Cécile Perret
email: cecile.perret@univ-savoie.fr

Maître De Conférences

Viable territorial development in Kabylia. A social capital approach

Using a decomposition of social capital nature, this research proposes an original analysis of the interactions between the social, the environmental, the governance and the economic sphere and their impact on a viable development in Kabylia, where the "art of association" is the expression of territoriality. When the governance is bad and/or when there is distrust in institutions, populations, according to their culture and to their territory, get organized to find solutions to the missing public goods. In Kabylia, the survival of an ancestral social organization anchored in the tradition and rooted values allow the local populations to overcome their difficulties.

Introduction

The territory is the result of a process of co-construction and demarcation by interacting actors. It is the fruit of history, myths and traditions but also exogenous or endogenous constraints (natural, human, economic …). It is a reality crystallized in various configurations (clusters, local productive systems …) and specific governance. Informal organizations, social orders inspired of ancient community structures can survive. They contain several levels in Kabylia (Algeria): the âarch (the tribe) and tajmaat 1 (popular assembly of the village), formerly adapted to geographical, economical and historical constraints [START_REF] Bourdieu | réédition 2012[END_REF], Lacoste-Dujardin, 2001, 75-76).

The specific ties which unite the actors of a territory can be examined into the light of the social capital concept which joins in three types of relations: inter-group, intra-group and environmental (for example ties to the ground, tamurt imazighen in Berber langage). If social capital indicates the norms and the values which govern collective action, a "quality created between actors" [START_REF] Burt | Le capital social, les trous structuraux et l'entrepreneur[END_REF], 601 In Lallement, 2003: 5) or an 'ability to be tied' (Alexis de Tocqueville spoke about the "art of association"), then it is also the expression of the territoriality. As underlined it Hanoteau and Letourneux in 1893, all the Kabyle community is embedded in the "art of association and solidarity" as well as in the slightest interests of the private life, the family, the village or the tribe. It's the reason why our analysis is based on this concept of social capital that can be viewed on both an individual and a communal resource.

According to Gagnon et al. (2008, 1), the "viable territorial development" can be defined as being the expression of a social change characterized by the rise of the partnership, the emergence of multiple actors, alternative solutions beside the economic superstructures and the introduction of social and cultural criteria next to purely economic rationalities. The interactions between the social, the environmental, the governance and thus the economic sphere are involved.

In this research, we concentrate on the subculture of the Mounts of Kabylia in the region of Djurdjura that Romain called Mons Ferratus and his inhabitants Quinque Gentii, what means five nations or five tribes. Those tribes have no political link between them: each of them establishes a kind of nominal federation, as so many independent units, tribes … subdivided in their turn into fractions, and free villages (Daumas et Fabar, 1847, 8-9). This already reveals, in this high antiquity, a kind of federalism (Daumas and Fabar, 1847, 10).

The first part of this article deals with the nature of social capital (concept developed by Weber (1921), [START_REF] Bourdieu | réédition 2012[END_REF][START_REF] Bourdieu | Le capital social. Notes provisoires[END_REF], [START_REF] Coleman | Social capital in the creation of human capital[END_REF] …): cognitive social capital (mental processes, values, norms … which predispose the agents to cooperate), structural social capital (organizations in which the individuals act), governmental social capital (institutions or organizations), the formal civil social capital (associations, NGO) and the informal civil social capital (traditional governance systems of oral transmission). A discussion about social ties nature completes the analysis [START_REF] Coleman | Social capital in the creation of human capital[END_REF][START_REF] Loudiyi | Capital social et développement territorial. Quel impact spatial des relations sociales ?[END_REF][START_REF] Angeon | Des liens sociaux à la construction d'un développement territorial durable : quel rôle de la proximité dans ce processus ?[END_REF] and characterizes the ties in Kabylia. In the second part, we analyze a possible link between environmental capital, social capital and viable territorial development [START_REF] Gagnon | Développement territorial viable, capital social et capital environnemental : quels liens ?[END_REF]. Our conclusions show that if inhabitants of tamurt Imazighen (Kabyle ground) possesses a consequent civil cognitive social capital, a distrust in the governmental structural social capital is negative for a viable territorial development. The survival of an informal governance system of the villages, anchored on ancestral traditions, offers missing public goods but is not sufficient to perpetuate a territorial development process 2 .

THE SOCIAL CAPITAL CONCEPT

The definitions

In its posthumous book Économie et société 3 published in 1921, Weber already pointed out that three types of resources improve living conditions: economic resources, political resources and symbolic resources (social relationships). However, until the end of 1970s, the concept of social capital was not used a lot. In 1980, Bourdieu gives the following definition: "social capital is the sum of the resources, actual or virtual, that accrue to an individual or a group by virtue of possessing a durable network of more or less institutionalized relationships of mutual acquaintance and recognition" (in Bourdieu et Wacquant, 1992, 119), or, in other words, in the membership to a community, as a set of elements which are not only endowed with common properties (susceptible to be perceived by the observer, by the others or by themselves) but are also close by permanent and useful connections (Bourdieu, 1980, 2).

Social capital is presented as the effect and as the cause of social institution. It is an individual resource inherited from a particular social class, an attribute of the group or of the social network (Kamanzi and al., 2007, 129). It explains the social class disparities. Here, social capital reflects the saying "It's not what you know, it's who you know" (Woolcock and Narayan, 2000, 225).

Coleman outlines "two broad intellectual streams in the description and explanation of social action" (Coleman, 1988, 95). The first one is a sociological approach: the individuals, in a social and cultural environment, are subject to norms, rules, and obligations (the actor is a product of his environment). The second one is an economic approach: independent individuals seek to fulfil their goals constrained by the social context (norms, interpersonal trust, social networks, social organizations are important in the functioning of the society).

Social capital is defined by interpersonal relations that characterize a social order. The social capital is not only seen as a stock held by powerful elites, but its value for all kinds of communities, including the marginalized or powerless. Coleman also highlights the links between micro and macro: the (private) resources that an individual removes from the valuation of his social capital have repercussions in the macroeconomic level.

Robert Putnam also presents remarkable works around this concept (1995,1996,2000). His definition underlines the utilitarian character of social relationships and behavior and the strategic attitudes of the individuals possessing a social capital. If [START_REF] Bourdieu | Le capital social. Notes provisoires[END_REF] points out the private use of this resource (he analyzes how "elites" can use their social networks to keep their privileges), Putman (1995Putman ( or 2000) ) underlines its collective aspect: within a group "irrigated" by interpersonal relations networks, trust and reciprocity facilitate the collective action (Houard and Jacquemain, 2006, 8). Putman (2000) argues that social capital "allows resolving collective problems, increasing individual benefits by mutual cooperation, ensuring compliance with established norms and alleviating the individual burden in carrying out their missions" (Kapucu, 2011, 25).

If the concept of social capital was, at first, associated with a strictly individual utility, thus he was quickly connected with more collective concerns (as governance and community development).

Social ties and organizations

We can distinguish two streams of the social capital: the first one in terms of values and norms and the other one in terms of relational networks. In other words, it's the distinction between the cognitive social capital, that is the mental processes of the individuals, the values, the norms, the faiths etc. which predispose the agents to cooperate and the structural social capital, that is the structures (organizations) in which the agents act [START_REF] Uphoff | Understanding Social Capital: Learning from the Analysis and Experience of Participation[END_REF]. Collier (1998) also differentiates two interdependent types of social capital: the governmental social capital (institutions, State, public sector) and the civil social capital (organizations emerging from the civil society). Once crossed, these two visions of social capital, allow us to raise a typology of the concrete nature of this concept in the style of [START_REF] Sirven | Capital social et développement : quelques éléments d'analyse[END_REF] (cf. figure 1). We could add an oral tradition informal civil structural social capital like, for example, the governance system of the Kabyle village. Even nowadays, this informal governance system survives through the "committees of village" (or "the committees of district" in urban area) and runs in parallel of local formal institutions (Perret and Abrika, 2013-a). These committees possess their own budget, fed by contributions (of the villagers and emigrants of the village), gifts … to finance collective utility projects (pavement of streets, pipelines of drinkable water …). However, before dashing into important public works, committees contact the local institutions (city hall, daïra …) to be in conformity (technical norms…). Traditionally, this "villages republic" (Camps, 2007, 297), contains several levels, of which the tajmaat (the "assembly" which is male) and the âarch (the tribe) (Lacoste- , 2001, 75). The âarch is a federation of several villages bound by an eponym ancestor (Direche-Slimani, 2006, 185). Tajmaat (where the elder or the heads of the family can take part in)

Dujardin

possesses the political, administrative and judicial power. Sheik Mohand 4 identifies the basements of the Kabyle community as being tajmaat, the value of the effort and the importance of the brothership (Adli, 2010b, 47-48). The chief of the village (Amghar or Amqran) is elected by the members of tajmaat, coomposed of representatives of the village's families. Tajmaat, real court of justice, applies its decisions to all the domains of life, according to the law of the village (azref tadart (tardart significates village)), a kind of Constitution. Azref tadart rests on a custom transmitted orally (Llâda (significates tradition)) by the memory of the elders (ouqal). Llâda is adapted by tajmaat in the various villages (taârfit) (Adli, 2010a, 181). Llâda and taârfit are based on the equality of citizens (Adli, 2010a, 181). The non compliance with these laws, today the internal rules of the village (which can be drafted in a series of articles), can give rise to a financial penalty or a social avoidance.

Nature of social ties

The nature of the ties is specified by the well-known World Bank (2000) typology: the bonding ties, the linking ties and the bridging ties. The bonding ties unite individuals of identical status (horizontal ties). They characterize community relations, that is people adhering to the same representation system, but aim towards a "relational closure" [START_REF] Coleman | Social capital in the creation of human capital[END_REF] … even can result in discrimination.

Those ties can be the ones existing within a family, a tribe, an ethnic group, a village, a community, close friends, etc. The linking ties characterize interactions between agents across a cleavage that typically divides society (vertical ties between social classes). They require to be reaffirmed and are characterized by reciprocity transactions. Their frequency tends to result in the convergence of representations. Finally, the bridging ties link distant agents; this distance can be geographical (an emigrated from the village) or the tie is not permanently activated. The bridging network is a virtual one. The relations can be punctual, intermittent and the respected rules are similar to "an agreement without mutual commitment" (Angeon et al., 2006, 10). Bonding, linking and bridging ties "can assist in the increasing of trust between individuals, and as such, promote a healthy and expanded social network" (Kapucu, 2011, 24). According to the World Bank, the bridging network between members of the same community or with the members of other communities can allow to obtain better public services, an improvement of the infrastructures or to access certain markets or the credit. Loudiyi et al. (2004, 10) define three categories of simplified actors: the first one is a collective of individuals without specific common objective (G), the second one is a productive group (PG), a collective having a finalized common goal and sharing common rules and the third one is the institutions (I). To these three simplified actors, the authors associate recognizable qualities, dominant nature of social capital. To G, they associate the "live together" (bonding tie), to PG they associate the "produce together" (linking tie) and to the institutional actors (I) they associate the "organize together"

(bridging tie). The evolution of an actor in another one results from the evolution of their role and to the presence of a dominant tie.

Bonding ties are primordial in Kabylia. The family is a social primary unit, its members share a deep vision of the life in society. The most important, is the "will to live together", shared by all the inhabitants (Adli, 2010-b, 20). Without the family and the respect to its norms, individuals "will front of huge disappointments". A popular expression specifies moreover: "in his isolation a man is always disqualified" ("awhid yenghat ccraâ") (Adli, 2010b, 20-21). In the Kabyle village, territorial and political primary unit which relies the citizens, horizontal ties unite the villagers who share the same system of representation. However, the distant ties (bridging ties) with the emigrant community are essential to the development of the villages because they financially contribute to the budget of the villages and they also transfer knowledge. We could say that it's "the strength of weak ties" [START_REF] Granovetter | The strength of weak ties[END_REF]. The committee of village is a productive group in the manner of [START_REF] Loudiyi | Capital social et développement territorial. Quel impact spatial des relations sociales ?[END_REF], a collective of actors having a finalized common objective, with common rules which can, if they are not respected, give rise to penalties (financial ones, social exclusion, etc.). The villagers are the core of the territorial informal governance.

The Kabyle community is characterized by the survival of tiwiza 5 . Tiwizi is an Amazigh (Berber) feminine word which comes from the verb iwiss (to help). Tiwizi is a voluntary work realized for the community or a particular family. The word tiwiza, which is a plural, is sometimes written touiza this type of organization arose because, historically, the Amazigh community (in Algeria, Morocco, …) has never known neither slaves, nor serfs, nor system of sharecropping. Tiwiza establishes a cooperation otherwise a community development. Thus, we can say that the "civil connectivity" (Gagnon and al., 2008, 1), who testifies of the presence of a dense meshing in a community, is strong in Kabylia.

SOCIAL CAPITAL AND VIABLE TERRITORIAL DEVELOPMENT IN KABYLIA

Social capital and trustworthiness

Ties intensity is often presented as an explanatory discriminating element of economic development. [START_REF] Putnam | Making democracy work: civic institutions in modern Italy[END_REF] underline that if the North of Italy is more developed than the South it is thanks to a better social capital endowment. In a search on Russia, Rose (2000) demonstrated that the leaders of organizations such as churches, clubs, political parties … mobilize to offer the missing public goods in case of failure of the State (In Fafchamps, 2007, 12). But, are civil social capital and governmental social capital substitutable or complementary? According to Sirven (2004, 136), they are complementary in case of good governance (participatory, consensus oriented, transparent …) because the implementation of ties with the civil society favors social cohesion. Nevertheless, dangers can appear, even in the presence of good governance, when the community is composed of isolated groups (the most powerful group can then monopolize the governmental structures, a latent conflict endangers the community). In case of bad governance (low corruption perception index …), the community is in conflict with itself and there is a possibility of civil war. The lack of efficiency of the State can be mitigated by an important "informalisation" of the economy or the organization of the civil society.

We adapted below (cf. figure 2) the various situations discribed by Sirven (2004, 137). To Arrow, if the study of the social ties is interesting, the word social capital in itself is subject to controversy. It supposes the duration, sacrifices in the present for future profits and the alienability (Arrow, 1999, 4). Yet, to him, the essence of social relationships is that they are built for other reasons than their economic value (Arrow, 1999, 3), which questions the second condition. Solow agrees, writing that social capital is a set of elements like trustworthiness, ability to cooperate and to coordinate, habit to contribute to the common effort even if nobody looks (Solow, 1999, 7). We should better discuss of "behavior pattern" embedded in a specific community. [START_REF] Dasgupta | Une question de confiance : capital social et développement économique[END_REF] agrees too, writing that it is the development of the trustworthiness between the people that conditions economic development. He defines the social capital as networks between people able to establish and maintain trust. But, if social capital operates in a bad environment, he can hinder the economic development, even engender an economic decline. In Algeria, the social solidarity is often of informal nature, it exists through the family in the broad sense, the village and the personal networks called "the knowledge" (tamusni or in the plural timusniwin in Berber langage). We hear frequently expressions like "I have some knowledge in such administration", "I am going to call my knowledge" ... This "knowledge" can get closer the "dark side" because it is also used to obtain privileges (to find a job, to quickly obtain a telephone line …). Even the World Bank recognises "that personnal connections can be used to unfairly discriminate, distort and corrupt" (Woolcock and Narayan, 2000, 226)..

The question of trustworthiness settles differently if we assume that Human is rather selfish, rational or if we admit that he is guided by considerations. The notion of "given word" is very important in Kabylia. Traditionally, it is by the said and its metaphors, the loyalty, the bravery, the science and the knowledge, the value of the work and the given word that the man can claim he is rich (Adli, 2010b, 18). These immaterial values (tirugza) define the argaz (tabargazt in the feminine), word which could be translated in courageous or honest man. This ethics are either transaction rules or the trustworthiness between people who don't know each other but belong to the same community.

To [START_REF] Dasgupta | Une question de confiance : capital social et développement économique[END_REF], if the institutions hasn't been reliable in the past, it is very difficult and take time to knock down the faiths. Then, the transactions will be developed at the individual level. It is undeniable that in Algeria the problem of the trustworthiness into the institutions is accurate, particularly in Kabylia, characterized by an history peppered with revolts (Berber Spring in 1980, Black Spring in 2001 …) and by its resistance during the colonizations (Ottoman, Arab, French…). In Algeria, country composed by different subcultures 7 , the corruption perception index is low (34/100 in 2012 according to International Transparency). An applied research of Perret and Abrika (2013-b) demonstrate that trustworthiness into institutions and representatives of the public institutions is extremely weak in Kabylia, particularly at the level of the closer institutions (Wilaya …). That could explain the importance of community bonding ties (we trust in the family, in close friends, in the inhabitants of our village).

The viable territorial development

As an individual resource, "social capital plays a prevalent role in economic performance as an individual, organization, and country through its support of increased informational flow and reduction of transaction costs" (Kapucu, 2011, 24). The viable territorial development (VTD) points out the relations between the social and environmental variables (quality of the human development, wellbeing) through the notion of capital (Gagnon, 2008, 2). If the social capital concept, although being polysemous, is today well known, that of environmental capital is less usual. The environmental capital refers to given and constructed natural, economic, human and cultural capital. The viability supposes the sustainability of a dynamic system, thus it deals with criteria of equity (an income distribution preserving the social peace for example) and criteria of ethics (virtuous behavior pattern between the actors and between the actors and their natural environment for example). Their non compliance generates tensions between the actors of the system, even open conflicts (strikes, revolts, civil wars …). A viable development supposes the respect of the "Alive" (respect of the Man and respect of natural environment). The respect for deeply rooted identities subject to evolution (tiwiza tends to get lost as evolve the forms of solidarities), must be taken into account to preserve the balances (avoid social tensions ….). The trustworthiness between the actors and between the actors and their institutions must be strong so that a territorial dynamics is viable. Gagnon and al. (2008, 5) identify three strategic factors of the territorial dynamics: the integration of a territory in a regional area (it sends back to the idea of the opening, the transactions …), the "civil connectivity" and, finally, in absence of favorable socioeconomic characteristics, the territories where the actors are dynamic, proud of their identity, where there is a participative governance and a longterm vision. The informal participative governance is undeniable in Kabylia (Hanoteau and Letourneux specified in 1893 that this political and administrative organization was one of the most democratic and, at the same time, one of the simplest we can imagine), the pride of the "kabylity" is obvious but a long-term vision is missing. We voluntarily use this word "kabylity" rather than "berberity" because Kabyle are not only proud to be Berber but are also proud to belong to a territory which has a long tradition of resistance. However, being pride of this membership in a community embedded in a specific territory should not be transformed into community enclosure dangerous to the integration into the regional or national area. The results of the applied research implemented by Perret and Abrika (2013b) demonstrate that there is not specific community enclosure in Kabylia. In the scheme presented below (Scheme 1), we synthesize the different variables (environmental capital and social capital desegregated in three parts) which could impact on the territorial development viability. We have, also, indicated through dotted line arrows, the values impacting a process of virtuous development.

Scheme 1 -Synthesis

Réalisation personnelle.

• The link between the cognitive civil social capital and the informal structural social capital

The values deeply anchored in the Kabyle community (tirugza) perpetuate the committees of villages which survive themselves only if the Kabyle identity remains alive.

• The link between the cognitive civil social capital and the environmental capital

We know that social capital "is one of the core values which have a positive effect on individuals as well as organizational effectiveness" (Kapucu, 2011, 25). The "connectivity", the "art of association • The link between the informal structural social capital and the environmental capital If local authorities and associations are usually involved in natural resources management, it is very insufficient in Kabylia. The collection of household waste is poor and the lack of garbage dumps is a real danger for the environment. It's why, for example, a village named Zouvga, thanks to the awareness of his committee, conducts actions to protect its natural environment (acquisitions of trash cans …). Zouvga is today considered as the cleanest village in Kabylia. The "connectivity" impacts on the level and on the quality of the environmental capital.

• The link between cognitive civil capital and structural social capital and the cognitive governmental social capital

If the civil social capital is strong in Kabylia (identity and values), the distrust in the governmental structural capital is obvious. This situation could explain the pessimistic speeches about the future of this territory we frequently hear on the ground. It seems to be necessary that the authorities propose a vision of the future and restore trust, while respecting a strong identity, to generate a viable development dynamic. If the last condition is not respected, the dynamics of the system could be impacted on by tensions/conflicts.

• The link between structural and cognitive government social capital and environmental social capital

The governance and the legislation impact on the environmental capital. For example, the quality of the public school impacts on the accumulation of human capital which can himself impact on the evolution of the governance.

• The link between informal structural social capital and the governmental structural or the cognitive governmental social capital Governmental informal and formal structural social capital can be complementary or subsidiary, but a complete substitution among one to another would mean or "the independence" of the Kabyle village (if the informal structural capital subsists only), or the disappearance, even the negation of the traditions of organization (if the governmental social capital subsists only).

CONCLUSION

In Kabylia, territory where an informal organization anchored in the tradition survives, the "art of association" or "the propensity to generate ties" is also the expression of the territoriality. In the case of bad formal governance and/or when there is reliable distrust in governmental institutions, the populations, according to their culture, their history and their territory, get organized to find solutions.

It is the survival of an ancestral social organization (tajmaat and âarch), underlain by certain values (democracy, mutual aid …), which sometimes, allowed the Kabyle population to overcome its difficulties. Informal social structural capital have implications for economic devlopment and poverty reduction.

In the village, territorial and political primary unit, citizens share the same system of representation and are proud of their identity and bonding ties and tiwiza are the core of the informal governance.

Local coordination, through the village committee and collective utility work who offers missing public goods, impact on the process of territorial development and environmental protection. The involvement in committees and tiwiza is the sign of the involvement of the inhabitants in the political debate of the village, mobilization to the village development and protection of the Kabyle culture (you are an argaz if you freely help the others…) thus perpetuating the traditions.

If ties are potential resources, they become a capital when they are activated. The activities of the committees demonstrate that there is a transformation of the "art to create ties" into social capital. The production of resources results from the long tradition of reciprocity, from a culture and from practices embedded in a specific territory. Social capital can be used alongside other resources such as human capital, physical capital or economic capital.

Governmental structural formal and informal social capital must be complementary. In case of total substitution from one to another, two situations can arise: either a situation of "independence" of the village -which could impact on the integration in the sense of [START_REF] Gagnon | Développement territorial viable, capital social et capital environnemental : quels liens ?[END_REF] -or in a situation of disappearance of the ancestral governance systems. In both cases, social tensions could appear and impact on the territorial development.

Figure 1 -

 1 Figure 1 -Concrete nature of the social capital Nature of the social capital Structural social capital Cognitive social capital Civil social capital Associations, NGO … Norms, values, faiths, "art of association and solidarity", trustworthiness … Governmental social capital State, local organizations… Laws, political system …

(

  arabized term), or twiza. It means mutual aid, solidarity, voluntary service, voluntary work or generosity. It deals with the sharing of the collective work to offer the missing public goods (pavement of a street, construction of water pipelines, …) or the solidarity towards those who need help (elders, orphans, an owner of olive ground in the season of the picking …). According to Mohamed Azergui 6 ,

Figure

  Figure 2 -Typical case Social gouvernemental capital Social civil capital

  modifies the importance granted to certain values (familial solidarity decreases when social redistribution increases).

  ", has an impact on the development of the environmental capital (new business start-up, creation of associations, positive externalities of the mutual aid …) in which Men live. Conversely, the

	REGION	COGNITIVE
		CIVIL SOCIAL
		CAPITAL
		Identity,
		connectivity,
		Berber language
		Associated values:
	Associated values: Tirugza …	ability to get organized, innovation …
	INFORMAL	ENVIRONMENTAL
	STRUCTURAL	CAPITAL
	SOCIAL CAPITAL	economic and human
	Committees of	capital,
	village	cultural and natural
		endowment
	Associated values: complementarity, subsidiarity	Associated values: ethics, good
		governance
		STRUCTURAL AND
		COGNITIVE
		GOVERNMENTAL
		SOCIAL CAPITAL
		Associations, laws …

Numerous authors use the word jemaâ instead of tajmaat. We shall use the word tajmaat. It means assembly or meeting place.

Tamurt Imazighen, terre kabyle" is the title of Martial Rémond's work republished by Annic Droz, Ibis Press. The word tamurt is translated on the ground.

[START_REF] Weber | Économie et société, réédité en janvier 2003 aux éditions Pocket[END_REF]. Économie and société, Paris, Plon.

Kabyle poet (1836Kabyle poet ( -1901)).

Tiwiza survives even if we can see a relative social disintegration due to the migrations, to the transformation of the family (more conjugal homes), to the consequences of the civil war during the 90's or to the urbanisation.

See the article "Tiwizi (or tradition amazigh of active solidarity)" By Azergui Mohamed on the site: http: // tawiza.x10.mx / Tawiza131 / tiwizi.htm [accessed 20 january 2014].

According toMercury and al. (1997, 26-29), eight subcultures can be listed in Algeria: coastal Zone, High plateaus, Steppe, the South, Mounts of Kabylia, Mounts of Aurès, Tlemcen, Maghnia-Sebdou and M'zab.


A F M Ter Elst

E M Ouhabaz

Partial spectral multipliers and partial Riesz transforms for degenerate operators

Keywords: AMS Subject Classification: 42B15, 45F05 Spectral multipliers, Riesz transforms, singular integral operators, degenerate operators, Gaussian bounds. Home institutions:

We consider degenerate differential operators

respectively, Ω a bounded Lipschitz domain) and suppose that (a kj ) ≥ µ > 0 a.e. on supp χ (resp., a.e. on Ω). We prove a spectral multiplier type result: if

). We also prove boundedness on L p for all p ∈ (1, 2] of the partial Riesz transforms M χ ∇(I +A) -1/2 M χ . The proofs are based on a criterion for a singular integral operator to be weak type (1, 1).

Introduction

Let A be a non-negative self-adjoint uniformly elliptic operator in divergence form. More precisely, let a kj = a jk : R d → R be a bounded measurable function for all j, k ∈ {1, . . . , d} and assume that there exists a µ > 0 such that (1) ∂ k (a kj ∂ j ), defined by quadratic form techniques, is self-adjoint on L 2 (R d ). It is a standard fact that -A is the generator of a strongly continuous semigroup (e -tA ) t>0 on L 2 (R d ). The well known Aronson estimates assert that e -tA is given an integral kernel p t (called the heat kernel of A) which satisfies the Gaussian upper bound:

(2) |p t (x, y)| ≤ Ct -d/2 e -c |x-y| 2 t for all t > 0 and x, y ∈ R d .

Here C and c are positive constants.

In recent years, harmonic analysis of operator of type A has attracted a lot of attention and substantial progress have been made in which upper bounds for the heat kernel play a fundamental role. We mention for example the theory of Hardy and BMO spaces associated with such operators (see for example [DY] and [HLMMY]), spectral multipliers [DOS] and Riesz transforms (see [START_REF] Duong | The L p boundedness of Riesz transforms associated with divergence form operators[END_REF], [Aus], [Ouh], [She] and the references therein). Concerning spectral multipliers, it is known that if F : [0, ∞) → C is a bounded measurable function then the operator F (A), which is well defined on L 2 by spectral theory, extends to a bounded operator on L p for all 1 < p < ∞ provided F satisfies the condition (3) sup t>0 ϕ(.)F (t.) C s < ∞ for some s > d/2 and some non-trivial auxiliary function ϕ ∈ C ∞ c (0, ∞). See Duong-Ouhabaz-Sikora [DOS], where a more general result is proved. Note that condition (3) is satisfied if F has [d/2] + 1 derivatives such that sup λ>0 λ k |F (k) (λ)| < ∞ for all k ∈ {0, 1, . . . , [d/2] + 1}.

As an example, one obtains polynomial estimates on L p for imaginary powers of type A is p→p ≤ C(1+|s|) βp for all β p > d| 1 2 -1 p |. Taking F (λ) := (1-λ R ) α + one obtains Bochner-Riesz summability for all α > d/2. Concerning Riesz transforms R k := ∂ k A -1/2 , it is an obvious consequence of the ellipticity assumption (1) that R k is bounded on L 2 (R d ) for all k ∈ {1, . . . , d}. As for multiplier results, the Gaussian bound (2) combined with recent developments on singular integral operators allow to prove that R k is bounded on L p (R d ) for all p ∈ (1, 2) with the sole assumption (1) and bounded measurable coefficients (see , Auscher [Aus], Ouhabaz [Ouh]). Under weak regularity assumption on the coefficients one obtains boundedness of R k on L p (R d ) for all p ∈ (2, ∞) (cf. Auscher [Aus], Shen [She]).

In the present paper we wish to study similar problems for degenerate operators. Instead of (1) we merely assume that (4) d k,j=1 a kj (x)ξ k ξ j ≥ 0 for all ξ = (ξ 1 , . . . , ξ d ) ∈ R d and x ∈ R d .

In this case, we define the form (5)

a 0 (u, v) = d k,j=1 R d a kj (∂ j u) (∂ k v)
with form domain D(a 0 ) = C ∞ c (R d ). If this form is closable, then A will be the self-adjoint operator associated with its closure. If not, we take the regular part and consider A as the operator associated with the closure of this regular part (see [Sim] and [AE]).

Proving results like the previous ones for these operators seems unattainable because Gaussian (or Poisson) upper bounds are not true in general. Even L 1 -L ∞ estimates of e -tA are not valid in general. What we will do is to restrict the operators to parts where the matrix (a kj ) is elliptic. It is proved by ter Elst and Ouhabaz [START_REF] Elst | Partial Gaussian bounds for degenerate differential operators[END_REF]

that if χ ∈ C ∞ b (R d
) and µ > 0 are such that (a kj (x)) ≥ µI for a.e. x ∈ supp χ, then M χ e -tA M χ has a Hölder continuous kernel K t which satisfies the Gaussian bound

(6) |K t (x, y)| ≤ Ct -d/2 e -c |x-y| 2 t (1 + t) d/2 for all t > 0 and x, y ∈ R d .
Here M χ is the multiplication operator by χ. The same result holds for P Ω e -tA P Ω if Ω is a bounded Lipschitz domain such that (a kj (x)) ≥ µI for a.e. x ∈ Ω for some µ > 0. Here P Ω is the multiplication operator by the indicator function 1 Ω of Ω.

Note that in general one cannot get rid of the extra term (1 + t) d/2 in the right hand side of (6). For example, if a kj = δ kj on a smooth bounded domain Ω, then A is the Neumann Laplacian on L 2 (Ω) and 0 on L 2 (R d \ Ω). It is then clear that the Gaussian bound is not valid without the additional term (1 + t) d/2 . Because of that additional term in (6), we shall consider in the sequel I + A instead of A (of course, one can take εI + A for any ε > 0 to absorb the factor (1 + t) d/2 ).

For spectral multipliers and Riesz transforms we will prove the following results

. Given χ ∈ C ∞ b (R d ) (resp.
, Ω a bounded Lipschitz domain) such that (a kj (x)) ≥ µI for a.e. x ∈ supp χ (resp., for a.e. x ∈ Ω) for some constant µ > 0. The main theorems of this paper are the following.

Theorem 1.1. Let F : [0, ∞) → C be a bounded function such that sup t>0 ϕ(.)F (t.) C s < ∞ for some s > d/2 and some non-trivial function ϕ ∈ C ∞ c (0, ∞). Then M χ F (I + A)M χ (resp., P Ω F (I + A)P Ω ) is bounded on L p (R d ) for all 1 < p < ∞.
Theorem 1.2. For all k ∈ {1, . . . , d} the Riesz transform type operator

M χ ∂ k (I + A) -1/2 M χ is bounded on L p (R d ) for all 1 < p ≤ 2.
Now we discuss how we prove these results. In the elliptic case, besides the Gaussian bound (2), the proof of boundedness of spectral multipliers or Riesz transforms rely on a criterion proved by Duong and McIntosh [DM2] for singular integral operators to be weak type (1, 1). This criterion says that if T is bounded on L 2 with a (singular) kernel K such that there exists a family of operators (A t ) t>0 given by integral kernels a t which satisfy Gaussian (or Poisson) bounds, T A t is also given by a (singular) kernel K t and there are C, δ > 0 such that

(7) |x-y|≥δ √ t |K(x, y) -K t (x, y)| dx ≤ C
for all t > 0 and a.e. y, then T is weak type (1, 1). In applications to spectral multipliers of elliptic operators we start with T = F (A) and one takes A t = e -tA . Therefore, K t is the kernel of the operator F (A)e -tA which can be seen as a regularization of F (A). In the degenerate case and because of (6), it is tempting to choose A t = M χ e -t(I+A) M χ . Then,

T A t = M χ F (I + A)M χ M χ e -t(I+A) M χ = M χ F (I + A)M 2 χ e -t(I+A) M χ .
Now, the presence of M 2 χ does not allow to regularize F (I + A) by e -t(I+A) . The simple fact that we do not have F (I + A) next to e -t(I+A) in the expression of T A t destroys this strategy. The same problem occurs for the Riesz transform M χ ∂ k (I + A) -1/2 M χ . To overcome this difficulty we prove a version of the Duong-McIntosh criterion that is suitable for our purpose. It reads as follows (see Theorems 2.1 and 2.3 together with Remark 2.2 for precise and quantitive statements).

Theorem 1.3. Let T be a bounded linear operator on L 2 and (A t ) t>0 a family of linear operators which satisfy L 1 -L 2 off-diagonal estimates. Suppose that there exists a bounded linear operator S on L 2 and δ, W > 0 such that

(8) |x-y|≥(1+δ)t |(T -SA t )u(y)| dy ≤ W u 1 for all x ∈ R d , t > 0 and u ∈ L 1 ∩ L ∞ supported in the ball B(x, t). Then T is weak type (1, 1).
Note that the estimate (8) is satisfied if T and SA t are given by (singular) kernels K and K t and there are C, δ > 0 such that

|x-y|≥δ √ t |K(x, y) -K t (x, y)| dx ≤ C
for all t > 0 and a.e. y ∈ R d . Theorem 1.3 gives more freedom by choosing any appropriate operator S and not necessarily S = T . Coming back to spectral multipliers for degenerate operators A, we had T = M χ F (I + A)M χ and we choose now S = M χ F (I +A) and A t = e -t(I+A) M χ . Then T A t = M χ F (A+I)e -t(I+A) M χ for which we can prove the estimate in Theorem 1.3. Similarly for the Riesz transforms where

T = M χ ∂ k (I + A) -1 M χ , we take S = M χ ∂ k (I + A) -1
which turns to be bounded on L 2 and A t = e -t(I+A) M χ . We emphasize that A t = e -t(I+A) M χ satisfies L 1 -L 2 off-diagonal estimates but it is not known whether it satisfies Gaussian upper bounds in general 1 . We believe that our version of the Duong-McIntosh criterion can be used in other circumstances in which a products of several operators come into play. Also, as in [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF], our version holds for operators on domains of spaces of homogeneous type.

Notation. We fix some notation which we will use throughout this paper. If (X, ρ, µ) is a metric measured space, x ∈ X, r > 0 and j ∈ N, then we denote by B(x, r) := {y ∈ X : ρ(x, y) < r} the open ball of X with centre x and radius r, the annulus C j (x, r) = B(x, 2 j+1 r) \ B(x, 2 j r) if j ≥ 2 and C 1 (x, r) = B(x, 4r). Let v(x, r) = µ(B(x, r)) be the volume of the ball B(x, r). Next, T p→q is the norm of T as an operator from L p to L q . If E is a measurable set, then P E denotes the multiplication operator by the indicator function 1 E of E. If s ∈ (0, ∞) \ N then we denote by C s the space of all Lipschitz functions on [0, ∞) of order s (i.e., functions which are continuously differentiable up to [s] and the derivative of order [s] is Hölder continuous of order s -[s]). By W r,p we denote the classical Sobolev spaces on R d .

All our operators are linear operators.

We emphasize that we shall use C, C , c, . . . for all inessential constants. A constant C may differ from line to line, even within one line.

Singular integral operators

Let (X, µ, ρ) be a metric measured space. We shall assume that 0 < v(x, r) < ∞ for all x ∈ X and r > 0 and that X is a space of homogeneous type. This means that it satisfies the following doubling condition

(9) v(x, 2r) ≤ C 0 v(x, r)
for some C 0 > 0, uniformly for all x ∈ X and r > 0. If (9) is satisfied then there exist positive constants C 1 and d such that

(10) v(x, λr) ≤ C 1 λ d v(x, r)
for all x ∈ X and r ≥ 1. Let Ω be an open subset of X. It is endowed with ρ and µ but (Ω, µ, ρ) is not necessarily a space of homogeneous type. Let T be a bounded linear operator on L p 0 (Ω) := L p 0 (Ω, µ) for some

p 0 ∈ [1, ∞).
We say that T is given by a kernel

K : Ω × Ω → C if K is measurable and (11) T u(x) = Ω K(x, y) u(y) dµ(y)
for all u ∈ L p 0 (Ω) with bounded support and a.e. x outside the support of u. We also say that K is the associated kernel of T . A classical problem in harmonic analysis is to find conditions on the kernel K which allow to extend the operator T from L p 0 (Ω) to other L p (Ω)-spaces. Several results are known in this direction. We refer the reader to [Ste], [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF], [BK], [Aus] and the references therein.

The main result in [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF] states that if there exists a family of bounded operators A t , with t > 0, which are given by integral kernels a t satisfying a Gaussian or Poisson estimate and if the associated kernel of T -T A t does not oscillate too much in a certain sense then T is weak type (1, 1). Here we prove by the same method that if there exists a bounded operator S on L q 0 (X) for some q 0 ∈ (1, ∞) such that the associated kernel of T -SA t does not oscillate too much then T is weak type (1, 1). As explained in the introduction, this new version gives more freedom by choosing any appropriate S which may not coincide with T . This extension turns out to be powerful to prove spectral multiplier type results as well as Riesz transforms for degenerate operators, whereas it is not clear how to apply the condition from [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF]. Note also that, following ideas from [BK] and [Aus] we can weaken the assumption on the kernel of A t . Instead of assuming a Gaussian or Poisson bound, we merely assume an L 1 -L q 0 off-diagonal estimate (see (12) below). This difference is again illustrated in our application to degenerate operators. In addition it is possible to formulate the result in [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF] without reference to the kernels (see also the remark immediately after the next theorem).

We first state and prove the result in the case Ω = X.

Theorem 2.1. Let T be a non-zero bounded linear operator on L p 0 (X) for some p 0 ∈ (1, ∞). Suppose there exists a bounded linear operator S on L q 0 (X) for some q 0 ∈ (1, ∞), a family of bounded linear operators (A t ) t>0 on L q 0 (X) and a sequence (g(j)) j∈N in R such that

(12) 1 v(x, 2 j+1 t) C j (x,t) |A t f | q 0 1/q 0 ≤ g(j) 1 v(x, t) B(x,t) |f |
for all x ∈ X, t > 0, j ∈ N and f ∈ L q 0 (B(x, t)), and ∞ j=1 2 jd g(j) < ∞. Finally, suppose there exist δ, W > 0 such that

(13) X\B(x,(1+δ)t) |(T -SA t )u| ≤ W u 1 for all x ∈ X, t > 0 and u ∈ L 1 (X) ∩ L ∞ (X) supported in the ball B(x, t).
Then T is a weak type (1, 1) operator with

(14) T L 1 →L 1,w ≤ C(1 + δ) d W + T p 0 →p 0 + S q 0 q 0 →q 0 T 1-q 0 p 0 →p 0 .
Here C is a constant depending only on the constants in (10). In particular, T extends to a bounded operator on L p (X) for all p ∈ (1, p 0 ).

Remark 2.2. Let p 0 , q 0 ∈ (1, ∞), T ∈ L(L p 0 (X)
) and for all t > 0 let S, A t ∈ L(L q 0 (X)). Suppose that T and S A t have a kernel K and K t , respectively. Let δ, W > 0 and assume that ( 15)

ρ(x,y)≥δt |K(x, y) -K t (x, y)| dµ(x) ≤ W < ∞, for all t > 0 and y ∈ X. Fix now x ∈ X, t > 0 and u ∈ L 1 (X) ∩ L ∞ (X) supported in the ball B(x, t). Then X\B(x,(1+δ)t) |(T -SA t )u(y)| dµ(y) = X\B(x,(1+δ)t) | B(x,t) K(y, z) -K t (y, z) u(z) dµ(z)| dµ(y) ≤ X ρ(y,z)≥δt |K(y, z) -K t (y, z)| dµ(y) |u(z)| dµ(z) ≤ W u 1 .
Thus, ( 13) is satisfied. The condition ( 15) is the direct analogue of the condition in .

We also observe that one does not need kernels for both operators T and T A t but a kernel H t (x, y) for the difference T -SA t . We may then replace K(x, y) -K t (x, y) in ( 15) by H t (x, y). On the other hand the local estimate (13) which does not appeal to kernels may have advantage of avoiding measurability questions with respect to x and y of the expected singular kernels.

Proof. As mentioned before, the arguments are similar to the arguments used in [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF] and [Aus]. We give the details for convenience. Recall we denote by C all inessential constants.

We begin by the classical Calderón-Zygmund decomposition. There exist c, N > 0 such that the following is valid.

Fix f ∈ L 1 (X) ∩ L ∞ (X) and α > f 1 µ(X) . There exist g, b 1 , b 2 , . . . ∈ L 1 (X) ∩ L ∞ (X) such that f = g + b = g + i b i and (i) |g(x)| ≤ cα for a.e. x ∈ X, (ii) each b i is supported in a ball B i = B(x i , r i ) and b i 1 v(x i ,r i ) ≤ cα, (iii) i v(x i , r i ) ≤ c f 1
α , and, (iv) there exists a constant N such that i 1 B i (x) ≤ N for a.e. x ∈ X.

See Section III.2 in [CW].

We proceed in several steps.

Step 1 Using the boundedness of T on L p 0 we have

µ({x ∈ X : |(T g)(x)| > α}) ≤ T p 0 p 0 →p 0 α p 0 g p 0 p 0 ≤ Cα p 0 -1 T p 0 p 0 →p 0 α p 0 g 1 ≤ C T p 0 p 0 →p 0 α g 1 .
It follows from (ii) and (iii) above that b 1 ≤ c f 1 and hence

g 1 ≤ (1 + c) f 1 . Therefore, (16) µ({x ∈ X : |(T g)(x)| > α}) ≤ C T p 0 p 0 →p 0 α f 1 .
Step 2 We shall prove that (17

) i A r i b i q 0 ≤ Cα 1-1 q 0 f 1/q 0 1
.

We follow similar arguments as in [Aus]. Fix u ∈ L q 0 with u q 0 = 1, where q 0 is the dual exponent of q 0 . Let i, j ∈ N and set C i,j := C j (x i , r i ). Then

C i,j |A r i b i | |u| ≤ C i,j |A r i b i | q 0 1/q 0 C i,j |u| q 0 1/q 0 ≤ g(j) v(x i , 2 j+1 r i ) 1/q 0 v(x i , r i ) |b i | C i,j |u| q 0 1/q 0 ≤ cαg(j)v(x i , 2 j+1 r i ) 1 v(x i , 2 j+1 r i ) C i,j |u| q 0 1/q 0 ,
where we have used (12) and property (ii) in the Calderón-Zygmund decomposition. Denote by M the Hardy-Littlewood maximal operator. Then

1 v(x i , 2 j+1 r i ) C i,j |u| q 0 ≤ M(|u| q 0 )(y)
for all y ∈ B i . Combining the previous inequalities and using the doubling condition (10) one estimates

C i,j |A r i b i | |u| ≤ Cα2 jd g(j)v(x i , r i ) M(|u| q 0 )(y) 1/q 0 .
Taking the integral over y ∈ B i gives

C i,j |A r i b i | |u| ≤ Cα2 jd g(j) B i M(|u| q 0 )(y) 1/q 0 dµ(y).
We sum over j, i and use j 2 jd g(j) < ∞ together with property (iv) in the Calderón-Zygmund decomposition to obtain

X | i A r i b i | |u| ≤ Cα X 1 ∪ i B i (y) M(|u| q 0 )(y) 1/q 0 dµ(y) ≤ Cα 1 ∪ i B i q 0 M(|u| q 0 ) 1/q 0 q 0 ,w ≤ Cα i v(x i , r i ) 1/q 0 |u| q 0 1/q 0 1 .
Note that we have used the fact that M is weak type (1, 1) to obtain the last inequality. Using now (iii) of the Calderón-Zygmund decomposition and u q 0 = 1, we obtain (17).

By assumption, S is bounded on L q 0 . Hence

µ({x ∈ X : |(S i A r i b i )(x)| > α}) ≤ 1 α q 0 S q 0 q 0 →q 0 i A r i b i q 0 q 0 . Now we use (17) to obtain (18) µ({x ∈ X : |(S i A r i b i )(x)| > α}) ≤ C α S q 0 q 0 →q 0 f 1 .
Step 3 Let δ be as in ( 13) and for all i ∈ N set Q i := B(x i , (1 + δ)r i ), the ball of centre x i and radius (1 + δ)r i . Then

µ({x ∈ X : | i (T -SA r i )b i (x)| > α}) ≤ i µ(Q i ) + µ({x ∈ X \ j Q j : | i ((T -SA r i )b i )(x)| > α}) ≤ C(1 + δ) d i v(x i , r i ) + 1 α X\ S j Q j | i ((T -SA r i )b i )(x)| dµ(x) ≤ C(1 + δ) d α f 1 + 1 α i X\Q i |((T -SA r i )b i )(x)| dµ(x) ≤ C(1 + δ) d α f 1 + W α i |b i (y)| dµ(y) ≤ C(1 + δ) d (1 + W ) α f 1 .
Note that the penultimate inequality follows from assumption ( 13) and the last one from properties (ii) and (iii) in the Calderón-Zygmund decomposition. Hence

(19) µ({x ∈ X : | i ((T -SA r i )b i )(x)| > α}) ≤ C(1 + δ) d (1 + W ) α f 1 .
Step 4 It follows from ( 16) that

µ({x ∈ X : |(T f )(x)| > α}) ≤ µ({x ∈ X : |(T g)(x)| > α 2 }) + µ({x ∈ X : |(T b)(x)| > α 2 }) ≤ C T p 0 p 0 →p 0 α f 1 + µ({x ∈ X : |(T b)(x)| > α 2 }).
For the second term we use ( 18) and ( 19) to estimate

µ({x ∈ X : |(T b)(x)| > α 2 }) = µ({x ∈ X : | i (SA r i b i )(x) + i ((T -SA r i )b i )(x)| > α 2 }) ≤ µ({x ∈ X : |(S i A r i b i )(x)| > α 4 }) + µ({x ∈ X : | i ((T -SA r i )b i )(x)| > α 4 }) ≤ C(1 + δ) d α S q 0 q 0 →q 0 + (1 + W ) f 1 .
We then conclude that T is of weak type (1, 1) with a weak type estimate (20)

T L 1 →L 1,w ≤ C(1 + δ) d 1 + W + T p 0 p 0 →p 0 + S q 0 q 0 →q 0 .
If we replace T and S by T -1 p 0 →p 0 T and T -1 p 0 →p 0 S we obtain (13) with T -1 p 0 →p 0 W instead of W . Thus applying (20) to T -1 p 0 →p 0 T , T -1 p 0 →p 0 S and T -1 p 0 →p 0 W yields ( 14). Finally, by Marcinkiewicz interpolation theorem the operator T extends to a bounded operator from L p (X) ∩ L p 0 (X) to L p (X) for all p ∈ (1, p 0 ).

Following again an idea in [START_REF] Duong | Singular integral operators with nonsmooth kernels on irregular domains[END_REF] we can prove a version of the previous theorem on arbitrary domains. Let Ω be an open subset of X and assume that T is bounded on L p 0 (Ω) and S and A t are bounded on L q 0 (Ω). We define T : L p 0 (X) → L p 0 (X) by

T f = 1 Ω T (1 Ω f )
and similarly for S and A t . If A t satisfies (21) below then A t satisfies (12). The operator T is weak type (1, 1) if and only if T is weak type (1, 1). Applying the previous theorem to T , S and A t gives the following result.

Theorem 2.3. Let T be a non-zero bounded linear operator on L p 0 (Ω) for some p 0 ∈ (1, ∞). Suppose there exists a bounded operator S on L q 0 (Ω) for some q 0 ∈ (1, ∞), a family of bounded operators (A t ) t>0 on L q 0 (Ω) and a sequence (g(j)) j∈N in R such that

(21) 1 v(x, 2 j+1 t) C j (x,t)∩Ω |A t f | q 0 1/q 0 ≤ g(j) 1 v(x, t) B(x,t)∩Ω |f |
for all ball x ∈ Ω, t > 0, j ∈ N and f ∈ L q 0 (B(x, t)∩Ω), and ∞ j=1 2 jd g(j) < ∞. Finally, suppose there exist δ, W > 0 such that

(22) Ω\B(x,(1+δ)t) |((T -SA t )u)(y)| dµ(y) ≤ W u 1 for all x ∈ X, t > 0 and u ∈ L 1 (Ω)∩L ∞ (Ω) supported in the ball B(x, t)∩Ω.
Then T is a weak type (1, 1) operator with

(23) T L 1 (Ω)→L 1,w (Ω) ≤ C(1 + δ) d W + T p 0 →p 0 + S q 0 q 0 →q 0 T 1-q 0 p 0 →p 0 .
Here C is a constant depending only on the constants in (10). In particular, T extends to a bounded operator on L p (Ω) for all p ∈ (1, p 0 ).

As in Remark 2.2 the condition ( 22) follows if the operators T and SA t are given by kernels K and K t (in the sense of ( 11)) and there are δ, W > 0 such that ( 24)

ρ(x,y)≥δt |K(x, y) -K t (x, y)| dµ(x) ≤ W < ∞,
for all t > 0 and y ∈ Ω. It suffices to note that the associated kernel of T is the extension by 0 outside Ω × Ω of the kernel of T where T f = 1 Ω T (1 Ω f ) as above. Similarly for the kernel of SA t .

We may replace in the previous theorems the annulus C j (x, r) by the annulus A(x, j, r) := B(x, (j +1)r)\B(x, jr). In that case, v(x, 2 j+1 r) has to be replaced by v(x, (j +1)r) and the condition on g becomes j j d g(j) < ∞.

Following [BK] it is proved in [Aus] that a bounded operator T on L 2 (X) is of weak type (r, r) if

(25) 1 v(x, 2 j+1 t) C j (x,t) |T (I -A t )f | 2 1/2 ≤ g(j) 1 v(x, t) B(x,t) |f | r 1/r and (26) 1 v(x, 2 j+1 t) C j (x,t) |A t f | 2 1/2 ≤ g(j) 1 v(x, t) B(x,t) |f | r 1/r
for all x ∈ X, t > 0, j ∈ N and f ∈ L 2 supported in B(x, t), and g(j)2 dj < ∞. One can prove a version of this result in which T -T A t in ( 25) is replaced by T -SA t as in Theorem 2.1. We do not write the details here since we have no concrete application. Theorem 2.1 is suitable for our purpose. Finally, let us mention that a Gaussian upper bound implies assumption (12). Indeed, assume that A t is given by a kernel a t such that

|a t (x, y)| ≤ C v(y, t 1/m ) exp{-c ρ(x, y) m/(m-1) t 1/(m-1) }
for all t > 0 and x, y ∈ X. Here m ≥ 2 and C, c are two positive constants. Fix x 0 ∈ X. Note that the operator 1 C j (x 0 ,t) A t m 1 B(x 0 ,t) has the kernel (x, y) → 1 C j (x 0 ,t) (x)a t m (x, y)1 B(x 0 ,t) (y). But

|1 C j (x 0 ,t) (x)a t m (x, y)1 B(x 0 ,t) (y)| ≤ C v(y, t) 1 C j (x 0 ,t) (x)1 B(x 0 ,t) (y) exp{-c ρ(x, y) m/(m-1) t m/(m-1) } ≤ C v(y, t) e -c2 jm/(m-1) 1 B(x 0 ,t) (y) ≤ C v(x 0 , t) e -c2 jm/(m-1) ,
where we used the doubling property in the last step. This an L 1 -L ∞ estimate. By interpolation, it implies an L 1 -L q 0 estimate which gives (12) for every q 0 ∈ (1, ∞).

A partial multiplier theorem for degenerate operators

Let the coefficients a kj , form a 0 and the self-adjoint operator A associated with a 0 be as in the introduction. For every bounded measurable function F : [0, ∞) → C, the operator F (A) is well defined by spectral theory and is bounded on L 2 (R d ). As mentioned in the introduction, if A is uniformly elliptic then F (A) extends to a bounded operator on L p (R d ) for all p ∈ (1, ∞) provided F has a finite number of derivatives on [0, ∞) which have good decay. We address here the same problem for degenerate operators. This is a difficult problem because no global Gaussian upper bounds are available for A in general. We prove a partial result by projecting on the part where the matrix (a kj ) is uniformly elliptic. There are two versions.

Theorem 3.1. Let Ω ⊂ R d be an open bounded set with Lipschitz boundary. Suppose there exists a µ > 0 such that (a kj (x)) ≥ µI for almost every x ∈ Ω and denote by P Ω the projection from

L 2 (R d ) onto L 2 (Ω). Set H = A + I. Let F : [0, ∞) → C be a bounded function such that (27) sup t>0 ϕ(.)F (t.) C s < ∞
for some s > d/2 and some non-trivial function ϕ ∈ C ∞ c (0, ∞). Then P Ω F (H)P Ω is of weak type (1, 1) and extends to a bounded operator on

L p (R d ) for all p ∈ (1, ∞). Theorem 3.2. Let χ ∈ C ∞ b (R d ), µ > 0 and suppose that (a kj (x)) ≥ µI for almost every x ∈ supp χ. Set H = A + I. Let F : [0, ∞) → C be a bounded function such that (28) sup t>0 ϕ(.)F (t.) C s < ∞
for some s > d/2 and some non-trivial function ϕ ∈ C ∞ c (0, ∞). Then M χ F (H)M χ is of weak type (1, 1) and extends to a bounded operator on L p (R d ) for all p ∈ (1, ∞).

The proof of both theorems is almost the same. It relies mainly on weighted estimates for the associated kernel of M χ F (H)M χ (or the kernel of P Ω F (H)P Ω ) together with Theorem 2.1. The proof of weighted estimates for the kernel of M χ F (H)M χ (or of P Ω F (H)P Ω ) is based on partial Gaussian bounds proved in [START_REF] Elst | Partial Gaussian bounds for degenerate differential operators[END_REF] and a similar strategy as in [DOS] and [Ouh].

In the sequel of this section we assume that there exists a constant µ > 0 such that (a kj (x)) ≥ µI for a.e. x ∈ Ω respectively for a.e. x ∈ supp χ ∪ supp χ. In the first case Ω is a bounded Lipschitz domain of R d and in the second case χ, χ ∈ C ∞ b (R d ). We denote by S t := e -tA the holomorphic semigroup generated by -A on L 2 (R d ). We recall the following result from [START_REF] Elst | Partial Gaussian bounds for degenerate differential operators[END_REF].

Theorem 3.3. There are C, c > 0 such that for all t > 0 the operator M e χ S t M χ (respectively P Ω S t P Ω ) is given by a kernel p t which satisfies

|p t (x, y)| ≤ Ct -d/2 e -c |x-y| 2 t (1 + t) d/2 for all t > 0 and x, y ∈ R d .
The theorem is stated in [START_REF] Elst | Partial Gaussian bounds for degenerate differential operators[END_REF] with χ = χ but the arguments work with different χ and χ. It is also proved there that (29)

M χ S t 2→∞ ≤ Ct -d/4 (1+t) d/4 respectively P Ω S t 2→∞ ≤ Ct -d/4 (1+t) d/4 . If z = t + is ∈ C with t = Re z > 0, then M χ S z M χ 1→∞ = M χ S t/2 S is S t/2 M χ 1→∞ ≤ Ct -d/4 (1 + t) d/4 S is S t/2 M χ 1→2 ≤ Ct -d/4 (1 + t) d/4 S t/2 M χ 2→∞ ≤ Ct -d/2 (1 + t) d/2 .
Similarly, (30)

P Ω S z P Ω 1→∞ ≤ C(Re z) -d/2 (1 + Re z) d/2
for all z ∈ C with Re z > 0. Using the Gaussian bounds of Theorem 3.3 for real t together with the uniform bounds ( 30) for complex z it follows as in Theorem 3.4.8 in [Dav] or Theorem 7.2 in [Ouh] that for all ε > 0 the kernel p (0)

z of M χ S z M χ respectively P Ω S z P Ω satisfies the bound (31) |p (0) z (x, y)e -εz | ≤ C ε (Re z) -d/2 exp{-c |x -y| 2 |z| cos(arg z)} for all x, y ∈ R d and z ∈ C with Re z > 0. Let H = A + I and define p z (x, y) = p (0)
z (x, y)e -z . Then p z is the kernel of M χ e -zH M χ . We shall formulate the results below for M χ F (H)M χ only, but all statements are also valid for P Ω F (H)P Ω . In the following lemmas, we shall always assume that (a kj (x)) ≥ µI for almost every x ∈ supp χ. Since associated kernels with several operators are involved in the sequel we shall denote by K T the kernel associated to a given operator T whenever it exists.

Lemma 3.4. For all s > 0 and ε > 0 there exists a C > 0 such that

R d |K MχF (H)Mχ (x, y)| 2 (1 + √ r|x -y|) s dx ≤ Cr d/2 δ r F 2 C s/2+ε
for all r > 0, y ∈ R d and F ∈ C s/2+ε supported in [0, r]. Here (δ r F )(λ) := F (rλ).

Proof. The arguments are very similar to those of Lemma 4.3 in [DOS]. Fix r > 0 and assume first that F is supported in [0, 1]. Set g(λ) := F (λ)e λ and H r := 1 r H. By (31), the kernel p z/r of M χ e -zHr M χ satisfies (32)

|p z/r (x, y)| ≤ Cr d/2 (Re z) -d/2 exp{-cr |x -y| 2 |z| cos(arg z)}
for all x, y ∈ R d and z ∈ C with Re z > 0, with constants C, c independent of r. We write g(λ) = R ĝ(ξ)e iλξ dξ, where ĝ is the Fourier transform of g. Then

F (H r ) = R ĝ(ξ) e -(1-iξ)Hr dξ, from which one obtains (33) K MχF (Hr)Mχ (x, y) = R ĝ(ξ) p (1-iξ)/r (x, y) dξ.
Let y ∈ R d . Using the estimate (32) with z = 1 -iξ gives

R d |p (1-iξ)/r (x, y)| 2 (1 + √ r|x -y|) s dx ≤ Cr d exp{-2cr |x -y| 2 1 + ξ 2 } (1 + √ r|x -y|) s dx ≤ Cr d (1 + ξ 2 ) s/2 exp{-cr |x -y| 2 1 + ξ 2 } dx ≤ Cr d (1 + ξ 2 ) s/2 1 + ξ 2 r d/2 = Cr d/2 (1 + ξ 2 ) (d+s)/2 .
It follows from ( 33), the continuous version of the Minkowski inequality and the previous estimate that

R d |K MχF (Hr)Mχ (x, y)| 2 (1 + √ r|x -y|) s dx 1/2 ≤ R |ĝ(ξ)| R d |p (1-iξ)/r (x, y)| 2 (1 + √ r|x -y|) s dx 1/2 dξ ≤ Cr d/4 R |ĝ(ξ)| (1 + ξ 2 ) (d+s)/4 dξ ≤ Cr d/4 g W (d+s+2)/2,2 ≤ Cr d/4 F W s/2+α,2 . ( 34
)
Here α = (d + 2)/2 and the constants are independent of r and y. On the other hand M χ F (H r )M χ = M χ g(H r )e -Hr M χ . It follows from ( 29) that

e -Hr M χ 1→2 ≤ Cr d/4 (1 + 1 r ) d/4 e -1/r ≤ Cr d/4
for all r > 0. Moreover,

M χ g(H r ) 2→2 ≤ e χ ∞ F ∞ . Therefore (35) R d |K MχF (Hr)Mχ (x, y)| 2 dx ≤ M χ F (H r )M χ 2 1→2 ≤ Cr d/2 F 2 ∞ .
This is valid for all F with support in [0, 1] and for all s > 0. The estimates (34) and ( 35) together with an interpolation argument (see [MM], p. 151 and [DOS], p. 455) give then that for all s > 0 there exists a C > 0 such that (36)

R d |K MχF (Hr)Mχ (x, y)| 2 (1 + √ r|x -y|) s dx ≤ Cr d/2 F 2 C s/2+ε .
Finally, if F has support in [0, r] we use the last estimate with δ r F and obtain the lemma.

Lemma 3.5. The operators A t := e -t 2 H M χ satisfy (12).

Proof. Let ψ ∈ W 1,∞ (R d , R) be such that |∇ψ| ≤ 1. For all ρ ∈ R define U ρ = M e ρψ and set S ρ t := U ρ e -tH U -ρ . It follows from [EO1]
Proposition 3.6 by duality and a limit n → ∞ that there exist C, ω > 0, independent of t, ρ and ψ, such that (37)

S ρ t M χ 1→2 ≤ Ct -d/4 e ωρ 2 t . Now fix two bounded open non-empty sets E and F of R d and choose ψ(x) := d(x, E) ∧ N , where N = sup{|x -y| : x ∈ E, y ∈ F } + 1. For all h ∈ L 2 (E)
and ρ ≥ 0 one has

M χ e -tH h = M χ U -ρ S ρ t h. Therefore M χ e -tH h L ∞ (F ) ≤ e -ρd(E,F ) M χ S ρ t h ∞ ≤ Ct -d/4 e -ρd(E,F ) e ωρ 2 t h 2 .
Choosing ρ = d(E,F ) 2ωt yields the Davies-Gaffney type estimate

(38) P F (M χ e -tH )P E 2→∞ ≤ Ct -d/4 e -d(E,F ) 2 4ωt .
In particular,

P C j (x,t) e -t 2 H M χ P B(x,t) 1→2 ≤ Ct -d/2 e -c4 j
for all x, y ∈ R d and j ∈ N. This shows the lemma.

Proof of Theorems 3.1 and 3.2. As mentioned above, the proof of both theorems is almost the same. We consider M χ F (H)M χ only. The proof is based on Theorem 2.1 and the previous lemmas. It is in the same spirit as in the elliptic case where a Gaussian bound holds (cf. [DOS], [Ouh]). Let

ϕ ∈ C ∞ c (0, ∞) be such that supp ϕ ⊂ [1/4, 1] and ∞ n=-∞ ϕ(2 -n λ) = 1
for all λ > 0. Then

F (λ) = ∞ n=-∞ ϕ(2 -n λ)F (λ) =: ∞ n=-∞ F n (λ).
We apply Theorem 2.1 to M χ F n (H)M χ for each fixed n ∈ Z. We choose

S := M χ F n (H) and A t := e -t 2 H M χ .
By Lemma 3.5, the operators A t satisfy (12). It remains to prove (15). For this we have to estimate for all y ∈ R d the integral

I n,t := |x-y|≥t |K MχGn,t(H)Mχ (x, y)| dx, where G n,t (λ) = F n (λ) -F n (λ)e -t 2 λ = ϕ(2 -n λ)F (λ)(1 -e -t 2 λ
). First, by the Cauchy-Schwarz inequality we have

I n,t ≤ R d |K MχGn,t(H)Mχ (x, y)| 2 (1 + 2 n/2 |x -y|) 2s dx 1/2 × × |x-y|≥t (1 + 2 n/2 |x -y|) -2s dx 1/2 (39)
We apply Lemma 3.4 with r = 2 n and obtain (40)

R d |K MχGn,t(H)Mχ (x, y)| 2 (1 + 2 n/2 |x -y|) 2s dx ≤ C2 nd/2 δ 2 n G n,t 2
C s+ε .
Simple computations show that there exists a C > 0, independent of n and t, such that

δ 2 n G n,t C s+ε = ϕ(.)F (2 n .)(1 -e -t 2 2 n . ) C s+ε ≤ C sup t >0 ϕ(.)F (t .) C s+ε min(1, t 2 2 n ). (41)
On the other hand (see [DOS] or (7.46) in [Ouh]) one estimates (42)

|x-y|≥t (1 + 2 n/2 |x -y|) -2s dx ≤ C2 -nd/2 min(1, (t2 n/2 ) d-2s ).
Using (39), ( 40), ( 41) and ( 42) we obtain

I n,t ≤ C min(1, t 2 2 n ) min(1, (t2 n/2 ) d 2 -s ) sup t >0 ϕ(.)F (t .) C s+ε . Hence ∞ n=-∞ I n,t ≤ C   n∈Z, t 2 2 n ≤1 t 2 2 n + n∈Z, t2 n/2 >1 (t2 n/2 ) d 2 -s   sup t >0 ϕ(.)F (t .) C s+ε
and the RHS is bounded by a constant independent t since s > d 2 . This proves Theorem 3.1.

As explained in the introduction, the reason why we consider H = A + I instead of A in the previous results comes from the fact the Gaussian upper bound in Theorem 3.3 is valid with the extra factor (1+t) d/2 . If one considers the case where a kj = δ kj on a smooth bounded domain Ω, then A is the Neumann Laplacian on L 2 (Ω) and 0 on L 2 (R d \ Ω). It is then easy to see that L 2 -L ∞ estimates (respectively, Gaussian bounds) for M χ e -tA or P Ω e -tA (respectively, M χ e -tA M χ or P Ω e -tA P Ω ) cannot hold without an extra factor (1 + t) d/4 (respectively, (1 + t) d/2 ). On the other hand we can replace in the previous theorems H = A + I by H = A + εI for any ε > 0.

It may be possible that if (a kj ) ≥ µI on a connected subset F of R d which is 'large enough' (in some sense) one obtains Theorem 3.3 without the extra factor (1 + t) d/2 in the Gaussian bound. This remains to be proved. We mention that if such bound holds, we obtain by the same proof Theorems 3.1 and 3.2 for F (A) rather than F (H).

We emphasize also that we consider here general degenerate operators with non-smooth coefficients. One may obtain global results for some specific operators which are degenerate at every point and the coefficients are not continuous at every point. For example, one might take a pure second-order subelliptic operator in divergence form with real measurable coefficients on a Lie group with polynomial growth. Then global Gaussian bounds are valid by [SS], Théorème 1 together with regularization argument (see, for example Section 2.1 in [ER]). Therefore a global spectral multiplier result for such operators follows directly from [DOS]. Note however that the order of smoothness required on the function F is larger than half of Euclidean dimension. On the other hand, the operators that we consider in this paper are allowed to vanish on big sets.

Examples 3.6. We give some examples which are direct applications of the previous theorems.

Imaginary powers. Set F (λ) = λ is where s ∈ R. Then Theorems 3.1 and 3.2 together with the Riesz-Thorin interpolation theorem imply that for all ε > 0 and p ∈ (1, ∞) there exists a C > 0 such that

M χ H is M χ p→p ≤ C ε (1 + |s|) (d+ε)| 1 2 -1 p | Proof. Let u ∈ D(a). Then there exists a sequence (u n ) n∈N in D(a 0 ) = C ∞ c (R d ) such that lim u n = u in L 2 (R d
) and a(u) = lim a 0 (u n ). By the ellipticity assumption on the support of χ one deduces

(43) µ R d χ 2 |∇u n | 2 ≤ d k,j=1 R d a kj (∂ k u n ) (∂ j u n ) χ 2 ≤ χ 2 ∞ a 0 (u n ) for all n ∈ N. Therefore (χu n ) n∈N is bounded in W 1,2 (R d ).
Hence it has a weakly convergent subsequence in W 1,2 (R d ). Since lim

χu n = χu in L 2 (R d ) it follows that χu ∈ W 1,2 (R d ). Then taking the limit n → ∞ in (43) one estimates µ R d χ 2 |∂ k u| 2 ≤ χ 2 ∞ a(u) ≤ χ 2 ∞ H 1/2 u 2 2
for all k ∈ {1, . . . , d}.

Lemma 4.3. Let χ, χ ∈ C ∞ b (R d
), µ > 0 and assume that (a kj (x)) ≥ µI for almost every x ∈ supp χ ∪ supp χ. Then for all β > 0 small enough we have

R d |(M e χ e -sH M χ u)(y)| 2 e β |x-y| 2 s dy ≤ Cs -d/2 e 2βt 2 /s e -s u 2 1 for all t > 0, s > 0, x ∈ R d and u ∈ L 2 (R d ) with supp u ⊂ B(x, t).
≤ Ce -s R d s -d/2 e -(c-β) |y-z| 2 s |u(z)| dz 2 e 2βt 2 /s ≤ Ce -s s -d/2 u 1 R d s -d/2 e -(c-β) |y-z| 2 s |u(z)| dz e 2βt 2 /s .
Taking β < 1 2 c and integrating over y yields the lemma. Since e -sH L 2 (R d ) ⊂ D(a) for all s > 0 we obtain from Lemma 4.2 the inclusion M χ ∇e -sH M χ (L 2 (R d )) ⊂ W 1,2 (R d ) for all s > 0. The following weighted L 2 -estimate is in the same spirit as weighted gradient estimates for heat kernels (see [CD], [Gri] and Theorem 6.19 in [Ouh]).

Lemma 4.4. For all β > 0 small enough we have

R d |(M χ ∇e -sH M χ u)(y)| 2 e β |x-y| 2 s dy ≤ Cs -d/2-1 e 6βt 2 /s u 2 1 for all t > 0, s > 0, x ∈ R d and u ∈ L 2 (R d ) with supp u ⊂ B(x, t).
Proof. In order to avoid domain problems of forms we shall proceed by approximation. First, we prove the lemma for uniformly elliptic coefficients with constants β and C depending only on µ > 0 such that (a kj (x)) ≥ µ I a.e. x ∈ supp χ.

Assume that there exists a µ 0 > 0 such that (a kj (x)) ≥ µ 0 I for a.e. x ∈ R d . In this case the form a has domain W 1,2 (R d ). We use similar ideas as in the proof of Theorem 6.19 in [Ouh] but we want to prove that the constants in the estimates are independent of µ

0 . Let ψ ∈ C ∞ c (R d ) be such that ψ(x) = 1 for all x ∈ B(0, 1) and 0 ≤ ψ ≤ 1. For all n ∈ N define ψ n ∈ C ∞ c (R d ) by ψ n (x) = ψ( 1 n x). Set I n := R d |χ(y) (∇e -sH M χ u)(y)| 2 e β |x-y| 2 s ψ n (y) dy
and define f := e -sH M χ u. Then,

I n ≤ 1 µ k,j R d a kj (y) (∂ k f )(y) (∂ j f )(y) e β |x-y| 2 s χ(y) 2 ψ n (y) dy = 1 µ k,j R d a kj (∂ k f ) ∂ j f e β |x-•| 2 s χ 2 ψ n + 1 µ k,j R d a kj (y) (∂ k f )(y) f (y) 2β(x j -y j ) s e β |x-y| 2 s χ(y) 2 ψ n (y) dy - 2 µ k,j R d a kj (y) (∂ k f )(y) f (y) e β |x-y| 2 s (∂ j χ)(y) χ(y) ψ n (y) dy - 1 nµ k,j R d a kj (y) (∂ k f )(y) f (y) e β |x-y| 2 s χ(y) 2 (∂ j ψ)( 1 n y) dy =: J 1,n + J 2,n + J 3,n + J 4,n . Since y → f (y)e β |x-y| 2 s χ(y) 2 ψ n (y) is an element of W 1,2 (R d ) we have J 1,n = 1 µ a(f, f e β |x-•| 2 s χ 2 ψ n ) = 1 µ R d (Ae -sH M χ u) (e -sH M χ u) e β |x-•| 2 s χ 2 ψ n ≤ χ ∞ µ He -sH M χ u 2 e β |x-•| 2 s M χ e -sH M χ u 2 .
The standard estimate He -sH 2→2 ≤ s -1 and Lemma 4.3 give

(44) J 1,n ≤ Cs -d/2-1 e 2βt 2 /s u 2 1 if β is small enough. Using the obvious inequality |x j -y j | s ≤ 1 √ ε s e ε |x-y| 2 s we have |J 2,n | ≤ C √ s k R d |∂ k e -sH M χ u| χ 2 |e -sH M χ u| e 2β |x-•| 2 s ≤ C √ s I n R d |(M χ e -sH M χ u)(y)| 2 e 3β |x-y| 2 s dy 1/2 . Therefore Lemma 4.3 implies (45) |J 2,n | ≤ C I n s -d 4 -1 2 e -s/2 e 3βt 2 /s u 1 .
We estimate the third term in a similar way.

|J 3,n | ≤ C k R d |∂ k e -sH M χ u| |χ ∂ j χ| |e -sH M χ u| e β |x-•| 2 s ψ n ≤ C I n R d |M ∂ j χ e -sH M χ u| 2 e β |x-•| 2 s 1/2 ≤ C I n s -d 4 -1 2 e -s/3 e βt 2 /s u 1 . (46) Finally, |J 4,n | ≤ C n k,j R d |(χ∂ k e -sH M χ u)(y)| |(M χ e -sH M χ u)(y)| e β |x-y| 2 s × × |(∂ j ψ)( 1 n y)| dy ≤ C n M χ ∇ e -sH M χ u 2 R d |(M χ e -sH M χ u)(y)| 2 e 2β |x-y| 2 s dy 1/2 ≤ C n M χ ∇ e -sH M χ u 2 s -d/4 e -s/2 e 2βt 2 /s u 1 . (47)
Therefore, we obtain from ( 44), ( 45), ( 46) and ( 47) that

I n ≤ Cs -d/2-1 e 6βt 2 /s u 2 1 + C n M χ ∇ e -sH M χ u 2 s -d/4 e 2βt 2 /s u 1 .
Letting n → ∞ and then use Fatou's lemma yields The constants C and β are independent of µ 0 . Now we prove the lemma for degenerate operators. For all n ∈ N set a This proves the lemma.

Proof of Theorem 4.1. It follows from Lemma 4.2 that the Riesz transform M χ ∂ k A -1/2 is bounded on L 2 (R d ).

In order to prove a weak type estimate for T = M χ ∂ k H -1/2 M χ we apply Theorem 2.1 with S = M χ ∂ k H -1/2 and A t = e -t 2 H M χ . These operators are bounded on L 2 (R d ) and by Lemma 3.5 the operators A t satisfy assumption (12). It remains then to check (13). By the formula

H -1/2 = 1 2 √ π ∞ 0 e -sH ds √ s
we have

H -1/2 e -t 2 H = 1 2 √ π ∞ 0 e -(s+t 2 )H ds √ s = 1 2 √ π ∞ 0 e -sH 1 {s>t 2 } ds √ s -t 2 .
Let β > 0 be as in Lemma 4. ν(s, t)s -1/2 e -γt 2 /s ds.

The latter integral is bounded by some constant M independent of t. This proves (13) and hence T = M χ ∂ k H -1/2 M χ is weak type (1, 1). By interpolation, it is bounded on L p (R d ) for all 1 < p ≤ 2.

As discussed at the end of the previous section, we note that if one proves a version of Theorem 3.3 without the extra factor (1 + t) d/2 if (a kj (x)) ≥ µI for a.e. x in a 'big' domain, then Theorem 4.1 holds with A in place of H. That is M χ ∂ k A -1/2 M χ is weak type (1, 1) and bounded on L p (R d ) for all 1 < p ≤ 2. In [START_REF]Partial Gaussian bounds for degenerate differential operators II[END_REF], we prove by a different method that if the coefficients a kj ∈ W 1,∞ (R d ), then M χ ∂ k (I + A) -1/2 and M χ ∂ k ∂ j (I + A) -1 are bounded on L p (R d ) for all p ∈ (1, ∞). Moreover, if a kj ∈ W ν,∞ (R d , C) then we show that M χ ∂ k (I + A) -1/2 M χ is bounded on L p for all p ∈ (1, ∞).

a

  kj (x)ξ k ξ j ≥ µ|ξ| 2 for all ξ = (ξ 1 , . . . , ξ d ) ∈ R d and x ∈ R d The operator A = -d k,j=1

  sH M χ u)(y)| 2 e β |x-y| 2 s = | B(x,t) e -s p s (y, z) u(z) dz| 2 e β |x-y| 2 s ≤ Ce -s B(x,t) s -d/2 e -c |y-z| 2 s e β |x-y| 2 2s |u(z)| dz 2

  χ ∇e -sH M χ u)(y)| 2 e β |x-y| 2 s dy ≤ Cs -d/2-1 e 6βt 2 /s u 2 1 .

  = a kj + 1 n δ kj . Then (a (n) kj (x)) ≥ 1n I for a.e. x ∈ R d and (a(n) kj (x)) ≥ µ I for a.e. x ∈ supp χ. Moreover, a (n) kj ∞ ≤ 1 + a kj ∞ .We denote by A n the elliptic operator with the coefficients a (n) kj and H n = I + A n . We apply (48) to H n and obtain (49)R d |(M χ ∇e -sHn M χ u)(y)| 2 e β |x-y| 2 s dy ≤ Cs -d/2-1 e 6βt 2 /s u 2 1 .for some constants C and β > 0 which are independent of n. Let k ∈ {1, . . . , d}. Then|(e -sHn M χ u, ∂ k M χ (ϕ e β |x-.| 2 2s ))| ≤ Cs -d 4 -1 2 e 3βt 2 /s u 1 ϕ 2 for all ϕ ∈ C ∞ c (R d ).On the other hand, e -tHn converges strongly in L 2 (R d ) to e -tH (see[AE] Corollary 3.9). It follows then that|(e -sH M χ u, ∂ k M χ (ϕ e β |x-.| 2 2s ))| ≤ Cs -d 4 -1 2 e 3βt 2 /s u 1 ϕ 2 .Since this is true for all ϕ ∈ C ∞ c (R d ) we have by density(M χ ∂ k e -sH M χ u) • e β |x-

.

  4 and letδ > 0. Fix x ∈ R d , t > 0 and let u ∈ L 2 (R d ) with supp u ⊂ B(x, t). Set ν(s, t) = 1 {s>t 2 } χ ∂ k e -sH M χ u)(y)| dy ν(s, t) ds ≤ χ ∂ k e -sH M χ u)(y)| 2 e β |x-y| 2Note that we have used Lemma 4.4 in the last inequality. NowR d \B(x,(1+δ)t) Cs d/2 e -β (1+δ) 2 t 2 2sChoosing δ ≥ 4 we obtain a positive constant γ such that R d \B(x,(1+δ)t) |(T -SA t )u(y)| dy ≤ C ∞ 0

under the additional assumption that a kj ∈ W 1,∞ (R d ), we proved recently in[START_REF]Partial Gaussian bounds for degenerate differential operators II[END_REF] that e -t(I+A) M χ has a kernel which satisfies a Gaussian bound.

Acknowledgements Most of this work was carried out whilst the second named author was visiting the Department of Mathematics at the University of Auckland. He wishes to thank both the University of Auckland and the CNRS for financial support. Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

and

for all s ∈ R. The Schrödinger group. Set F (λ) = (1 + λ) -α e itλ with t ∈ R and α > d/2. The operators M χ (I + H) -α e itH M χ and P Ω (I + H) -α e itH P Ω are bounded on L p (R d ) for all p ∈ (1, ∞). Their L p -norms are estimated by C(1 + |t|) α . By interpolation, we obtain boundedness on L p for all α > d| 1 2 -1 p | and t ∈ R. Remark. Using the same proof as in [CCO], these results can be obtained directly from the Gaussian upper bound of Theorem 3.3 without appealing to Theorems 3.1 and 3.2.

Riesz transforms

The aim in this section is to prove boundedness on L p (R d ) of a type of Riesz transform operator M χ ∇(I +A) -1/2 M χ . We keep the same notation as in the previous section. The main result of this section is the next theorem.

Here ∂ k denotes the distributional derivative. The proof is based on Theorem 2.1 and uses some ideas from [CD], [START_REF] Duong | The L p boundedness of Riesz transforms associated with divergence form operators[END_REF] and Chapter 7 in [Ouh] in the uniformly elliptic case. We start with the following lemma. Let a be the closure of the regular part of the form a 0 defined in (5).