |v| 2 ds ≤ 2 γ R |v| 2 ds log R |v ′ | 2 ds R |v| 2 ds + γ -1 4 .
Consider the γ-dependent terms, i.e.

2 γ R |v| 2 ds log R |v ′ | 2 ds R |v| 2 ds + γ -1 4 -2 γ log γ R |v| 2 ds = 2 R |v ′ | 2 ds - 1 2 R |v| 2 ds f (t) , with f (t) := 1 t log(t + 1) and t = 1 4 γ 4 R R |v ′ | 2 ds R R |v| 2 ds -1 .
An elementary analysis shows that f is decreasing so that, in terms of γ, the minimum of the right hand side is always achieved at γ = 1/4. In this case we recover the one-dimensional logarithmic Sobolev inequality written in the scale invariant form [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF]. On the other hand, if we send γ to ∞ which implies that t → 0 and f (t) → 1, we recover the logarithmic Sobolev inequality in the standard euclidean form (see [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF]):

(29) R |v| 2 log |v| 2 R |v| 2 ds ds + 1 2 R |v| 2 ds log 2π e 2 ≤ 2 R |v ′ | 2 ds .
We can also recover Hardy's inequality from (25) by taking the limit γ → +∞ and observing that lim γ→+∞ K(γ, σ)/(2 γ) = 2 log σ. The radial function u ∈ D 1,2 a (R d ) given in terms of w by the inverse of the Emden-Fowler change of coordinates ( 9) satisfies (30)

1 4 (d -2 -2 a) 2 R d |u| 2 |x| 2 (a+1) dx ≤ R d |∇u| 2 |x| 2 a dx .
This holds true for any d ∈ N, d ≥ 2 and any a < 0. For d ≥ 3, if we define f (x) := |x| -a u(x), x ∈ R d , then Inequality ( 30) is equivalent to the usual Hardy inequality (with a = 0), namely (31)

1 4 (d -2) 2 R d |f | 2 |x| 2 dx ≤ R d |∇f | 2 dx , f ∈ D 1,2 (R d ) .
Using Schwarz' symmetrization, it is then straightforward to see that optimality is achieved for radial functions, thus showing that, with σ = (d -2 -2 a)/2, Inequality (25) implies Inequality [START_REF] Davies | Heat kernels and spectral theory[END_REF] for any function u ∈ D 1,2 a (R d ) (and not only for radial functions), that is Hardy's inequality if a = 0 and all Caffarelli-Kohn-Nirenberg inequalities with b = a + 1, a < 0, otherwise.

Summarizing, the family of inequalities [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] of Theorem B' implies as extreme cases the logarithmic Sobolev inequality (28) at the endpoint γ = 1/4, and the euclidean logarithmic Sobolev inequality [START_REF] Cianchi | Hardy inequalities with non-standard remainder terms[END_REF] and the Hardy inequality [START_REF] Dávila | Hardy-type inequalities[END_REF] as γ tends to +∞. In both cases, one-dimensional versions of the inequalities are involved. On C, it is possible to recover the optimal logarithmic Sobolev inequality from the logarithmic Hardy inequality as follows.

Let dµ and and dν σ (t) := (2πσ 2 ) -1/2 exp(-t 2 /(2σ 2 )) dt be respectively the uniform probability measure on S d-1 induced by Lebesgue's measure on R d and the gaussian probability measure on R. Using the tensorization property of the logarithmic Sobolev inequalities (see for instance [START_REF] Ané | Panoramas et Synthèses [Panoramas and Syntheses[END_REF]), we obtain the Lemma 5. For any d ≥ 2, the following inequality holds

(32) C |w| 2 log |w| 2 C |w| 2 dy dy + K σ d C |w| 2 dy ≤ max 2 d -1 , 2σ 2 C |∇w| 2 dy ∀ w ∈ H 1 (C)
with optimal constant

K σ d = 1 2 log 2 π e 2 σ 2 |S d-1 | 2 = 1 + 1 2 log 8 π d+1 σ 2 Γ(d/2) 2 .
Proof. The sharp logarithmic Sobolev inequality on the sphere S d-1 is

S d-1 |w| 2 log |w| 2 S d-1 |w| 2 dµ dµ ≤ 2 d -1 S d-1 |∇w| 2 dµ ∀ w ∈ H 1 (S d-1 )
where dµ is the uniform probability measure on S d-1 induced by Lebesgue's measure in R d . It can be recovered as the limit as q → 2 + of sharp interpolation inequalities stated in [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF], namely

2 q -2 S d-1 |w| q dµ 2 q - S d-1 |w| 2 dµ ≤ 2 d -1 S d-1 |∇w| 2 dµ ,
and optimality is easily checked by considering the sequence of test functions w n = 1 + ϕ 1 /n, where ϕ 1 is a spherical harmonic function associated to the first non-zero eigenvalue of the Laplace-Beltrami operator on the sphere. On R, the logarithmic Sobolev inequality associated to the gaussian probability measure dν σ has been established by L. Gross in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF]:

R |w| 2 log |w| 2 R |w| 2 dν σ dν σ ≤ 2 σ 2 R |w ′ | 2 dν σ ∀ w ∈ H 1 (R) .
Again the constant 2σ 2 is optimal as can be checked considering the sequence of test functions w n = 1 + ψ 1 /n, where ψ 1 (t) = t exp(-t 2 /(2σ) 2 ) is the first non constant Hermite function, up to a scaling.

The tensorization property of the logarithmic Sobolev inequalities shows that

C |w| 2 log |w| 2 C |w| 2 dµ ⊗ dν σ dµ ⊗ dν σ ≤ max 2 d -1 , 2σ 2 C |∇w| 2 dµ ⊗ dν σ .
Taking into account the normalization of dµ and dν σ , |S d-1 | = 2 π d/2 /Γ (d/2) and rewriting Gross' inequality with respect to Lebesgue's measure, we end up with [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. The constant K σ d is optimal as can be shown again by considering a sequence of test functions based either on spherical harmonics or on Hermite functions.

As a special case, for σ 2 = 1/(d -1) and K d := K σ d , we have the following inequality on the cylinder.

Corollary 6. For any d ≥ 2, with

K d = 1 + 1 2 log 8 π d+1 (d-1) Γ(d/2) 2 , the following inequality holds (33) C |w| 2 log |w| 2 C |w| 2 dy dy + K d C |w| 2 dy ≤ 2 d -1 C |∇w| 2 dy ∀ w ∈ H 1 (C) .
Using log(1 + X) ≤ α -1log α + α X for any α > 0, X > 0, which amounts to write log Y ≤ Y -1 with Y = α (X + 1), and applying it in [START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF] with

X = σ -2 C |∇w| 2 dy/ C |w| 2 dy, σ = (d -2 -2 a)/2, α = σ 2 /(γ (d -1)), we deduce that C |w| 2 log |w| 2 C |w| 2 dy dy -2 γ log (γ (d -1) C GLH ) + σ 2 γ (d-1) -1 C |w| 2 dy ≤ 2 d -1 C |∇w| 2 dy
for any w ∈ H 1 (C). We may observe that

K d + 2 γ log (γ (d -1) C * GLH ) + σ 2 γ (d -1) -1 = 4 γ -1 2 [Z -1 -log Z] with Z = 4 σ 2 (4 γ -1) (d -1)
.

Hence, if 4 σ 2 = (4 γ -1) (d -1) and if (for this specific value) C GLH = C * GLH , then the optimal logarithmic Sobolev inequality [START_REF]The optimal Euclidean L p -Sobolev logarithmic inequality[END_REF] is a consequence of [START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF].

We may observe that 4 σ 2 = (4 γ -1) (d -1) means Z = 1 and exactly corresponds to the threshold for the symmetry breaking result of Theorem C. Notice that proving that

C GLH = C *
GLH is an open question.

Symmetry breaking. Proof of Theorem C

In this section we study the symmetry breaking of the Caffarelli-Kohn-Nirenberg interpolation inequality as well as of the logarithmic Hardy inequality. To achieve this, we use a technique introduced Catrina and Wang in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] and later improved by Felli and Schneider in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]. Also see [START_REF] Byeon | Symmetry breaking of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Lin | Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF]. The method amounts to consider a functional made of the difference of the two sides of the inequality, with a constant chosen so that the functional takes value zero in the optimal case, among radially symmetric functions, when the inequality is written for functions on R d . Equivalently, we can consider functions depending only on one real variable in the case of the cylinder. By linearizing around the optimal radial function, we obtain an explicit linear operator and can study when the eigenvalue corresponding to the subspace generated by the first non-trivial spherical harmonic function becomes negative. It is then clear that the functional can change sign, so that optimality cannot be achieved among radial functions. This proves the symmetry breaking. We will apply the method first to the interpolation inequality [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF], thus generalizing the results of Felli and Schneider to a more general Caffarelli-Kohn-Nirenberg interpolation inequality than the one they have considered, and then to the logarithmic Hardy inequality [START_REF] Avkhadiev | Unified Poincaré and Hardy inequalities with sharp constants for convex domains[END_REF].

5.1. Symmetry breaking for the interpolation inequality. Based on [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], consider on H 1 (C) the functional

J [w] := C |∇w| 2 + 1 4 (d -2 -2 a) 2 |w| 2 dy -[C * (θ, p, a)] -1 θ C |w| p dy 2 p θ C |w| 2 dy 1-θ θ .
Among functions w ∈ H 1 (C) which depend only on s, J [w] is nonnegative, its minimum is zero and it is achieved by w(y) := cosh(λ s)

-2 p-2 , y = (s, ω) ∈ R × S d-1 = C , with λ := 1 4 (d -2 -2 a) (p -2)
R |w| 2 ds = 1 λ I 2 , R |w ′ | 2 ds = λ J 2 = 4 λ p 2 -4 I 2 , R |w| p ds = 1 λ I p = 4 p + 2 1 λ I 2 .
In terms of p and θ, we investigate the symmetry of optimal functions for (15) or, equivalently, for [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] in the range

0 < p -2 ≤ 4 d -2 and ϑ(p, d) ≤ θ ≤ 1 .
Consider now J [w + ε φ] and Taylor expand it at order 2 in ε, using the fact that w is a critical point and assuming that C w p-1 φ dy = 0:

1 ε 2 J [w + ε φ] = C |∇φ| 2 dy -κ C w p-2 |φ| 2 dy + µ C |φ| 2 dy -ν C w φ dy 2 + o(1)
as ε → 0, with

κ := p -1 θ 1 I p λ 2 J 2 + 1 4 (d -2 -2 a) 2 I 2 , µ := 1 4 (d -2 -2 a) 2 + 1 -θ θ 1 I 2 λ 2 J 2 + 1 4 (d -2 -2 a) 2 I 2 , ν := 1 -θ 2 θ 2 λ I 2 2 λ 2 J 2 + 1 4 (d -2 -2 a) 2 I 2 .
Spectral properties of the operator L := -∆+κ w p-2 +µ are well known. Eigenfunctions can be characterized in terms of Legendre's polynomials, see for instance [47, p. 74] and [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]. Using spherical coordinates and spherical harmonic functions, see [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], the discrete spectrum is made of the eigenvalues

λ i,j = µ + i (d + i -2) - λ 2 4 1 + 4 κ λ 2 -(1 + 2 j) 2 ∀ i , j ∈ N ,
as long as 1 + 4 κ/λ 2 ≥ 2 j + 1. The eigenspace of L corresponding to λ 0,0 is generated by w. Next we observe that the eigenfunction φ (1,0) associated to λ 1,0 is not radially symmetric and such that C w φ (1,0) dy = 0 and C w p-1 φ (1,0) dy = 0. Hence, if λ 1,0 < 0, optimal functions for (20) cannot be radially symmetric and, as a consequence, C(θ, p, a) > C * (θ, p, a).

A lengthy computation allows to characterize for which values of p, θ, a and d, the eigenvalue λ 1,0 takes negative values. Using the fact that for any a < (d -2)/2 the quantity 2 + p (2 θ -1) is positive, the corresponding condition turns out to be

4 p (d -1) p 2 + 2 p + 8 θ -8 -d 2 + 4 a 2 -4 a (d -2) (p -2) (p + 2) 2 < 0 .
This is never the case in the admissible range of our parameters if 0 ≤ a < (d -2)/2. On the other hand, for a < 0 there is always a domain where symmetry breaking occurs. More precisely let

ϑ(p, d) = d (p -2) 2 p , Θ(a, p, d) := p -2 32 (d -1) p (p + 2) 2 (d 2 + 4 a 2 -4 a (d - 2
ϑ(p, d) ≤ θ < Θ(a, p, d) when a ≥ d -2 2 - 2 √ d -1 (p -2)(p + 2) or ϑ(p, d) ≤ θ ≤ 1 when a < d -2 2 - 2 √ d -1 (p -2)(p + 2) .
In other words, symmetry breaking occurs for the optimal functions of (15) if a, θ and p are in any of the two above regions. Moreover, if a < -1/2, there exists ε > 0, γ 1 > d/4 and γ 2 > γ 1 such that symmetry breaking occurs if θ = γ (p -2) for any γ ∈ (γ 1 , γ 2 ) and any p ∈ (2, 2 + ε).

An elementary computation shows that, Θ(a, p, d) > 1 amounts to [START_REF] Bartier | Improved intermediate asymptotics for the heat equation[END_REF]. This condition is the symmetry breaking condition for Caffarelli-Kohn-Nirenberg inequalities found in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF].

Symmetry breaking for the weighted logarithmic Hardy inequality. Proof of Theorem C.

We now consider the weighted logarithmic Hardy inequalities of Theorem A' in the equivalent form of Theorem A". As we have seen in Section 2, after the Emden-Fowler transformation these inequalities take the equivalent form:

C |w| 2 log |w| 2 C |w| 2 dy dy + 2 γ log C * GLH C |w| 2 dy ≤ 2 γ C |w| 2 dy log C |∇w| 2 dy C |w| 2 dy + σ 2 with σ = (d-2-2 a)/2.
It is an open question to give sufficient conditions for which optimality is achieved among functions depending on s only, so that 2 γ log C * GLH = K(γ, σ) + log |S d-1 |. We recall that equality among radial functions in ( 26) is achieved by

w(s, ω) := |S d-1 | -1/2 w(s) , y = (s, ω) ∈ R × S d-1 = C , where w(s) = 4 σ 2 2 π (4 γ -1) 1/4 exp - σ 2 s 2 4 γ -1 ∀ s ∈ R .
We note that w(s, ω) is normalized to 1 in L 2 (C). As a consequence, it follows that

K(γ, σ) + log |S d-1 | = 2 γ log C |∇ w| 2 dy + σ 2 - C | w| 2 log | w| 2 dy .
After these preliminaries, consider the functional

F [w] := C |∇w| 2 dy C |w| 2 dy + σ 2 -|S d-1 | 1 2 γ exp K(γ, σ) 2 γ + 1 2 γ C |w| 2 C |w| 2 dy log |w| 2
C |w| 2 dy dy .

We know that F

[ w] = 0. Let G[φ] := lim ε→0 F [ w + ε φ] 2 ε 2 .
We have in mind to consider an angle dependent perturbation of w, so we shall assume that After some elementary but tedious computations, one finds that

G[φ] = C (L φ) φ dy , with L φ := -∆ φ + 1 4 A 2 |s| 2 φ -3 2 A φ and A := 4 σ 2 4 γ-1
. By separation of variables, it is straightforward to check that the spectrum of L is purely discrete and made of the eigenvalues

λ i,j = i (d + i -2) + A (j -1) ∀ i , j ∈ N . It follows that λ 1,0 < 0 if (34) d 4 ≤ γ < 1 4 1 + (d -2 a -2) 2 d -1 .
Hence symmetry breaking occurs provided that

d 4 < 1 4 1 + (d -2 a -2) 2 d -1 ,
which is equivalent to a < -1/2. This concludes the proof of Theorem C.

We recover the limit range for symmetry breaking in the interpolation inequalities studied in Section 5.1. Condition [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF]

Concluding remarks

The purpose of this paper is to establish a new family of inequalities in the Euclidean space R d and in the cylinder R × S d-1 . These inequalities are stronger than Hardy's inequality and the logarithmic Sobolev inequalities, but are related to both of them and, for this reason, we have called them logarithmic Hardy inequalities. They are invariant term by term under scaling, which distinguishes them from usual logarithmic Sobolev inequalities. On R d , they are written for unbounded measures and, as far as we know, cannot be easily reduced to inequalities written for probability measures or for Lebesgue's measure. They also appear as an endpoint of a family of Caffarelli-Kohn-Nirenberg inequalities, which is more general than the subfamily studied for instance by Catrina and Wang in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF].

A very natural question is to determine whether optimal functions on R d are radially symmetric or not. Using the method introduced by Catrina and Wang in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] and extended in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF] by Felli and Schneider, we prove that optimal functions in R d are not radially symmetric functions in the case of the general Caffarelli-Kohn-Nirenberg inequalities and in the corresponding logarithmic Hardy inequalities, for parameters in a certain range. The method is rather simple. It amounts to linearize the inequality around an optimal function among radial functions and study the sign of the first eigenvalue of an associated operator. A negative eigenvalue then means that optimality is achieved among non-radial functions. The results of symmetry breaking that we obtain by this method are fully consistent with previously known results. They allow us to characterize a whole region where the weights are strong enough to break the symmetry that would naturally arise from the nonlinearity in the absence of weights (and can then be proved either by symmetrization techniques or by moving plane methods as in [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in R n[END_REF]). The symmetry region is by far less understood, although it has recently been established in [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] that both regions are separated by a curve (in the case of the subfamily considered by Catrina and Wang). In the general form of the Caffarelli-Kohn-Nirenberg inequalities, there is an additional term which competes with the nonlinearity to break the symmetry, thus making the analysis more difficult. Hence the main challenge is now to establish the range for symmetry of the optimal functions. This would have some interesting consequences. As mentioned in Section 4, if, for instance, symmetry holds in the complementary region of the one for which symmetry breaking has been established, then we would recover the optimal logarithmic Sobolev inequality on the cylinder as a direct consequence of the logarithmic Hardy inequality.

  2 and b ∈ [a, a + 1] when d ≥ 3. In addition, for any d ≥ 1, we assume that

3. 1 .

 1 The sharp interpolation inequality in the one-dimensional cylindric case. If w depends only on s =log |x|, Inequality (20) can be reduced to its one-dimensional version,[START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF]

3. 2 .

 2 The sharp Logarithmic Hardy inequality in the one-dimensional cylindric case. With 2 γ = p/(p -2) and θ = γ (p -2), we observe that the condition θ ∈ [ϑ(p, 1), 1] is equivalent to γ ∈ [1/(2 p), 1/(p -2)] and that 2 + (2θ -1) p = (2 γ p -1)(p -2) is positive for any p > 2 since γ > 1/(2 p).

3. 3 .∀

 3 The sharp inequalities for radial functions. Let d ≥ 2 and consider the interpolation inequality (15) restricted to the subset D * a (R d ) of radial functions in D a (R d ), i.e. u ∈ D * a (R d ) where C * (θ, p, a) denotes the best constant. Let σ = (d -2 -2 a)/2. By the Emden-Fowler change of coordinates (9), the above inequality is equivalent to w ∈ H 1 (C) depending only on s =log |x|. Up to a normalization factor depending on |S d-1 | = 2 π d/2 / Γ (d/2), (27) is equivalent to the one-dimensional inequality (21) with best constant K (θ, p, σ). It is straightforward to check that C * (θ, p, a) = |S d-1 | -(p-2)/p K (θ, p, σ). We also note that the range of θ is as in Lemma 3, that is, θ ∈ [ϑ(p, 1), 1].

p+2 2 p.

 2 θ-(p-2) . See the proof of Lemma 3 for more details. We can notice that[C(θ, p, a)] -1 θ = C |∇w| 2 + 1 4 (d -2 -2 a) 2 |w| 2 dy C |w| 2 dyWith a slight abuse of notations, we shall write w as a function of s only, which solves the ODE λ 2 (p -2) 2 w ′′ -4 w + 2 p |w| p-2 w = 0 and, as in the proof of Lemma 3,

C

  |∇ w| 2 dy + σ 2 • 2 -C | w| 2 log | w| 2 dy C |φ| 2 dy + C log | w| 2 |φ| 2 dy .

Figure 1 .Figure 2 .

 12 Figure 1. Plot of the admissible regions (gray areas) with symmetry breaking region established in Theorem 7 (dark grey) in (η, θ) coordinates, with η := ba, for various values of a, in dimension d = 3: from left to right, a = 0, a = -0.25, a = -0.5 and a = -1. The two curves are η → ϑ(p, d) = 1 -η and η → Θ(a, p, d), for p = 2 d/(d -2 + 2 η). In the range a ∈ (-1/2, 0), they intersect for a = a-(p), i.e. η = 2 a (1 -d)/(d + 2 a). They are tangent at (η, θ) = (1, 0) for a = -1/2. The symmetry breaking region contains a cone attached to (η, θ) = (1, 0) for a < -1/2, which determines values of γ for which symmetry breaking occurs in the logarithmic Hardy inequality.

  Theorem A". Let d ≥ 1, a < (d -2)/2, γ ≥ d/4 and γ > 1/2 if d = 2.Then, for any w ∈ H 1 (C) normalized by C w 2 dy = 1, we have

	(10)	C	|w| 2 log |w| 2 dy ≤ 2 γ log C GLH	C	|∇w| 2 dy +	1 4	(d -2 -2 a) 2	.
	The optimal constant C GLH is the same in Theorems A' and A". Similarly, to the case of radial functions
	depending only on |x| corresponds the case of functions depending only on s = -log |x|.
	Theorem B". Let d ≥ 1, a < (d -2)/2 and γ ≥ 1/4. If w ∈ H 1 (C) depends only on s ∈ R and is normalized by C w 2 dy = 1, then

  )) -4 p (p + 4) (d -1) .

	Symmetry breaking occurs if θ < Θ(a, p, d). We observe that, for p ∈ [2, 2 * ), we have ϑ(p, d) < Θ(a, p, d) if and only if a < a -(p) with a -(p) := d -2 2 -2 (d -1) p + 2 .
	We note that the condition ϑ(p, d) < 1 is always satisfied under the assumption p ∈ [2, 2 * ). On the other hand the condition Θ(a, p, d) ≤ 1 is equivalent to a ≥ d -2 2 -2 √ d -1 (p -2)(p + 2) .

Finally, we notice that

d/4 = ∂ϑ(p, d)/∂p |p=2 < [1 + (d -2 a -2) 2 /(d -1)]/4 = ∂Θ(a, p, d)/∂p |p=2 if a < -1/2.

Summarizing these observations, we arrive at the following result. Theorem 7. Let d ≥ 2, 2 < p < 2 * and a < a -(p). Optimality for (15) (resp. for (20)) is not achieved among radial (resp. s-dependent) functions, that is, C(θ, p, a) > C * (θ, p, a) if either

  asymptotically defines a cone in which symmetry breaking occurs (see Theorem 7) given by d/4 = ∂ϑ(p, d)/∂p |p=2 < γ < [1 + (d -2 a -2) 2 /(d -1)]/4 = ∂Θ(a, p, d)/∂p |p=2 . 5.3. Plots.

	1				1					1					1
	0.8				0.8					0.8					0.8
	0.6				0.6					0.6					0.6
	0.4				0.4					0.4					0.4
	0.2				0.2					0.2					0.2
	0.2	0.4	0.6	0.8	1	0.2	0.4	0.6	0.8	1	0.2	0.4	0.6	0.8	1	0.2	0.4	0.6	0.8	1

Acknowledgments. J.D. has been supported by the ECOS contract no. C05E09 and the ANR grants IFO, EVOL and CBDif, and thanks the department of Mathematics of the University of Crete and the Departamento de Ingeniería Matemática of the University of Chile for their warm hospitality. This work is also part of the MathAmSud NAPDE project (M.d.P. & J.D.). Plots have been done with Mathematica tm . c 2009 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.