Jérôme Denis
email: denis@telecom-paristech.fr

David Pontille
email: pontille@ehess.fr

Materiality, Maintenance and Fragility The Care of Things

Keywords:

This communication attempts to go beyond a definition of materiality, largely accepted in organization studies, that only focusses on consistency and stability. Drawing on the results of an ethnography of subway signs maintenance it highlights the importance to consider material fragility and vulnerability, particularly in the case of ordering devices. Following maintenance workers helps to understand that the stability of artifacts is not an ontological property but the ongoing result of an exploratory and improvised work that consists in constantly taking care of things and being aware of their material multiplicity.

ʻWhen transported to other sites and situations, the logic of care articulated here will have to be translated'. (Mol 2006: 91) Materiality and its role in organization processes are an important but somehow hazy issue. As [START_REF] Orlikowski | Technology and institutions: What information systems research and organization studies can learn from each other[END_REF] notably point out, the main problem about the majority of theories that highlight material properties of technical systems is twofold: they draw on an abstract definition of both materiality and technology, without dealing with concrete artifacts; and they tend to reduce materialism to determinism. Furthermore, these perspectives (contengency theory or media richness theory for instance) tend to draw clear distinctions between ʻmaterial' and ʻsocial' properties, which are not satisfying [START_REF] Leonardi | Materiality and Change: Challenges to Building Better Theory about Technology and Organizing[END_REF]. These reproaches made in organization studies are more or less similar to the those addressed by Ingold to material culture studies, where the use of the notion of materiality is generally too encompassing, and where ʻthe polarity of mind and matters remains' (Ingold 2007, p. 3).

Beyond accusations of determinism, materiality raise another issue in organization studies. It seems, indeed, that ever in the more cautious studies tackling the question, materiality is almost exclusively discussed in terms of consistency, or at least tangibility [START_REF] Leonardi | Digital materiality? How artifacts without matter, matter[END_REF]. Paradigmatic of such a posture is the use of such notions as ʻobjectification' and ʻreifification'. Material properties of artifacts are most of the time described as components that bring some resistance: whether they give to artifacts the ability to constraint users' action or, on the contrary, wether they provide practical resources. Such a ʻmateriality-as-stability' approach is particularly present in management studies, where post-giddensian researches emphasize materiality as a key feature of structuring processes. Artifacts and technologies are studied here mainly as inanimate objects within which social relations or political categories are gradually frozen. One can find more or less the same vocabulary in what J. Law termed a ʻmanagerialist' appropriation of Actor-Network Theory (Law 1999: 5) which studies artifacts agentivity with a narrow and non-relational definition of such notions as ʻblack boxes' or ʻimmutable mobiles'. In this perspective, successful and viable objects are essentially seen as the material result of the stabilization of a sociotechnical network.

The problem with such approaches lies in the fact that materiality is somehow taken for granted, at least by actors, and often by researchers themselves. Artifacts and technologies are described as distinct and solid objects, and their frontiers or robustness are rarely questioned [START_REF] Pels | The Status of the Object. Performances, Mediations, and Techniques[END_REF]. This is not only the case in ʻperspectivism' and ʻrepresentationalism', for which materiality goes hand in hand with univocity, and plurality comes only from actors' interpretations [START_REF] Mol | Ontological politics. A word and some questions[END_REF][START_REF] Barad | Posthumanist Performativity: Toward an Understanding of How Matter Comes to Matter[END_REF], but also in the ʻnaive formalism' (Star 1995; Berg 1997) adopted by the numerous researches that treat material properties of information technologies as intrinsic structuring components.

Recently, W. [START_REF] Orlikowski | Sociomaterial Practices: Exploring Technology at Work[END_REF] made a plea for a new consideration of materiality in organization. Even if it was mainly turned against the lack of discussion about materiality in organization studies, her position can also be seen as a response to the approaches discussed above. Orlikowski and her colleagues indeed refuse to specify material aspects of technologies and to isolate their action. Instead, they defend a situated and practice-based point of view [START_REF] Suchman | Plans and Situated Actions: The Problem of Human-Machine Communication[END_REF][START_REF] Gherardi | From Organizational Learning to Practice-Based Knowing[END_REF], from which what counts is 'technology-in-use' rather than technology itself [START_REF] Orlikowski | Using Technology and Constituting Structures: A Practice Lens for Studying Technology in Organizations[END_REF]. Here then, matter matters in terms of process and hybridity and the ʻentwining of the material and the social' [START_REF] Leonardi | Materiality and Change: Challenges to Building Better Theory about Technology and Organizing[END_REF] is the main issue that have to be explored.

In a communicational perspective, a similar move has been made by researchers who have emphasized the role of texts within organizations, not simply as powerful material vehicles for information, but as full agent. This is notably what F. Cooren and his colleagues have done, urging organization (and communication) scholars to take into account the plurality of such an agentivity. Cooren notably shows that ambiguity and undecidability are the roots of texts' agency within organizations [START_REF] Cooren | Action and Agency in Dialogue: Passion, Incarnation and Ventriloquism[END_REF]: it is precisely because the meaning and the force of documents and technologies are unpredictable that they do really count. Such a consideration focuses on the place of texts and their materiality within the constant dialogues that punctuate organizing processes.

These streams of research defend an ecological view of materiality within organizations. They certainly point out the importance of objects and artifacts in the dynamics of ordering and structuring, but they refuse to impute stability to material properties only: what is at stake here is the ʻinherent sociomaterial nature of organizational practices' [START_REF] Orlikowski | Sociomaterial Practices: Exploring Technology at Work[END_REF].

This displacement is essentially due to an interest in agency as an essential theoretical issue (more than social structures or organizations). The numerous studies conducted within this framework highlight the active role of hybrid assemblages within which humans and nonhumans are closely connected [START_REF] Barad | Posthumanist Performativity: Toward an Understanding of How Matter Comes to Matter[END_REF][START_REF] Latour | Reassembling the Social: An Introduction to Actor-Network-Theory[END_REF]). To question agency in these terms prompts scholars to ignore the vocabulary of materiality-as-stability (or tangibility), and to develop instead a more dynamic and processual approach. As Leonardi recently wrote, some researchers even voluntarily use ʻmateriality' and ʻmaterial' in a very broad sense, which simply refers to performativity as a sociomaterial process [START_REF] Leonardi | Digital materiality? How artifacts without matter, matter[END_REF].

In this communication, we propose to complement such a program in pointing another way to question the place of materiality in organizational processes and to challenge the stability and tangibility version. For that, we will step aside from the traditional organization studies: instead of investigating the way people work, and act, with artifacts and objects, we will take a close look at the way some people directly work on things. This is especially the case of repair and maintenance practices, which have been largely overlooked in organization studies. Because they deal with the very vulnerability of things, focusing on these practices gives the opportunity to explore a very different side of the material features of artifacts. A side where materiality can definitively not been considered as a synonym of consistency.

Drawing on the results of an ethnography of Parisian subway signs repair and maintenance (Denis and Pontille 2010a), we address three main issues. First, we show that subway signs can be analyzed, in the materiality-as-stability framework, as ordering devices (1). Second, through the exploration of maintenance workers practices, we focus on heterogeneity (2) and fragility (3) as important material features of signboards, however normalized they are. Third, we go back to the question of sociomaterial agency in order to highlight the particular contribution of repair and maintenance studies. Because maintenance practices imply a specific engagement with matter, which emphasizes vulnerability, we draw a parallel between maintenance and what A. [START_REF] Mol | The Logic of Care[END_REF] terms the ʻlogic of care' in medical practices, assuming that ʻthe care of things' could be a promising domain of research.

Ordering

Wayfinding systems are essential devices amongst the numerous immobile infrastructures that support and organize mobility (Amin and Thrift 2002, Urry 2007[START_REF] Knox | Enacting Airports: Space, Movement and Modes of Ordering[END_REF]. They provide a graphical layer dedicated to the accountability of places and the everyday organization of public settings. Street nameplates, traffic lights or signboards inform passersby about their own location in the city (ʻthis' neighborhood, ʻthis' street) and give indications about the suitable behavior they have to adopt (ʻstop here', ʻturn left'). By marking sites, giving places a name, designating directions, these signs are what [START_REF] Garfinkel | Ethnomethodology's program[END_REF] terms ʻterritorial organizational things'.

In the case of Paris, subway signs became a central issue for the Régie Autonome des Transports Parisiens (RATP) at the beginning of the 1990s. At this time, a team of specialists (designers, cartographers and architects) was brought together with the aim of reconsidering the whole organization of subway signs. Surveys, field studies and experiments were conducted. This resulted in the creation of a complete wayfinding system and the writing of a particularly ambitious policy. This policy was organized around two main principles. The first one is multimodality. Before the 1990's, each transportation mode had its own particular wayfinding system with its own typography and colors. The new wayfinding system has been conceived as a means to transcend these differences. The same set of signs has been designed and implemented for each transportation mode provided by the carrier: subway, regional trains, bus, tramway, and so forth. The main goal was to offer a seamless service to riders: from then on, they could circulate through the whole city by following the same kind of signs from one transportation mode to another.

The second principle is omnipresence. Numerous new boards have been created and signage occupies now every possible positions in the transportation network. This is a very specificity of the Parisian subway system compared to other wayfinding systems in the world. In Paris, each junction, corridor, hall, stairs, and plate-form has been fully equipped with a board. Such a policy contributes to the production of an hybrid environment which is no more an architectural space only and where ʻthe distinction between the building and its signs, between the text and the territory, becomes indistinct' (Fuller 2002: 236).

These two principles, multimodality and omnipresence, were supported by the writing of an ambitious standardization policy, which was a big first in the history of Parisian subway signs. The previous wayfinding systems used to stand in few documents and used to be transmitted or recalled on the basis of an oral tradition: since someone needed to change a board or to get a new one, someone else would explain it to her with more or less precisions. Conversely, new signboards have come out with extremely precise guidelines, detailed in voluminous files. Every dimensions of signboards are now strictly standardized: content, shape, size, colors and emplacement. Standardization then turned the whole wayfinding system into a powerful means of homogenization and stabilization of subway spaces.

Regardless the location, signage is endless repeated and declined, always perfectly the same, according to the established rules. Repetition creates unity, promotes the recognition, and enhance the comfort of the riders. (Internal document, RATP 1997: 5) Such a normalization policy has had consequences beyond the signboards themselves. For instance, the uses of names themselves have been homogenized in a document entitled ʻofficial denomination' that specifies the rules to follow in order to identically designate places through the variety of settings and media. Subway signs standardization has thus lead to the public display of a stabilized version of names that remained more or less changing until then. Sometimes an unofficially denomination has even been created.

The name used to indicate a place has to be as short as possible, and close to everyday language. Therefore, official designation will not be systematically taken up. (For instance: ʻCentre Georges Pompidou' instead of ʻCentre National d'Art et de Culture Georges Pompidou'). (Guide Information Voyageurs, RATP 2007: 22) Focusing on the standardization process clearly prompts to analyze subway signs in terms of material consistency and stability. Actually, emplacement, shape, size and content normalization fits perfectly with the vocabulary of immutability as emphasized by the first version of Actor-Network Theory [START_REF] Law | After ANT: complexity, naming and topology[END_REF]. Because of their immutability and their recurrence, subway signs perform a graphical environment that not only smooths out the architectural differences but also shapes several easily identifiable zones in each stations: halls, corridors, platforms... From this standpoint, subway signs are powerful ordering devices: a set of immutable artifacts whose standardization and material consistency mold a stabilized infrastructure dedicated to mobility. Such an assertion is perfectly acceptable and gives an accurate account of the Parisian signage policy. An account that insists on the role of material consistency in ordering processes, and draws a centralized and strategic version of Parisian subway signs, a foucaldian one, some would say.

Assuming such an account amounts to perform a specific reality [START_REF] Law | Performing Technology's Stories: On Social Constructivism, Performance, and Performativity[END_REF], where consistency and materiality are two synonymous dimensions of agency. But what do we miss with it? Which parts of the reality of subway signs slip away from us when we describe such an orderly phenomena? A lot, obviously. But, mainly, what this version of subway signs does not give any chance to is mess [START_REF] Law | After Method: Mess in Social Science Research[END_REF]. And yet, even if it is better known in workplace studies than in organization studies, mess is a very common aspect of organizations. Moreover, mess is a very material phenomena: just think of all you have to handle, to move or to throw away when you decide to tidy up your office. Dirt also is material: you will probably wash your hands after that.

So, how can we appreciate the messy version of subway signs, and more generally, the messy materiality of organizational processes? Following the precursory works of [START_REF] Orr | Talking About Machines: An Ethnography of a Modern Job[END_REF], [START_REF] Henke | The Mechanics of Workplace Order : Toward a Sociology of Repair[END_REF] and [START_REF] Graham | Out of Order: Understanding Repair and Maintenance[END_REF], we think that studying cleaning or repairing practices can be extremely useful for that purpose. Indeed, to understand the mess, we should ask with S.L. Star : ʻWho is doing the dishes? Where does the garbage go?' (Star 1995, p. 3). This is a way to question the ʻmaterial basis for practices' (p. 3), not focusing on the ʻnoble' side of matter. It is also a way to consider the role of people who take care of the garbage and repair stuffs in the ongoing production of social order. In Parisian subway system, these people are numerous. However, few workers are fully dedicated to subway signs: they are gathered in the signage maintenance department. In this communication, we focus on the work of the four maintenance operators who were in charge of signs installation, repair and replacement at the time of our ethnography. We already pointed out the richness of their practices, which implies specific competencies (Denis andPontille 2010a, 2010b). We only investigate here their engagement with signboards material properties, which allows us to highlight two main aspects of signs: their material heterogeneity and their fragility.

Heterogeneity

We saw that, in the designers discourse and in the guidelines, the signs that compose the Parisian wayfinding system are clearly identified as stabilized material objects whose size, shape, and materials have been normalized. But in the day-to-day maintenance practices, things are not that clear: when it comes to repair them or even put them up, signboards do not appear so consistent. Above all, for maintenance workers, subway signs are never completely closed and finite objects. During their work, they change and go through distinct states. In the course of an intervention, a sign can for instance switch in few minutes from an electric board to a pile of screws next to a plastic sheet and a transparent display. What was present in the situation as a simple artifact suddenly turns into a list of various materials.

Nathan and Stephan have to change a board in order to update its information. The amount of handling they have to do for this seemingly simple operation is surprising. Sixteen small screws have to be removed from the metal frame in order to take down the broken board (fig. 1). Once it is done, they put up the new plaque very carefully: a false move could damage it. Then they screw back the frame before hanging it up on the ceiling. A large part of the daily activity of the maintenance operators consists in dismantling or jointing various signage components. In their hands, signboards are not the single objects sought by designers: they appear as heterogeneous assemblages whose form evolves according to the steps of their work. What is important here is the transitivity of the process. Disassembling is of course essential in repair and maintenance practices, but not more than assembling. As repaired and maintained objects, signboards can change from a mode of existence to another in both directions: from a unified entity to scattered heterogeneous parts, and back to a consolidated artifact. These changes never occur at the same time, they are essential steps in what we could term, imitating [START_REF] Pickering | The Mangle of Practice: Time, Agency, and Science[END_REF], the ʻdance of maintenance'.

There is another part of maintenance practices that highlights the material heterogeneity workers deal with. Repairing or installing a signage component does rarely imply a sole signboard, even if the latter keeps its original form. In almost every situations, maintenance workers have to take other things into consideration, and particularly the material layers that interpose between the enameled board and the concrete of the wall. To put up a new board or to repair an old one, they have to deal at least with tiles, plaster, glue, screws and metal brackets. Thus, during their interventions, what appear for the mundane observer as a seamless association of two distinct objects (the board and the wall) is both unfurl and blurred. Maintenance workers go through what one would see as the ʻnatural' boundaries of things, and explore and test the relations of components that can not be assigned to a specific ʻobject'.

These insights show that a very particular version of subway signs occurs during maintenance work. Here, signboards are not handle as stabilized objects and their material properties are not ʻthose tangible resources that provide people with the ability to do old things in new ways and to do things they could not do before' (Leonardi and Barley 2008: 161). Moreover, this is not the ʻmateriality' of signs that counts for maintenance operators (since it literally spreads out in heterogeneous components), but the diverse materials that are no more, or not yet, crystallized in clearly distinguishable objects [START_REF] Ingold | Materials against materiality[END_REF].

Two things are important here. First, the mess itself. It is clear that during maintenance work, signboards, bindings and walls materials, in their very diversity, can not be seen as agents of an ordering process. They do not provide a solid support for the workers action either. On the contrary, they generally are objects of worries: will the glue hold? Should the concrete be covered? Will we be able to remove these screws? How much dry is the plaster? This particular version of signboards does not emphasizes their stability, but the shifting of their boundaries and their fluidity. In the hands of maintenance workers, signboards constitute unsteady multi-faceted objects (Denis and Pontille 2010b), they are agents of a potential mess. And workers deal with such an heterogeneity every day: each intervention inevitably goes with uncertainty about materials. Instead of being stable resources, the material properties of signboards are important issues of the maintenance work itself.

Second, heterogeneity must not be seen as a weakness. This would be the case if we thought about signboards as ʻblack-boxes' whose success would depend on their ability to remain closed and consistent. Conversely, the success of the wayfinding system rests on the capacity for signboards to be opened and closed by the people in charge of their maintenance, and on the ability of the latter not to focus on the primary boundaries of artifacts. In other words, the more the material properties resist to be splitted into bits and pieces and to be extended to others, the less is the system operative. This is precisely what is at stake when maintenance operators talk about signboards ʻmaintenability' (or lack of it): a quality that describes their easiness to be unfolded, and their diverse components to be accessed, disassembled and then reassembled.

Fragility

If signboards can not be seen as mere black-boxes, they are not eternal either. As time goes by, they get worn, as every other mundane object. Their colors fade out, the plastic sheets sometimes come unstuck (fig. 2). Moreover, as some subway lines go under the river (the ʻSeine'), dampness is a strong issue in certain stations. There are leaks in some ceilings, and walls may get wet. As a consequence, plaster may crumble and boards may fall down or get themselves rusty (fig 3 .).

Therefore, steadiness staged in the guidelines does not mean that, once put up on walls and ceilings, signboards remain inert. Shadowing maintenance workers all day long within the Parisian subway system, we actually face one mutation after another. In their eyes and in their hands, signboards are thus mutable objects. Wear and tear is not the only driving force of such a mutability. Signboards are also exposed to subway users and their various practices. While the majority of riders 'use' signboards only glancing at them, some are more active and become violent against them. For instance, some people intentionally hit or spoil them to express their dissatisfaction to the transportation carrier (or just to have fun). Some others are a lot more peaceful, but not less problematic, cherishing signboards so much that they steal them, as collectors or simply to decorate their own apartment. And among these unofficial practices, graffiti writers take of course also an important place. Not only they do graffiti on trains and on walls, but they regularly overlap signboards with tags or stickers.

Therefore, the mutability of signboards also comes from the various forms of violence they are exposed to. As public settings, subway spaces are more or less hostile environment within which the consistency of wayfinding system components, and even their very presence, are endangered.

Considering both wear and the various attacks they are subjected to, what strokes the observer of the maintenance work is the vulnerability of subway signs. Disassembling and reassembling practices never threaten signboards consistency: they are conducted by maintenance workers in the sake of the wayfinding system policy, in order to insure its dayto-day enforcement. Conversely, rust or blows go against it. They constitute unavoidable risks in the mundane life of publicly exposed artifacts.

What can we learn from this other foray in maintenance and its mess? First, we saw that it strongly contrasts with the version of standardized signboards as it appear in guidelines. When repaired and maintained, signboards are a lot more changing and unsteady than in the pages that detail their normalized measurements. But, more generally, it invites to seriously broaden our definition of matter by opening it to the various entities that maintenance workers take more or less directly into account. This is precisely what T. Ingold proposes when he insists, against material culture studies, to leave the very notion of materiality behind, in order to fully explore the diversity of materials, their relationships and their mutations.

It seemed to me that the concept of materiality, whatever it might mean, has become a real obstacle to sensible enquiry into materials, their transformations and affordances. (Ingold 2007: 12) To endorse such a program, we have to examine the lushness of the material ecology in which signboards and their diverse components take part. Moreover, we have to acknowledge -against a simplistic focus on immutability -the vitality of things [START_REF] Ingold | Materials against materiality[END_REF], their liveliness [START_REF] Barad | Posthumanist Performativity: Toward an Understanding of How Matter Comes to Matter[END_REF]. Enamel, ink, paint, plaster compose living organisms in constantly interacting with each others and with others. And during these interactions, these organisms mute in more or less dramatic ways.

What our inquiry into maintenance highlights is the very frangibility that goes with this vitality. Searching for the messy side of an ordering apparatus, we have eventually encountered its fragility. But does this result invalidates our first assertion about the Parisian subway wayfinding system? Does it mean that subway signs have nothing to do with any ordering processes? That their normalized immutability is just an illusion? Not necessarily. We can instead accept to break with a definition of order as a ʻonce and for all' stabilized state. This is what [START_REF] Henke | The Mechanics of Workplace Order : Toward a Sociology of Repair[END_REF], and [START_REF] Graham | Out of Order: Understanding Repair and Maintenance[END_REF] do when they insist on the importance of repair and maintenance. Drawing on Goffman's and Garfinkel's conception of mundane social order, these scholars show that material order is also subject to numerous reparations. It should be seen as the ongoing result of a permanent upkeep work: not a prior order, but a continuous ordering process.

Such considerations about maintenance work do not deny the existence of ordering devices at all, neither do they leave all interest for matter behind. On the contrary, they prompt to seriously investigate the way such a work is accomplished and to analyze its place within organizations. [START_REF] Mol | The Logic of Care[END_REF] has recently identified specific practices in healthcare services, which she describes as the main features of a ʻlogic of care'. We think maintenance work has a lot to do with such a logic, and we suggest to follow her invitation -which appears in the epigram with which we started this paper -to transport, and translate, her analysis from healthcare services to other sites and situations. Indeed, we assume such a movement can help to more precisely understand what maintenance is. Our goal is also to get rid of the vocabulary of materiality-as-stability and to set the basis for a research program that would explore the possibilities of a ʻcare of thing'.

Care

Mol articulates the ʻlogic of care' in contrast to what she terms the ʻlogic of choice', which emphasizes the capacity for patients to make decisions, ʻindividual choices' that empower them against medical expertise and its somehow inhumanity. The first, and essential, dimension of the ʻlogic of care' lies in the way it treats disease and fragility. Contrarily to the logic of choice, which is entirely focused on the autonomy of patients and the return to normality, ʻthe logic of care starts out form the fleshiness and the fragility of life' (Mol 2008: 11). Care practices accept the vulnerability of patients, they exist through it. We saw that such an assumption lies also at the core of maintenance practices. The latter are dedicated to material fragility, which is not considered as a temporary result of an extraordinary event (an accident or a breakdown) but as the mundane mode of existence of things. Maintenance, as care, ʻtakes failures to be an unavoidable part of life' (Mol 2008: 92). Studying maintenance as an occurrence of care thus allows to abandon a normative definition of material life implicitly based on presupposed health and order. For maintenance and care, there is no such thing as a ʻnormal situation'.

There is another element about care pointed by Mol about the work of repair technicians, that can bring a useful light on the maintenance work. Care is grounded on watchfulness and attention: it can not be normalized. As what is wrong can not be completely settled in advance, it demands constant adaptation. This is why maintenance gives so much contrast with designers guidelines. Improvisation is the main fuel of maintenance operators [START_REF] Henke | The Mechanics of Workplace Order : Toward a Sociology of Repair[END_REF] whose interventions always overwhelm the normalized procedures [START_REF] Orr | Talking About Machines: An Ethnography of a Modern Job[END_REF].

Finally, the confrontation of maintenance with the logic of care invites to explore another aspect, which highlights important differences. Describing the logic of care, Mol insists on the distribution of practices through large collectives. Contrarily to the logic of choice, which stages and isolates individual patients and medical experts, care are shared between professionals (practicians, nurses…) and lay persons (patients, family, friends…) who act hand in hand. This is not exactly how maintenance works. Within the Parisian subway system, only the operators we followed are committed to the maintenance of signboards. This has a lot to do with their ability to notice tiny mutations before anybody else. This is basically their job: taking care of things before everyone even notices any change and so, avoiding breakdowns and incidents. Indeed, maintenance workers share what [START_REF] Goodwin | Professional Vision[END_REF] terms a ʻprofessional vision': not a stock of knowledge or a gift, but collective and situated accomplishments made of gazes, discursive practices, and material representations. As [START_REF] Orr | Talking About Machines: An Ethnography of a Modern Job[END_REF] showed it, such a professional know how is built not only in front of the objects that have to be repaired, but also during pauses, lunches, where stories and advices continuously circulate. These (mostly secret) stories and the vision they nourish, are the lively roots of occupational communities such as repair technicians and maintenance operators.

In the case of subway signs, there is however a dimension that brings back maintenance close to the care practices that Mol depicted. The professional vision thanks to which they can be aware of signboards mutabilities is by no means considered by operators as their exclusive domain. On the contrary, they regularly try to make others sensitive to it. One of their leitmotiv is to get people aware of the necessity of taking care of signboards. This is particularly the case of stations superintendants who are supposed to report anomalies when opening the station. During our observations, we witnessed several harsh discussions between maintenance operators and superintendants about reports that have not be done correctly (or not at all), the former explaining in details the way the latter should do to look at boards and to notice their transformation, if not their absence.

There is thus some distribution. Maintenance is not entirely taken in charge by an occupational community [START_REF] Orr | Talking About Machines: An Ethnography of a Modern Job[END_REF]). Yet, this distribution has its limits. Riders themselves are not at all involved in signboards maintenance. As they are the ones towards who subway signs are addressed, a strong boarder seems to be drawn between ʻthem' and the different workers. We think such a separation is not a specificity of the care of things. Indeed we know situations where users are explicitly enrolled in maintenance practices. For instance, this is crucial in information security management [START_REF] Denis | Organization, Materiality, Maintenance[END_REF]. However it raises an important issue about care, and especially material one: its relationship with visibility and invisibility. In the case of Mol, care practices are visible to (and accomplished by) almost everyone. Conversely, the care of such normalized and standardized things as subway signs seems to go with the performation of two regions: the users one and its backrooms. Maintenance here enacts a two-sided object, both consistent and vulnerable, both immutable and mutable.

Conclusion

Examining the way maintenance workers treat signs is a useful means to avoid the flattening effect of the materiality-as-stability approach. The stability and immutability of organizational artifacts, even 'ordering' ones, are not ontological properties: they are the result of an exploratory and improvised maintenance work that consists in constantly taking care of things and being aware of their material multiplicity. What matters for maintenance workers, what their action is mainly oriented toward to, is by no means the 'materiality' of things. Rather, in their daily practices they deal with both the very heterogeneity of materials and the fragility of organisms.

In many respects, this logic of care enacts a singular version of matter. By following the people in charge of the maintenance of Parisian wayfinding system, we saw how important this logic is. But we also highlighted its relative nature. The logic of care is supported by certain actors only, and it clearly contrasts with another one within the organization: the logic of normalization and standardization. So, even though everyone in the carrier (and probably in the subway spaces themselves) assumes they are dealing with the same signboards, the logic of ordering and the logic of care can be seen as two specific 'modes of mattering' [START_REF] Law | The Materials of STS[END_REF], two distinct forms of enacting matter. While, in the first one, material properties of signboards are meant to constitute reliable resources for riders, they compose, in the second one, assemblages whose consistency and solidity remain always uncertain. In other words, as ordering devices, the material agentivity of signboards is a matter of fact, while as object of maintenance work it is a matter of care (Puig de la Bellacasa 2010).

Post-ANT studies generally analyze such diversity of versions of reality as the result of the coexistence of distinct networks that involve different places, rhythms, persons, artifacts and devices [START_REF] Gad | On the Consequences of Post-ANT[END_REF]. Depending on the nature of the networks, the relationships of the versions vary: multiple versions may be completely separated from one another, two versions may contradict each other, or one version may include others [START_REF] Law | The Materials of STS[END_REF]). In our case, the different material versions of signboards do not only coexist: they are strongly relational. Not only they occur in the same place (the subway spaces), but one version directly depends on the other. The possibility for riders to rely on subway signs as immutable artifacts that perform a stabilized environment is tied to the workers' ability to handle the same subway signs as mutable objects that have to be constantly repaired and replaced. In other terms, one version is dedicated to the other. The material ordering of the world goes through the care of things.

Figure 1 .

 1 Figure 1. Disassembling a signboard

Figure 2 .

 2 Figure 2. A discolored signboard


R Alexandre

Y Morimoto
email: morimoto@math.h.kyoto-u.ac.jp

S Ukai
email: ukai@kurims.kyoto-u.ac.jp

AND C.-J Xu

T Yang

UNIQUENESS OF SOLUTIONS FOR THE NON-CUTOFF BOLTZMANN EQUATION WITH SOFT POTENTIAL

Keywords: 2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40 Boltzmann equation, singular change of velocity variables, uniqueness of solution

In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.

Introduction

Consider the Cauchy problem for the spatially inhomogeneous Boltzmann equation, (1.1)

∂ t f + v • ∇ x f = Q(f, f ), f (0, x, v) = f 0 (x, v),
where f = f (t, x, v) is the density distribution function of particles with position x ∈ R 3 and velocity v ∈ R 3 at time t. The right hand side of (1.1) is given by the Boltzmann bilinear collision operator

Q(g, f ) = R 3 S 2 B (v -v * , σ) {g(v ′ * )f (v ′ ) -g(v * )f (v)} dσdv * ,
which is well-defined for suitable functions f and g specified later. Notice that the collision operator Q(• , •) acts only on the velocity variable v ∈ R 3 . In the following discussion, we will use the σ-representation, that is, for σ ∈ S 2 ,

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ,
which give the relations between pre-and post-collisional velocities. The nonnegative cross section B(z, σ) depends only on |z| and the scalar product z |z| • σ. As in our previous works, we assume that it takes the form

B(|v -v * |, cos θ) = Φ(|v -v * |)b(cos θ), cos θ = v -v * |v -v * | • σ , 0 ≤ θ ≤ π 2 ,
where

(1.2) Φ(|z|) = Φ γ (|z|) = |z| γ , b(cos θ)θ 2+2s → K when θ → 0+,
for some γ > -3, 0 < s < 1 and K > 0. The angle θ is the deviation angle, i.e., the angle between pre-and post-collisional velocities. The range of θ is a full interval [0, π], but it is customary [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] to restrict it to [0, π/2], replacing b(cos θ) by its "symmetrized" version [b(cos θ) + b(cos(π -θ))]1 0≤θ≤π/2 , which is possible due to the invariance of the product f (v ′ )f (v ′ * ) in the collision operator Q(f, f ) under the change of variables σ → -σ.

We will use the following weighted function spaces: For p ≥ 1 and β ∈ R, we set

f L p β = R 3 | v β f (v)| p dv 1/p
, and for m ∈ R

f H m β (R 3 v ) = R 3 | D v m v β f (v) | 2 dv 1/2
,

where v = (1 + |v| 2 ) 1/2 and D v = (1 -△ v ) 1/2 . f (t, x, v
) is called a weak solution of the Cauchy problem (1.1), if

f ∈ C(R + ; S ′ (R 6 x,v )) ∩ L 1 ([0, T ]; L ∞ (R 3 x , L 1 2+γ + (R 3 v )
)), and it satisfies (1.1) in the following weak sense:

R 6 f (t, x, v)η(t, x, v)dxdv - R 6 f 0 (x, v)η(0, x, v)dxdv - t 0 dτ R 6 f (τ, x, v)(∂ τ + v • ∇ x )η(τ, x, v)dxdv (1.3) = t 0 dτ R 6 Q(f, f )(τ, x, v)η(τ, x, v)dxdv, where η ∈ C 1 (R; C ∞ 0 (R 6 )).
Here, the right hand side of the last integral can be defined by

R 3 Q(f, g)(v)η(v)dv = R 6 S 2 B f (v * )g(v)(η(v ′ ) -η(v))dvdv * dσ.
For the uniqueness of weak solutions, we consider the function space with polynomial decay in the velocity variable. More precisely, for m ∈ R, ℓ ≥ 0 and T > 0, set

P m ℓ ([0, T ] × R 6 x,v ) = f ∈ C 0 ([0, T ]; S ′ (R 6 x,v )); s.t. f ∈ L ∞ ([0, T ] × R 3 x ; H m ℓ (R 3 v ))
. Our theorem is concerned with the uniqueness of solutions for the case when γ ≤ 0 in the cross-section that includes the soft potential and Maxwell molecule for the inverse power law.

Theorem 1.1. For 0 < s < 1 and max{-3, -3/2 -2s} < γ ≤ 0, suppose that the Cauchy problem (1.1) admits two weak solutions

f 1 (t), f 2 (t) ∈ P 2s ℓ0 ([0, T ]×R 6 x,v ) with 0 < T < +∞ and ℓ 0 ≥ 14 having the same initial datum f 0 ∈ L ∞ (R 3 x ; H 2s ℓ0 (R 3 v )). If one solution is non-negative, then f 1 (t) ≡ f 2 (t).
Remark 1.2. The above result holds true for the spatially homogeneous Boltzmann equation. Moreover, according to the proof of the above theorem, the uniqueness holds also true for the cutoff Boltzmann equation in the function space P 0 ℓ0 ([0, T ] × R 6

x,v ).

Let us now review the previous results on the cutoff spatially inhomogeneous Boltzmann equation. First of all, there is an extensive literature on the existence of classical and weak solutions, which is verified basically in two settings, that is, as a small perturbation of a profile or a global Maxwellian and as a large perturbation of vacuum. For the small perturbation problem, the uniqueness usually follows from the construction of the solutions, cf. [START_REF] Guo | The Boltzmann equation in the whole space[END_REF][START_REF] Liu | Energy method for Boltzmann equation[END_REF][START_REF] Ukai | Solutions of the Boltzmann equation[END_REF] and references therein. Here, we would like to mention that the weak perturbation solution in L ∞ β ∩ L 2 around a global Maxwellian was proved to be unique by the fixed point theorem, [START_REF] Ukai | The Boltzmann equation in the space L 2 ∩L ∞ β : Global and time-periode solutions[END_REF]. However, for large perturbation solutions, even though the uniqueness of classical solution can be well justified, the uniqueness for weak solutions, such as the renormalized solutions introduced by [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: global existence and weak stability[END_REF], remains unsolved as a challenging open problem in this area. A preliminary result is found in [START_REF] Lions | Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, III[END_REF][START_REF] Lu | A result on uniqueness of mild solutions of Boltzmann equation[END_REF] that if the Cauchy problem (1.1) has one renormalized solution and one classical solution, then they should coincide.

On the other hand, for the Boltzmann equation without angular cutoff, the uniqueness problem was studied in our joint works [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF][START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions[END_REF] for solutions with exponential decay in the velocity variable. Therefore, the uniqueness result proved in this paper for solutions with polynomial decay in the velocity variable can be viewed as one step forward in the study on the uniqueness for the weak solutions. Finally, we would like to mention that there are also some interesting results on the uniqueness for the spatially homogeneous Boltzmann equation, for example, for the Maxweillian case in [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas[END_REF] for entropy solution; and for the mild singularity, that is, 0 < s < 1 2 , in [START_REF] Desvillettes | Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF] in the function space W 1,1 ℓ . Throughout this paper, we will use the following notation: f g means that there exists a generic positive constant C such that f ≤ C g.

The rest of the paper will be arranged as follows. In the next section, we will give the strategy in the proof. Some basic properties of the weight function in (x, v) will be given in Section 3. The two main estimates, one on the commutator of the weight function between the collision operator and another one on the upper bound of the collision operator with weight, will be given in the last section. These two main estimates lead to the completion of the proof of our uniqueness Theorem 1.1.

Outline of the proof of Theorem 1.1

Set F = f 1 -f 2 . Then it follows from (1.1), in the weak sense of (1.3), that (2.1)

F t + v • ∇ x F = Q(f 1 , F ) + Q(F, f 2 ) , F | t=0 = 0, which is equivalent to, for any t, t ′ ∈ [0, T ], R 6
F (t, x, v)η(x, v)dxdv - R 6 F (t ′ , x, v)η(x, v)dxdv - t t ′ dτ R 6 F (τ, x, v)(v • ∇ x )η(x, v)dxdv (2.2) = t t ′ dτ R 6 Q(f 1 , F ) + Q(F, f 2 ) (τ, x, v)η(x, v)dxdv,
where the test function η is chosen to be independent of t. Now we choose a mollification of the function F and take it as a test function.

Let S(τ ) ∈ C ∞ 0 (R) satisfy 0 ≤ S ≤ 1 and S(τ ) = 1, |τ | ≤ 1 ; S(τ ) = 0, |τ | ≥ 2.
Then, for any N ∈ N and any m ∈ R, we have

S N (D x ) = S(2 -2N |D x | 2 ) : H m (R 3 ) → H ∞ (R 3 ),
and for any f ∈ H m , lim

N →∞ S N (D x )f -f H m = 0. For ℓ ∈ R, we set also ϕ(v, x) = 1 + |v| 2 + |x| 2 and W ℓ (v) = v ℓ , W ϕ,ℓ = W ℓ (v) ϕ(v, x) = (1 + |v| 2 ) ℓ/2 1 + |v| 2 + |x| 2 . Then for F = f 1 -f 2 ,
with f 1 and f 2 given as in the statement of Theorem 1.1, we have

η(t, x, v) = W ϕ,ℓ S N (D x ) 2 W ϕ,ℓ F ∈ L ∞ ([0, T ]; H ∞ (R 3 x ; H 2s ℓ0-2ℓ (R 3 v ))
) . Similarly to Lemma 4.3 of [START_REF] Alexandre | Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff[END_REF], by taking η( t, x, v), for a fixed t, as a test function in (2.2), we can prove that

S N (D x )W ϕ,ℓ F ∈ Lip([0, T ]; H ∞ (R 3 x ; L 2 ℓ0-2ℓ (R 3 v )))
. Hence, for any 0 < t < T , we have

S N (D x )W ϕ,ℓ F (t) 2 L 2 (R 6 ) = 2 t 0 v • ∇ x (ϕ -1 )W ℓ F (τ ), S 2 N (D x )W ϕ,ℓ F (τ ) L 2 (R 6 ) + W ϕ,ℓ Q(f 1 (τ ), F (τ )) + W ϕ,ℓ Q(F (τ ), f 2 (τ )) , S 2 N (D x )W ϕ,ℓ F (τ ) L 2 (R 6 ) dτ, because v • ∇ x S N (D x )W ϕ,ℓ F (τ ) , S N (D x )W ϕ,ℓ F (τ ) L 2 (R 6 ) = 0 .
Taking the limit N → ∞, we get that, for any 0 < t < T ,

W ϕ,ℓ F (t) 2 L 2 (R 6 ) = 2 t 0 (v • ∇ x (ϕ -1 )W ℓ F (τ ), W ϕ,ℓ F (τ )) L 2 (R 6 ) (2.3) + W ϕ,ℓ Q(f 1 (τ ), F (τ )) + W ϕ,ℓ Q(F (τ ), f 2 (τ )) , W ϕ,ℓ F (τ ) L 2 (R 6 ) dτ.
The first term on the right hand side is estimated by

W ϕ,ℓ F 2 L 2 (R 6 ) because |v • ∇ x (ϕ -1 )| ϕ -1 .

If we admit the following two estimates

W ϕ,ℓ Q(f 1 , F ) , W ϕ,ℓ F L 2 (R 6 ) f 1 L ∞ (R 3 x ,H 2s 2ℓ+3/2+ε (R 3 v )) W ϕ,ℓ F 2 L 2 (R 6 ) , (2.4) and W ϕ,ℓ Q(F, f 2 ) , W ϕ,ℓ F L 2 (R 6 ) f 2 L ∞ (R 3 x ,H 2s ℓ+2s (R 3 v )) W ϕ,ℓ F (t) 2 L 2 (R 6 ) , (2.5) we can obtain (2.6) W ϕ,ℓ F (t) 2 L 2 (R 6 ) ≤ B t 0 W ϕ,ℓ F (τ ) 2 L 2 (R 6 ) dτ , with B = C f 1 L ∞ ([0,T ]×R 3 x ,H 2s 2ℓ+3/2+ε (R 3 v )) + f 2 L ∞ ([0,T ]×R 3 x ,H 2s ℓ+2s (R 3 v ))
, which concludes the proof of Theorem 1.1.

It remains to prove the two estimates (2.4) and (2.5). Set

D(f, h) = B(v -v * , σ)f * (h -h ′ ) 2 dvdv * dσ .
From here and now on, we will use the notations

f = f (v), f * = f (v * ), f ′ = f (v ′ ) and f ′ * = f (v ′ * ). The estimate (2.4
) is a consequence of the following two propositions.

Proposition 2.1. Let 0 < s < 1 and 0 ≥ γ > max{-3, -2s -3/2}. Then we have

Q(f, h), h L 2 (R 6 ) ≤ - 1 2 D(f, h)dx + C f L ∞ (R 3 x ,H 2s ′ 3/2+ε (R 3 v )) h 2 L 2 (R 6 ) ,
where s ′ ≥ 0 satisfies γ + 2s ′ > -3/2 and s ′ < min{s, 3 4 }. Proof. Regarding x as a parameter we have

Q(f, h), h L 2 (R 3 v ) = 1 2 -D(f, h) + Bf * (h ′2 -h 2 )dvdv * dσ = 1 2 -D(f, h) + R(f, h) .
It follows from the cancellation lemma [START_REF] Alexandre | Entropy dissipation and longrange interactions[END_REF] that

|R(f, h)| |v -v * | γ |f * ||h| 2 dvdv * {|v-v * |>1} |f * ||h| 2 dvdv * + {|v-v * |≤1} |v -v * | γ |f * ||h| 2 dvdv * f L 1 (R 3 v ) h 2 L 2 (R 3 v ) + {|v-v * |≤1} |v -v * | 2(γ+2s ′ ) dv * 1/2 |f * | 2 |v -v * | 4s ′ dv * 1/2 |h| 2 dv f L 2 3/2+ε (R 3 v ) h 2 L 2 (R 3 v ) + f H 2s ′ (R 3 v ) h 2 L 2 (R 3 v )
, where we have used Hardy inequality.

Remark that if 0 ≥ γ > -3/2, then we can get

(2.7) Q(f, h), h L 2 (R 6 ) ≤ - 1 2 D(f, h)dx + C f L ∞ (R 3 x ,L 2 3/2+ε (R 3 v )) h 2 L 2 (R 6 ) .
The next result takes care of commutator's estimates.

Proposition 2.2. Let ℓ ≥ 6. If max{-3, -2s -3/2} < γ ≤ 0 and 0 < s < 1, then W ϕ,ℓ Q(f, g) -Q(f, W ϕ,ℓ g) , h L 2 (R 6 ) (2.8) f L ∞ (R 3
x ;H

(2s-1) + 3/2+ε (R 3 v )) h L 2 (R 6 ) W ϕ,ℓ g L 2 (R 6 ) + D(|f |, h)dx 1/2 f 1/2 L ∞ (R 3 x ;L 2 2ℓ+3/2+ε (R 3 v )) W ϕ,ℓ g L 2 (R 6 ) . Remark 2.3. If 0 < s < 1/2, we have W ϕ,ℓ Q(f, g) -Q(f, W ϕ,ℓ g) , h L 2 (R 6 ) f L ∞ (R 3 x ;L 2 3+3/2+ε (R 3 v )) W ϕ,ℓ g L 2 (R 6 ) + f L ∞ (R 3 x ;L 2 ℓ (R 3 v )) W ϕ,3/2+ε g L 2 (R 6 ) h L 2 (R 6 )
. When f 1 is non-negative, the combination of (2.7) and (2.8) gives (2.4) by using the Cauchy-Schwarz inequality.

Together with the last result, the estimate (2.5) will be a consequence of the following proposition.

Proposition 2.4. If max{-3, -2s -3/2} < γ ≤ 0 and 0 < s < 1, then for ℓ ≥ 6, we have

W ϕ,ℓ Q(f, g) , h L 2 (R 3 v ) g H 2s ℓ+2s (R 3 v )) W ϕ,ℓ f L 2 (R 3 v ) h L 2 (R 3 v ) .
Thus we obtain (2.6) with ℓ = 6 if

f 1 , f 2 ∈ L ∞ ([0, T ] × R 3 x ; H 2s 14 (R 3 v )
). The rest of this paper is devoted to the proof of the above two Propositions 2.2 and 2.4.

Preliminary lemmas

For the estimation on the commutator between the collision operator and the weight function W ϕ,ℓ , we prepare some technical lemmas. Lemma 3.1. For ℓ ≥ 4, we have

(3.1) |W ϕ,ℓ (v) -W ϕ,ℓ (v ′ )| sin θ 2 W ℓ (v) + W ℓ (v * ) ϕ(v, x) θ W ϕ,ℓ (v)W ℓ (v * ),
and

|W ϕ,ℓ (v) -W ϕ,ℓ (v ′ )| ≤ C sin(θ/2) W ℓ (v) + W ℓ-3 (v)W 3 (v * ) ϕ(v * , x) + sin ℓ-3 (θ/2)W ϕ,ℓ (v * ) θ W ℓ (v)W ϕ,3 (v * ) + θ ℓ-2 W ϕ,ℓ (v * ) . (3.2) Remark 3.2. Remark that we can improve (3.2) to (3.3) |W ϕ,ℓ (v) -W ϕ,ℓ (v ′ )| θ W ℓ (v)W 3 (v * ) + θ ℓ-2 W ℓ (v * ) 1 + |v| 2 + |v * | 2 + |x| 2 .
Proof. For k ≥ 0, a ≥ 0, set

F k (λ) = λ k λ + a , λ ∈ [1, ∞[.
Then, for j = 1, 2, we have

d j dλ j F k (λ) ≥ 0 if k ≥ j. Thus if k ≥ 2, it follows from the mean value theorem that for λ, λ ′ ≥ 1 |F k (λ) -F k (λ ′ )| ≤ d dλ F k (λ + |λ -λ ′ |) |λ -λ ′ | , because d dλ F k (λ) is positive and increasing on [1, ∞[ for k ≥ 2. Setting λ = v 2 , λ ′ = v ′ 2 , then | v 2 -v ′ 2 | ≤ 2|v -v ′ ||v| + |v -v ′ | 2 ≤ |v| 2 + 2|v -v ′ | 2 .
So that we have

|F k ( v 2 ) -F k ( v ′ 2 )| ≤ d dλ F k 2( v 2 + |v -v ′ | 2 ) 2|v| + |v -v ′ | |v -v ′ | ≤ 2kF k-1/2 2( v 2 + |v -v ′ | 2 ) |v -v ′ |, because √ λ d dλ F k (λ) ≤ kF k-1/2 (λ). Therefore, choosing a = |x| 2 and k = ℓ 2 ≥ 2, we get |W ϕ,ℓ (v) -W ϕ,ℓ (v ′ )| v ℓ-1 |v -v ′ | v 2 + |v -v ′ | 2 + |x| 2 + |v -v ′ | ℓ v 2 + |v -v ′ | 2 + |x| 2 |v -v ′ | v ℓ-3 F 1 ( v 2 ) + |v -v ′ | ℓ v 2 + |v -v ′ | 2 + |x| 2 (3.4) = B 1 + B 2 . Note that v 2 ≤ 2 v * 2 + 2|v -v * | 2 .
Then the increasing property of F 1 implies

B 1 |v -v ′ | v l-3 v * 2 + |v -v * | 2 v * 2 + |v -v * | 2 + |x| 2 sin θ 2 W ℓ (v) + W ℓ-3 (v)W 3 (v * ) |v| 2 + ϕ(v * , x) ,
where we have used

|v -v * | 2 ≥ 1 2 |v| 2 -|v * | 2 and |v -v ′ | 2 = sin 2 θ 2 |v -v * | 2 .
This implies also

B 2 |v -v * | ℓ sin ℓ θ 2 1 + 1 -sin 2 θ 2 |v| 2 + 1 2 sin 2 θ 2 |v * | 2 + |x| 2 sin ℓ-2 θ 2 W ℓ + W ℓ, * |v| 2 + ϕ(v * , x)
.

Hence, we get the desired estimate (3.1), (3.2) and (3.3) .

When the change of variables is singular (see below), we need also a high order moment estimate. Lemma 3.3. For l ≥ 6, we have

W ϕ,ℓ (v) -W ϕ,ℓ (v ′ ) -∇ v W ϕ,ℓ (v ′ ) • (v -v ′ ) = 1 0 (1 -τ )∇ 2 W ϕ,ℓ v ′ + τ (v -v ′ ) dτ (v -v ′ ) 2 sin 2 θ 2 W ℓ (v) + W ℓ-4 (v)W 4 (v * ) ϕ(v * , x) + sin ℓ-2 θ 2 W ϕ,ℓ (v * ) (3.5) θ 2 W ℓ (v)W ϕ,4 (v * ) + θ ℓ-2 W ϕ,ℓ (v * ),
and

∇ v W ϕ,ℓ (v) -∇ v W ϕ,ℓ (v ′ ) • (v -v ′ ) sin 2 θ 2 W ℓ (v) + W ℓ-4 (v)W 4 (v * ) ϕ(v * , x) + sin l-2 θ 2 W ϕ,ℓ (v * ) (3.6) θ 2 W ℓ (v)W ϕ,4 (v * ) + θ ℓ-2 W ϕ,ℓ (v * ).
We also have

W ϕ,ℓ (v ′ ) -W ϕ,ℓ (v) -∇ v W ϕ,ℓ (v) • (v ′ -v) sin 2 θ 2 ϕ(v, x) -1 (W ℓ (v) + W ℓ (v * ))1 {|v-v * |≥1} (3.7) + (W ℓ-2 (v) + W ℓ-2 (v * ))|v -v * | 2 1 {|v-v * |<1} θ 2 W ℓ (v * )W ϕ,ℓ (v) 1 {|v-v * |≥1} + |v -v * | 2 1 {|v-v * |<1} .
Proof. As for (3.5), we use the Taylor expansion of second order

W ϕ,ℓ (v)-W ϕ,ℓ (v ′ ) -(∇W ϕ,ℓ )(v ′ ) • (v -v ′ ) = 1 0 (1 -τ )(∇ 2 W ϕ,ℓ )(v ′ + τ (v -v ′ ))dτ (v -v ′ ) 2 = I 2 (v, v ′ ) .

We have with v

τ = v ′ + τ (v -v ′ ) |I 2 (v, v ′ )| |v -v ′ | 2 1 0 F ′ ℓ/2 ( v τ 2 ) + v τ 2 F ′′ ℓ/2 ( v τ 2 ) dτ |v -v ′ | 2 F ℓ/2-1 (2( v 2 + |v ′ -v| 2 )) v ℓ-2 |v ′ -v| 2 + |v ′ -v| ℓ v 2 + |v ′ -v| 2 + |x| 2 , (3.8) because F ′ k (λ), F ′′ k (λ) are positive in [1, ∞), and
F ′ k ≤ CF k-1 , F ′′ k ≤ CF k-2 and F k-1 is increasing there, if k = ℓ/2 ≥ 3.
Here we have used the fact that

v τ 2 ≤ 2 v + 2|v ′ -v| 2 . Noticing again that v 2 ≤ 2 v * 2 + 2|v -v * | 2 and F 1 is increasing, we have |I 2 (v, v ′ )| |v -v * | 2 sin 2 (θ/2) v ℓ-4 F 1 ( v 2 ) + B 2 sin 2 (θ/2) W ℓ (v) + W ℓ-4 (v)W 4 (v * ) ϕ(v * , x) + sin ℓ-2 (θ/2)W ϕ,ℓ (v * ) ,
which yields (3.5). The proof of (3.6) is similar. The last inequality (3.7) follows easily from (3.8).

Proofs of Propositions 2.2 and 2.4

In this section, we regard (t, x) as a parameter.

Proof of Proposition 2.2 :

First of all, we have

W ϕ,ℓ Q(f, g) -Q(f, W ϕ,ℓ g) , h L 2 (R 3 v ) = B W ′ ϕ,ℓ -W ϕ,ℓ f * gh ′ dvdv * dσ = B W ′ ϕ,ℓ -W ϕ,ℓ f * g(h ′ -h)dvdv * dσ + B ∇ v W ϕ,ℓ (v) • (v ′ -v)f * ghdvdv * dσ + B W ′ ϕ,ℓ -W ϕ,ℓ -∇ v W ϕ,ℓ (v) • (v ′ -v) f * g h dvdv * dσ = D 1 + D 2 + D 3 .
By the Cauchy-Schwarz inequality, we get in view of (3.1),

|D 1 | D(|f |, h) 1/2 B|W ′ ϕ,l -W ϕ,ℓ | 2 |f * |g 2 dvdv * dσ 1/2 D(|f |, h) 1/2 f 1/2 L 1 2ℓ W ϕ,ℓ g L 2 (R 3 v ) D(|f |, h) 1/2 f 1/2 L 2 2ℓ+3/2+ε W ϕ,ℓ g L 2 (R 3 v ) . Using (3.7) gives |D 3 | B θ 2 W ℓ, * W ϕ,ℓ 1 |v-v * |≥1 + |v -v * | 2 1 |v-v * |<1 |f * gh| dvdv * dσ |(W ℓ f ) * |(W ϕ,ℓ g)h| dvdv * + 1 |v-v * |<1 |v -v * | γ+2 |(W ℓ f ) * (W ϕ,ℓ g)h| dvdv * f L 1 l (R 3 v ) W ϕ,ℓ g L 2 (R 3 v ) h L 2 (R 3 v ) + |v-v * |<1 |v -v * | 2(γ+2) dv * 1/2 W ℓ f L 2 l (R 3 v ) (W ϕ,ℓ g)h| dv f L 2 ℓ+3/2+ε (R 3 v ) W ϕ,ℓ g L 2 (R 3 v ) h L 2 (R 3 v )
. We now consider the estimate of the term D 2 by first noticing that

v ′ -v = 1 2 (v * -v)(1 -k • σ) + 1 2 |v * -v| σ -(k • σ)σ , where k = (v -v * )/|v -v * |.
It follows from the symmetry on σ that

S 2 b(k • σ) σ -(k • σ)σ dσ = 0. Hence, (1 -k • σ) = 2 sin 2 (θ/2) implies |D 2 | = 1 2 B ∇ v W ϕ,ℓ (v) • (v * -v)(1 -k • σ)f * g h dvdv * dσ b θ 2 ∇ v W ϕ,ℓ (v)|v * -v| γ+1 |f * g h| dvdv * dσ . Since ∇ v W ϕ,ℓ |v * -v| γ+1 W ϕ,ℓ W 1, * 1 {|v-v * |≥1} + W ϕ,ℓ-1 |v * -v| γ+1 1 {|v-v * |<1} ,
we get

|D 2 | |(W ℓ f ) * |(W ϕ,ℓ g)h| dvdv * + 1 {|v-v * |<1} |v -v * | γ+1 |(W ℓ f ) * (W ϕ,ℓ g)h| dvdv * f L 1 l (R 3 v ) W ϕ,ℓ g L 2 (R 3 v ) h L 2 (R 3 v ) + |v-v * |<1 |v -v * | 2(γ+2s) dv * 1/2 (W ℓ f ) 2 * |v -v * | 2(2s-1) dv * 1/2 |(W ϕ,ℓ g)h| dv f L 2 ℓ+3/2+ε (R 3 v ) W ϕ,ℓ g L 2 (R 3 v ) h L 2 (R 3 v ) + f H (2s-1) + ℓ (R 3 v ) W ϕ,ℓ g L 2 (R 3 v ) h L 2 (R 3 v )
. By summing up the above estimates and integrating with respect to x, we finish the proof of Proposition 2.2.

We now turn to Proof of Proposition 2.4 :

Here we need to use the mollification of the function. In the estimate stated in Proposition 2.4, we put the weight on the first function in the collision operator. To estimate this, we need a singular change of the variables between the pre and post collision velocities as follows:

v * → v ′ = v + v * 2 + |v -v * | 2 σ ,
where the Jacobian is

∂v * ∂v ′ = 8 I -k ⊗ σ = 8 |1 -k • σ| = 4 sin 2 (θ/2) , θ ∈ [0, π/2],
where again k = (v -v * )/|v -v * |. Note that this change of variables is singular when θ = 0. After this change of variables, k = (v -v * )/|v -v * | is a function of v, v ′ , σ, so that θ no longer plays the role as the polar angle. In fact, "pole k" moves with σ and hence the measure dσ is no longer given by sin θdθdφ. Hence, we need to choose a new pole which is independent of σ. Choose k ′′ = (v ′ -v)/|v ′ -v|, then the polar angle ψ defined by cos ψ = k ′′ • σ satisfies,

ψ = π 2 - θ 2 , dσ = sin ψdψdφ, ψ ∈ [ π 4 , π 2 ].
Note that now the angular singularity in b(cos θ)dσ becomes θ -2-2-2s , which is stronger than (1.2) where it is of order θ -1-2s . On the other hand, there is another singularity in the kinetic factor of the cross section for soft potential. To study this, we decompose the kinetic factor of collision operator Φ γ (v -v * ) = |v -v * | γ in two part by using a cutoff function. Let 0 ≤ φ(z) ≤ 1 be a smooth radial function with value 1 for z close to 0, and 0 for large values of z.

Set Φ γ (z) = Φ γ (z)φ(z) + Φ γ (z)(1 -φ(z)) = Φ sing (z) + Φ reg (z). Then we write Q(f, g) = Q sing (f, g) + Q reg (f, g),
where the kinetic factor in the collision operator is defined according to the decomposition respectively. We consider firstly the regular part.

Since Φ reg (z) is smooth, and Φ reg (z) ≤ Φγ (z) = (1 + |z| 2 ) γ/2 , Q reg (f, g) has similar upper bound and commutator estimates as for Q Φγ (f, g).

Let us recall several propositions obtained in [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF]. For 0 < s < 1, γ ∈ R, we proved the following upper bound estimate (Theorem 2.1 of [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF])

(4.1) |(Q Φγ (f, g), h)| ||f || L 1 α + +(γ+2s) + ||g|| H m+s (α+γ+2s) + h H s-m -α
, for any m, α ∈ R, and the estimate of commutators with weight (Lemma 2.4 of [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF]) (4.2)

W ℓ Q Φγ (f, g) -Q Φγ (f, W ℓ g), h f L 1 ℓ+(2s-1) + +γ + g H (2s-1+ǫ) + ℓ+(2s-1) + +γ + h L 2 .
It also follows from Lemma 2.8 of [START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF] that

(4.3) b |f * | (g ′ -g) 2 dσdvdv * f L 1 2s g 2 H s s .
We now study the estimate of Proposition 2.4 for the regular part.

Proposition 4.1. Let ℓ ≥ 6. If max{-3, -2s -3/2} < γ ≤ 0 and 0 < s < 1, then

W ϕ,ℓ Q reg (f, g), h L 2 (R 3 v ) g H 2s ℓ+2s (R 3 v )) W ϕ,ℓ f L 2 (R 3 v ) h L 2 (R 3 v ) . Proof. Write W ϕ,ℓ Q reg (f, g) , h L 2 (R 3 ) = Q reg (f, W ϕ,ℓ g) , h L 2 (R 3 ) + W ϕ,ℓ Q reg (f, g) -Q reg (f, W ϕ,ℓ g) , h L 2 (R 3 ) = A + B .
By using the upper bound estimate (4.1) with m = s, α = 0, we have

|A| f L 1 (γ+2s) + (R 3 v ) W ϕ,ℓ g H 2s (γ+2s) + (R 3 v ) h L 2 (R 3 v ) ϕ x 2 f ϕ L 2 2s+3/2+ε (R 3 v ) x 2 ϕ W ℓ g H 2s (γ+2s) + (R 3 v ) h L 2 (R 3 v ) W ϕ,2s+7/2+ε f L 2 (R 3 v ) g H 2s ℓ+2s (R 3 v ) h L 2 (R 3 ) .
Here we have used ϕ

x 2 v 2 . For the term B, we have

B = W ϕ,ℓ Q reg (f, g) -Q reg (f, W ϕ,ℓ g), h L 2 (R 3 v ) = b Φ reg W ϕ,ℓ ′ -W ϕ,ℓ f * g h ′ dvdv * dσ = b Φ reg ∇ v W ϕ,ℓ (v ′ ) • (v ′ -v) f * g h ′ dvdv * dσ + b Φ reg W ϕ,ℓ ′ -W ϕ,ℓ -∇ v W ϕ,ℓ (v ′ ) • (v ′ -v) f * g h ′ dvdv * dσ = B 1 + B 2 .
By using (3.5), we have

|B 2 | b θ 2 | W ϕ,4 f * || W ℓ g ||h ′ |dvdv * dσ + b θ ℓ-2 W ϕ,ℓ f * |gh ′ |dvdv * dσ (4.4) =M 1 + M 2 ,
where we have used the fact that γ ≤ 0. By the Cauchy-Schwarz inequality, we get

M 2 1 b θ 2 W ϕ,4 f * W ℓ g 2 dvdv * dσ × b θ 2 W ϕ,4 f * (h ′ ) 2 dvdv * dσ W ϕ,4 f 2 L 1 (R 3 v ) g 2 L 2 ℓ (R 3 v ) h 2 L 2 (R 3
v ) . Here we have used the regular change of variables

v → v ′ = v + v * 2 + |v -v * | 2 σ ,
whose Jacobian is given by

∂v ∂v ′ = 8 I + k ⊗ σ = 8 |1 + k • σ| = 4/ cos 2 (θ/2) ≤ 8 .
On the other hand, by the Cauchy-Schwarz inequality again, we have

M 2 2 b θ ℓ-7/2 |g| W ϕ,ℓ f 2 * dv * dvdσ × b θ ℓ-1/2 |g||h ′ | 2 dv * dvdσ , if we choose ℓ so that ℓ -7/2 -(1 + 2s) = ℓ -1/2-(2 + 2s+2) > -1 .
Then a direct calculation reduces the first integral to

b θ ℓ-2-3/2 |g| W ϕ,ℓ f 2 * dv * dvdσ g L 1 (R 3 v ) W ϕ,ℓ f 2 L 2 (R 3
v ) . For the second integral, we now use the singular change of variables v

* → v ′ whose Jacobian is ∂v * ∂v ′ = 4 sin 2 (θ/2)
.

Then, we have

R 3 v ×R 3 v * ×S 2 b(cos θ) θ ℓ-1/2 |g| |h ′ | 2 dv * dvdσ R 3 v ×R 3 v ′ π/2 0 θ ℓ-1/2-(2+2s+2) |g| |h ′ | 2 dv ′ dvdθ g L 1 (R 3 v ) h 2 L 2 (R 3 v ) . Thus, M 2 g L 1 (R 3 v ) W ϕ,ℓ f L 2 (R 3 v ) h L 2 (R 3 v ) .
For the term B 1 , we decompose it further into

B 1 = Φ reg b ∇ v W ϕ,ℓ (v ′ ) -∇ v W ϕ,ℓ (v) • (v -v ′ )f * gh ′ dvdv * dσ + Φ reg b ∇ v W ϕ,ℓ (v)g(v) -∇ v W ϕ,ℓ (v ′ )g(v ′ ) • (v -v ′ )f * h ′ dvdv * dσ + Φ reg b f * gh ∇ v W ϕ,ℓ (v ′ ) • (v -v ′ )dvdv * dσ = B (1) 1 + B (2) 1 + B (3)
1 . It follows from the symmetry of σ variable that B

|B (2) 1 | 2 b|f * ||v -v * | 2 ∇ v W ϕ,ℓ (v)g(v) -∇ v W ϕ,ℓ (v ′ )g(v ′ ) 2 dvdv * dσ × b sin 2 (θ/2)|f * ||h ′ | 2 dvdv * dσ = B (2,1) 1 × B (2,2) 1
.

Using the regular change of variables v → v ′ , we get

B (2,2) 1 f L 1 (R 3 v ) h 2 L 2 (R 3 v ) f L 2 3/2+ε (R 3 v ) h 2 L 2 (R 3 v ) . Putting G = ∇ v W ϕ,ℓ g, in view of |v -v * | |v ′ -v * | ≤ v * + v ′ we have B (2,1) 1 b| v * 2 f * | G(v) -G(v ′ ) 2 dvdv * dσ + b|f * | v G(v) -v ′ G(v ′ ) 2 dvdv * dσ + b|f * | | v -v ′ )G(v)| 2 dvdv * dσ .
We apply (4.3) to the first two terms. Note that

| v -v ′ | |v -v ′ | v * v sin(θ/2),
in the last term. Then we get

B

(2,1) 1

f L 1 2 W ϕ,ℓ g 2 H s s (R 2 v
) . On account of (3.6), we have We finally turn to the singular part Q sing (f, g). As shown in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions[END_REF], the singular part Q sing requires fairly long computations. For our use, we now recall some estimates in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions[END_REF]. The following upper bound estimate is a consequence of Proposition 2.1 from [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions[END_REF]: for 0 < s < 1, γ > max{-3, -2s -3/2} and m ∈ [s -1, s],

(4.5) |(Q sing (f, g), h) L 2 (R 3 v ) | f L 2 (R 3 ) ||g|| H s+m (R 3 ) h H s-m (R 3 ) .
And the following commutator estimate is implied by Proposition 2.5 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions[END_REF]: letting 0 < s < 1, γ > max{-3, -2s -3/2}, for any ℓ, β, δ ∈ R,

(4.6) W ℓ Q sing (f, g) -Q sing (f, W ℓ g), h f L 2 ℓ-1-β-δ g H (2s-1+ǫ) + β ||h|| L 2 δ .
For the estimate of singular part with weight introduced in this paper, we now want to prove Proposition 4.2. Let ℓ ≥ 2. If max{-3, -2s -3/2} < γ ≤ 0 and 0 < s < 1, then W ϕ,ℓ Q sing (f, g) , h L 2 (R 3 ) g H 2s ℓ (R 3 ) W ϕ,ℓ f L 2 (R 3 ) h L 2 (R 3 ) . (4.7)

Proof. Write W ϕ,ℓ Q sing (f, g) , h

L 2 (R 3 ) = Q sing f x 2 , W ℓ g , x 2 ϕ(v, x) h L 2 (R 3 v ) + W ℓ Q sing f x 2 , g -Q sing f x 2 , W ℓ g , x 2 ϕ(v, x) h L 2 (R 3 v ) = A 1 + A 2 .
It follows from (4.5) that with m = s,

|A 1 | ϕ(v, x) x 2 f ϕ(v, x) L 2 (R 3 v ) W ℓ g H 2s (R 3 v ) x 2 ϕ(v, x) h L 2 (R 3 v ) g H 2s ℓ (R 3 ) W ϕ,2 f L 2 (R 3 ) h L 2 (R 3 ) , because x 2 ≤ ϕ(v, x) ≤ x 2 v 2 .
Using (4.6) with β = ℓ -1, δ = 0, we have

|A 2 | ϕ(v, x) x 2 f ϕ(v, x) L 2 (R 3 v ) g H (2s-1+ε) + ℓ-1 (R 3 v ) x 2 ϕ(v, x) h L 2 (R 3 v ) g H 2s ℓ (R 3 ) W ϕ,2 f L 2 (R 3 ) h L 2 (R 3 ) .
The above two estimates complete the proof of Proposition 2.4.

Figure 1 .

 1 Figure 1. Symmetry of σ 1 and σ 2 , v ′ → v = ψ σ (v ′ ) Since |v -v ′ | = |v -v * | sin(θ/2) and |Φ reg | 1 for γ ≤ 0, by the Cauchy-Schwarz inequality, we have

|B

b θ 2 |

 2 W ϕ,4 f * || W ℓ g ||h ′ |dvdv * dσ + b θ ℓ-2 W ϕ,ℓ f * |gh ′ |dvdv * dσ ,which is the same as for B 2 in (4.4). We have proved Proposition 4.1.

Acknowledgements : The research of the first author was supported in part by the Zhiyuan foundation and Shanghai Jiao Tong University. The research of the second author was supported by Grant-in-Aid for Scientific Research No.22540187, Japan Society of the Promotion of Science. The research of the fourth author was supported partially by " the Fundamental Research Funds for the Central Universities". The last author's research was supported by the General Research Fund of Hong Kong, CityU No.103109, and the Lou Jia Shan Scholarship programme of Wuhan University.


Salla Savela

Pentti Koistinen

Reijo S Tilvis

Arto Y Strandberg

Kaisu H Pitkälä

Veikko V Salomaa

Tatu A Miettinen

Timo E Strandberg
email: timo.strandberg@oulu.fi

A Tilvis

Miettinen Strandberg

Leisure-time physical activity, cardiovascular risk factors and mortality during a 34-year follow-up in men

Keywords: Body mass index CI, Confidence interval CVD, Cardiovascular disease HR, Hazard ratio

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Given that physical activity affects several cardiovascular disease (CVD) risk factors, it is possible that exercise prevents deaths simply through risk factor modification. [START_REF] Johansson | Change in lifestyle factors and their influence on health status and all-cause mortality[END_REF] However, the known risk associations may be confounded through several pathways including lifestyle choices, social class [START_REF] Kim | Contrasting socioeconomic profiles related to healthier lifestyles in China and the United States[END_REF] and varying adherence -or more broadly the "healthy user effect". In other words, people who initiate and adhere to preventive habits may be systemically healthier and more health-seeking than others. [START_REF] Brookhart | Adherence to lipid-lowering therapy and the use of preventive health services: an investigation of the healthy user effect[END_REF] Furthermore, the physical activity of work and leisure time, lifestyle and other stress factors differ in both their prevalence and modifiability in various socioeconomic groups, and consequently, population-based studies may not reflect the effects of physical activity as such. In order to reduce some of these caveats, we studied the associations between leisure-time physical activity and 34-year mortality in an initially healthy and socioeconomically homogeneous cohort of middle-aged men in Helsinki, Finland.

Methods

Baseline examinations in 1974 and questionnaire surveys in 1985-1986 and 2000 The cohort and examinations have been described earlier [START_REF] Strandberg | Explaining the obesity paradox: cardiovascular risk, weight change, and mortality during long-term follow-up in men[END_REF]. The study procedures have been approved by the Ethics Committee of the Department of Medicine, University of Helsinki. In brief, initially healthy men of high socioeconomic status, mostly business executives born in 1919-1934, had participated in structured health check-ups during the 1960s and early 1970s at the Institute of Occupational Health in Helsinki, Finland. They were evaluated with questionnaires and clinical and laboratory examinations in 1974, whereupon 1,815 men were found to be actively working and healthy without diabetes, clinical cardiovascular disease or regular medications. Of the 1,815 men, a structured questionnaire about details of their physical activity is available for 782 men. In 1974, this detailed information was mostly (90.7%, n=709) gathered from men who had at least one of the following risk factors: [START_REF] Johansson | Change in lifestyle factors and their influence on health status and all-cause mortality[END_REF] serum cholesterol level 7.0 mmol/L (270mg/dL) or higher; (2) serum triglyceride level 1.7mmol/L (150mg/dL) or higher; (3) systolic blood pressure 160mmHg or higher; (4) diastolic blood pressure 95 mmHg or higher; [START_REF] Miettinen | Multifactorial primary prevention of cardiovascular diseases in middle-aged men. Risk factor changes, incidence, and mortality[END_REF] smoking more than ten cigarettes per day;

(6) relative body weight (weight relative to mean weight for height) 120% or higher (indicating overweight); [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF] one-hour glucose tolerance 9.0 mmol/L (162mg/dL) or higher. These men were assessed to be at higher cardiovascular risk [START_REF] Miettinen | Multifactorial primary prevention of cardiovascular diseases in middle-aged men. Risk factor changes, incidence, and mortality[END_REF]. However, their age distribution as well as long-term mortality were not statistically different from those men without detailed physical activity assessment.

Although part of the men had participated in a controlled multifactorial prevention trial during the 1970s [START_REF] Miettinen | Multifactorial primary prevention of cardiovascular diseases in middle-aged men. Risk factor changes, incidence, and mortality[END_REF], our preliminary analyses showed that participation in the 1970s trial did not affect the present long-term results. Therefore, all the 782 men with detailed exercise information form the population of the present analyses.

In 1974, weight and height were measured and body mass index (BMI) was calculated as weight (kg) divided by height (m) squared. In 1974, the men also reported their weight at 25 years of age, which was used to calculate midlife weight gain (weight in 1974 minus reported weight at 25 years of age). In 1974 the men were also asked questions on how they rated their present health and physical fitness on a 5-step scale ("very good", "good", "fair", "poor", "very poor"). To reflect composite risk of coronary artery disease, the Keys' risk score [START_REF] Keys | Probability of middle-aged men developing coronary heart disease in five years[END_REF] including age, BMI, smoking, cholesterol and systolic blood pressure was calculated in 1974.

The Framingham risk score could not be calculated because HDL cholesterol was not available in 1974.

The men were also asked to give a global description of their leisure-time physical activity on a 4-step scale: 1) Reading, watching television, going to the cinema or other sedentary activity.

2) Walking, cycling, skiing, gardening, bowling, fishing or other light exercise weekly.

3) Jogging, running, skiing, swimming, tennis, badminton, heavy gardening or similar exercise weekly on a regular basis. 4) Regular vigorous/competitive exercise several times a week on a regular basis.

Because only 11 men reported competitive activity, groups 3 and 4 were combined in the analyses. Men answering yes to question 1 were categorized as low activity; 2 as moderate; and 3 and 4 as high activity group. Preliminary analyses showed that global description differentiated mortality risk most consistently, and therefore more detailed analyses were performed comparing these groups.

Of the 782 men, 80.7% (n=631) and 70.6% (n=552) could be re-assessed with questionnaires in the 11-year (1985-1986) and 26-year surveys (2000), respectively, but questions about exercise were less comprehensive than in 1974: In the 11-year survey, men were asked whether they exercised regularly and weekly, and if yes, how many hours per week; in the 26year survey, men were asked whether they exercised regularly and weekly, and if yes, how many hours per week. However, in the 26-year survey men were also asked how many times per week they exercised with an intensity leading to sweating and breathlessness ("vigorous" activity).

Mortality follow-up

Total mortality of the study cohort up to 31 March 2007 was retrieved from the Central Population Register, which keeps registry of all Finnish citizens. According to the Register, assessment of vital status is very reliable for people having their permanent place of residence in Finland (over 95% of the present cohort) irrespective of whether they die in Finland or abroad. Moreover, the assessment of the vital status is also quite reliable for Finnish citizens living permanently abroad. Differences in survival curves were analyzed with log rank test. Hazard ratios (HR) with their 95% confidence intervals (CI) for mortality associated with physical activity at baseline were calculated using Cox's proportional hazards regression. Other risk factors were adjusted for in respective models. In statistical analyses 2-tailed tests were used, and P values < 0.05 were taken as significant.

Results

The baseline characteristics of the study population are shown in Table 1. Of the various CVD risk factors, physical activity was inversely associated with weight gain during adulthood, BMI in midlife, and with one-hour post-load glucose concentration. Sedentary men smoked more than other groups, but serum lipid levels and blood pressure were only weakly associated with physical activity.

As a global description of physical activity, 148 (18.9%), 398 (50.9%), and 236 (30.2%) men reported low (sedentary), moderate, and high leisure-time physical activity, respectively. Most men (80.5%, n=630) had sedentary work and commuted to work by car or public transportation. No association was observed between physical activity groups and the type of work.

Physical activity was measured three times: at baseline and in the 11-and 26-year surveys, yet the questions about physical activity were most detailed at baseline, as described above.

Tracking of the baseline physical activity was good during the follow-up in the 11-year and 26-year surveys. (Table 2). The reported regular weekly exercise in the 11-year and 26-year surveys was 50-70%, 69-80% and 92-85% for low, moderate and high physical activity groups, respectively (Table 2). The continuity of initially reported regular weekly exercise was significantly higher among the most physically active men in both subsequent surveys (P<0.001 and <0.002, respectively). "Vigorous" exercise leading to sweating and breathlessness was reported by 56%, 68% and 79% men in the low, moderate and high physical activity groups in the 26-year survey, while 35%, 45% and 60%, respectively, performed vigorous exercise more than twice a week.

During the 34-year follow-up, 295 men died (37.8%), with the cause of death available for 282 deaths (95.6% of total). Mortality was not significantly associated with type of work (sedentary, mobile) or type of commuting to work (walking or cycling).

Global description of leisure-time physical activity significantly predicted death: mortality per 1,000 person years was 16.5, 12.9, and 10.8 (P=0.01) in the low, moderate and high physical activity groups, respectively. Unadjusted survival curves are shown in Figure 2, demonstrating the consistent difference between the groups. Causes of deaths are shown in Table 4, suggesting that especially CVD deaths were associated with physical activity.

Multivariable analyses (Table 4) indicate that the protective effect of physical activity was independent of traditional CVD risk factors (age, BMI, cholesterol, smoking and systolic blood pressure) reflected by Keys' risk index. The result did not change after one-hour glucose and triglycerides were added to the model.

Discussion

The self-reported "global" leisure time physical activity was strongly associated with 34-year mortality independently of traditional CVD risk factors in a socioeconomically homogeneous cohort of men. Comparison of the causes of death showed that especially CVD mortality was associated with physical activity, supporting the findings of earlier Finnish [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF][START_REF] Barengo | Low physical activity as a predictor for total and cardiovascular disease mortality in middle-aged men and women in Finland[END_REF][START_REF] Pekkanen | Reduction of premature mortality by high physical activity: a 20-year follow-up of middle-aged Finnish men[END_REF] and other population studies in Western countries [START_REF] Johansson | Change in lifestyle factors and their influence on health status and all-cause mortality[END_REF][START_REF] Manini | Daily activity energy expenditure and mortality among older adults[END_REF][START_REF] Sandvik | Physical fitness as a predictor of mortality among healthy, middle-aged Norwegian men[END_REF][START_REF] Slattery | Leisure time physical activity and coronary heart disease death. The US Railroad Study[END_REF][START_REF] Byberg | Total mortality after changes in leisure time physical activity in 50 year old men: 35 year follow-up of population based cohort[END_REF][START_REF] Chave | Vigorous exercise in leisure time and the death rate: a study of male civil servants[END_REF][START_REF] Haveman-Nies | Dietary quality and lifestyle factors in relation to 10-year mortality in older Europeans: the SENECA study[END_REF][START_REF] Sui | Cardiorespiratory fitness as a predictor of nonfatal cardiovascular events in asymptomatic women and men[END_REF][START_REF] Schnohr | Changes in leisure-time physical activity and risk of death: an observational study of 7,000 men and women[END_REF][START_REF] Paffenbarger | The association of changes in physical-activity level and other lifestyle characteristics with mortality among men[END_REF][START_REF] Paffenbarger | Physical activity, all-cause mortality, and longevity of college alumni[END_REF][START_REF] Leon | Leisure-time physical activity levels and risk of coronary heart disease and death. The Multiple Risk Factor Intervention Trial[END_REF][START_REF] Chakravarty | Reduced disability and mortality among aging runners: a 21-year longitudinal study[END_REF][START_REF] Kodama | Cardiorespiratory fitness as a quantitative predictor of all-cause mortality and cardiovascular events in healthy men and women: a meta-analysis[END_REF][START_REF] Talbot | Changes in leisure time physical activity and risk of all-cause mortality in men and women: the Baltimore Longitudinal Study of Aging[END_REF][START_REF] Hakim | Effects of walking on mortality among nonsmoking retired men[END_REF].

While the difference in total work load showed the largest predictive power between the sedentary and active tertile, significant differences in mortality rates became evident after approximately 15 years of surveillance, and were interestingly shown to increase with time through the 34-year follow-up. The weekly physical activity was assessed twice during the follow-up and was shown to increase in all activity groups in both the 11-year and the 26-year survey. These findings can be accounted for by the positive effect of increasing exercise, and increasing selection through excessive mortality of sedentary participants. At the 26-year survey, vigorous exercise on a weekly basis was reported by 59% and 79% and the frequency of three times or more per week by 35% and 60% in the low and high activity groups, respectively. The exercise habit seemed to track best among the initially most active men in both surveys. An increase in both intensity and amount of exercise in all groups can be suggested to account for the dose-related decrease in mortality in relation to increased exercise.

Our results indicating that approximately 15 years of surveillance is needed before the differences between mortality rates of the low and high activity groups become evident are in accordance with a recent Swedish follow-up study [START_REF] Byberg | Total mortality after changes in leisure time physical activity in 50 year old men: 35 year follow-up of population based cohort[END_REF]. It showed that an average of 10 years of increased physical activity after midlife is required to reduce excessive mortality in physically inactive men to the level of those who are physically active. Additionally, our results suggest that training has an independent dose-response effect in reducing mortality, which can even increase with time at least up to 34 years.

Both aerobic fitness and physical activity have been shown to associate with mortality. In a 16-year follow-up of Norwegian men the relative risk of death from any cause in the highest fitness quartile as compared with the lowest quartile was significantly reduced by 46% after adjustment for age, smoking status, serum lipids, blood pressure, resting heart rate, vital capacity, BMI, level of physical activity, and glucose tolerance. A high level of leisure time physical activity was strongly associated with fitness, but was not found to have an independent prognostic value. [START_REF] Sandvik | Physical fitness as a predictor of mortality among healthy, middle-aged Norwegian men[END_REF] In another 5.5-year prospective study with age-adjusted quartiles, a multivariate analysis including clinical characteristics, risk factors, exercise test data and activity patterns revealed that exercise capacity (HR per quartile 0.62; CI: 0.47-0.82) and energy expenditure from adulthood recreational activity (HR per quartile 0.72; CI: 0.58-0.89, P=0.002) were the sole significant predictors of mortality [START_REF] Myers | Fitness versus physical activity patterns in predicting mortality in men[END_REF]. It should be noted, however, that in addition to exercise training, physical fitness level is markedly influenced by hereditary factors, which can affect survival. On the other hand, regular exercise is a prerequisite for the maintenance of individual fitness level. The constancy and even increasing tendency of regular physical activity during our 34-year follow-up probably have a crucial impact on the observed mortality benefit, emphasizing the importance of a life-long sporty lifestyle.

In the present study, physical activity was inversely associated with weight gain during adulthood, BMI in midlife, and with one-hour post-load glucose concentration. Nevertheless, BMI or even one-hour glucose did not show an independent effect on mortality, in evidence against reverse causality. Our findings are in line with a follow-up study of 9.8 years in middle-aged Finnish women and men, which showed that physical inactivity has an independent effect on increased CVD risk, while obesity was concluded to increase the risk partly through the modification of other risk factors [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF]. The education level included in multivariate analysis was not found to have an independent effect on CVD risk, even though it was speculated that unmeasured components of socio-economic status may strengthen the positive effect of leisure-time activity and weaken the protective effects of occupational activity [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF].

It has been suggested that public health programs may be of benefit by advising each socioeconomic status group separately while considering the country's level of development [START_REF] Kim | Contrasting socioeconomic profiles related to healthier lifestyles in China and the United States[END_REF]. A combination of high stress levels and smoking or physical inactivity has been revealed to be especially harmful among individuals with low socioeconomic status [START_REF] Krueger | Being poor and coping with stress: health behaviors and the risk of death[END_REF]. The present study is unique in terms of a very long follow-up in a socioeconomically homogeneous cohort of men of high socioeconomic status, and it can be considered as being dated in a phase of constant and quite rapid social upswing and increasing awareness of health concerns in the Finnish society. The similarity of the results of our study with earlier Finnish population studies [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF] suggests that the adoption of life-style and physical activity habits among higheconomy social class of men were not markedly different from the whole population.

The distinct strength of this primary prevention study is the long follow-up time, which reduces the confounding effect of subclinical diseases at baseline. The fact that our cohort consisted of healthy men of high socioeconomic class can be considered to further reduce confounding. Most studies concerning physical activity and mortality are based on a single estimate of activity [START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference and waist-to-hip ratio with the risk of cardiovascular disease among middle-aged Finnish men and women[END_REF][START_REF] Barengo | Low physical activity as a predictor for total and cardiovascular disease mortality in middle-aged men and women in Finland[END_REF][START_REF] Chave | Vigorous exercise in leisure time and the death rate: a study of male civil servants[END_REF][START_REF] Haveman-Nies | Dietary quality and lifestyle factors in relation to 10-year mortality in older Europeans: the SENECA study[END_REF][START_REF] Leon | Leisure-time physical activity levels and risk of coronary heart disease and death. The Multiple Risk Factor Intervention Trial[END_REF][START_REF] Hakim | Effects of walking on mortality among nonsmoking retired men[END_REF]27,28], while in our study physical activity was also assessed during follow-up. Response rates to the surveys were high.

Our study also has some limitations. Physical activity was determined by self-report, which is considered to overestimate actual amounts of physical activity and thus underestimate the benefits of physical activity among older adults [START_REF] Manini | Daily activity energy expenditure and mortality among older adults[END_REF]. Despite this, our study shows a graded and independent association between physical activity and reduced long-term mortality.

Another limitation is that the questionnaires sent during the follow-up were not as detailed as the questionnaire at baseline. Generally, physical activity nevertheless seems to track well during the decades in our cohort. The fact that the physical activity pattern remained largely unchanged suggests that the physical activity habits were correctly reported. A clear limitation is that our cohort consisted only of men of high socioeconomic status. However, relations between physical activity and mortality have been reported to be considerably similar between sexes [START_REF] Barengo | Low physical activity as a predictor for total and cardiovascular disease mortality in middle-aged men and women in Finland[END_REF]. Another recent study showed no significant association between leisure-time physical activity and mortality in women, but the statistical power was limited by the small number of deaths among women [START_REF] Talbot | Changes in leisure time physical activity and risk of all-cause mortality in men and women: the Baltimore Longitudinal Study of Aging[END_REF].

Conclusions

In our 34-year follow-up study, leisure-time physical activity in midlife was associated with better long-term prognosis in a stepwise manner, irrespective of CVD risk factors and BMI among men with a homogeneous socioeconomic status. Whether this longer life achieved by physical activity is also associated with better quality of life in old age is important from the compression of morbidity viewpoint and needs to be investigated further.

  Causes of death up to 31 December 2006 were determined from the countrywide computerized Cause-of-Death Register of Statistics Finland, in which trained nosologists code the causes of death. The causes were categorized into 5 groups: cardiovascular, cancer, accidents, suicides, and other causes. Statistical methods NCSS statistical software (2004 version, www.ncss.com) was used for the analyses. In the analyses, global level of physical activity was categorized as described above. T-tests, nonparametric tests and analyses of covariance (ANCOVA) were used where appropriate to compare continuous variables; chi-square and trend tests were used to compare proportions.

( 27 )Figure 1 .Figure 2 .

 2712 Figure 1. Flow chart of the study

  126.0 (56.0-252.0) 126.0 (42.0-252.0) 126.0 (70.0-252.0) 136.0 (63.0-280.0) median (interquartile range) Units in parentheses: Mean (SD) unless otherwise stated

Table 1 .

 1 Characteristics of the study group according to the level of physical activity at baseline in 1974

		Whole group,	Low physical	Moderate physical	High physical
	Variable	n=782	activity, n=148	activity, n=398	activity, n=236
	Age, yr	47.8 (4.1)	48.4 (3.9)	47.9 (4.1)	47.4 (4.1)
	Body mass index at 25 years of age, kg/m 2	22.2 (2.2)	22.0 (2.1)	22.2 (2.3)	22.4 (2.2)
	Body mass index in 1974, kg/m 2	25.8 (2.9)	26.0 (3.0)	26.1 (2.9)	25.2 (2.6)
	Weight gain from 25 years of age until 1974, kg	11.3 (8.4)	12.6 (8.9)	12.3 (8.8)	8.8 (6.6)
	Perceived health, n (%)
	-very good or good	410 (53.9)	69 (48.6)	196 (50.5)	145 (63.0)
	-fair	320 (42.1)	69 (48.6)	175 (45.1)	76 (33.0)
	-poor or very poor	30 (3.9)	4 (2.8)	17 (4.4)	9 (3.9)
	Perceived fitness, n (%)
	-very good or good	231 (30.3)	24 (16.9)	86 (22.1)	121 (52.6)
	-fair	420 (55.2)	77 (54.2)	243 (62.5)	100 (43.5)
	-poor or very poor	110 (14.5)	91 (28.9)	60 (15.5)	9 (3.9)
	Keys' risk index,
	%/5 years, median (interquartile range)	1.7 (1.1-2.8)	1.9 (1.3-3.1)	1.7 (1.0-2.8)	1.6 (1.1-2.7)
	Blood pressure, mm Hg
	Systolic	145.7 (18.6)	145.9 (17.4)	144.8 (19.2)	147.0 (18.2)
	Diastolic	94.1 (10.7)	94.3 (10.1)	93.9 (10.9)	94.2 (10.8)
	Pulse rate, per min	64.6 (10.8)	67.8 (11.8)	64.8 (10.

Table 2 .

 2 Reported regular weekly exercise in the11-year (1985) and26-year (2000) surveys, n (%)

		Low physical	Moderate physical	High physical
		activity in 1974,	activity in 1974,	activity in 1974,
		n=148	n=398	n=236 P-value
	Weekly exercise in the 11-
	year survey, n (%)	59 (49.6)	220 (69.2)	176 (91.7) <0.001
	Weekly exercise in the 26-
	year survey, n (%)	64 (70.0)	216 (80.0)	157 (85.3)	0.002
	"Vigorous" exercise *
	in the 26-year survey, n (%)	51 (56.0)	184 (68.1)	146 (79.3) <0.001
	"Vigorous" exercise *
	≥3 times/week in the 26-year
	survey	18 (35.3)	82 (44.6)	88 (60.3) <0.001

* Reported exercise intensity leading to sweating and breathlessness