Thomas Debrosse

Maurice Pillet

Jean Luc Maire

Nathalie Baudet

SYMME Debrosse Thomas

Sensory perception of surfaces quality -Industrial practices and prospects

In this paper we carry out a critical analysis of the industrial practices used by the partner firms to control a product's aspect. The group of partners represents many different sectors of activity such as: luxury goods industry, furniture industry, medical equipment (prosthesis), plastic injection and watch-making industry. These practices deal with the identification of aspect anomalies, the evaluation of anomalies severity as well as the decisions about the product's conformity. We will show that current practices do not allow us to decrease the variability of the results frequently observed, because the subjectivity often associated with this kind of control is not eliminated using such methods. In every sector studied, we find the same dissatisfaction about the results obtained. We will then present our approach to sharply decrease this variability. This approach, tested in a famous Swiss watch-making company, is based on sensory analysis concepts. This approach is original due to the breakdown of the visual-control process into three sub-processes: the detection of aspect anomalies, the evaluation of those anomalies and the conformity decision. This approach also includes a metrological organisation and some tools which allow us to measure the efficiency reached. The paper will show how our approach succeeds in meeting the different aims of the partner company group and proposes an initial structured approach for a generic metrological organisation adapted to control the variability during the surface quality control by humans.

INTRODUCTION

Product quality is usually defined as "the characteristics which allow the implicit and expressed needs to be satisfied" [1]. So, the level of product quality strongly depends on its functional skills, its reliability, and its cost or even on the quality of the associated services (after-sales, guarantee…). However, the quality also depends on the way the customer perceives the product [2]. This perceived quality, which is evaluated by visual, tactile, acoustic, gustatory and olfactory impressions [START_REF] Liu | Voluntary attention enhances contrast appearance[END_REF], plays a very important role in order to reach the best quality of a product proposing more than functional efficient products but also esthetic efficient products. In the same time K. Forslund [4] shows the effect of geometric variation on perceived quality. The visual robustness of geometrical variations is so presented as a side of perceived quality.

Our work deals with one criterion of the perceived quality: the aspect of product's surfaces. This study is part of a European research program « INTERREG » putting together two laboratories from the two universities, university of Savoy and the Lausanne federal polytechnic school (EPFL), two institutional partners: CTDEC and CETEHOR and six industrial firms. The aim of this project is to create methodological support and the tools needed to improve the visual control of high added value products. First of all, in this paper we carry out a critical analysis of the industrial practices used by the partner firms to control product aspect. The group of partners represents many different sectors of activity such as: luxury goods industry, furniture industry, medical equipment (prosthesis), plastic injection and watch-making industry. We will then present the tools we propose to answer to those problematic.

INDUSTRIAL PROBLEMATICS

This diversity shows the universality of the perceived quality problematic. However the objectives of aspect-control differ depending on the product specificities and its market positioning. Indeed, for the luxury goods industry and furniture industry the way to reach a quasi perfect esthetic imposes its own criterion. For the plastic products, the aspect control permits to show out the deviance of the fabrication process in order to guarantee the product's functionalities. And in the medical field (prosthesis fabrication) impacting a lot the patient life, the aspect quality is the esthetic way to transmit the global quality level of the whole product (resistance, functionality, reliability). Those different faces of aspect-control provide specifics problems. Despite of this diversity, all the partner firms meet behind the same problematic: how to build a sensory evaluation process which is repeatable, reproducible and stable in time.

The subjectivity of the Human judgment in terms of esthetic and perception introduce variance in the control results. The problem of repeatability and reproducibility is due, for the main part to this variability [5]. This variability can be explained by the variance of the measure conditions, of the semantics and cognitive process of perception between controllers [6, 7, 8] and the perception and judgement performances of one controller at a different time, due to his emotion or his tiredness [9].

More than punctual disagreement about the product's conformity, this omnipresent subjectivity induces difficulties to maintain a precise quality level. Experience shows that controllers tend to be more and more strict, which increase the quality level.

PRACTICES ASSESMENT

Due to the lack of recognized universal approaches, each firm set up special practices in order to solve the visual-aspect-control problems. But in all firms, the direction committees are not satisfied by the control efficiency and the controllers complains about how hard it is to respond to the task. Here is the list of the most heard complaints: We noticed that all those complaints are coming from the repeatability and reproducibility problem:

The Repeatability is defined as the ability of the appraiser to "repeat" his/her decisions (agree with him/herself). Calculated as (# agreements / # parts inspected)

And the Reproducibility: Ability of all the appraisers as a whole to "repeat" their decisions among them. Calculated as (# agreements among all appraisers / # parts inspected)

The reproducibility result is made of three components:

Concordance: (# parts that agree with the standard / # parts inspected).

False Alarms: (# parts classified as "defective" when in fact is perfect / # of perfect parts)

Wrong Classification: (# parts classified as "perfect" when in fact is defective / # of defective parts)

In this paper we want to make the list of the most significant practices we meat in the different firms. And we propose an analysis of theirs relevence and efficiency. This repeatability and reproducibility problem is one of the most important points. In this paper we also want to highlight the origins of this problem.

A rich and varied vocabulary

To harmonize judgements, the use of a common and shared vocabulary is very important. However we noticed that the controllers in the partner firms use too many terms to describe the different kind of anomalies (from few dozen till one hundred fifty terms). Moreover the terms used, often mix the nature, the causes or even the consequences of the anomaly. The table 2 groups few example of terms or expression used in the firms: This fuzzy about the anomaly description is, once again source of variability in anomaly judgement. [10,11] Some sensory analysis method named "free choice profiling" does not impose any constraints concerning the terms to be used: « descriptors » [12]. Those methods are easier to set up but provide a judgement more personal (no common criterion) and subjective (no references) without any justification.

In the case of visual-aspect-control we try to evaluate and organize the sensory perceptions to give a shared judgement comparing to common criterion (even if they are not the controller's criterion). That is why using the conventional profiling method (finite numbers of descriptors) permit to reduce judgements variability, getting easier the anomaly description and categorization task insuring a good communication between controllers. But the efficiency of this method comes from the good choice of descriptors respecting some conditions of pertinence, independence, exhaustiveness and discrimination [13].

One important objective of this study is to evaluate the possibility of building a generic descriptor list which could correspond to all fields, to be used by all our industrial partners. In her work with a well-known Swiss watch maker firm, A.S.Guerra proposed a short-list of four descriptors respecting the previous named characteristics [5].

Use of anomaly panoply

As an answer to the quality level stability, and judgement reproducibility, the firms had to keep in memory all the anomalies detected and the conformity judgement associated. This wish leads the firms to build a whole database named "panoply". Those databases illustrate each anomaly by a picture or a real product having this anomaly. The main advantage of this tool is the knowledge capitalization of all anomalies within a firm depending on processes and products. That permits to set up easily formation programs, or bring them a lot more efficient and to compare the nature of anomalies meat from a firm to another depending on theirs activities.

On the other hand building such panoply is corresponding to make the list of the possible spelling errors. We thus realize how width could be those panoplies which sometimes gather hundreds of illustrations (pictures or products). Moreover an anomaly could have different impact according to his location on the product, so several pictures are necessary to define the conformity level.

Panoplies represent the reference of the expected quality level and is often used as comparison tool to evaluate the aspect quality. Indeed in some firms the panoplies stand in the working station, it permits the controllers to compare, treat and provide a conformity opinion. But it is very difficult to compare anomalies with different intensity (deeper, longer, larger…), observed on different product (colour, shape, material…). And thinking to build exhaustive panoplies illustrating all intensity of all anomalies on every kind of product is just amazing because the panoplies are product oriented whereas they should be an illustration of the firm's quality level. But even the firms don't have such a quality level standard, it has to be defined, created and shared.

Evaluation criterion and decision process

In addition to the anomaly detection problems, evaluation and decision steps could be source of variability and conflict. Every firm has different way to take care of this point.

The firms from the luxury industry have more sensibility and experience in visual control than others as furniture industry firms for example. And the esthetic specifications impose to treat smaller anomalies. The size of luxury product anomalies makes the comparison of visualimpact harder and it becomes hard to judge the conformity of the product. So in this kind of industry, panoplies are used as a comparison-evaluation tool which allows no to formalize the criterion based on visual perception. This method is available because of the experience of the controllers and sensitivity about visual quality of products.

In the firms more industrial, with less qualified employees, more automated machines and faster production rhythm; the evaluation can not be based on people experience. Those employees from the "hyper-industrial" firms are generally less sensitive and formed to product quality and visual control. Those conditions increase the anomaly evaluation difficulties. The binary criterion ("I see/ I don't see") shortcuts the evaluation task. The decision thus depends directly on the control conditions (light orientation, detection distance, time…). By using this criterion, control the environment becomes mandatory.

visual control process formalization

In order to transmit the visual-control instructions, the firms created the "control-procedures" which often gather the dimensional and aspects specifications of the product. Here we deal with the part of those "control-procedures" specifying the visual-aspect-control.

Concretely this is a paper document specifying the firm requirement and guides the controller by listing the different kind of anomaly. The tolerance is often specified by giving the allowed number of non acceptable anomalies on the product. These documents also give the control frequency and the maximum number of defective products in case of batch control. The accuracy and relevance of those documents can vary a lot from a firm to another.

As evaluation criterion, the control-procedure mentions the different kind of anomalies to detect, through the descriptors defined. On the other hand, no anomaly quantification systems are set up, and we mainly noticed two different decisions making process:

The worst case: no rules are defined and the controller, depending on his own perception, is in charge to judge the visual impact of the anomaly and provide an acceptance decision referring to the products in the panoply.

The second significant practice we noticed is to define a simple rule about the visual impact of the anomaly. This rule is named "I see / I do not see". The principle is to define all conditions of the control and cancelled the evaluation step. So respecting those observing conditions, if I see an anomaly, it is considered as a defect and the product is classified as non acceptable. In one of the partners, the control conditions are based on using conditions (location, orientation, illumination, distance…). This approach tries to bring the controller closer to the customer perception, but the diversity of using conditions makes the judgment no representative of the visual impact of a anomaly on the customer himself.

By analysing the control-procedures used in every firm, we noticed that they always define « what » to control whereas they should define "how" to control by specifying a standard on the evaluation criterion and describe and impose the environment conditions of the control. Mastering the detection process will permit to guarantee the control exhaustiveness and permit to insure the access to the necessary information for the evaluation or decision step.

A detailed analysis of the Ishikawa diagram applied to visual-control permit to highlight the visual-control most influent factors. We could also separate the factors in two categories: influencing the detection step or the evaluation step. Even if several factors are easy to control, a lot depends on the product, the material, the process or even on the firm organisation. The following table gathers some of the most influent factors: Finally we noticed that control-procedures mix three different concepts of visual-control: the anomaly detection, anomaly evaluation and acceptance decision. The decision rule previously mentioned "I see/I don't see" translates this confusion between the three appraise. A.S. Guerra [5] shows how necessary it is to separate those three appraises [5], we will detail this point in the next chapter of this paper.

Tool aided detection

As we just said, visual-control holds in three distinct steps: detection -evaluation -decision. The evaluation step is often absent in the visual-control process and the detection and decision step are confused. The anomalies detection problem is often identified by the firms, they thus set up some tools to help controllers to detect. The two factors mainly identified by are the anomaly's size and the light used during the control.

Enlargement tool

The norm concerning the watch making industry [14] specifies to control the products with the normal vision (without enlargement tools). In spite of this norm, several firms propose or impose to use binocular or flunks to improve controller's detection skills. In some cases those tools can be useful or even necessary. Some products may have very small dimensions and a lot of details, even smaller. For example the case of plastic injection products with very small holes: diameter less than one millimetre. That makes the control harder and extremely tiring for the eyes.

The second case concerns the size of the anomaly whereas the size of the product. For example in the luxury industry, the product tends to perfection. This search of perfection induces the detection of the smallest anomaly (even if it is acceptable). The enlargement tools permit to detect every anomaly, and thus permit to set up a performing evaluation step.

The light

Once again the norm [14] gives some recommendations about the illumination of the work station. It recommended in particular using an 800 lux illumination source. But this norm contains no specifications about more important points as the position and the orientation of the light source(s).

The experience shows that defining the light intensity is not enough to insure a good detection. To decrease the variability of the control and make sure the work conditions are comfortable, specify the intensity, position and orientation of the light is mandatory. A.S.Guerra shows impacts of light orientation on anomaly detection and perception [5]. The wrong practices within the firms are due to the lack of communication about this information

Practices conclustion

The most important points: every firm developed its own tools depending on its own problematic no specification of the visual-control task, and conditions confusion of the three concepts: detection, evaluation, decision (in every firm) many subjective tools: panoply

The work with the partners firms shows the variability of the practices to satisfy the visualcontrol. This variability is due to the product produced, which impose very different fabrication processes, production organisation and different level on esthetic and aspect properties specifications. Thus the lack of recommendation, references and exchange on this subject, each firm acquired its own experience. That explains the variability between the partner firms.

Within a firm the variability can be explain by product diversity, especially: size, colour and shape. But it is mainly due to the freedom granted to controllers. This freedom in the task comes from the lack of definition, specification, formalisation and recommendation about visualcontrol. Thus every controller proceed by his own way, which could be different from a day to another depending on the mood, the place, the time of the day, the motivation, the attention. Once again, this is the reputability problem. The impossibility to set up a reproducible control comes from the difficulty to share the same criterion of evaluation and control conditions [15].

A STRUCTURED APPROACHE

The Human subjectivity is the main source of variability in the visual control task. The food industry is used to set up sensory tests thanks to the sensory analysis method. Even if the industrial problematic and application fields are different, those sensory analysis method permits to reduce the variability in a product evaluation by human senses (mainly taste and smell but also sight and touch). So the idea is to use the sensory analysis tools, adapted to the partners' fields and products.

Sensory analysis : application to manufactured product

In the food industry, sensory analysis is used to classified products depending on a well defined criterion to emit a personal judgment often independent of experimentation conditions. The different methods proposed in literature [12,16] highlight the different kind of classification scale and present the most efficient comparison processes. One example of the food industry problematic: a study consists of classifying biscuit depending on his perceived sugar rate [16]. In this kind of studies, it is known that the test morphology: scale used (free or structured) and the product-presentation-order influences the results. Or this kind of influences is harmful in the case of industrial manufactured products.

A good aspect of the sensory analysis method as it is applied in food industry is the metrological structure. This structure is composed of experts who define the evaluation criterion, the acceptance limits, and they set up the references and the control morphology (as we described previously). This structure is useful to set up the experimentation objectives and deduce the conditions (time, tools, light) permitting to access to the mandatory information for the evaluation. Moreover, formation, calibration and accompaniment are necessary to make sure the controllers use the tools correctly.

Split the concepts : detection, evaluation and decision

A previous study [5] realised in a luxury watch making company permitted to evaluate the sensory analysis tools relevance applied in the context of manufacture product. This study shows in particular the interest of separating the three concepts of visual control: anomaly detection, evaluation of perceived anomaly (work on descriptors) and final acceptance decision.

A good control comes from a good vision

To detect correctly, it is necessary to know « what » to see. It is proved that visual attention permit to increase the concentration level and the performances of visual detection (in particular the contrast perception) [17]. That is why it is important to identify the possible anomalies we want to detect on the product. Of course those possible anomalies depend on the product, the fabrication process… (Even within a firm). Defining anomalies' characteristics and properties is necessary to use the right tools and method to be able to detect. To share this definition of anomalies, it is necessary to use a common vocabulary.

The definition and identification of anomalies is the first step. The next step is the controlprocess standardisation by procedures specifying:  Necessary skills (vision accuracy, colour detection, common vocabulary…)  Tools to be used (flunks, light…)  Product, body and eyes movements This standardisation helps the controller to be exhaustive during the detection task and permit the access to information permitting the evaluation task. It also permits to reduce practices variability during the detection.

A good evaluation comes from quantification and discrimination

The evaluation step consists on quantifying the visual impact of the detected anomalies. This visual impact is not measurable so the quantification is not easy. However it is possible to know the discriminating properties thanks to anomaly definition. For example a small stripe (small geometrical dimensions) on a black piece will have a worst visual impact than a big stripe on a white piece. So the geometrical properties are not discriminating. Finding those properties is part of the whole project.

The quantification is an important step to associate a magnitude level to the visual impact of each anomaly. Panoplies are often used as a quantification scale. But part of the method is to give a common meaning to some word to be able to set this quantification without references. Finding factual evaluation criterion is one important objective of the project. Those criterions could be based on anomaly visibility properties: which conditions are necessary for me to perceive the anomaly.

The work on detection helping tools permits to see more details and more anomalies faster than before. But more anomalies do not mean more defects. We thus noticed the importance of the evaluation step which allows us to make the difference between anomalies (acceptable) and defects (non acceptable).

A good decision takes into account all parameters

The decision step is the last one, and gives the final answer about the product conformity. The anomaly localisation is essential to determine its criticality. The decision, as in the dimensional metrology depends on tolerances fixed by the expert group (often integrating the marketing decisions). The same gap between the reference surface and the real surface does not have the same visual impact depending on its location. Actually the localisation analysis permits to evaluate the risk for the customer to see the anomaly. Thus, the most visible zones are considered as the most critical.

Finally the decision step consists on putting together results from each step:  Detection: which anomaly  Evaluation: which visual impact  Localisation: where stand the anomaly The decision, as it is build permits to emit an objective judgement about the conformity based on subjective data from the previous steps.

Finally, some concepts used in food industry are suitable to be used to control manufactured products. The difference between emitting a personal judgement (sensory analysis) and an objective evaluation based on defined criterion impose an evolution of the evaluation tools. So the tools should allow more than a comparison, an evaluation based on common references, criterions and acceptance limits fixed by the expert group.

GENERAL CONCLUSION

In this study we noticed that the lack of specification, recommendation or even referent studies about visual control, each firms developed its own practices and tools. We noticed a great variability of those practices. So we noticed two different variability sources: the intern variability, the people's practices variations within a firm and the extern variation which defines the practices variability between several firms.

The first answer of this study is to reduce the intern variability by the definition of control environment and the second one is to set up a generic method suitable for every fields and firms. So in several firms we are setting up a new kind of control procedures, specifying the control task. Those procedures take into account the fabrication process and the risk to create one anomaly in particular to specify what to detect. In addition to get the control easier and more efficient, most of the new procedures impose an eyes course and the amount of time for the task. To control and reduce the environment variability, we are working on the illumination conditions. Several light sources configurations are studied to find the best conditions independent from the natural light sources.

The approach needs the creation of the metrological structure. It is composed of a representative anomaly panel, an expert group emitting the "right" judgement and a formation an evaluation of measurement tool (the controllers). In every firm this group of expert has been created, and a panoply illustrating the most significant defects is being created. The next step is to create a tool illustrating the decisions taken by the expert group. To set up this tool we need to know the criterion of each expert and make them agree about the "generic criterion" to be understood and shared by everyone.

Finally, one of the most important points is to split the control along the three visual control concepts. The detection step impose to control the observation conditions, to evaluation correctly, a definition of shared evaluation criterion is mandatory and the definition of acceptance limits and critical zones in the product permit to take the right decision automatically.

Table 1 :

 1 Complaints list

	Too many defects are not detected	Very long learning period
	Not stable quality level in time	Different judgment depending on controllers
	Unjustified rejects	Communication difficulties in the supply chain

Table 2 : exemple de descripteurs rencontrés au sein des entreprises

 2

	Deformation, blur, stripe,	Run out welding, scraping,	Aspect, brightness, engraving,
	bubble, sandpaper traces	sanding, laser tackles	defects, painting drop

Table 3 :

 3 most influent factors from Ishikawa diagram

	Material	Colour / size, product shape / product position in final
		assembly / completion, texture (shiny, dully, rough, wet,
		soft…)
	Man Power	Vision accuracy / controller's mental and physical shape /
		instruction and anomaly interpretation / knowledge of the
		product
	Means	Products handling
	Method	Knowledge of fabrication process / production pace /
		complementary task
	Mother nature	Illumination (intensity, orientation and dazzling) / work
		station ergonomics / noise / comfort of work /


Digital and FM demodulation of a doubly-clamped single wall carbon nanotube oscillator: towards a nanotube cell phone Vincent Gouttenoire, † Thomas Barois, † Sorin Perisanu, † J.-L. Leclercq, ‡ Stephen T. Purcell, † Pascal Vincent, † Anthony Ayari, † * Université de Lyon, F-69000, France; Univ. Lyon 1, Laboratoire PMCN; CNRS, UMR 5586; F69622 Villeurbanne Cedex, INL, UMR5270/CNRS -ECL, LYON University, F69134 Ecully Cedex, France anthony.ayari « at »lpmcn.univ-lyon1 «dot »fr Electromechanical resonators are a key element in radio-frequency telecommunications devices and thus new resonator concepts from nanotechnology can readily find important industrial opportunities. In this paper, we report the successful experimental realization of AM, FM and digital demodulation with suspended single wall carbon nanotube resonators in the field effect transistor configuration. The crucial role played by the mechanical resonance in demodulation is clearly demonstrated. The FM technique is shown to lead to the suppression of unwanted background signals and the reduction of noise for a better detection of the mechanical motion of nanotubes. The digital data transfer rate of standard cell phone technology is within the reach of our devices.

Nanoelectromechanical systems (NEMS) are attracting increasing attention from researchers for their potential as novel, ultrasensitive, high speed and low power devices 1,2 as well as for diverse new physical phenomena in the thermal 3 and quantum limits 4 , or in the self oscillating regime 5,6 . Thanks to their nanometer-scale size and exceptional electrical and mechanical properties, individual single wall carbon nanotubes (CNTs) are ideal candidates as the central vibrating element for NEMS 7 working in the MHz and GHz range 7 . Despite recent progress, direct measurements of the intrinsic high frequency electrical response of individual CNTs are still very challenging due to impedance mismatching and parasitic capacitance. In particular, in the first studies of the response of CNT field effect transistor (without a mechanical degree of freedom), the operating frequencies response were limited to below 100 MHz 8 or within a narrow high frequency band 9 . It is only very recently that direct electrical detection in the GHz range has been achieved 10 .

High frequency response can be detected indirectly through signal mixing. Mixing is a widespread technique used in most high frequency instruments that allows varying the frequency of any signal without losing information about its amplitude. A high frequency signal can easily be converted into to a low frequency signal which simplifies tremendously the detection. (The necessary mathematical basis is developed in some detail below). The first mixing measurement of individual nanotube transistors reached a carrier frequency of 500 MHz 11 and shortly afterwards several tens of GHz were detected 12,13 . This technique is also presently the only one applied to detect electrically the mechanical resonances of CNT NEMS [14][15][16][17][18] . application. However the AM radio of ref. 20 was with a transistor using a fixed, non-suspended, nonresonating carbon nanotube and the mixing was only due to the non-linearity of the device. This lacked two important ingredients: i) Frequency tuning, i.e. no selection of a radio station was possible or in other words all the radio stations are received at the same time. ii) FM demodulation. An FM signal needs an additional stage called a discriminator that converts frequency into an amplitude signal. The radio of ref. 21 used the tunable mechanical vibrations of CNT field emitters and could manage AM and FM demodulation. However the drawback of this configuration are: i) it requires high voltage for field emission (FE), ii) it is less compatible with the microelectronic technology than a transistor, iii) the FE current is rather unstable creating noise that passes through the down mixing to the sound signal. In this paper we unified these two configurations to perform AM and FM demodulation on suspended single wall carbon nanotube in a transistor geometry where the tunable mechanical resonances play an essential role. Moreover, we go beyond this and demonstrate digital demodulation, a crucial point for cell phones and modern communications.

Signal mixing has been an essential ingredient in radio transmission

The device fabrication was as follows. CNTs were grown by chemical vapor deposition on a degenerated silicon substrate used as a back gate with a 300 nm SiO 2 layer. Contact pads for source and drain were fabricated by optical lithography, evaporation of 20 nm of Cr and 250 nm of gold and lift off.

The spacing between electrodes is 2 µm and the diameters of the tubes (measured by atomic force microscopy) were between 1 and 3 nm. Finally, the tubes are suspended by a wet etching in buffered hydrofluoric acid followed by a critical point drying in CO 2 . Figure 1a shows a scanning electron microscope (SEM) image of a suspended nanotube at the end of the process.

Measurements were performed in a home-made ultra high vacuum probe station (base pressure below 2*10 -10 Torr). A signal from a lock-in amplifier at low frequency (ω L ) was sent to the FM input of a radio frequency (RF) generator. We typically used ω L/ /2π=616.3 Hz though audio signal frequencies were also tried. An FM signal V FM (t)=V c *cos(ψ(t)), where ψ(t)=ω c t+(ω Δ /ω L )sin(ω L t), was applied to the source electrode of the nanotubes where V c was the applied voltage, ω c the carrier frequency, ω Δ the frequency deviation and t the time. Finally, the low frequency signal from the drain electrode was measured by the current input of the lock-in. In a standard FM radio, ω Δ is proportional to the amplitude of the sound to be carried and ω L is its frequency. Measurements were carried out on 7 different devices. We swept ω c over large frequency spans and recorded the magnitude (R) and phase (θ) of the lock-in signal for different fixed DC gate voltages.

Peaks in current were found at various frequencies. These peaks were proven to be due to the vibrations of the nanotube by varying the DC gate voltage which tunes the frequencies of the mechanical resonances by mechanical strains induced by electrostatic forces 22,14 . We also performed some measurements with the same mixing technique as ref. 14 to check that the resonance peaks of our sample are at the same frequency with both methods. After coarse identification of the resonances, finer frequency scans were made to determine their exact shape (figure 2a). R showed a rather unusual form with one central peak and two satellites while θ had abrupt and very clear 180° jumps when R was a minimum (figure 2b). To achieve a first understanding of these forms, for example the jump in θ, we also performed "X and Y" measurements with the lock-in, where X=Rcos(θ) is the in-phase current, and For comparison, we also performed AM demodulation at the resonance (see figures 2e and2d) by applying at the drain an AM signal V AM (t)=V c *(1+m*cos(ω L t))*cos(ω c t) where m is the modulation depth. The electromechanical signal is superimposed on a 25 pA background current and the signal is noisier. FM demodulation is a clear improvement in terms of mechanical detection as FM systems are better in rejecting noise than AM systems [START_REF] Rutledge | The electronics of radio[END_REF] . Indeed, noise is generally related to amplitude variation and is nearly independent of frequency as long as the signal is not too close to DC.

In the following we will show that X in figure 2c is the derivative of the real part of the complex Lorentzian response function of the resonator and that the cancelation of Y is a specificity of FM detection. The source current I(V sd ,x,V g ) depends on three parameters: the source drain voltage V sd , the gate voltage V g and x, the position of the middle of the nanotube perpendicular to the substrate. In our experiment, with a fixed DC gate voltage V g DC and an AC source drain voltage V(t), the leading terms of the current are given by a second order Taylor expansion around V sd =0 and x=x 0 . x 0  0 due to the electrostatic bending toward the gate induced by V g DC :

2 0 0 0 0 ) ( ) , , 0 ( ) ( ) , , 0 ( ) , , 0 ( ) ), ( ), ( ( I t x V x x I t V V x V I V x I V t x x t V I DC g DC g sd DC g DC g            (1)
where δx(t) is the instantaneous mechanical displacement of the nanotube due to electrostatic actuation and I 2 is the current coming from higher order terms in the Taylor expansion responsible for the low frequency signals that can be detected by the lock-in. The first and third terms on the right hand side are zero because no current flows in the absence of source drain voltage, for any x 0 or V g DC . The second term gives only high frequency signals.

I 2 is at the origin of the signal mixing technique that we mentioned in the introduction. Mixing in its simplest form is the multiplication of two harmonic signals that enter a two terminal mixer device. A signal proportional to their product is generated at the output due the non-linearity of the I/V characteristics. From trigonometry the output signal has a harmonic component at the difference in the frequencies of the two incoming signals with amplitude related to the high frequency intrinsic response of the device. So this technique can be used to shift the frequency of the input signal. For instance, if the two incoming signals are very close in frequency, the output signal is at very low frequency and can easily be detected.

If we just include the second order terms in I 2 we get:

2 2 2 2 2 2 2 2 )) ( ( 2 / 1 ) ( ) ( )) ( ( 2 / 1 t x x I t x t V V x I t V V I I sd sd             (2)
where for the same reason as for eq. 1 the last term is zero. The first term is the same as the one of ref.

20. For an applied AM signal, this term gives a DC rectified signal, several high frequency signals and a signal at the modulation frequency. For FM it is useful to decompose the applied signal with the Jacobi-Anger expansion:

  )t) cos(( ) 1 ( )t) n - ( cos( ) ( J t) cos( ) ( J Vc (t) V 1 L c L c L n c L 0 FM                  n n n        
where J n is the n-th Bessel function. It is straightforward to show that the square of this expression, needed for the V(t) 2 factor in (2)), has no signal at the modulation frequency ω L but still has DC and high frequency signals. So this demonstrates that pure electrical non linearities, as used in ref. 20, cannot demodulate an FM signal, i.e. no sound can be heard from this term in the FM mode.

Let's now take a look at the remaining term: the electromechanical current for an FM signal. It is usually assumed that  2 I sd /xV sd is proportional to the transconductance and to C', the space derivative of the capacitance between the tube and the gate, but it could also be related to a piezoresistive effect. Such piezoresistive effects are possible either due to a strain-induced intrinsic conductance change [START_REF] Cao | [END_REF][25][26] or a variation of the contact resistance. For generality we won't make any assumption here about this term.

The argument of the cosine in the V FM signal can be rewritten ψ(t+Δt)=ψ(t)+ψ/tΔt where Δt is a time of the order or smaller than the time scale of the oscillator Q/ω 0 , where Q is the quality factor of the resonator. A sufficient condition to neglect higher order terms and transient is ω c >> ω Δ and ω 0 /Q >> ω L .

The electromechanical current is only significant at the resonant frequency ω 0 so we consider here ω c ~ ω 0 . These conditions are satisfied in our experiment, so we can consider that the oscillator is submitted at each time t to an harmonic forcing of frequency ω i = ψ/t = ω c +ω Δ cos(ω L t), namely the instantaneous frequency, with an additional phase term ψ(t) that is constant over the time scale of interest. Thus in the approximation of a single mechanical degree of freedom, the complex frequency response function of the displacement δx*(ω) is:

Q i V m V V C x i i c eff offset DC g i       0 2 2 0 2 ) ( *     
where V offset is an offset voltage due to some residual charges at zero voltage, m eff is the effective mass of the nanotube. The instantaneous mechanical displacement of the nanotubes is:

)) ( sin( )) ( * Im( )) ( cos( )) ( * Re( ) ( t t x t t x t t x i i i i                 
Substituting this into Eq. ( 2) gives the low frequency term of the electromechanical current, I LF FM :

)) ( * Re( 2 / 1 2 i c sd FM LF x V V x I I      
Finally a Taylor expansion of Re(δx*(ω )) for ω Δ << ω c gives :

... ) cos( *) Re( 2 1 ) cos( *) Re( )) ( * Re( )) ( * Re( 2 2 2 2           t x t x x x L i L i c i             (3)
Remarkably there is no term in sin(ω L t) and no purely electrical term with cos(ω L t). Consequently, contrary to other detection techniques, there is no undesirable background current that interferes with the electromechanical current. For example in a two source setup as used by all previous work 14,18 there is an additional purely electrical background current from a  2 I sd /V sd V g term. In AM detection, a similar analysis shows that this background is superimposed on the real part of δx*(ω c ). In FM detection the current detected by the lock-in at frequency ω L is then proportional to:

  2 2 0 2 2 2 0 2 0 2 0 2 2 0 2 0 2 ) )( ( 2 )
(                        Q Q Q F c c c c c c             (4)
where F is the derivative of the real part of δx*(ω), without unnecessary multiplicative constants.

Interestingly, the zeroes of F(ω c ) gives the quality factor. We fitted our data by this function (see figure 2c) and obtained a good agreement. A peculiarity of the FM signal is that the power transmitted is independent of the value of ω Δ which follows from the identity:

         1 2 n 2 0 ) ( J 2 ) ( J 1 n u u
, where u is any real number. As the electromechanical current is proportional to ω Δ , it should increase linearly with ω Δ . We confirmed experimentally figure 3a) that the demodulated FM current follows this linear dependence for low ω Δ before peaking at a ω Δ slightly smaller than the Lorentzian width, because of higher order terms in equation ( 3). This maximum (of the order of  2 I sd /xV sd V c 2 Q/2ω 0 ) is comparable but larger by at least a factor of 2 to the signal intensity obtained by the two source mixing technique for the same amount of power. Thus there is a non-negligible gain of signal by using FM excitation. For too high ω Δ , the cos(ω L t) component of the current is no longer proportional to F(ω c ), its zeroes move further apart and they no longer give the quality factor directly (see figure 3a). From the experimental data at low ω Δ , we extracted the parameter of F(ω c ) and calculated numerically the value of the cos(ω L t) component of I LF FM for various ω Δ and got an excellent agreement (see figure 3a). Finally in this regime, detectable higher order terms implies the appearance of harmonics from the expansion of cos(ω L t) n terms (where n is for the n th order). Each harmonic is proportional, to first order, to the n th derivative of Re(δx*(ω i )) (see equation ( 3)). For ω Δ = 500 kHz, we observed up to 9 harmonics (figure 3c shows harmonics 2, 3 and 4) and could hear clearly distinguishable noise distortion when the signal was sent to a loudspeaker. However, for radio applications, where ω Δ < 200 kHz, our device is still in the linear regime. An important point about the cancellation of the Y component is that this detection technique is not only sensitive to the frequency ω L , but also to its phase. This is of some interest for digital data transfer since some common modulation schemes such as QPSK (quadrature phase shift keying) uses the phase of the signal to enhance data transmission by sending more than 1 bit at the same time. The principle is to code, for example, 2 bits by changing the phase φ by π/2 steps. This gives 4 different states (φ = 0 coding 00 in bits, φ = π/2 for 01, φ = π for 10 and φ = 3π/2 for 11). Experimentally, we used an independent low frequency generator to send the cos(ω L t+φ) signal that FM modulates the carrier frequency of the high frequency generator. We record the demodulated signal in the lock-in using its internal source. The carrier frequency is fixed at the resonance of the nanotube. For φ = 0, we set the phase of the lock-in, in order to have a large negative signal in X and zero in Y, then we keep constant the phase of the lock-in and only change the phase of the external low frequency generator. So for φ = π/2 we got zero in X and a large negative signal in Y i.e. the resonance is detected by the out of phase component. For φ = π (respectively φ = 3π/2) we got a sign reversal in X (respectively Y) compared to φ = 0 (respectively φ = π/2). Figure 4a clearly demonstrates that such digital demodulation is possible for nanotube NEMS. The maximum transfer rate about 60 bps (bit per second) is limited here by the 30 ms time transfer of our GPIB connection between the lock-in and the computer. To push the data transfer rate even further we i) used a sample with a resonant electromechanical current above 1nA to lower the time constant of the lock-in and still have a reasonable signal to noise ration and ii) did the measurement of the current with the analog 100 kHz bandwidth output of the lock-in sent to a high speed oscilloscope. Figure 4b shows a rise time of 200 µs with a signal to noise ration of 8 when passing from 00 to 01. Thus we have achieved a transfer rate of 10 kbps thanks to this 2 bit at a time transfer method.

This means that our device reached the requirement for the widely used GSM (global system for mobile communication) cell phone standard where the voice is digitally transported at a 9.6 kbps rate. In conclusion we showed that FM and digital demodulation is possible using carbon nanotubes NEMS in a transistor geometry. We theoretically demonstrated that the shape of the resonance peak is related to the derivative of the real part of the response function of the resonator. With this technique, it is possible to reduce the noise and unwanted back ground signals as well as to detect derivative of the response function up to the ninth order. The noise reduction of FM demodulation might be useful for NEMS applications such as ultimate mass sensing. Our device achieved a transfer rate of 10 kbps compatible with GSM requirement.

  since the birth of the industry. The low frequency electrical transduction of the sound is first carried over large distances at high frequency by up-mixing (also called modulation) and then recovered by down mixing (i.e. demodulation) in a standard radio set. So, an interesting avenue to explore has been the use of CNTs for the demodulation of signals generated by amplitude modulation (AM) or frequency modulation (FM). AM detection was mentioned in ref. 19 for the study of mechanical resonances of doubly clamped carbon nanotubes. Later, refs. 20 and 21 emphasized the potential of CNTS in a "nano-radio"

Figure 1 .

 1 Figure 1. (a) SEM image of one of our devices with a suspended nanotube between 2 gold electrodes

Figure 2 .

 2 Figure 2. (a) and (b) Magnitude and phase of the lock-in current versus carrier frequency in FM mode

Figure 3 .

 3 Figure 3. (a) (■) Absolute value of the maximum current at cos(ω L t) and () frequency difference

Figure 4 .

 4 Figure 4. (a) Digital demodulation (■ in phase and  out of phase component) for ω Δ /2π = 200 kHz,

ACKNOWLEDGMENT. This work is supported by French National Agency (ANR) through Nanoscience and Nanotechnology Program (Projects NEXTNEMS n°ANR-07-NANO-008-01), through "Jeunes Chercheuses et Jeunes Chercheurs" Program (Project AUTONOME n°ANR-07-JCJC-0145-01) and cluster micro-nano of the région Rhône Alpes. It was carried out within the framework of the Group Nanowires-Nanotubes Lyonnais. The authors acknowledge the support of the "Centre Technologique des Microstructures de l'Université Lyon 1". We thank the OMNT experts, L. Montes and G. Bremond for fruitful discussions, P. Cremilleu and R. Mazurczyk in the Institut NanoLyon technological platform for help in sample fabrication.


P Auscher
email: pascal.auscher@math.u-psud.fr

Sobolev spaces on multiple cones

Keywords: 2000 Mathematics Subject Classification. 46B70, 46E35, 42B20. Key words and phrases. Interpolation, Sobolev spaces, Poincaré inequality, Doubling property, Metric-measure spaces, Calderón-Zygmund decomposition

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from R n . The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.

Introduction

The theory of Sobolev spaces on domains of the Euclidean spaces is well developed and numerous works and books are available. For multi-connected open sets, there is apparently nothing to say. However, depending on the topology of the boundary, the closure of the space of test functions (ie compactly supported in R n ) might be a subtle thing. We propose here to investigate the Sobolev spaces on multiple cones with common vertex as unique common point of their boundaries. Surprisingly, we did not find a treatment in the literature.

Our motivation comes from Badr's PhD thesis where interpolation results for Sobolev spaces on complete metric-measure spaces are proved upon the doubling property and families of Poincaré inequalities. A question remained unsettled, namely whether the result is sharp, that is whether the conclusion is best possible given the hypotheses. Multiple (closed) cones are sets where doubling (for Lebesgue measure) holds and L p -Poincaré inequalities hold for some but not all p, more precisely for p greater than dimension. Cones are therefore simple but important examples for this matter. The study of Sobolev spaces on such sets provides us with the positive answer to our question and, in addition, we complete the interpolation result in this specific situation. As we shall see, these Sobolev spaces can be identified with the closure of test functions in the classical Sobolev space on open multiple cones.

1. The space W 1 p (Ω) coincides with W 1 p (Ω) if 1 ≤ p ≤ n but is of codimension 1 in W 1 p (Ω) for n < p ≤ ∞. (Section 2) 2. The spaces W 1 p (Ω), 1 ≤ p ≤ ∞, form a real interpolation family (Section 3)

3. The spaces W 1 p (Ω), 1 ≤ p ≤ ∞, do not form a real interpolation family. To obtain such a family, one needs to replace W 1 n (Ω) by a strict and dense subspace of it described in the text. (Sections 3, 4) 4. For p ∈ [1, ∞], p = n, the restriction operator to Ω maps W 1 p (R n ) continuously onto W 1 p (Ω) and there exists a common linear continuous extension operator from W 1 p (Ω) to W 1 p (R n ). For p = n, these results hold with W 1 n (Ω) replaced by the strict and dense subspace mentioned above. In particular, Ω has the extension property for W 1 p if and only if 1 ≤ p < n. (Section 5)

Of course, some of these results are known and we give references along the way when we have been able to locate them. But some results, like the interpolation results, are new. We also point out that we give two proofs of the interpolation result. Although the one presented in Section 5 using restriction/extension looks more natural to users of Sobolev spaces on subsets of the Euclidean space, we prefer the one done in Sections 3 and 4, because it is more in the spirit of analysis on metric spaces and contains ideas that we believe could be used in this context elsewhere. In particular, a special feature is that it allows to pass below the Poincaré exponent threshold by using Hardy type inequalities.

We shall make use of the Sobolev space H 1 p (X) arising from geometric measure theory on X = Ω. It is defined as the completion for the W 1 p (Ω) norm of the space of Lipschitz functions with compact support for p < ∞ and as the space of bounded and Lipschitz functions in X for p = ∞. It is easy to show it agrees with W 1 p (Ω) and it turns out that it will be easier to work with the former in Section 3. Finally, we make a connection with the Hajlasz-Sobolev space M 1,p (X). In particular, we will show (Section 5) that the Hajlasz-Sobolev space M 1,n (X) is a strict subspace of H 1 n (X), which can be surprising. In Section 6, we shortly describe the situation pertaining to these questions for homogeneous Sobolev spaces.

Density

Let 1 ≤ p ≤ ∞ and O an open set of R n . Define W 1 p (O) as the space of functions 1 f ∈ L p (O) such that f W 1 p (O) = f L p (O) + ∇f L p (O) < ∞.
The gradient is defined in the distributional sense in O. For p < ∞, denote by W1 p (O) the closure of the space of C ∞ 0 (R n ) (the subscript 0 means compact support) functions restricted to O in W 1 p (O). Among classical texts, we quote [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Evans | Measure theory and fine properties of functions[END_REF][START_REF] Maz'ya | Sobolev spaces[END_REF][START_REF] Maz'ya | Differentiable functions on bad domains[END_REF][START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF][START_REF] Ziemer | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF].

If n < p < ∞, recall that the Morrey-Sobolev embedding implies that if f ∈ W 1 p (Ω) then f is Hölder continuous on each connected component Ω ± of Ω, the half-cones defined by x ∈ Ω and sign(x n ) = ±1. Hence f has limits in 0 from Ω + and Ω -. These limits, which we call f (0 + ) and f (0 -), may be different.

This lemma is classical and we include a proof for convenience.

Lemma 2.1. Let 1 ≤ p < ∞ and f ∈ W 1 p (Ω). Assume 1 ≤ p ≤ n or n < p < ∞ and f (0 + ) = f (0 -) = 0. Then there exists a sequence of C ∞ 0 (R n ) functions (ϕ k ) with support away from 0 such that f -ϕ k W 1 p (Ω) tends to 0. Proof. First, it is enough to consider f ∈ W 1 p (Ω + ), with f (0) = 0 if p > n.
Second, we may also assume f bounded by using the truncations

f N = h N (f ) and N → ∞, where h N (t) = -N if t ≤ -N , h N (t) = t if -N ≤ t ≤ N and h N (t) = N if t ≥ N .
Next, we claim that we can approximate f by a function in g ∈ W 1 p (Ω + ) supported away from a ball centered at 0. Assuming this claim, it suffices to convolve this approximation with a smooth mollifying function which has compact support inside Ω + to conclude.

It remains to prove the above claim. Take χ ∈ C ∞ 0 (R n ) a positive, radial function, bounded by 1, supported in the unit ball with χ = 1 in the half-unit ball. For ǫ > 0, define χ ǫ (x) = χ( x ǫ ) and take f ǫ (x) = f (x)(1χ ǫ (x)). Every f ǫ = 0 on the ball of radius ǫ/2. We distinguish between 3 cases:

Case 1 ≤ p < n: By dominated convergence f ǫ converges to f in L p (Ω + ).
For the gradient, as f is bounded and ∇χ ǫ p ≤ Cǫ n/p-1 , we conclude that ∇f ǫ converges to ∇f in L p (Ω + ).

Case p = n: The function f ǫ does not converge to f in this case and we have to modify the construction. For 0 < δ < 1, we introduce the function

η δ (x) = | ln δ| | ln |x|| if |x| ≤ δ and η δ (x) = 1 if |x| > δ. Take f ǫ,δ = f η δ (1 -χ ǫ ) = f ǫ η δ with δ k =
ǫ and k > 0 large. It is easy to show that f ǫ,δ converges to f in L n (Ω + ) for ǫ → 0 and any k fixed. For the gradient, using |1χ ǫ | ≤ 1, we have

|∇(f -f ǫ,δ )| ≤ |(1 -η δ )∇f | + |χ ǫ ∇f | + |f ∇η δ | + |f η δ ∇χ ǫ |.
We observe that we assumed f bounded. A computation shows that η δ ∇χ ǫ n is bounded by C/k. So we pick and fix k big enough. Next, ∇η δ n goes to 0 as ǫ → 0 and the remaining term (1η δ )∇f n + χ ǫ ∇f n → 0 by dominated convergence.

Case p > n: By dominated convergence, f ǫ converges to f in L p (Ω + ). For the gradient, we have (1χ ǫ )∇f converges to ∇f in L p (Ω + ) by dominated convergence. It remains to prove that f ∇χ ǫ L p (Ω + ) tends to 0. By Morrey's theorem and recalling that f (0) = 0 we have for every

x ∈ Ω + , |x| < ǫ, f (x) ǫ ≤ C |x| ǫ 1-n/p {|y|<2ǫ}∩Ω + |∇f | p (y)dy 1/p . ( 2.1)
This implies

Ω + |f (x)∇χ ǫ (x)| p dx ≤ {|x|≤ǫ}∩Ω + f (x) ǫ p dx ≤ {|x|≤ǫ}∩Ω + |x| ǫ p-n dx 1 ǫ n {|y|≤2ǫ}∩Ω + |∇f | p (y)dy ≤ C {|y|≤2ǫ}∩Ω + |∇f | p (y)dy.
We conclude noting that the last integral converges to 0 when ǫ → 0 by the dominated convergence theorem.

Remark. The density of functions in W 1 n (Ω ± ) supported away from a ball centered at 0 was also proved in [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF]Lemma 2.4] for the special case of dimension n equals 2 (We are thankful to Monique Dauge for indicating this work). Their proof applies mutatis mutandis for dimensions higher.

Corollary 2.2. Let 1 ≤ p ≤ ∞. If p ≤ n, W 1 p (Ω) = W 1 p (Ω) and if n < p, W 1 p (Ω) = {f ∈ W 1 p (Ω) ; f (0 + ) = f (0 -)},

and hence is of codimension 1 in W 1

p (Ω).

Proof. For 1 ≤ p ≤ n, the equality follows immediately from Lemma 2.1. Assume now n < p ≤ ∞.

Trivially W 1 p (Ω) ⊂ f ∈ W 1 p (Ω); f (0 + ) = f (0 -) . Con- versely let f ∈ W 1 p (Ω), f (0 + ) = f (0 -) := f (0). Then g = f -f (0)χ, with χ ∈ C ∞ 0 (R n
) supported in the unit ball with χ ≡ 1 in a neighborhood of 0 verifies g(0 + ) = g(0 -) = 0. By lemma 2.1 for p < ∞ and by definition for p = ∞, this yields g ∈ W 1 p (Ω) and therefore f = g + f (0)χ.

Real interpolation

As far as

W 1 p (Ω) is concerned, we have if 1 ≤ p ≤ ∞ that W 1 p (Ω) = W 1 p (Ω + ) ⊕ W 1
p (Ω -) using restriction to Ω ± and extension by 0 from Ω ± to Ω. That is, if

f ∈ W 1 p (Ω), we write f = 1 Ω + f + 1 Ω -f. (3.1)
Since Ω ± is a Lipschitz domain, it is known [START_REF] Devore | Interpolation of linear operators on Sobolev spaces[END_REF] that the family of Sobolev spaces (W 1 p (Ω ± )) 1≤p≤∞ forms a scale of interpolation spaces for the real interpolation method. Hence the same is true for (W 1 p (Ω)) 1≤p≤∞ . There is a second chain of spaces appearing in the axiomatic theory of Sobolev spaces on a metric-measure space ( [START_REF] Hajlasz | Sobolev met Poincaré[END_REF], [START_REF] Hajlasz | Sobolev spaces on a arbitrary metric space[END_REF], [START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF]). Let X be the closure of Ω. Then X equipped with Euclidean distance and Lebesgue measure, which we denote by λ, is a complete metric-measure space. The balls are the restriction to X of Euclidean balls centered in X. For 1 ≤ p < ∞, we denote by H 1 p (X) the completion for the norm W 1 p (Ω) of Lip 0 (X), the space of Lipschitz functions in X with compact support. For p = ∞, we set

H 1 ∞ (X) = Lip(X) ∩ L ∞ (X). Identifying a Lipschitz function on Ω with its unique extension to X, H 1 ∞ (X) = f ∈ W 1 ∞ (Ω); f (0 + ) = f (0 -) .
There are also other Sobolev spaces of interest, like the Hajlasz spaces M 1,p (X). We shall come back to this in Section 5.

We recall the definitions of doubling property and Poincaré inequality:

Definition (Doubling property). Let (E, d, µ) be a metric-measure space. One says that E satisfies the doubling property (D) if there exists a constant C < ∞ such that for all x ∈ E, r > 0 we have

0 < µ(B(x, 2r)) ≤ Cµ(B(x, r)). (D)
Definition (Poincaré Inequality). A (complete) metric-measure space (E, d, µ) admits a q-Poincaré inequality for some 1 ≤ q < ∞, if there exists a positive constant C < ∞, such that for every continuous function u and upper gradient g of u, and for every ball B of radius r > 0 the following inequality holds:

B |u -u B | q dµ 1 q ≤ Cr B g q dµ 1 q . (P q )
There are weaker ways of defining the Poincaré inequalities but it amounts to this one when the space is complete. See [START_REF] Hajlasz | Sobolev spaces on a arbitrary metric space[END_REF] for more on this and definition of upper gradients. On X, |∇u| is an upper gradient of u.

Let us recall Badr's theorem in this context ([3], Theorem 7.11). On a metric-measure space there is a definition H 1 p (E) for p ≤ ∞ which, for X = E, is equivalent to the one given here. Theorem 3.1 (Badr). Let 1 ≤ q 0 < ∞. Assume (E, d, µ) is a complete metricmeasure space with the doubling property and q-Poincaré inequalities with q > q 0 . Then for q 0 < p 0 < p 1 ≤ ∞ and 1/p = (1θ)/p 0 + θ/p 1 ,

(H 1 p 0 (E), H 1 p 1 (E)) θ,p = H 1 p (E). (3.2)
The space (X, d, λ) has the doubling property and, as shown in [START_REF] Hajlasz | Sobolev spaces on a arbitrary metric space[END_REF] p.17, it supports a q-Poincaré inequality if and only if n < q. Thus, (H 1 p (X)) n<p≤∞ is a scale of interpolation spaces for the real interpolation method. As observed and proved in [2], Chapter 4 (see also Section 9 of [START_REF] Badr | Real interpolation of Sobolev Spaces[END_REF]), with arguments we reproduce here, H 1 p (X) = W 1 p (Ω) when 1 ≤ p < n, and this allowed her to identify H 1 p (X) as the interpolation space (H 1 p 0 (X), H 1 p 1 (X)) θ,p when 1 ≤ p 0 < p < p 1 ≤ ∞ and 1/p = (1θ)/p 0 + θ/p 1 with the restriction that either n < p or p 1 < n.

The missing cases are somehow intriguing and for the sake of curiosity we provide a complete picture in the following result. More interestingly, we provide two proofs that cover all cases at once.

Theorem 3.2. If 1 ≤ p 0 < p < p 1 ≤ ∞ and 1/p = (1 -θ)/p 0 + θ/p 1 , then (H 1 p 0 (X), H 1 p 1 (X)) θ,p = H 1 p (X), if p = n, H 1 n (X), if p = n. (3.3)
We shall see that H 1 n (X) is a strict subspace of H 1 n (X). This implies in particular that Badr's interpolation result is sharp in the class of Sobolev spaces on metric-measure spaces: in this example, the infimum of Poincaré exponents is also the smallest exponent p 0 for which the family (H 1 p (X)) p 0 <p≤∞ is a scale of interpolation spaces for the real interpolation method. Hence, she could not get a better conclusion in general. See section 5 for a further discussion on this.

The space H 1 n (X) will incorporate a sort of Hardy inequality with respect to the vertex point. To describe it, we need the following definition.

Definition. For a function f : X → R, we define its radial part f r and its antiradial part f a as follows: f r (x) is the mean of f on the sphere S |x| of radius |x| restricted to Ω with respect to surface measure and

f a (x) = f (x) -f r (x).
The number f r (x) depends only on the distance of x to the origin, hence the terminology radial (even if Ω is not invariant by rotations). But note that both f r and f a depend on Ω. Note that f → f r is a contraction on H 1 p (X). Denote by r : R n → R, r(x) = |x|.

Definition. H 1 n (X) = {f ∈ H 1 n (X) ; f a /r ∈ L n (X)} with norm f H 1 n (X) = f H 1 n (X) + f a /r L n (X) .
The following example shows that H 1 n (X) is a strict subspace of H 1 n (X). Assume n = 2 and β > 0, and consider the function f on X, supported on

r ≤ 1/2, C ∞ away from 0, which is sign(x 2 )| ln r| -β for r ≤ 1/4. It is easy to check that f ∈ H 1 2 (X) for all β > 0. Clearly, f = f a and f /r ∈ L 2 (X) if and only if β > 1/2. Hence for 0 < β ≤ 1/2 we have f / ∈ H 1 2 (X)
. Before we move on, the relation between H 1 p (X) and W 1 p (Ω) is the following.

Lemma 3.3. For 1 ≤ p ≤ ∞, H 1 p (X) = W 1 p (Ω) with the same norm. Proof. The equality at p = ∞ is obvious. Assume next that p < ∞. It is clear that W 1 p (Ω) ⊂ H 1 p (X) ⊂ W 1 p (Ω). Thanks to Corollary 2.2, we have our conclusion if 1 ≤ p ≤ n. Assume further n < p. Then functions in Lip 0 (X) satisfy f (0 + ) = f (0 -). Since f → f (0 ± ) are continuous on W 1 p (Ω), this passes to H 1 p (X). Applying again Corollary 2.2, we deduce that H 1 p (X) ⊂ W 1 p (Ω).
To prove our theorem, we first introduce the following spaces.

Definition. For 1 ≤ p ≤ ∞, set H 1 p (X) = {f ∈ H 1 p (X) ; f /r ∈ L p (X)} with norm f H 1 p (X) = f H 1 p (X) + f /r L p (X) = f W 1 p (Ω) + f /r L p (Ω) . Lemma 3.4. For 1 ≤ p ≤ ∞, H 1 p (X) is a Banach space which can be identified isometrically to {f ∈ W 1 p (Ω) ; f /r ∈ L p (Ω)}.
Proof. There is nothing to prove if 1 ≤ p ≤ n thanks to Corollary 2.2 and Lemma 3.

3. Assume next n < p ≤ ∞. Let f ∈ H 1 p (X), then the restriction of f to Ω belongs to {f ∈ W 1 p (Ω) ; f /r ∈ L p (Ω)}. Conversely if f ∈ W 1 p (Ω) and f /r ∈ L p (Ω), then f has a unique extension to a Hölder (Lipschitz if p = ∞) continuous function in both Ω ± . The condition f /r ∈ L p (Ω) forces f (0 + ) = f (0 -) = 0. Hence this extension is in H 1 p (X) and thus in H 1 p (X).
The next result is the main step.

Theorem 3.5. The family ( H 1 p (X)) 1≤p≤∞ is a scale of interpolation spaces for the real interpolation method.

This result is proved in the next section. We continue with Proposition 3.6.

If 1 ≤ p < n, H 1 p (X) = H 1 p (X). If n < p ≤ ∞, H 1 p (X) = {f ∈ H 1 p (X) ; f (0) = 0}
and has codimension 1 in H 1 p (X). Before we prove this proposition we need the following Hardy type inequality (we thank Michel Pierre for indicating a simple proof):

Lemma 3.7. Let 1 ≤ p ≤ ∞ with p = n. Then there exists a constant C = C(p, Ω) such that Ω f r p dx ≤ C Ω |∇f | p dx (3.4) for every f ∈ H 1 p (X) with, in addition, f (0) = 0 if p > n, and (3.4) is under- stood with L ∞ norms if p = ∞.
The example above shows that the lemma is false when p = n.

Proof. Assume first 1 ≤ p < n. Take f ∈ Lip 0 (X). We have

Ω + f r p dx = Ω + ∩S 1 ∞ 0 r n-1-p |f (r, θ)| p drdσ(θ) = Ω + ∩S 1 1 n -p r n-p |f (r, θ)| p ∞ 0 dσ (θ)
-

Ω + ∩S 1 ∞ 0 1 n -p r n-p p|f | p-1 sign f ∂f ∂r drdσ(θ) = - p n -p Ω + ∩S 1 ∞ 0 f r p-1 sign f ∂f ∂r r n-1 drdσ(θ) ≤ p n -p Ω + ∩S 1 ∞ 0 f r p r n-1 drdσ(θ) p-1 p × Ω + ∩S 1 ∞ 0 ∂f ∂r p r n-1 drdσ(θ) 1 p
.

After simplification, we get (3.4) on Ω + . We do the same for the integral on Ω -and therefore (3.4) holds for every f ∈ Lip 0 (X). By density, (3.4) holds for every

f ∈ H 1 p (X). Assume next n < p < ∞. Let f ∈ Lip 0 (X) such that f (0) = 0. We denote A = Ω + ∩{|x|>ǫ} f r
p dx, where ǫ > 0. By Morrey's theorem, we have for every

x ∈ Ω, |f (x)| ≤ C |∇f | p |x| α with α = 1 -n/p.
Repeating the computation of (3.4) and since f has a compact support, one obtains

A = Ω + ∩S 1 ∞ ǫ r n-1-p |f (r, θ)| p drdσ(θ) = Ω + ∩S 1 1 n -p r n-p |f (r, θ)| p ∞ ǫ dσ(θ) - Ω + ∩S 1 ∞ ǫ 1 n -p r n-p p|f | p-1 sign f ∂f ∂r dσ(θ)dr = ǫ n-p p -n Ω + ∩S 1 |f (ǫ, θ)| p dσ(θ) + p p -n Ω + ∩S 1 ∞ ǫ f r p-1 sign f ∂f ∂r r n-1 drdσ(θ) ≤ C p ∇f p p + p p -n Ω + ∩S 1 ∞ ǫ f r p r n-1 drdσ(θ) p-1 p Ω + ∩S 1 ∞ ǫ ∂f ∂r p r n-1 drdσ(θ) 1 p . This yields A ≤ C p |∇f | p p + p p -n A p-1 p |∇f | p . Plugging A p-1 p |∇f | p ≤ δ p ′ A p ′ + 1 pδ p |∇f | p p
for every δ > 0, with p ′ = p p-1 , one obtains

A(1 - pδ p ′ (p -n)p ′ ) ≤ (C p + 1 (p -n)δ p ) |∇f | p p .
Choosing δ small enough, we deduce that

Ω + ∩{|x|>ǫ} f r p dx ≤ C Ω + |∇f | p dx.
We then let ǫ → 0. We do the same for the integral on Ω -and therefore (3.4) holds for every f ∈ Lip 0 (X) such that f (0) = 0. By density, (3.4) holds for every f ∈ H 1 p (X) such that f (0) = 0.

When p = ∞, (3.4) is a direct consequence of the definition of H 1 ∞ (X) and that f (0) = 0 with the mean value theorem.

Proof of Proposition 3.6. When 1 ≤ p < n, Lemma 3.7 shows that H 1 p (X) ⊂ H 1 p (X) and the proposition follows. Now, when p > n, Lemma 3.7 yields

f ∈ H 1 p (X); f (0) = 0 ⊂ H 1 p (X). Conversely if f ∈ H 1 p (X)
, by the continuity of f at 0 and the L p integrability of f /r we easily see that f (0) = 0.

It remains to prove that

H 1 p (X) is of codimension 1 in H 1 p (X). This follows by writing f ∈ H 1 p (X) as f = f -f (0)χ + f (0)χ, where χ ∈ C ∞ 0 (R n )
, supp χ ⊂ B(0, 1) and χ = 1 in a neighborhood of 0, and using the above characterization of H 1 p (X). Although this is a simple description of H 1 p (X), the jump at p = n does not allow us to use this result to conclude for Theorem 3.2. We need to further analyze the radial and antiradial parts of a function.

Lemma 3.8. Let 1 ≤ p ≤ ∞.

For a function f depending only on the distance to the origin

, f ∈ H 1 p (X) ⇐⇒ f ∈ W 1 p (R n ) with same norm up to a constant. 2. Assume p = n. For a function f : X → R and f a = f -f r , we have f a ∈ H 1 p (X) ⇐⇒ f a ∈ H 1 p (X) with comparable norms.
Proof. The first item is trivial. The constant is the ratio of the surface measure of Ω inside the unit sphere divided by the surface measure of the unit sphere.

As for the second item, it follows from the previous proposition directly if p < n and by observing that f a (0) = 0 if p > n.

Let us recall the following definition:

Definition. Let f be a measurable function on a measure space (X, µ). The decreasing rearrangement of f is the function f * defined for every t ≥ 0 by

f * (t) = inf {λ : µ({x : |f (x)| > λ}) ≤ t} .
The maximal decreasing rearrangement of f is the function f * * defined for every t > 0 by

f * * (t) = 1 t t 0 f * (s)ds.
Remark. It is known that (Mf ) * ∼ f * * with M the Hardy-Littlewood maximal operator, f * * p ∼ f p for all p > 1 (see [START_REF] Stein | Introduction to Fourier Analysis in Euclidean spaces[END_REF], Chapter V, Lemma 3.21, p.191 and Theorem 3.21, p.201) and µ({x : |f (x)| > f * (t)}) ≤ t for all t > 0. We refer to [START_REF] Bennett | Interpolation of operators[END_REF], [START_REF] Bergh | Interpolation spaces, An introduction[END_REF] for other properties of f * and f * * .

We can now complete the proof of Theorem 3.2.

Proof. Let us examine the case where neither p 0 , p 1 is n. By the reiteration theorem, this reduces further to

p 0 = 1, p 1 = ∞. Set F p = (H 1 1 (X), H 1 ∞ (X)) θ,p with θ = 1 -1/p. Let f ∈ F p . Since f → f r is contracting on H 1 q (X
) for all 1 ≤ q ≤ ∞ and using Lemma 3.8, one has that

K(f r , t, W 1 1 (R n ), W 1 ∞ (R n )) ≤ CK(f, t, H 1 1 (X), H 1 ∞ (X)).
K is the K-functional of interpolation defined as in [START_REF] Bennett | Interpolation of operators[END_REF], [START_REF] Bergh | Interpolation spaces, An introduction[END_REF]. Hence

f r ∈ W 1 p (R n ) by classical interpolation for the W 1 p (R n ). Thus f r ∈ H 1 p (
X) by Lemma 3.8. We also have by Lemma 3.8 again,

K(f a , t, H 1 1 (X), H 1 ∞ (X)) ≤ CK(f, t, H 1 1 (X), H 1 ∞ (X)). Theorem 3.5 shows then that f a ∈ H 1 p (X). We conclude that f ∈ H 1 p (X) if p = n and f ∈ H 1 n (X) if p = n. Reciprocally, let f ∈ H 1 p (X) if p = n and f ∈ H 1 n (X) if p = n. By Lemma 3.8, whatever p is, we have that f r ∈ W 1 p (R n ) and f a ∈ H 1 p (X). By Theorem 3.5, f a ∈ ( H 1 1 (X), H 1 ∞ (X)) θ,p with θ = 1 -1/p. Hence f a ∈ F p .
For the radial part, for each t > 0, one can find a decomposition f r = g t +h t almost minimizing for

K(f r , t, W 1 1 (R n ), W 1 ∞ (R n ))
and one can assume both g t and h t are radial. Thus Lemma 3.8 implies that g t ∈ H 1 1 (X) and

h t ∈ H 1 ∞ (X), hence f r ∈ F p .
It remains to study the case where p 0 or p 1 is equal to n. Let us consider the case p 1 = n as the other one is similar. It is also enough to look at the result when p 0 = 1. As we know all interpolation spaces between H 1 1 (X) and H 1 ∞ (X), by the reiteration theorem, if 1 < p < n and

1 p = 1 -θ + θ n we have (H 1 1 (X), H 1 n (X)) θ,p = H 1 p (X)
. Hence, we have

H 1 p (X) = (H 1 1 (X), H 1 n (X)) θ,p ⊂ (H 1 1 (X), H 1 n (X)) θ,p ⊂ H 1 p (X).
The last inclusion is the easy part of the interpolation: we recall that

K(f, t, H 1 1 (X), H 1 n (X)) ≥ K(f, t, L 1 , L n ) + K(|∇f |, t, L 1 , L n )
and that

K(f, t, L 1 , L n ) ∼ t α 0 f * (u) du + t ∞ t α f * (u) n du 1 n
,

where 1 α = 1-1 n .
Integrating, using the definition of the maximal rearrangement function and its properties in the Remark above, we deduce that

f p (H 1 1 (X),H 1 n (X)) θ,p = ∞ 0 t -θ K(f, t, H 1 1 (X), H 1 n (X)) p dt t ≥ C f p H 1 p (X)
and therefore (H

1 1 (X), H 1 n (X)) θ,p ⊂ H 1 p (X)
. This concludes the proof.

Remark. The inclusion

H 1 n (X) ⊂ H 1 n (X)
is dense. This is due to the fact that as H 1 n (X) = W 1 n (Ω), the space of restrictions to X of smooth functions on R n with compact support in R n \ {0}, a subspace of H 1 n (X), is dense in H 1 n (X). Note that the last part of the argument shows that (H 1 p 0 (X), H 1 p 1 (X)) θ,p = H 1 p (X) for the appropriate p whenever p 0 or p 1 is equal to n. In particular, when p 0 = n this furnishes an endpoint to Badr's result for the case of E = X.

Remark. As X is symmetric with respect to S : x → -x, we can define H 1 n (X) differently by doing an analysis with even and odd parts. Define the even and odd parts f e and f o of a function f :

X → R as f e = 1 2 (f + f • S) and f o = 1 2 (f -f • S). We have that H 1 n (X) = {f ∈ H 1 n (X) ; f o /r ∈ L n (X)}. Let f ∈ H 1 n (X). Write f o = (f r ) o + (f a ) o and easily (f r ) o = 0. Hence f a ∈ H 1 n (X) =⇒ (f a ) o ∈ H 1 n (X) =⇒ f o ∈ H 1 n (X).
Next, write f a = (f e ) a + (f o ) a . We claim that (f e ) a /r ∈ L n (X). Hence,

f o ∈ H 1 n (X) =⇒ (f o ) a ∈ H 1 n (X) =⇒ f a ∈ H 1 n (X).
To see the claim, we observe that the evenness of f e implies that (f e ) r (x) is also equal to the mean of f e on Ω + ∩ S |x| . Thus (f e ) a is an even function with mean value 0 on each Ω ± ∩ S |x| . Applying the classical (with gradient instead of upper gradient) Poincaré inequalities (P n ) for spherical caps (that is geodesic balls) with respect to surface measure on S |x| , we obtain

Ω ± ∩S |x| |(f e ) a | n dσ(θ) ≤ C(n, Ω ± )|x| n Ω ± ∩S |x| |∇ θ (f e ) a | n dσ(θ)
where ∇ θ is the tangential gradient on S |x| . Notice that |∇ θ (f e ) a | ≤ |∇(f e ) a | on Ω ∩ S |x| . So adding the two inequalities, multiplying by r -n = |x| -n and integrating with respect to dr we obtain

Ω |(f e ) a | n r n dx ≤ C(n, Ω) Ω |∇(f e ) a | n dx.
Hence (f e ) a /r ∈ L n (X) as claimed.

Remark. We have used Poincaré inequalities (P n ) for spherical caps on spheres. Equipped with geodesic distance and surface measure, they are spaces of homogeneous type. Actually, (P p ) hold for all p ≥ 1 and all geodesic balls (including the sphere itself):

B |f (θ) -m B f | p dσ(θ) ≤ c(p, n) diam(B) p B |∇ θ f (θ)| p dσ(θ).
We 4 Proof of Theorem 3.5

For the proof of Theorem 3.5, we need a Calderón-Zygmund decomposition as in [START_REF] Badr | Real interpolation of Sobolev Spaces[END_REF]. We incorporate here a further control to take care of the vertex point. Let 1 < p < ∞ and f ∈ H 1 p (X). Identifying f to its restriction to Ω, write

f = f | Ω + + f | Ω -= f + + f -.
We establish the following Calderón-Zygmund decomposition for f + and the same decomposition holds for f -. Proposition 4.1 (Calderón-Zygmund lemma). Let α > 0. Then one can find a collection of balls (B i+ ) i of Ω + , functions b i+ and a Lipschitz function g + such that the following properties hold:

f + = g + + i b i+ on Ω + (4.1) |g + (x)| + |g + (x)| |x| + |∇g + (x)| ≤ Cα λ -a.e x ∈ Ω + (4.2) supp b i+ ⊂ B i+ , B i+ |b i+ | + |b i+ | |x| + |∇b i+ | dx ≤ Cα (4.3) i λ(B i + ) ≤ Cα -p Ω + |f + | + |f + | |x| + |∇f + | p dx (4.4) i χ B i+ ≤ N. (4.5)
The constants C and N only depend on p and on the constants in (D) and (P 1 ) in Ω + .

A ball of Ω + is the restriction to Ω + of an open ball of R n having center in Ω + .

Proof. To simplify the exposition, we omit the index + keeping it only for Ω + . For x ∈ R n , denote r(x) = |x|. Consider

U = x ∈ Ω + : M Ω + (|f | + |f | r + |∇f |)(x) > α with M Ω + f (x) = sup B: x∈B 1 λ(B) B |f |dx
where B ranges over all balls of Ω + . Recall that M Ω + is of weak type (1, 1) and bounded on

L p (Ω + , λ), 1 < p ≤ ∞. If U = ∅, then set g = f , b i = 0 for all i so that (4.
2) is satisfied according to the Lebesgue differentiation theorem. Otherwise the maximal theorem gives us

λ(U ) ≤ Cα -p Ω + |f | + |f | r + |∇f | p dx < ∞ (4.6)
In particular U = Ω + as λ(Ω + ) = ∞. Let F be the complement of U in Ω + . Since U is an open set distinct of Ω + , we use a Whitney decomposition of U ( [START_REF] Coifman | Analyse harmonique sur certains espaces homogènes[END_REF]): one can find pairwise disjoint balls B i of Ω + and two constants

C 2 > C 1 > 1, such that 1. U = ∪ i B i with B i = C 1 B i and the balls B i have the bounded overlap property; 2. r i = r(B i ) = 1 2 d(x i , F ) and x i is the center of B i ; 3. each ball B i = C 2 B i intersects F (C 2 = 4C 1 works).
Recall that the above balls are balls of Ω + , that is Since λ(B i ) ≤ Cλ(B i ) (the doubling property for Ω + ) and B i ∩ F = ∅ for all i, we have

B i = B(x i , r i /C 1 ) ∩ Ω + , B i = B(x i , r i ) ∩ Ω + , B i = B(x i , r i C 2 ) ∩ Ω + and x i ∈ Ω + where B(x,
B i (|f | + |f | r + |∇f |)dx ≤ B i (|f | + |f | r + |∇f |)dx ≤ Cαλ(B i ). (4.7)
Let us derive some useful properties. For x ∈ U , denote I x = {i : x ∈ B i }. By the bounded overlap property of the balls B i , we have that ♯I x ≤ N . Fixing j ∈ I x and using the properties of the B i 's, we easily see that 1 3 r i ≤ r j ≤ 3r i for all i ∈ I x . In particular, B i ⊂ 7B j for all i ∈ I x . We can deduce from that |f B jf B i | ≤ Cr j α with C independent of i, j ∈ I x and x ∈ U . Indeed, we use that B i and B j are contained in 7B j , Poincaré inequality (P 1 ) on balls of Ω + (here 7B j ), the comparability of r i and r j , and the control of the gradient term in (4.7).

Let us now define the functions b i and prove (4.3). Let (χ i ) i be a partition of unity of U subordinated to the covering (B i ), such that for all i, χ i is a Lipschitz

function supported in B i with |∇χ i | ∞ ≤ C r i . To this end it is enough to choose for x ∈ Ω + , χ i (x) = ψ C 1 d(x i , x) r i k ψ( C 1 d(x k , x) r k ) -1
, where ψ is

a smooth function, ψ = 1 on [0, 1], ψ = 0 on [ 1+C 1 2
, +∞[ and 0 ≤ ψ ≤ 1. We declare B i of type 1 if 4r i ≤ d(B i , 0) and of type 2 otherwise. Here d(B i , 0) is the distance from B i to 0. Indeed, it could well be that one B i even touches the origin. We set

b i = (f -f B i )χ i if B i is of type 1 and b i = f χ i if f is of type 2. It is clear that supp b i ⊂ B i .
We first begin the proof of the estimates on b i by assuming B i of type 1. We remark that for all x ∈ B i we have r i ≤ |x|/4 = r/4 from the type 1. We have

B i |b i |dx = B i |(f -f B i )χ i |dx ≤ 2 B i |f |dx ≤ Cαλ(B i ).
We applied (4.7) in the last step. For |b i | r , we use 1/r ≤ 1/4r i on B i and the Poincaré inequality (P 1 ) on B i :

B i |b i | r dx ≤ B i |f -f B i | 4r i dx ≤ C B i |∇f |dx ≤ Cαλ(B i ), using (4.7) again. For ∇b i , since ∇ (f -f B i )χ i = χ i ∇f + (f -f B i )∇χ i , the
Poincaré inequality (P 1 ) on B i and (4.7) yield

B i |∇b i |dx ≤ C B i |χ i ∇f |dx + B i |f -f B i | |∇χ i |dx ≤ C χ i ∞ αλ(B i ) + C ∇χ i ∞ r i B i |∇f |dx ≤ Cαλ(B i ). Therefore (4.3) is proved if B i is a type 1 ball. If B i is a type 2 ball, the control of B i |b i |dx and B i |b i |
r dx is direct from definition and (4.7). For B i |∇b i |dx, the only term requiring an argument is B i |f ∇χ i |dx. We remark that the type 2 implies |x| ≤ 2r i + d(B i , 0) ≤ 6r i for x ∈ B i . Hence, we have |∇χ i (x)| ≤ C/r i ≤ 6C/r on B i and we can use (4.7) again. Therefore (4.3) is proved if B i is a type 2 ball. Remark that we proved

B i |b i |dx ≤ Cαr i λ(B i ) (4.8)
for all i and also |f B i | ≤ Cαr i λ(B i ) for type 2 balls.

Set g = fi b i so that (4.1) is granted and it remains to establish (4.2). We begin by some observations. Since the sum is locally finite on U , g is defined almost everywhere on Ω and g = f on F . Observe that g is a locally integrable function on Ω. Indeed, let ϕ ∈ L ∞ with compact support. Since d(x, F ) ≥ r i for x ∈ supp b i and λ(B i ) ≤ Cλ(U ) by using doubling and the disjointness of the balls B i , we obtain, using (4.8),

i |b i | |ϕ| dx ≤ i |b i | r i dx sup x∈Ω + d(x, F )|ϕ(x)| ≤ Cαλ(U ) sup x∈Ω + d(x, F )|ϕ(x)| . Since f ∈ L 1 loc (Ω + ), we deduce that g ∈ L 1 loc (Ω + ) 2 .
We also note that by Lebesgue differentiation theorem, we have

|f | + |f | r + |∇f | ≤ α, λ -a.e. on F. (4.9)
We turn to proving the estimate g ∞ ≤ Cα. Using that i χ i = 1 on U and 0 on F , we have

g = f 1 1 F + B i type 1 f B i χ i .
By (4.9), it remains to estimate the series. By (4.7), |f B i | ≤ Cα and we conclude using i χ i ≤ 1.

We continue with the estimate g/r ∞ ≤ Cα. We use the same decomposition for g as above. On F , |g|/r ≤ α. Let x, y ∈ B i with B i a type 1 ball. We have r i ≤ |y|/4 by type 1 definition. We also have |x -y| ≤ 2r i . Hence we deduce that |y| ≤ 2r i + |x| ≤ |y|/2 + |x|, so that |y| ≤ 2|x| and

|f B i | |x| ≤ 2 B i |f (y)| |y| dy ≤ Cα.
Then let x ∈ U . Considering only the balls B i containing x, we have:

|g(x)| |x| ≤ i∈Ix:B i type 1 |f B i | |x| χ i (x) ≤ Cα i∈Ix χ i (x) = Cα.
It remains to prove ∇g ∞ ≤ Cα. For that, we use the original representation of g, differentiate in the sense of distributions and calculate

∇g = ∇f - i ∇b i = ∇f - i χ i ∇f -h = 1 1 F (∇f ) -h with h = B i type 1 (f -f B i )∇χ i + f B i type 2 ∇χ i .
By (4.9), |1 1 F (∇f )| ≤ α λ-a.e.. We claim that a similar estimate holds for h, i.e. |h(x)| ≤ Cα for all x ∈ Ω + . For this, note first that h vanishes on F .

Then fix x ∈ U . Observe that i ∇χ i (x) = 0, and by the definition of I x , the sum reduces to i ∈ I x . Pick j ∈ I x with B j of type 2 if there is one such ball, otherwise any j ∈ I x will do. We have

h(x) = i∈Ix:B i type 1 (f B j -f B i )∇χ i (x) + f B j i∈Ix:B i type 2 ∇χ i (x)
because the difference with the previous equation is

(f (x) -f B j ) i∈Ix ∇χ i (x) = 0.
We have seen that |f B jf B i | ≤ Cr j α ≤ 3Cr i α with C independent of i, j ∈ I x and x ∈ U . Since |∇χ i (x)| ≤ C/r i and I x has cardinal bounded by N , we are done for the the first term in the right hand side. For the second term, either no i ∈ I x are such that B i is of type 2, in which case this term is 0.

In the opposite case, we know that |f B j | ≤ Cαr j since B j is of type 2 and we conclude using |∇χ i (x)| ≤ C/r i ≤ 3C/r j for i ∈ I x and that I x has cardinal bounded by N . From these estimates we deduce that |∇g(x)| ≤ Cα λ-a.e..

We are now able to characterize the K-functional of interpolation between H 1 1 (X) and H 1 ∞ (X). Theorem 4.2. We have that

K(f, t, H 1 1 , H 1 ∞ ) ∼ t f * * (t) + |f | r * * (t) + |∇f | * * (t)
for every f ∈ H 1 1 (X) + H 1 ∞ (X) and t > 0. The implicit constants are independent of f and t.

Proof. The lower bound follows from the fact that K(g, t, L 1 , L ∞ ) ∼ tg * * (t) for g ∈ L 1 + L ∞ . Now for the upper bound, consider first the case when f ∈ H 1 p (X). Identifying f to its restriction to Ω, write f = f + + f -and take the above Calderón-Zygmund decomposition for each f + and f -for α > 0 to be chosen. We obtain open subsets U ± and functions g ± , b ± . We assume that U ± are nonempty; the easy modifications otherwise are left to the reader.

Here is the point of working with the H spaces instead of the H spaces. As g + (0 + ) = g -(0 -) = 0, if we define g = g + on Ω + and g -on Ω -, then g can be extended to a Lipschitz function on X = Ω with g r ∞ ≤ Cα. Hence g ∈ H 1 ∞ (X) with norm controlled by Cα.

Therefore we can write

f ∈ H 1 p (X) as f = g + b with b ∈ W 1 1 (Ω) = H 1 1 (X) and g ∈ H 1 ∞ (X). We have g H 1 ∞ (X) ≤ Cα and b H 1 1 (X) ≤ Cα(λ(U + )+λ(U -)). Let α ± (t) = M Ω ± (|f ± | + |f ± | r + |∇f ± |) * (t), α = max(α + (t), α -(t)).
Remark that

α + (t) |f + | * * + |f + | r * * + |∇f + | * * (t) |f | * * + |f | r * * + |∇f | * * (t)
where the implicit constant depends only on the doubling constant of Ω + . We used the fact that {x ∈ Ω

+ ; |f + (x)| > λ} ⊂ {x ∈ Ω; |f (x)| > λ}, hence f * + (t) ≤ f * (t). Similarly, ( f + r ) * (t) ≤ ( f r ) * (t) and |∇f + | * (t) ≤ |∇f | * (t). As U + is contained in x ∈ Ω + ; M Ω + (|f + | + |f + | r + |∇f + |)(x) > α + (t)
we have λ(U + ) ≤ t. Similarly we get λ(U -) ≤ t. This yields

K(f, t, H 1 1 , H 1 ∞ ) ≤ b H 1 1 + t g H 1 ∞ ≤ Ct f * * (t) + |f | r * * (t) + |∇f | * * (t) .
For the general case when f ∈ H 1 1 (X)+ H 1 ∞ (X), we apply a similar argument to that of [START_REF] Devore | Interpolation of linear operators on Sobolev spaces[END_REF] to obtain the upper bound. We omit details.

Proof of Theorem 3.5. Set H 1 p,1 (X) = ( H 1 1 (X), H 1 ∞ (X)) 1-1/p,p
. By the reiteration theorem, it suffices to establish H 1 p,1 (X) = H 1 p (X) with equivalent norms. First, from the Calderón-Zygmund decomposition, we have

H 1 p (X) ⊂ H 1 1 (X)+ H 1
∞ (X) for 1 < p < ∞ where the inclusion is continuous.

From the previous results we have that for

f ∈ H 1 1 (X) + H 1 ∞ (X) f 1-1/p,p ∼ ∞ 0 |f | * * (t) + |f | r * * + |∇f | * * (t) p dt 1/p ∼ f * * p + |f | r * * p + + |∇f | * * p ∼ f p + f r p + |∇f | p ∼ f H1 p ,
where we used that for l > 1, f * * l ∼ f l .

5 Restriction/Extension from/to R n

We study the restriction operator onto Ω and construct an extension that is p independent. The subject of restriction and extension has been very studied for Sobolev spaces on domains (that is, connected open sets). For some definite answers see [START_REF] Hajlasz | Sobolev embeddings, extensions and measure density condition[END_REF] and the references therein. But recall that the double cone is not a domain. For references closer to what we are doing here, see [START_REF] Rychkov | Linear extension operators for restrictions of function spaces to irregular open sets[END_REF] which considers the Bessel-Sobolev spaces in subsets of R n and [START_REF] Shvartsman | On extensions of Sobolev functions defined on regular subsets of metric measure spaces[END_REF] which treats the Hajlasz-Sobolev spaces in spaces of homogeneous type. Let us first state our result.

Theorem 5.1. Let 1 ≤ p ≤ ∞.

• The restriction operator is bounded from

W 1 p (R n ) into H 1 p (X)
. Further, it is onto for p = n and for p = n, its range is H 1 n (X).

• There exists a linear extension operator E that is bounded from

H 1 p (X) to W 1 p (R n ) if p = n and from H 1 n (X) to W 1 n (R n ).
The interesting part of this result is p = n. Observe also that this shows that H 1 n (X) does not have the extension property. In accordance with the cited references, the second item is closely related to ontoness in the first. It could well be a direct consequence but we prefer producing an explicit extension operator.

In [START_REF] Shvartsman | On extensions of Sobolev functions defined on regular subsets of metric measure spaces[END_REF], the author studies restrictions onto regular sets in a space of homogeneous type. Note that the double cone is a regular set in R n . For such sets, he proves the following interesting theorem (Theorem 1.3). For all 1 < p ≤ ∞, the restriction of M 1,p (R n ) to Ω equals M 1,p (X) and there exists a linear continuous extension from the latter space to the first one. Here, M 1,p is the Hajlasz-Sobolev space. Given the fact that W 1 p (R n ) = M 1,p (R n ) with equivalent norms for 1 < p ≤ ∞ ( [START_REF] Hajlasz | Sobolev spaces on a arbitrary metric space[END_REF]), this implies that M 1,p (X) interpolate for 1 < p ≤ ∞. But combining this with our result gives the following corollary.

Corollary 5.2. For 1 < p ≤ ∞, M 1,p (X) = H 1 p (X) if p = n and M 1,n (X) = H 1 n (X).
The result for p > n is known (see [START_REF] Hajlasz | Sobolev met Poincaré[END_REF]): it follows from the fact that X is complete and satisfies Poincaré inequality for any p > n. Not much more can be said in general without these two conditions so this corollary is seemingly new for p ≤ n. The interesting case is the identification for p = n: M 1,n (X) is a strict subspace of H 1 n (X). This study of restriction/extension properties uses the decomposition into radial and anti-radial parts defined earlier. It also possible to reprove the interpolation property of the H 1 p (X) spaces by this method. We first study the restriction operator, then construct the extension. We next prove the ontoness and conclude with the application to interpolation.

Restriction

The restriction operator R is defined by R

(f ) = f | Ω . Let 1 ≤ p < ∞. It is obvious that if f ∈ W 1 p (R n ) then R(f ) ∈ W 1 p (Ω), and that R : W 1 p (R n ) → W 1 p (Ω) is bounded. As C ∞ 0 (R n ) is dense in W 1 p (R n ), the range is contained in W 1 p (Ω) = H 1 p (X). For p = n, we show that R maps into H 1 n (X). Let f ∈ W 1 n (R n
) and let g = R(f ). Since we already know that g ∈ H 1 n (X), it remains to show that g a /r ∈ L n (X). Write f = f ρ + f α where f ρ (x) is here the average of f on the whole sphere of radius |x|. Identifying f ρ with its restriction to Ω, we see that (f ρ ) r = f ρ and (f ρ ) a = 0. Thus, if we write g = g r + g a , we conclude that

g a = (f α | Ω ) a , i.e g a (x) = f α (x) -Ω∩S |x| f α dσ(θ) for x ∈ Ω. Thus Ω∩S |x| |g a | n dσ(θ) ≤ 2 n Ω∩S |x| |f α | n dσ(θ) ≤ 2 n S |x| |f α | n dσ(θ) ≤ C|x| n S |x| |∇ θ f α | n dσ(θ)
where the last inequality is Poincaré inequality (P n ) on the sphere S |x| since f α has mean value 0 on it, and ∇ θ is the tangential gradient (see the last remark in Section 3). Since r(x) = |x| it follows that

Ω g a r n dx = +∞ 0 Ω∩Sr g a r n dσ(θ)dr ≤ C +∞ 0 Sr |∇ θ f α | n dσ(θ)dr = C +∞ 0 Sr |∇f α | n dσ(θ)dr ≤ C R n |∇f | n dx < ∞.
We used that ∇ θ f α = ∇ θ f since f ρ does not depend on θ and then

|∇f α | ≤ |∇f |. Therefore R(f ) = g ∈ H 1 n (X). For p = ∞, it is obvious that R is bounded from W 1 ∞ (R n ) into H 1 ∞ (
X) (and it is onto by Whitney's extension theorem).

Extension

Let 1 ≤ p ≤ ∞. Let f ∈ H 1 p (X) if p = n (resp. f ∈ H 1 n (X)). Write f = f r + f a as in Section 3. Lemma 3.8 yields f r ∈ W 1 p (R n ) and f a ∈ H 1 p (X). It remains to extend f a . We write f a = f a | Ω + + f a | Ω -= f a+ + f a-.
We treat f a+ , the same analysis applying to f a-. Our strategy is to enlarge Ω + to a slightly bigger half-cone Ω + , then we map them with a smooth bilipschitz map onto the upper half-space R n + and a slightly bigger version R n + . We use the reflection principle for Sobolev spaces to extend from the upper-half space to the full space, localise on R n + with a homogeneous cut-off and finish by mapping back onto Ω + . There are many ways to do that. Here are the details.

Let w < π/2 be the half-angle of Ω + with respect to the vertical axis and let Ω + be an open half-cone with half-angle ω + ε < π/2 for some small ε with same vertex point and rotation axis. Using the spherical angle θ ∈ [0, π) defined by θ = arccos xn |x| by writing x = (x ′ , x n ), x ′ ∈ R n-1 , x n ∈ R, the map ψ + (x) = y with y = sin(2ωθ/π) sin θ

x ′ , cos(2ωθ/π) cos θ

x n is a smooth bilipshitz (this is somewhat tedious to check carefully but it reduces to a planar estimate) map, leaving the norm invariant (|ψ + (x)| = |x|), from R n + onto Ω + and from R n + onto Ω + where R n + is a half-cone with half-angle π(w+ε) 2ω > π 2 . We consider now the even extension ζ + :

W 1 p (R n + ) → W 1 p (R n ). Let m + ∈ C ∞ (R n \ {0}) ∩ L ∞ (R n ),
homogeneous of degree 0, such that m + = 1 on R n + and supp m + ⊂ R n + ∪ {0}. With these ingredients we define the extension ξ + (f a+ ) of f a+ as

ξ + (f a+ ) = m + ζ + (f a+ • ψ + )] • ψ -1 + .
It readily follows from the properties of m + that m + g r for all g ∈ W 1 p (R n ). Using this fact, that bilipschitz maps preserve Sobolev spaces and density of Lipschitz functions, we obtain that ξ + (f a+ ) ∈ W 1 p (R n ) with supp ξ + (f a+ ) ⊂ Ω + ∪ {0} and ξ + (f a+ ) W 1 p (R n ) ≤ C f a+ H 1 p (X) . We conclude that ξ(f a ) = ξ + (f a+ )+ξ -(f a-) ∈ W 1 p (R n ) is an extension of f a to W 1 p (R n ). Therefore, E defined by

E(f ) = f r + ξ(f a )
is an extension of f to W 1 p (R n ). We have shown that the map E is

H 1 p (X) → W 1 p (R n )-bounded if p = n and H 1 n (X) → W 1 n (R n )-bounded if p = n.

Ontoness and relation to interpolation for H 1 p (X)

From the previous subsections, we deduce that R • E operates boundedly on H 1 p (X) for p = n and on H 1 n (X) as the identity map. In particular, R acting on W 1 p (R n ) is onto H 1 p (X) for p = n and onto H 1 n (X) for p = n. Using the preservation of interpolation properties for retract diagrams, it follows that

(H 1 1 (X), H 1 ∞ (X)) 1-1/p,p = R(W 1 1 (R n ), W 1 ∞ (R n )) 1-1/p,p ) = R(W 1 p (R n )).
Therefore (H 1 1 (X), H 1 ∞ (X)) 1-1/p,p = H 1 p (X) for p = n and H 1 n (X) for p = n.

Homogeneous versions

Homogeneous Sobolev spaces are defined up to a constant, removing control on the L p norms on f . Since the vertex point plays a specific role, it is best here to fix the floatting constant by imposing control at this vertex point. We adopt the following definitions. Let Lip 0 (X) be the space of Lipschitz functions in X vanishing at 0. For 1 ≤ p ≤ ∞, we set E p = {f ∈ Lip 0 (X) ; |∇f | L p (X) < ∞}, E p = {f ∈ Lip 0 (X) ; |∇f | L p (X) + f /r L p (X) < ∞}.

Then E p and E p are normed spaces and we call H 1 p (X) and H 1 p (X) their completions. Clearly

E ∞ = H 1 ∞ (X) = E ∞ = H 1 ∞ (X) = Lip 0 (X).
Remark. (central role of the vertex) The analysis in this article does not use the fact that the cone Ω is symmetric under x → -x and also does not use the specific opening angle. This means that the upper and lower (open) halfcones can be replaced by two half-cones located independently of one another provided they share the same vertex and that they are strictly separated by a hyperplane passing through the vertex and not containing any direction of the boundaries. Also the (finite) number of disjoint half-cones is not limited to 2 provided each pair satisfies the above requirements.

Remark. (other types of cones) The half-cones can be replaced by R * + ×N where N is a Lipschitz domain on the unit sphere. On such domains, one has Poincaré inequalities with any exponents (adapt the proof sketched in the last remark of Section 3) and this allows to adapt the arguments.

Remark. (local geometry) Of course, the analysis done with inhomogeneous norms is stable by (smooth) truncation of the cone away from the vertex point. For example, if one wants to work on a truncated cone by requiring r < 1, then one can use local variants as in Badr's thesis [2]. Details are left to the reader.

  r) denotes an Euclidean open ball in R n . Condition (4.5) is nothing but the bounded overlap property of the B i 's and (4.4) follows from (4.5) and (4.6).

We consider real functions but everything is valid for complex functions.

Note that since b ∈ L 1 in our case, we can say directly that g ∈ L 1 loc . However, this way of doing applies to the homogeneous case presented in Section 6.

It is easy to show that H 1 p (X) is composed of locally p-integrable functions. For p > n, one has (2.1) from the Morrey embedding and f (0) = 0.

It is clear that H 1 p (X) ⊂ H 1 p (X) but for 1 ≤ p ≤ n the inclusion is strict 3 . Indeed a Lipschitz function supported away from 0 which agrees with e i|x| -α for |x| ≥ 1 satisfies f /r p = ∞ and belongs to E p if α > 0 is large enough. For p > n, the inclusion is an equality as we shall see.

Remark. From there, one can see that the restrictions to Ω of functions in C ∞ 0 (R n ) that vanish at 0 form a dense subspace of H 1 p (X).

Corollary 6.2.

Proof. Assume first that 1 ≤ p < n. Then by the previous lemma, one can assume that f ∈ Lip 0 (X) for which the argument of (3.4) applies.

p dx. Then argue as in the proof of (3.4). In the integration by parts, one picks an extra term which has a negative sign because np < 0. Thus one can cancel it and obtain A ≤ C |∇f | p L p (X + ) with C independent of ǫ, R. Taking limits and doing the same thing on Ω -shows that f ∈ E p and we are done.

The first item also show that the closure in

Theorem 6.3. The family ( H 1 p (X)) 1≤p≤∞ is an interpolation family for the real method. Hence the same is true for (H 1 p (X)) n<p≤∞ .

The proof for the spaces H 1 p (X) is a minor adapatation of the one of Theorem 3.5 and is left to the reader. The second point follows from the above corollary.

Interpolation for the spaces H 1 p (X) for p ≤ n is unclear.

Some remarks and generalizations

Remark. (some explicit extensions) There are many extension operators. The following example was communicated to us by Michel Pierre. For the (double) cone of R 2 consisting of the 2 quadrants defined by xy > 0, then one can take Ef (x, y) =    f (x, y), if xy > 0, x 2 f (x, -y) + y 2 f (-x, y)

x 2 + y 2 , if xy < 0.

3 In contrast with the inhomogeneous case for p < n.