Adrien Dartiguelongue

Antoine Leybros
email: antoine.leybros@cea.fr

CEA, DEN, DTCD, SPDE Agnès Grandjean

Solubility and solution enthalpy of a cesium-selective calixarene in supercritical carbon dioxide

Keywords: CalixOctyl, supercritical CO 2, solubility, solution enthalpy, cesium extraction

come

Solubility and solution enthalpy of a cesium-selective calixarene in supercritical carbon dioxide

1.

Introduction

Calixarenes can be defined as cyclic oligomers produced by the base or acid-induced condensation of a phenol with an aldehyde. This reaction was carried out for the first time in 1872 by Baeyer [START_REF] Baeyer | Ueber die verbindungen der aldehyde mit den phenolen[END_REF], who obtained a cement-like substance by mixing miscellaneous phenols and aldehydes. Over the following century, studies of this kind of reaction continued for the synthesis of resinous then crystalline materials for applications in the petroleum and pharmaceutical industry. Following the first synthesis of calixarenes by Gutsche [START_REF] Gutsche | Calixarenes: An Introduction[END_REF], these compounds were used as enzyme mimics in the early 1970s. Gutsche's project was to design specific receptors for a given molecular substrate and functionalize these receptors with chemical groups able to interact with the substrate. Subsequent advances in calixarene chemistry now allow the synthesis of many functionalized compounds with highly selective yields. Calixarenes can be used to prevent the oxidation of organic polymers [START_REF] Perrin | The potential industrial applications of calixarenes[END_REF] and functionalized compounds have been employed as phase-transfer agents for reactions in nonpolar media [4], as ion scavengers in resins for the encapsulation of electronic devices [START_REF] Taniguchi | Catalytic activity of an octopus-type calixarene on the formation of ethers[END_REF], as accelerators for instant adhesives [START_REF] Harris | Encapsulating compositions[END_REF], as films for chemical sensing applications [START_REF] Harris | Functionalised oxacalixarenes, their preparation and use in instant adhesive compositions[END_REF]8] and as catalysts for hydrolysis [START_REF] Erdoğan | Swelling behaviour of calixarene film exposed to various organic vapours by surface plasmon resonance technique[END_REF] or polymerization reactions [START_REF] Arimura | Host-guest properties of new watersoluble calixarenes derived from p-(chloromethyl) calixarenes[END_REF].

As mentioned in several reviews however [START_REF] Kneafsey | Polymerization catalysts and polymerization process using such catalysts[END_REF][START_REF] Atanassova | Synergism as a phenomenon in solvent extraction of 4felements with calixarenes[END_REF][START_REF] Mokhtari | Binding abilities and extractive applications of nanobaskets of calixarenes[END_REF][START_REF] Mokhtari | A review of calixarene applications in nuclear industries[END_REF][START_REF] Ludwig | Calixarenes in analytical and separation chemistry[END_REF], the main application of calixarenes is the chelation of metallic cations and neutral species. Calixarenes have notably been used to extract the following species from aqueous solutions: α-amines and α-amino acid methylesters [START_REF] Tabakci | Design and synthesis of new chiral calix[4]arenes as liquid phase extraction agents for α-amino acid methylesters and chiral α-amines[END_REF], gold [START_REF] Torgov | Complexation and reduction of gold (III) during extraction from HCl by calix[4,6]arenes upper-Rim functionalized with alkyl-or tolylthiamethyl groups[END_REF], copper [START_REF] Kostin | Complexation and extraction of non-ferrous metals by calix[n]arene phosphine oxides[END_REF], technetium [START_REF] Ludwig | Solvent extraction of Tc(VII) by calixarenes bearing pyridino groups[END_REF], americium and europium [START_REF] Sansone | and calix[END_REF], uranium [START_REF] Leydier | Sequestering agents for uranyl chelation: new calixarene ligands[END_REF], alkaline earth metals [START_REF] Park | Effect of conformation on metal ion extraction by calix[END_REF] and alkali metal cations [START_REF] Ohto | Allosteric coextraction of sodium and metal ions with calix[4]arene derivatives 1: role of the first-extracted sodium ion as an allosteric trigger for self-coextraction of sodium ions with calix[END_REF]. Cesium extraction is of particular interest because of the huge volumes of 137 Cs-polluted waste awaiting decontamination.

Calix [4]arenes-crown-6 [START_REF] Dozol | Calix[4arènes-couronnes, leur procédé de préparation et leur utilisation pour l'extraction sélective du césium et des actinides[END_REF]25] (consisting of four phenol units functionalized by a crown ether with six oxygen atoms on the lower rim) are the most efficient and cesium-selective of these compounds. They have been used extensively for the liquid-liquid extraction of cesium in both acid and basic media [START_REF] Luo | Extraction of cesium ions from aqueous solutions using calix[4]arene-bis(tertoctylbenzo-crown-6) in ionic liquids[END_REF][START_REF] Pierce | Performance testing of the nextgeneration CSSX solvent with actual SRS tank waste[END_REF][START_REF] Rais | Extraction of radioactive Cs and Sr from nitric acid solutions with 25,27-bis(1-octyloxy)calix[4]-26,28-crown-6 and dicyclohexyl-18-crown-6: effect of nature of the organic solvent[END_REF][START_REF] Moyer | Next generation solvent development for caustic-side solvent extraction of cesium[END_REF][30][START_REF] Wang | Verification of the calixcrown process for partitioning cesium from high-level liquid waste (HLLW)[END_REF][START_REF] Zhou | Highly lipophilic, mono-ionizable calix[4]arenebenzocrown-6 extractants for removal of radiocesium from nuclear wastes[END_REF][START_REF] Mohapatra | Evaluation of calix-crown ionophores for selective separation of radio-cesium from acidic nuclear waste solution[END_REF]. In particular, 25,27-Bis(1-octyloxy)calix [4]arenecrown-6, 1,3-alternate (CalixOctyl, CAS n°161282-97-9) diluted in 1,2-nitrophenyl hexyl ether allowed a complete recovery of cesium from nitric acid solutions with a very high selectivity over sodium [START_REF] Dozol | Calix[4arènes-couronnes, leur procédé de préparation et leur utilisation pour l'extraction sélective du césium et des actinides[END_REF].

The high diffusivity and solvent power of supercritical CO 2 (SC-CO 2 ) make it a promising candidate for the depollution of Cs-contaminated soils [START_REF] Leybros | Cesium removal from contaminated sand by supercritical CO 2 extraction[END_REF]. An additional benefit of this approach is that it allows to minimize harmful and flammable solvents or strong oxidants quantities. Thereby, Kanekar et al. [START_REF] Kanekar | Online recovery of radiocesium from soil, cellulose and plant samples by supercritical fluid extraction employing crown ethers and calix-crown derivatives as extractants[END_REF] combined calix [4]arenes-crown-6 (or crown ethers) and organic modifiers to recover cesium from soils in SC-CO 2 . Based on environmental, economic and safety considerations however, the solubility of a selective extractant alone or with an appropriate modifier is one of the key parameter of such a process.

Very little is known though about the solubility of calixarenes in SC-CO 2 . Graham et al. [START_REF] Graham | Solubility of p-t-butylcalixarenes in supercritical carbon dioxide: a comparison of static and dynamic measurement techniques[END_REF] found that simple p-t-butylcalix [4,[START_REF] Harris | Encapsulating compositions[END_REF]8]arenes were only weakly soluble in SC-CO 2 (at best 1.03•10 -6 mol•mol -1 at 20.6 MPa and 323.1 K). The lack of data available on the use of calixarenes to extract cesium lead us to investigate solubility in SC-CO 2 of CalixOctyl, the calix [4]arene-crown-6 that previous studies suggest is the most efficient and selective for cesium extraction [START_REF] Dozol | Calix[4arènes-couronnes, leur procédé de préparation et leur utilisation pour l'extraction sélective du césium et des actinides[END_REF][START_REF] Kanekar | Online recovery of radiocesium from soil, cellulose and plant samples by supercritical fluid extraction employing crown ethers and calix-crown derivatives as extractants[END_REF]. This paper reports solubility measurements performed at 10.3-27.4MPa at 314, 324 and 334 K. In order to better our understanding of the solubilization process, we also measured the solution enthalpy of CalixOctyl-a quantity rarely determined experimentally-at the same three temperatures and at a fixed pressure of 24.0 MPa.

2.

Materials and methods

Chemicals

CalixOctyl chemical structure is shown in Fig. 1. It was synthesized following the original scheme of Casnati et al. [25] and its purity (> 95%) was verified by measuring its melting point by 1 H and 13 C NMR, IR spectroscopy and size-exclusion chromatography. The remaining chemicals and the CO 2 (Air Liquide, ≥ 99.99%) were used without further purification: dibenzo-24-crown-8 (Aldrich, 98%, CAS n°14174-09-5), ethyl acetate (Fluka, LC-MS Chromasolv ® , ≥ 99.7%) and acetonitrile (Fluka, LC-MS UltraChromasolv ® , ≥ 99.9%) were respectively used to validate the experimental device, for the extraction of CalixOctyl from SC-CO 2 , and for UV analysis. The DSC sensor was calibrated in temperature and enthalpy using indium standard (Sigma-Aldrich) and benzoic acid (Merck Millipore, Emsure ® Reag. Ph. Eur., ≥ 99.9%,).

Solubility measurements

A dynamic method was developed, involving the solubilization of CalixOctyl in flowing SC-CO 2 and the continued recovery of the solute from the CO 2 using ethyl acetate. This approach is similar to the one used by Cross et al. [START_REF] Cross | Determination of metal-chelate complex solubilities in supercritical carbon dioxide[END_REF] to measure the solubility of solid compounds in SC-CO 2 . A diagram of the experimental setup is shown in Fig. 2 and the global procedure to acquire solubility data is detailed hereafter.

A high pressure pump (PU-2080-CO 2 Plus, Jasco, uncertainty on the CO 2 flow rate: ± 3.3•10 -5 cm 3 •s -1 ) was used to deliver CO 2 at up to 1.7•10 -1 cm 3 •s -1 at 263 K. The device was heated using a heat exchanger (labeled HE in Fig. 2) attached to a stove (S in Fig. 2, FED 115, Binder) enclosing the whole device. An HPLC switching valve (SWV, model 7000, Rheodyne) enabled the CO 2 flow to be directed at either the reactor or the back-pressure regulator. The titanium solubilization reactor (R, 38 cm 3 , Top Industrie) was capped with two 20 µm sintered stainless steel disks to prevent mechanical entrainment of the undissolved solute. The reactor was loaded with glass beads (0.75-1.00 mm in diameter) in which CalixOctyl was dispersed. Supercritical CO 2 saturated with CalixOctyl was then brought into contact with an ethyl acetate flow delivered by a solvent pump (PS, PU-4180, Jasco, standard uncertainty on the ethyl acetate flow rate: ± 3.3•10 -5 cm 3 •s -1 ). Ethyl acetate does not react with SC-CO 2 and the two compounds are fully miscible [START_REF] Byun | High-pressure phase behavior and modeling of binary mixtures for alkyl acetate in supercritical carbon dioxide[END_REF], allowing the formation of a mixed flow. In the other hand, ethyl acetate totally solubilizes CalixOctyl in these conditions.

A back-pressure regulator (BPR, Tescom, 26-1700 series) was used to control the operating pressure in the device. Downstream of the BPR, CO 2 and ethyl acetate were separated by adjusting the solvent power. The ethyl acetate containing CalixOctyl was gathered in collectors C1 and C2. The total mass of CO 2 flowing through the device (mCO 2 ) was measured using a thermal mass flow meter (FM, SLA5860, Brooks, standard uncertainty on the CO 2 mass flow: ± 1.7•10 -7 kg•s -1 ). The operating temperature and pressure were measured in the solubilization reactor using a pressure transmitter (PA-33X, Keller, relative uncertainty: 0.1%) and a K type thermocouple (HMPV-KI-M, USA, relative uncertainty: 0.75%). All masses were determined with a precision balance (DV214C, Ohaus, standard uncertainty: ± 10 -7 kg).

The solubility measurements were conducted as follows:

 The reactor R was loaded with a known mass (m 0 ) of CalixOctyl.

 The reactor R was by-passed using the switching valve and the device was heated and pumped up to operating conditions.

 The CO 2 flow rate, , was kept fixed at 1.7•10 -2 cm 3 •s -1 and ethyl acetate was injected into the device at = 3.3•10 -2 cm 3 •s -1 .

 After a short equilibration time, the injection of ethyl acetate was stopped and the valve was switched back.  Ethyl acetate was injected once more with a flow rate, = 3.3•10 -2 cm 3 •s -1 .

 Equilibrium was maintained for 1 hour in the solubilization reactor. The pump was then stopped, the CO 2 inlet valve V1 was closed, and the contents of the collectors C1 and C2 and the reactor R were weighed. In the equations below, the mass of solution in C1 is denoted m 1 and the mass of CalixOctyl remaining in R, m 2 . No trace of ethyl acetate was found in C2.

Quantitative analysis

The quantitative analysis of CalixOctyl was performed by UV/visible spectrophotometry (UV-1800, Shimadzu) at 268 nm (an absorbance maximum), with quartz UV cells (10 mm path length, Hellma Analytics). Because ethyl acetate also absorbs at 268 nm, it was completely removed from the collected solution, first using a rotary evaporator and then a vacuum gas manifold. The solid residue was then dissolved in an appropriate volume of acetonitrile, , and the absorbance of this new solution was determined. The following relationship was obtained by regression analysis ( ) between the absorbance at 268 nm, , and the molar concentration of CalixOctyl in acetonitrile, (in mol•L -1 ):

(

The main advantage of this set-up is that it avoids solute plugging [START_REF] Cross | Determination of metal-chelate complex solubilities in supercritical carbon dioxide[END_REF]. Indeed, CalixOctyl is highly soluble in ethyl acetate (> 1.2•10 -2 mol•L -1 ) at 298.2 K according to preliminary tests) and therefore remains dissolved in the liquid solvent downstream of the back-pressure regulator (where CO 2 is gaseous). This avoids any clogging of the instrument.

Moreover, CalixOctyl is more precisely quantified in solution than by simple weighing, especially for small amounts of solute, using UV/visible spectrophotometry or another analytical technique (e.g. refractometry or mass spectrometry).

Solubility calculations, uncertainties and mass balances

The solubility of CalixOctyl (mol•mol -1 ) was calculated using Eq. ( 2):

(

where is the molar concentration (mol•L -1 ) in CalixOctyl determined by UV/visible spectrophotometry analysis using Eq. ( 1), is the volume of acetonitrile (L), M CO2 is the molar mass of CO 2 , and is the density of CO 2 (kg•m -3 ) at the experimental pressure and temperature, obtained from the NIST database [START_REF]Thermophysical Properties of Fluid Systems[END_REF]. Each experimental point was triplicated and the standard uncertainties for the solubility values were calculated. The standard uncertainties for the pressure and temperature during an experiment were estimated to be 1.3 K and 0.08 MPa, respectively.

The mass balances of CalixOctyl, CO 2 and ethyl acetate were also calculated to evaluate the reliability of the experimental data. The mean percentage deviations of the mass balances for each triplicated point were calculated using Eqs. (3-5):

(

) (4) 3
where and are the total masses of CO 2 and ethyl acetate delivered during the entire experiment, calculated using Eqs. ( 6) and ( 7):

(6) (7)
Here, is the duration of the experiment (in seconds), the density of injected CO 2 (in kg•cm -3 , at 263 K) and , the density of ethyl acetate.

Solution enthalpy measurements

The experimental setup (Fig. 3, Setaram Instrumentation) based on differential scanning calorimetry (DSC) used to measure solution enthalpies in SC-CO 2 is similar to the one described by Stassi and Schiraldi [START_REF] Stassi | Solubility of vegetable cuticular waxes in supercritical CO 2 : isothermal calorimetry investigations[END_REF]. Experiments were performed at constant temperature and pressure and variable CO 2 flow rate. The calorimetric analysis involved the following steps:

 Cell B was filled with the solid sample to be analyzed.

 The device was heated up to the operating temperature.

 The device was pumped up to the operating pressure, which was then maintained using the BPR. The CO 2 flow rate was set to 1.7•10 -2 cm 3 •s -1 .

 Once the DSC signal (heat flow) had stabilized, the CO 2 flow rate was lowered in a stepwise fashion, with a different plateau for the signal at each flow rate.

Solution enthalpy experiments were carried out at 314, 324 and 334 K, under 1.7, 1.5, 1.3 and 1.1•10 -2 cm 3 •s -1 CO 2 at a fixed pressure of 24.0 MPa. The DSC signal stabilizes rapidly at these flow rates. Fig. 4 presents a typical DSC profile obtained in this way. To correct for the effects of asymmetry in the system (i.e. intrinsic differences between cells A and B), the values obtained for CO 2 alone under the same operating conditions (cell B remaining empty) were subtracted from the raw signal. Thus, one characteristic plateau value of the DSC signal (s in W) was collected for each CO 2 flow rate tested at each temperature and pressure. Each measurement was triplicated in order to calculate the standard uncertainty associated with each value. The standard uncertainties for the pressure and temperature were estimated to be 0.02 K and 0.08 MPa, respectively.

Eq. 8 was used to relate the DSC signal at constant pressure to the solution enthalpy.

According to Riva and Schiraldi [START_REF] Riva | Kinetic parameterization of transitions and reactions in food systems from isothermal and non-isothermal DSC traces[END_REF] indeed, the observed signal depends on the solubility of CalixOctyl in CO 2 ( , mol•mol -1 ), the CO 2 flow rate ( , g•s -1 , converted from cm 3 •s -1 using the CO 2 densities in the NIST database [START_REF]Thermophysical Properties of Fluid Systems[END_REF]), the molar mass of CO 2 (M CO2 , g•mol -1 ), and the molar solution enthalpy of CalixOctyl in CO 2 ( , J•mol -1 ) as follows:

(8)

Results and discussion

Device validation

The reliability of the device used for solubility measurements (Fig. 2) was assessed by comparing the values obtained for dibenzo-24-crown-8 (DB24C8) in SC-CO 2 with those published by Shamispur et al. [START_REF] Shamispur | Solubility of large crown ethers in supercritical carbon dioxide[END_REF]. The following regression equation ( ) was obtained between the absorbance at the 275.4 nm maximum, , and the molar concentration of dibenzo-24-crown-8 in acetonitrile, (in mol•L -1 ):

(9)
Table 1 shows the solubilities measured at 16.2, 20.3 and 24.3 MPa with the temperature kept fixed at 318 K (using Eq. 2), along with the corresponding mean deviations of the mass balances (Eqs. [START_REF] Perrin | The potential industrial applications of calixarenes[END_REF](4)[START_REF] Taniguchi | Catalytic activity of an octopus-type calixarene on the formation of ethers[END_REF]). The two datasets are in good agreement. As expected, the solubility of dibenzo-24-crown-8

increases with the pressure at constant temperature. These results show that our experimental device is a reliable means to measure the solubility of solid compounds in SC-CO 2 .

Solubility of CalixOctyl in SC-CO 2

The solubility of CalixOctyl was determined in terms of equilibrium mole fraction of solute in SC-CO 2 at 314, 324 and 334 K for pressures ranging from 10.3 to 27.4 MPa, as summarized in Table 2.

Table 2

Solubility of CalixOctyl (25,27-Bis(1-octyloxy)calix [4]arene-crown-6, 1,3-alternate) in supercritical CO The solubility of CalixOctyl is low, varying from (0.2 ± 0.02)•10 -6 mol•mol -1 at 334 K and 10.4 MPa to (72.8 ± 0.3)•10 -6 mol.mol -1 at 324 K and 27.4 MPa. Fig. 5 shows how the solubility of CalixOctyl evolves as a function of pressure and temperature. The solubility isotherms in Fig. 5 intersect at a pressure of 21.4 MPa. In other words, the solubility is temperature independent at this pressure. At lower pressures, the solubility of CalixOctyl increases with the pressure but decreases as the temperature increases; i.e. the solubility of CalixOctyl increases as the CO 2 becomes denser. This is the trend usually observed for liquid and solid solutes in SC-CO 2 [START_REF] Gupta | Solubility in supercritical carbon dioxide[END_REF]. Above the crossover pressure however, whereas the solubility still increases with the pressure, the solubility is highest at the intermediate temperature (324 K). This may be because the temperature influences both the CO 2 density and the vapor pressure of the solute and its potential interactions with CO 2 .

The solubilities of p-t-butylcalix [4,8]arenes and calix [4]arene in SC-CO 2 are also maximal at a similar intermediate temperature (323 K) but there is no crossover pressure for either of these compounds [START_REF] Graham | Solubility of p-t-butylcalixarenes in supercritical carbon dioxide: a comparison of static and dynamic measurement techniques[END_REF]. Crossover pressure points are observed for crown ethers however-at around 16.2 MPa for dibenzo-24-crown-8, dibenzo-27-crown-9 and dibenzo-30crown-10 [START_REF] Shamispur | Solubility of large crown ethers in supercritical carbon dioxide[END_REF]. In qualitative terms, since CalixOctyl combines the structures of a calixarene and of a crown ether, its solubility curves should also reflect the presence of these two components. Quantitatively, CalixOctyl and crown ethers are roughly one hundred times more soluble in SC-CO 2 than are simple calixarenes. This is consistent with the observations of Sarbu et al. [START_REF] Sarbu | Non-fluorous polymers with very high solubility in supercritical CO 2 down to low pressures[END_REF] and Kilic et al. [START_REF] Kilic | Effect of grafted Lewis base groups on the phase behavior of model poly(dimethyl siloxanes) in CO 2[END_REF] that the addition of ether chains to a macromolecule can dramatically increase its solubility in SC-CO 2 .

Solution enthalpy of CalixOctyl in SC-CO 2

Fig. 6 shows the evolution of the corrected DSC signal as a function of the CO 2 flow rate at the three studied temperatures. Table 3 lists the correlations obtained between the DSC signal and the CO 2 flow rate at the three temperatures, along with the corresponding values of CalixOctyl's enthalpy of solution in CO 2 , calculated using Eq. 9 and the solubility data in Table 2.

Table 3.

Correlation between the differential scanning calorimetry signal (s) and the CO 2 flow rate ( ) at 24 MPA and three different temperatures, and the corresponding solution enthalpies of CalixOctyl (25,27-Bis(1octyloxy)calix [4]arene-crown-6, 1,3-alternate) in supercritical CO 2 calculated using Eq. 9.

T (K) s (W) (kJ.mol

-1 ) 314 ( ) 324 ( ) 334 ( )
The temperature dependence of CalixOctyl's enthalpy of solution is also linear at 24.0 MPa:

-

with . As is equal to zero at the crossover pressure (see Appendix, 21.4 MPa in this case), if one assumes that its variation with pressure is linear (following Stassi and Schiraldi [START_REF] Stassi | Solubility of vegetable cuticular waxes in supercritical CO 2 : isothermal calorimetry investigations[END_REF]), the enthalpy of solution at a fixed temperature for any given pressure can be obtained from using Eq. ( 11):

Using the values of determined at 314, 324 and 334 K, the estimated pressure dependences of the solution enthalpy at these temperatures are:

This experimental method thus enables solution enthalpies of solid compounds to be determined in SC-CO 2 at any pressure and temperature.

As explained by Chrastil [START_REF] Chrastil | Solubility of solids and liquids in supercritical gases[END_REF], the solution enthalpy of an organic solid in SC-CO 2 is the sum of its sublimation enthalpy and its solvation enthalpy therein:

, with and . At a fixed pressure of 24.0 MPa, the increase with temperature of CalixOctyl's enthalpy of solution (see Table 3) is expected because as the temperature increases, the density and thus the solvent power of CO 2 decrease, leading to weaker solvation. On the other hand, the sublimation enthalpy is known to increase with temperature.

A pressure increase at fixed temperature has a more ambiguous effect on the solution enthalpy. At 314 K, CalixOctyl's enthalpy of solution decreases as the pressure increases. This is probably because the density and thus the solvation power of CO 2 increase (through increased coordination between CO 2 and the solute). For the two other temperatures, a pressure increase leads to an increase in the solution enthalpy. Zhang et al. [START_REF] Zhang | Direct measurements of the enthalpy of solution of solid solute in supercritical fluids: study on the CO 2 -naphthalene system[END_REF] reported the same behavior for the solution enthalpy of naphthalene in SC-CO 2 . Using quantum chemistry calculations, these authors showed that the CO 2 energy level decreases as the pressure increases. As a result, the heat released during the formation of the solvent-solute clusters decreases as the pressure increases, which explains the decrease in the solvation enthalpy and thus the increase in the solution enthalpy. Further investigations combining in situ analytical methods to characterize the presence (or not) of clusters in the SC-CO 2 phase (such as smallangle X-ray scattering and small-angle neutron scattering) and molecular simulations would be necessary to completely elucidate this solubilization mechanism.

Conclusions

We have determined the solubility of CalixOctyl in SC-CO 2 . Although calixarenes are promising selective extractants of metals, there are very few studies devoted to the solubilities of calixarenes in SC-CO 2 in the literature. Information on the solubility of CalixOctyl in SC-CO 2 will also help to optimize existing processes. The solubility of CalixOctyl in SC-CO 2 varies from (0.20 ± 0.02)•10 -6 mol.mol -1 at 334 K and 10.4 MPa, to (72.78 ± 0.31)•10 -6 mol.mol -1 at 324 K and 27.4 MPa, always increasing with pressure at fixed temperature.

Although the CalixOctyl molecule combines the structures of a calixarene and of a crown ether, similar to the latter, its solubility is 100 times higher than that of simple calixarenes.

However, the solubility curves of CalixOctyl have properties typical of both components, exhibiting a crossover pressure (as those of crown ethers do) and a maximum at an intermediate temperature of 324 K (as also observed for p-t-butylcalix [4]arene). Its good selectivity towards cesium [START_REF] Dozol | Calix[4arènes-couronnes, leur procédé de préparation et leur utilisation pour l'extraction sélective du césium et des actinides[END_REF] and far greater solubility in SC-CO 2 thus make CalixOctyl a more promising extractant than simple calixarenes.

An experimental setup based on differential scanning calorimetry was also developed in this study to measure the solution enthalpies of solid compounds in SC-CO 2 . As expected, CalixOctyl's enthalpy of solution increases linearly with temperature at fixed pressure as the CO 2 density and solvation power decrease. Likewise, the solvation weakens and the solution enthalpy increases for pressure increases at fixed temperatures of 324 and 334 K. This is likely due to a decrease of the energy released by the formation of solvent-solute clusters. At

Fig. 1 .

 1 Fig. 1. Chemical structure of CalixOctyl (25,27-Bis(1-octyloxy)calix[4]arene-crown-6, 1,3-alternate).

Fig. 2 .

 2 Fig. 2. Diagram of the experimental setup used to measure the solubility of CalixOctyl (25,27-Bis(1octyloxy)calix[4]arene-crown-6, 1,3-alternate) in supercritical CO 2 . V1, V2 and V3: stop valves; PU: CO 2 pump; HE: heat exchanger; R: solubilization reactor; SWV: switching valve; CV: check valve; SF: solvent flask; PS: solvent pump; BPR: back-pressure regulator; S: stove; C1: glass collector; CB: cooling bath; C2: steel collector; FM: mass flow meter.



  The operating pressure was readjusted.

Fig. 3 .

 3 Fig. 3. Diagram of the device used to measure solution enthalpies. SYP: syringe pump, HJ: heating jacket, DSC-S: DSC sensor, CB: cooling bath, A: reference cell, B: measurement cell, RD: rupture disks, PS: pressure sensor, TIC: thermally insulated chamber, BPR: back-pressure regulator.

Fig. 4 .

 4 Fig. 4. Differential scanning calorimetry signal measured for CalixOctyl (25,27-Bis(1-octyloxy)calix[4]arenecrown-6, 1,3-alternate) at CO 2 flow rates of 1.7, 1.5, 1.3 and 1.1•10 -2 cm 3 •s -1 at 314 K and 24.0 MPa.

Fig. 5 .

 5 Fig. 5. Solubility of CalixOctyl (25,27-Bis(1-octyloxy)calix[4]arene-crown-6, 1,3-alternate) as a function of pressure at 314, 324 and 334 K.

Fig. 6 .

 6 Fig. 6. Differential scanning calorimetry signal intensity measured for CalixOctyl (25,27-Bis(1-

Table 1

 1 Solubility of dibenzo-24-crown-8 in supercritical CO 2 as measured in this study and as reported byShamispur et

		al. [42].
	Pressure	Temperature	10 6 •y (this work)				10 6 •y [42]
	(MPa)	(K)	(mol•mol -1 )	(%)	(%)	(%)	(mol•mol -1 )
	16.2	318	16.2 ± 0.1	4.5	9.9	6.2	15.8
	20.3	318	24.1 ± 0.2	8.9	3.5	7.3	22.4
	24.3	318	38.2 ± 0.1	7.1	9.3	7.6	33.4

Acknowledgments

The authors wish to acknowledge the ANR DEMETERRES project and AREVA for their financial support and Marine Cordray, Amélys Venditti and Pierre Venditti for their technical help.

probably because of stronger solvation.

In the context of future thermodynamic studies of metal extraction systems, the data obtained in this work could be incorporated in a global investigation of ternary metal extraction systems involving CalixOctyl, a cationic exchanger and SC-CO 2 . A better understanding of the physicochemical properties of these complex systems would indeed facilitate the scaling up of experimental processes for industrial applications.

Appendix

At the crossover pressure ( ), the molar solution enthalpy of CalixOctyl in SC-CO 2 can be written:

where is the molar enthalpy of CalixOctyl in SC-CO 2 and is the molar enthalpy of pure CalixOctyl.

The chemical potential of CalixOctyl in SC-CO 2 at is given by Eq. (A.2):

in which is the activity of CalixOctyl in SC-CO 2 . Eq. (A.2) can be rewritten in terms of , and , respectively the activity coefficient and solubility of CalixOctyl in SC-CO 2 , (A.

As the solubility of CalixOctyl in SC-CO 2 is low ( ), one can assume that , hence:

Differentiating with respect to temperature yields:

As pointed out by Foster et al. [START_REF] Foster | Significance of the crossover pressure in solid-supercritical fluid phase equilibria[END_REF], the mathematical criteria for the crossover pressure is so:

Using the Gibbs-Helmoltz equation , Eq. 6 becomes:

(A.7)

And finally, substituting Eq. (A.7) into Eq. (A.1) yields:

(A.8)


Nadji Hermas
email: hermasnadji@yahoo.fr

About the intrinsic symbolic calculus for pseudodi¤erential operators on manifolds

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About the intrinsic symbolic calculus for pseudodifferential operators on manifolds

Nadji Hermas

Introduction

In this paper, we present a simple calculus for pseudodi¤erential operators on an arbitrary smooth manifold M . This symbolic calculus has two characters: First, it is intrinsic, this means that it doesn't depend on the di¤erent coordinate systems of M . Secondly, it works well in the case where we use the classes of Hörmander S m ; with the condition 0 max( ; 1=3) < 1, whereas the usual calculus based on the coordinate systems which we …nd in the classic books as [H], [Ta] and [Tr], works well only under the condition 1 < 1 (which implies 1=2 < 1). The language of linear connections has a fundamental role in the construction of the symbolic calculus for pseudodi¤erential operators on manifolds, for this reason, I used it in this paper.

H. Widom discovered in the seventies that the use of linear connections to de…ne the intrinsic symbols of pseudodi¤erential operators is essential. By using these mathematical beings he constructed a version of symbolic calculus on the manifolds [W], but he didn't utilize a global phase function in his construction and was satis…ed by a local utilization of the standard phase function. On the contrary, Safarov gave a complete intrinsic symbolic calculus in its article [Sa], he used global phase functions and presented the distribution kernels of the operators as oscillatory integrals de…ned in open neighborhoods of the diagonal M . 0 2000 Mathematics Subject Classi…cation: 47G30, 53B05, 53C05, 58J40 0 Key words: Densities, density tensors, linear connections, pseudodi¤erential operators, intrinsic symbols. 1 In this paper I followed the footsteps of Safarov which treated the case where the symbols of operators are functions de…ned on the cotangent bundle T M . I treated here the case where these symbols are "densities over M " de…ned on T M , and to make this I generalized some operations of derivation de…ned on the sets of functions and tensors to the case of densities and density tensors.

To construct an intrinsic symbolic calculus on the manifold M , we have need of that which we call the phase functions of pseudodi¤erential type. These classes of the functions are generally de…ned on sets of the form W = f(y; x; ) : (x; y) 2 W; 2 T x M g where W

M M are open neighborhoods of the diagonal M , and since long time, it became clear that these classes are bound to the family of the connections on M . In fact, for all connection on M , there is a family of the phase functions associated to this connection. For example, the phase function which we …nd in the theory of pseudodi¤erential operators in R n , is the phase function associated with the Levi-Civita connection on R n .

The symbolic calculus on the open sets of R n appear simple because it is written in the standard Cartesian system which is de…ned in all R n . All normal coordinate systems corresponding to the Levi-Civita connection on R n , are bound linearly to this system. But this is not the case when we deal with the pseudodi¤erential operators on arbitrary manifolds. Because in the last case, a favorite connection doesn't exist, and the normal coordinate systems corresponding to the family of the connections on M , are not universal and depend on the points of M . I signal that the last property appear in R n if we replace the Levi-Civita connection by a non- ‡at connection in the survey of the symbolic calculus.

Most results exposed in this article are natural generalizations of those exposed in [Sa], and I added here the two results 5.2 and 6.7 which we can consider as natural generalizations of the classic theorems 3.1 and 4.2 in the book [Sh].

In this paper, I used some symbols and amplitudes of Hörmander's classes corresponding to the family of the connections on M, and it is clear that one can generalize easily this work to contain other classes of symbols and amplitudes.

The section 2 contains elementary results about the properties of the connections that we can …nd in any book dedicated to the survey of the connections as [KN]. But these results are very important to comprehend the language used here in the determination of the symbols, for this reason I was obliged to present them in this article.

Finally, concerning the operators and the symbols, I utilized the same notations of the classic book [H], and concerning the densities, the tensors and the density tensors, I used the most usable notations in the books of the di¤erential geometry as [KN], [P], [Sp], [Ste] and [Str].

Recalls of some properties of connections on manifolds

1. Let M be a smooth n-dimensional manifold and be an a¢ ne connection on M . For (x; y) 2 M M , we denote by x;y : [0; 1] ! M the geodesic (of course if it exists) joining x and y such that x;y (0) = x and x;y (1) = y. This geodesic is uniquely determined as soon as it exists. We put V = (x; y) 2 M M : x;y exists .

The theory of ODE assures that V is an open neighbourhood of the diagonal

M = f(x; x) : x 2 M g in M M
. Also, we can see easily that V is symmetric with respect to M .

Let ' = (x 1 ; : : : ; x n ) be a coordinate system de…ned in U M . Then W = (x; y) 2 U U \ V : x;y ([0; 1]) U is an open set in U U and in the case where '(U ) is convex in R n , we can use the Taylor formula to prove that

_ k x;y (0) = y k x k + 1 2 X ij k ij (x)(y i x i )(y j x j ) + X 3 j j N P k (x)(y x) +(N + 1) 1 Z 0 (1 s) N X j j=N +1
(y x) P k (x + s(y x))ds;

(2.1) for all (x; y) 2 W , where

x k = x k (x), y k = x k (y), _ x;y (0) = (x k x;y ) 0 (0), k ij
are the Christo¤el symbols of , and P k are some polynomials in the Christo¤el symbols and their derivatives. If (x i ) is a normal coordinate system with origin x, then _ k x;y (0) = _ k x;y (t) = y k x k for all y 2 V (x) = fz 2 M : (x; z) 2 V g and t 2 [0; 1]. Therefore we have P k (x) = 0, 8k,8 6 = 0, in this system. In particular, k ij (x) + k ji (x) = 0 in any normal coordinate system with origin x.

Let (x;

) 2 T M . The horizontal lift of the vector v 2 T x M at the point (x; ) is given by

r v (x; ) = X j v j @ x j (x; ) + X ijk k ij (x)v i k @ @ j (x; )
where (x 1 ; : : : ; x n ; 1 ; : : : ; n ) is a coordinate system de…ned in a neighbourhood of (x; ), v = P j v j @ @x j (x) and = P j j dx j (x). If X is a vector …eld on M , then its horizontal lift in T M is given by

r X (x; ) = r X(x) (x; ); (x; ) 2 T M .
The horizontal distribution and the vertical distribution are de…ned by

HT M = [ (x; )2T M f(x; )g H (x; ) T M and V T M = [ (x; )2T M f(x; )g V (x; ) T M
respectively, where H (x; ) T M = fr v (x; ) : v 2 T x M g, V (x; ) T M = ker T (x; ) P and P : T M ! M is the canonical surjection.

3. Let (x; y) 2 V . We denote here by (x; y) the parallel displacement from T x M to T y M along the geodesic x;y . Obviously, (y; x) = (x; y) 1 , and if the points x, y and z belong to the same geodesic, then (x; z) = (y; z) (x; y). Also, the theory of ODE assures that the map is of class C 1 from V to the bundle S

(x;y)2M M f(x; y)g Hom(T x M; T y M ).
Let x 2 M and let (y k ) be a normal coordinate system with origin x. Let us denote by x (y) the matrix of (x; y) in this system. Based on the de…nition of the parallel displacement, we get

X k (y k x k )@ y k ( x ) i j (y) = X pq (y p x p ) q pj (y)( x ) i q (y); y 2 V (x) (2.2)
where x k = y k (x) and y k = y k (y). And from here, we deduce that

@ y k ( x ) i j (x) = i kj (x) = 1 2 T i kj (x), @ y `@y k ( x ) i j (x) = 1 2 @ y k i `j (x) + @ y ` i kj (x) + X p p kj (x) i `p(x) + X p p `j (x) i kp (x) ! where n T i kj (x)
o is the torsion tensor of . So, if is symmetric, then we obtain

@ y k ( x ) i j (x) = 0, (2.3) @ y `@y k ( x ) i j (x) = 1 2 (@ y k i `j (x) + @ y ` i kj (x)) = 1 6 (R i kj`( x) + R i `jk (x)) (2.4) where n R i kj`( x)
o is the curvature tensor of . If M is a pseudo-Riemannian manifold with metric tensor fg ij g and is the Levi-Civita connection on M , then we have

t x (y) G 1 (y) x (y) = G 1 (x); y 2 V (x)
where G(y) = (g ij (y)). And in this case, (2.3) and (2.4) immediately implies

@ y k G(x) = @ y k G 1 (x) = 0, @ y `@y k g ij (x) = 1 3 (R ikj`( x) + R i`jk (x)) (2.5)
where R ijk`= P p g ip R p jk`.

Densities and Density Tensors

1. Densities. Let M be a smooth n-dimensional manifold and be a real number. By de…nition, a smooth density on M of order is a C 1 section of the complex line bundle (M ). But I prefer here to identify each density with its components, in this way, one says that u : M ! C is a density on M of order if it veri…es the following condition:

ũ(x) = det @x i @ xj (x) u(x); x 2 U
where (x i ) and (x i ) are two coordinate systems de…ned on U M , and u(x) and ũ(x) are the values of u in these two systems. We often denote by C 1 (M ; ) and C 1 c (M ; ) the spaces of smooth -densities and smooth -densities with compact supports respectively.

Also, one says that u : T M ! C is a density of order on M if we have

ũ(x; ) = det @x i @ xj (x) u(x; ); (x; ) 2 T U
where (x i ) and (x i ) are always two coordinate systems de…ned on U M , and u(x; ) and ũ(x; ) are the values of u in these two systems. We denote here by C 1 (T M ; ) the spaces of smooth densities u : T M ! C of order , and it is necessary to note that

C 1 (T M ; ) = C 1 (T M ) C 1 (M ; ) 6 = C 1 (T M ; ) C 1 (T M ).
Now, let M 1 ,..., M r be smooth manifolds, and let 1 ,..., r be real numbers. We say that u is a density of order ( 1 ; : : :

; r ) on M = M 1 M r if it veri…es the following ũ(x) = det @x i 1 @ xj 1 (x 1 ) ! 1 det @x i r @ xj r (x r ) r u(x); x = (x 1 ; : : : ; x r ) 2 U 1 U r M
where (x i `) and (x i `) are two coordinate systems de…ned on U ` M `(1 ` r), and u(x; ) and ũ(x; ) are the values of u in the two systems ((x i1 1 ) i1 ; : : : ; (x ir r ) ir ) and ((x i1 1 ) i1 ; : : : ; (x ir r ) ir ) respectively.We will denote by C 1 (M 1 M r ; 1;:::; r ) and C 1 c (M 1 M r ; 1;:::; r ) the spaces of smooth ( 1 ; : : : ; r )-densities and smooth ( 1 ; : : : ; r )-densities with compact supports respectively. Obviously, if

= 1 = = r then C 1 (M 1 M r ; 1;:::; r ) = C 1 (M 1 M r ; ) and C 1 c (M 1 M r ; 1;:::; r ) = C 1 c (M 1 M r ;
). Using the local charts, one can construct two appropriate topologies on the two spaces C 1 (M 1 M r ; 1;:::; r ) and C 1 c (M 1 M r ; 1;:::; r ) (See for example [CP]). We will denote here by D 0 (M 1 M r ; 1;:::; r ) the topological dual of the space C 1 c (M 1 M r ; 1 1;:::;1 r ). Let M be a smooth n-dimensional manifold and be an a¢ ne connection on M . We de…ne the density % as follows: %(x; y) = det (x; y); (x; y) 2 V where (x; y) is the parallel displacement from T x M to T y M along the geodesic x;y . The density %(x; y) (2 C 1 (V ;

1;1 )) depends on and will play a fundamental role in the following. Now, let (y k ) be a normal coordinate system with origin x. From (2.2), it follows that

X k (y k x k )@ y k %(x; y) = X k`( y k x k ) k`( y)%(x; y); y 2 V (x)
where x k = y k (x) and y k = y k (y). And from this, we obtain

X k (y k x k )@ y k %(x; y) + X k`( y k x k )(y ` x `)@ y `@y k %(x; y) = X kp (y k x k ) p kp (y)%(x; y) + X k`( y k x k )(y ` x `) X p @ y k p `p(y) + X pq p kp (y) q kq (y) ! %(x; y), y 2 V (x). (3.1) Therefore @ y k %(x; y) =y=x = X p p kp (x) = 1 2 X p T p kp (x), (3.2)
@ y `@y k %(x; y) =y=x = 1 2 X p (@ y k p `p(x) + @ y ` p kp (x)) + 1 2 X pq p kp (x) q `q (x).
So, if is symmetric, then we have

@ y k %(x; y) =y=x = 0, (3.3) @ y `@y k %(x; y) =y=x = 1 2 X p (@ y k p `p(x) + @ y ` p kp (x)) = 1 6 (R k`( x) + R `k(x)) (3.4) where fR k`g is the Ricci tensor of de…ned by R k`= P j R j kj`.
Let's suppose that M is a pseudo-Riemannian manifold and is the Levi-Civita connection on M . Then we have

%(x; y) = g 1 (x)g(y); (x; y) 2 V
where g is the canonical density of M . In this case, (3.3) and (3.4) become as follows @ y k g(y) =y=x = 0, @ y `@y k g(y

) =y=x = 1 3 R k`( x)g(x), (3.5)
since the tensor fR k`g is symmetric.

2. Density Tensors. We denote by T p q (M ; ) the space of the C 1 sections of T p q (M ) (M ). The elements of this space are called the density tensors of order and type (p; q) on M . And to simplify this exposition, I will identify each density tensor with its components and this in every local chart of M .

Let be a connection on M . We want to prolong the action of to the spaces of the density tensors. First, let X be a vector …eld on M and (x i ) be a coordinate system de…ned on U M . If T 2 T p q (M ; ), we put D X (x)T i1;:::;ip j1;:::;jq (x) = X j X j (x)@ x j T i1;:::;ip j1;:::;jq (x)

+ X ji 0 1 i1 ji 0 1 (x)X j (x)T i 0 1 ;:::;ip j1;:::;jq (x) + + X ji 0 p ip ji 0 p (x)X j (x)T
i1;i2;:::;i 0 p j1;:::;jq (x)

X jj 0 1 j 0 1 jj1 (x)X j (x)T
i1;:::;ip j 0 1 ;:::;jq (x)

X jj 0 q j 0 q jjq (x)X j (x)T
i1;:::;i 0 p j1;j2;:::;j 0 q (x); x 2 U where T i1;:::;ip j1;:::;jq (x) are the components of T with respect to (x i ) and

X = P j X j @ @x j in U . We know that if = 0, then D X T = n D X (x)T i1;:::;ip j1;:::;jq (x)
o is a tensor of the type (p; q) on M which is called the covariant derivative of T with respect to X. If 2 R, we de…ne two density tensors D X T and Ð X T as follows: in any coordinate system (x i ) (de…ned on U M ), the components of D X T and Ð X T are given by the relations i1;:::;ip j1;:::;jq (y) =y=x where y = (y i ) is the normal coordinate system with origin x 2 U associated to (x i ). The two density tensors D X T and Ð X T are called the covariant derivatives of T with respect to X, and we can notice easily that if = 0, then these two derivatives coincide with the usual covariant derivative of T .

Let T 2 T p q (M ; ). The two density tensors with the components belong to the space T p q+r (M ; ) and are called the r-th covariant di¤erentials of T (here (y i ) is the normal coordinate system with origin at the point x associated to the system used in this point, and D k (y) =D @=@y k (y)). The symmetrizations of these density tensors with respect to k 1 , ..., k r are called the r-th symmetric covariant di¤erentials of T , and we will denote them respectively by D r T = n D T . We now give the following proposition which its simple proof is based on (3.1) and (3.2). Proposition 3.1. Let T 2 T p q (M ; ). If is symmetric, then D X T =Ð X T for all vector …eld X on M , and if is ‡at (T 0;R 0), then D r T =Ð r T and D r 0 (D r T ) =D r (D r 0 T ) =D r+r 0 T for all (r; r 0 ) 2 N 2 .

Let's suppose that M is a pseudo-Riemannian manifold and is the Levi-Civita connection on M . In this case if T 2 T p q (M ; ), then F = g T 2 T p q (M ), and we can verify easily that D X T = Ð X T = g D X F and D r T = g D r F for all vector …eld X on M and all r 2 N.

3. Horizontal Di¤erentials. Let a(x; ) 2 C 1 (T M; ) and let (x i ) be a coordinate system de…ned on U M . We put d a(x; ) = @ y (% (x; y)a(y; (x; y) )) =y=x , r a(x; ) = @ y (a(y; (x; y) )) =y=x where y = (y i ) is the normal coordinate system with origin x 2 U associated to the system (x i ) and 2 N n . If p 2 N, the two density tensors fd a(x; )g j j=p and fr a(x; )g j j=p will be called the p-th symmetric horizontal di¤erentials of a. The following proposition justi…es the use of this terminology.

Proposition 3.2. Let a(x; ) 2 C 1 (T M; ) and p 2 N. Then the density tensor fd a(x; )g j j=p is the symmetrization of the density tensor with the components r

(x) i1 r (x)
ip (% (x; y)a(y; )) =(y; )=(x; ) , and the tensor fr a(x; )g j j=p is the symmetrization of the density tensor n r , where y = (y i ) is the normal coordinate system with origin at the point x associated to the system used in this point and r

(x) k = r @=@y k .
Proof. Let x 2 M and let y = (y i ) be a normal coordinate system with origin x. From (2.2) we have

X j j=p p! ! (y x) @ y (a(y; (x; y) )) = X i1;:::;ip (y i1 x i1 ) (y ip x ip )r (x) i1 r (x) ip a(y; (x; y) ); y 2 V (x),
where y i (x) = x i and y i (y) = y i . Therefore

X j j=p p! ! (y x) @ y (a(y; (x; y) )) =y=x = X i1;:::;ip (y i1 x i1 ) (y ip x ip )r (x) i1 r (x) ip a(x; ) + O(jy xj p+1 ).
Putting y i = x i + "X i and passing to the limit as " ! 0, we obtain

X j j=p p! ! X @ y (a(y; (x; y) )) =y=x = X i1;:::;ip X i1 X ip r (x) i1 r (x) ip a(x; ):
This equality immediately gives the proposition.

Remark 3.3. Let a(x; ) 2 C 1 (T M; ). In any coordinate system (x i ) de…ned on U M we have

r k a(x; ) = 2 a(x; ) X i ( i ik (x) + i ki (x)) + r x k a(x; ), d k a(x; ) = 2 a(x; ) X i T i ki (x) + r k a(x; ) = a(x; ) X i i ki (x) + r x k a(x; ),
where (x; ) 2 T U and r x k = r @=@x k . From this we obtain

(r k r ` r `rk )a(x; ) = X ij R i jk`( x) i @ j a(x; ) a(x; ) X i R i ik`( x) + 1 2 X i D k T i i`( x) 1 2 X i D `T i ik (x) + 1 2 X ip T p k`( x)T i ip (x) + 1 4 X ip T i kp (x)T p `i(x)
1 4

X ip T p ki (x)T i `p(x) 1
A ;

(d k d ` d `dk )a(x; ) = X ij R i jk`( x) i @ j a(x; ) a(x; ) X i R i ik`( x) 1 4 X ip T i kp (x)T p `i(x) + 1 4 X ip T p ki (x)T i `p(x) 1 A ; (d k r ` r `dk )a(x; ) = X ij R i jk`( x) i @ j a(x; ) a(x; ) X i R i ik`( x) + 1 2 X i D k T i i`( x) + 1 4 X ip T p k`( x)T i ip (x) 1 A ; (@ k r ` r `@ k )a(x; ) = (@ k d ` d `@ k )a(x; ) = 1 2 X j
T k `j (x)@ j a(x; ):

Remark 3.4. Let a(x; ) 2 C 1 (T M; ). By Proposition 3.2 we deduct that if is ‡at, then d a(x; ) = r a(x; ) = @ y a(y; ) =(y; )=(x; ) ; 8 . So r r a(x; ) = r r a(x; ) = r + a(x; ); 8 ; 8 ; @ r a(x; ) = r @ a(x; ) = @ @ y a(y; ) =(y; )=(x; ) ; 8 ; 8 :

In view of Remark 3.3 these equalities are not true in the general case.

Remark 3.5. It is clear that if a(x; ) 2 C 1 (T M ), then d a(x; ) = r a(x; ); 8 ,
and in this case, a is constant along any horizontal curve in T M if and only if all its symmetric horizontal di¤erentials are equal to zero.

Remark 3.6. Let T 2 T p 0 (M ; ). We put a(x; ) = X i1;:::;ip

T i1;:::;ip (x) i1 ip = X j j=p p! ! T (x)
where fT g j j=p is the symmetrization of the density tensor T . We can verify easily that if (x i ) is a coordinate system de…ned on U M , then

r x k a(x; ) = X i1;:::;ip D k (x)T i1;:::;ip (x) i1 ip = X j j=p p! ! D k (x)T (x) ; (x; ) 2 T U .
From this equality and Proposition 3.2 it follows that

d a(x; ) = X j j=p p! ! D T (x) ; (x; ) 2 T U , r a(x; ) = X j j=p p! ! Ð T (x) ; (x; ) 2 T U ,
where 2 N n . Also, we deduct that if X is a vector …eld on M , then r X (y; )(% (x; y)a(y;

)) =(y; )=(x; ) = X j j=p p! ! D X T (x) ; (x; ) 2 T U; r X (y; )a(y; ) =(y; )=(x; ) = X j j=p p! ! Ð X T (x) ; (x; ) 2 T U;
where y = (y i ) is the normal coordinate system with origin x associated to the system (x i ).

Some Classes of Symbols and Amplitudes

1. Let M be a smooth n-dimensional manifold, and let m, and be real numbers (0 , 1). The classic class of symbols S m ; (T M ; ) consists of densities a(x; ) 2 C 1 (T M ; ) such that in any coordinate system (x i ) de…ned on U M for all compact set K U and all ( ; ) 2 N n N n @ @ x a(x; ) const K; ; h i m+ j j j j x ; 8(x; ) 2 T K;

where h i x = q 1 + 2 1 + + 2 n ; = P j j dx j (x).
Also, if N is another smooth n 0 -dimensional manifold, the classic class of amplitudes S m ; (N T M ; ; ) consists of densities a(y; x; ) 2 C 1 (N T M ; ; ) such that in any coordinate systems (x i ) and (y i ) de…ned (respectively) on U M and W N for all compact set K 1 K 2 U W and all ( ; ; ) 2 N n N n N n 0 @ y @ @ x a(y; x; ) const K1;K2; ; ; h i m+ (j j+j j) j j x ; (x; ) 2 T K 1 ; y 2 K 2 :

We signal that the properties of the classes S m ; are widely-known and we can …nd them in any book dedicated to the survey of the pseudodi¤erential operators.

2. M is always a smooth n-dimensional manifold. Let be a connection on M and let m, and be real numbers (0 , 1). The class of symbols S m ; (T M ; ; ) consists of densities a(x; ) 2 C 1 (T M ; ) such that in any coordinate system (x i ) de…ned on U M for all compact set K U and all 2 N n and (i 1 ; : : : ; i p ) @ r x i 1 r x ip a(x; ) const K; ;i1;:::;ip h i m+ p j j x ; 8(x; ) 2 T K where r x i = r @=@x i . Safarov introduced these classes in his article [Sa].

If N is another smooth n 0 -dimensional manifold, the class of amplitudes S m ; (N T M ; ; ; ) consists of densities a(y; x; ) 2 C 1 (N T M ; ; ) such that in any coordinate systems (x i ) and (y i ) de…ned (respectively) on U M and W N for all compact set K 1 K 2 U W and all ( ; ) 2 N n N n 0 and (i 1 ; : : : ; i p ) @ y @ r x i 1 r x ip a(y; x; ) const K1;K2; ; ;i1;:::;ip h i

m+ (j j+p) j j x ; (x; ) 2 T K 1 ; y 2 K 2 : (4.1)
For the sake of simplicity we will designate by S m ; ( ; ; ) the space S m ; (N T M ; ; ; ) and at the same time the space of density tensors T 2 T p q (N T M ; ; ) (p; q 2 N) which all their components verify (4.1).

Let (N T M ; ;0 ; ); where X 1 ; : : : ; X p are vector …elds on M . By using the partitions of unity on M , we can prove this result: If a j 2 S mj ; ( ; ) with a j ! 1 as j ! +1, then there exists a density tensor a 2 S m ; ( ; ), m = max m j , such that a ( ; ). In particular, we have fd a(x; )g j j=p ; fr a(x; )g j j=p 2 S m+p ; ( ; ); 8p:

Proof. It su¢ ces to see that, in any coordinate system (x i ) de…ned on U M , we have

r (x) i1 r (x) ip a(y; ) =(y; )=(x; ) = 0 (x)a(x; ) + X i i (x)r x i a(x; ) + X i1;i2 i1;i2 (x)r x i 1 r x i 2 a(x; ) + + X
i1;:::;ip i1;:::;ip (x)

r x i 1 r x ip a(x; ); (x; ) 2 T U (4.2)
where 0 , i , i1;i2 ,..., i1;:::;ip are functions of class C 1 in U .

Remark 4.2. Let a(x; ) 2 S m ; (T M ; ; ). From the previous proposition, it follows that if X 1 ; :::; X p are vector …elds on M , then (V ; ; ); where x = (x i ) is the normal coordinate system with origin x associated to the system (x i ), ỹ = (ỹ i ) is the normal coordinate system with origin y associated to the system (y i ) and (z t ; z s ) = ( x;y (t); x;y (s)).

Some Classes of Oscillatory Integrals

1. Phase Functions. Let M be an open set of R n . We know well that if A is a pseudodi¤erential operator on M , then its distribution kernel is given by an oscillatory integral of the form

K A (x; y) = Z e i(x y) a(x; y; ) † ; (x; y) 2 M M (5.1)
where a 2 S S m ; (M M R n ) and † = (2 ) n d . But when M is a general manifold, the writing of the pseudodi¤erential kernels as in (5.1) is not possible except in the local charts of M or in small neighborhoods of the diagonal M = f(x; x) : x 2 M g. We study here some oscillatory integrals which we can consider pseudodi¤erential kernels, and to make this, I need a class of the phase functions. We signal that the families of the phase functions of pseudodi¤erential type bind to the family of the connections on M ; in fact, for all connection on M , there is a family of the phase functions associated to this connection. For example, the phase function that we …nd in the theory of pseudodi¤erential operators in R n is the phase function associated with the Levi-Civita connection on R n .

Let M be a smooth manifold and a connection on M . The phase functions of pseudodi¤erential type associated to are

' t (x; ; y) = < _ x;y (t); >; (x; y) 2 V ; ; 2 T zt M
where t 2 [0; 1] and z t = x;y (t) (These functions are similar to the phase functions introduced by Drager [D]). Based on the de…nitions of the geodesic and the parallel displacement, we deduct easily that we have ' t (x; ; y) = ' 1 t (y; ; x); 8(x; y) 2 V; 8t 2 [0; 1] ; 8 2 T zt M; ' t (x; ; y) = ' s (x; (z t ; z s ) ; y); 8(x; y) 2 V; 8(t; s) 2 [0; 1] 2 ; 8 2 T zt M:

Let (x i ) be a coordinate system de…ned on U M . The formula (2.1) shows that for all point x 0 2 U , there exists an open neighborhood W U of

x 0 such that the expression of ' 0 with respect to the system (x i ) takes in W the following form

' 0 (x; ; y) = (x y) (A(x; y) ); 8(x; y) 2 W W; 8 2 T x M (5.2)
where A 2 C 1 (W W ; M n n (R)) and A(x; x) = I n ; det A(x; y) 6 = 0 for all (x; y) 2 W W .

Let y = (y i ) be a normal coordinate system with origin x 2 M . We have in this system ' t (x; ; y) = (x y) ; 8y 2 V (x); 8t 2 [0; 1] ; 8 2 T zt M: 2. Oscillatory Integrals. Let M be a smooth n-dimensional manifold and a connection on M . Let a(y; x; ) 2 S m ; (V ; ; ; ) and 2 R. Under the condition ( ; ) 2 ]0; 1] [0; 1[ we can consider the oscillatory integral

K(x; y) = % 1 (x; z t )% (x; z s )% 1 (x; y) Z e i' t (x; ;y) a(z s ; z t ; ) † ; (x; y) 2 V
like an element of the dual space D 0 (V ; + + ;1 ) where its action on the elements of the space D(V ; 1 ; ) is determined as follows < K(x; y); '(x; y) >= X jk`Z e i' t (x; ;y) % 1 (x; z t )% (x; z s )% 1 (x; y) a(z s ; z t ; )'(x; y) j (x) k (y) `(z t )dxdy †

where ' 2 D(V ; 1 ; ) and ( j ) is a partition of unity associated to some atlas of M . Let's notice that the integrals in this sum are usual oscillatoy integrals. So K(x; y) 2 C 1 (V M ; + + ;1 ) (this is the fundamental property which characterizes the pseudodi¤erential Kernels).

We signal that we can write the oscillatory integral K(x; y) as follows K(x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) a 0 (z s ; z t ; ) † ; (x; y) 2 V with a 0 (y; x; ) = % (x; y)a(y; x; ). Because of that we are going to study all integrals K(x; y) under this form.

Proposition 5.1. Let a(y; x; ) 2 S m ; (V ; 0; ; ) and (t; s; r; ) 2 [0; 1] 3 R. Under the condition 0 < 1 we can write the oscillatory integral

K(x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) a(z s ; z t ; ) † ; (x; y) 2 V as follows K(x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) t (z t ; ) † + f t;s; (x; y) = % (x; z r )% 1 (z r ; y) Z e i' r (x; ;y) b(z s ; z r ; ) † + f t;s;r; (x; y)
where f t;s; ; f t;s;r; 2 C 1 (V ; + ;1 ), b(y; x; )

P (t r) j j ! D d
x a(y; x; ) and t (x; ) P (s t) j j ! D r y a(y; x; ) =y=x . In particular we have

K(x; y) = % (x; z s )% 1 (z s ; y) Z e i' s (x; ;y) s (z s ; ) † + C 1 (V ; + ;1 )
where s (x; )

P (t s) j j ! D d x a(y; x; ) =y=x .
Proof. Let us prove the …rst equality in the proposition. By using Taylor's formula, we immediately get the following expansion

a(y; x; ) = X j j N _ x;y ! r y a(y; x; ) =y=x + X j j=N +1 _ x;y ! a (y; x; ); (x; y) 2 V where N 2 N , a (y; x; ) = N +1 ! R 1 0 (1
) N @ y a(y; x; ) =y=z d and _ x;y = ( _ 1

x;y (0)) 1 ( _ n x;y (0)) n . So, the oscillatory integral K(x; y) becomes as follows

K(x; y) = % (x; z t )% 1 (z t ; y) X j j N (t s) j j ! Z D (e i' t (x; ;y) ) r y a(y; z t ; ) =y=zt † + % (x; z t )% 1 (z t ; y) X j j N (t s) j j Z
D (e i' t (x; ;y) )a (z s ; z t ; ) † since _ zt;zs = (s t) _ x;y (t). Now by integrating by parts with respect to , we obtain

K(x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) 0 @ X j j N (s t) j j ! D r y a(y; z t ; ) =y=zt + X j j=N +1 (s t) j j D a (z s ; z t ; ) 1 A † : If one chooses t (x; ) 2 S m ; (T M ; ; ) such that t (x; ) X (s t) j j ! D r y a(y; x; ) =y=x ; then one gets K(x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) t (z t ; ) † +% (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) b N (z s ; z t ; ) † with b N (y; x; ) = t (x; ) X j j N (s t) j j ! D r y a(y; x; ) =y=x + X j j=N +1
(s t) j j D a (y; x; ):

From here we deduce the wanted since b N 2 S m+( )(N +1) ; 0 (V ; 0; ), < and the choice of N is arbitrary.

To prove the second equality in the proposition one makes the change of variables = (z r ; z t ) in the oscillatory integral K(x; y), so it becomes as follows

K(x; y) = % (x; z r )% 1 (z r ; y) Z e i' r (x; ;y) % (z r ; z t )a(z s ; z t ; (z r ; z t ) ) † ,
and for arriving to the wanted we use the following expansion

% (x; y)a(z; y; (x; y) ) = X j j N _ x;y ! d x a(z; x; ) + X j j=N +1 _ x;y ! a (z; y; x; ); (z; y; x; ) 2 V where a (z; y; x; ) = N + 1 ! 1 Z 0
(1 ) N @ y (% (x; y)a(z; y; (x; y) )) =y=z d and V = f(z; y; x; ) : (x; y) 2 V ;

(x; z) 2 V ; (y; z) 2 V ; 2 T x M g.
The following proposition is demonstrated exactly as the previous proposition.

Proposition 5.2. Let a(y; x; ) 2 S m ; (V ; 0; ; )(0 < 1). We choose a function 2 C 1 (M M ) such that supp V , 1 in a small neighborhood of M and the two projections 1 ; 2 :supp ! M are proper maps. We de…ne a density (x; ) as follows: If (x i ) is a coordinate system de…ned on U M , the value of in this system is given by

(x; ) = Z e i' 0 (x; ;y) (x; y)a(y; x; )dy † ; (x; ) 2 T U
where y = (y i ) is the normal coordinate system with origin x associated to (x i ).

Then (x; ) 2 S m ; (T M ; ; ) and (x; ) P 1 ! D r y a(y; x; ) =y=x . So, we have Z e i' 0 (x; ;y) a(y; x; ) † Z e i' 0 (x; ;y) (x; ) † 2 C 1 (V ; +1;0 ).

Corollary 5.3. Let a(x; ) 2 S m ; (T M ; ; )(0 < 1) and (t; ) 2 [0; 1] R. If % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) a(z t ; ) † 2 C 1 (V ; + ;1 ) then a(x; ) 2 S 1 (T M ; ).
Proof. According to Proposition 5.1 we have Z e i' 0 (x; ;y) a(x; ) † 2 C 1 (V ; +1;0 ).

Then there exist a neighborhood W V of M and a density b(y; x; ) 2 S 1 (W ; 0; ) such that Z e i' 0 (x; ;y) (a(x; ) b(y; x; )) † = 0; 8(x; y) 2 W , where W = f(y; x; ) : (x; y) 2 W; 2 T M g. From this identity and Proposition 5.2, it follows that

0 a(x; ) X 1 ! D r y b(y; x; ) =y=x .
So a(x; ) 2 S 1 (T M ; ).

I end this section by the following remark.

Remark 5.4. Let a(y; x; ) 2 S m ; (V ; 0; ; ) with ( ;

) 2 ]0; 1] [0; 1[. If K(x; y) = % (x; z r )% 1 (z r ; y) Z e i' r (x; ;y) a(z s ; z r ; ) † ; (x; y) 2 V , then t K(x; y) = % 1 (x; z 1 r )% + (z 1 r ; y) Z e i' 1 r (x; ;y) a(z 1 s ; z 1 r ; ) † ; K (x; y) = % 1 (x; z 1 r )% + (z 1 r ; y) Z e i' 1 r (x; ;y) a(z 1 s ; z 1 r ; ) † ;
where (x; y) 2 V .

6. Pseudodi¤erential Operators acting in the Spaces of Densities for all local amplitude a(y; x; ) of A (every density veri…es this condition is called a principal symbol of A). We can also verify that we have (T M ;

):

Now, let be a connection on M . If A 2 m ; (M ; ; ), then there exist a neighborhood W

V of M and a density a(y; x; ) 2 S m ; 0 (W ; 0; ) ( 0 = max( 1; )) such that

K A (x; y) = % 1 (x; y)
Z e i' 0 (x; ;y) a(y; x; ) † ; (x; y) 2 W .

Based on this simple remark, we give the following de…nition (this de…nition is the same de…nition which Safarov presented in its article [Sa]).

De…nition Let be a connection on M and Let m, , , and be real numbers; 0 < 1, 0 < 1. We say that the linear continuous operator A : C 1 c (M ; ) ! D 0 (M ; ) belongs to the set m ; (M ; ; ; ) if its distribution kernel K A (x; y) veri…es the two following conditions: 2. Symbols of Di¤erential Operators. We know well that all di¤erential operator A : C 1 (M ; ) ! C 1 (M ; ) belongs to the space m 1;0 (M ; ; ), therefore we have the right to ask the following question: If is a connection on M , how can we determine the symbols of A with respect to ? Let's notice that, according to Remark 6.2, it su¢ ces to determine the symbol A (x; ) = A;0 (x; ).

(i) K A (x; y) 2 C 1 (M M M ; ;
First we assume that A 2 m ; (M ; ; ; ) (0 < 1). If A (x; ) is a symbol of A, then K A (x; y) = % 1 (x; y) Z e i' 0 (x; ;y) A (x; ) † + f (x; y); (x; y) 2 V with f (x; y) 2 C 1 (V ; ;1 ). Now let u(x) 2 C 1 (M ; ) and (x; y) 2 C 1 (M M ) such that supp V and 1 in a small neighborhood of M . From the previous equality it follows that if (x i ) is a coordinate system de…ned on U M , then the value of Au in this system is given by

Au(x) = A (y; D y )(% 1 (x; y) (x; y)u(y)) =y=x + Z K A (x; y)(1 (x; y))u(y)dy + Z f (x; y) (x; y)u(y)dy
where x 2 U and y = (y i ) is the normal coordinate system with origin x associated to (x i ). So if A is a di¤erential operator of order m on M and A (x; ) = P j j m a (x) in the system (x i ), then we have

Au(x) = A (y; D y )(% 1 (x; y)u(y)) =y=x = X j j m a (x)D y (% 1 (x; y)u(y)) =y=x . (6.1)
From here we deduct that the value of A (x; ) in the system (x i ) is given by A (x; ) = A(y; D y )(e i(y x) % 1 (x; y)) =y=x ; (x; ) 2 T U: (6.2) I will denote by % n (x; y) the value of %(x; y) in any coordinate system normal (with respect to ) at the point x and by %n (x; y) the value of %(x; y) in any coordinate system normal (with respect to ~ ) at the same point x. Also, I put J ; ~ (x; y) = det @y j =@ ỹk (y) ; (x; y) 2 V = V \ Ṽ ;

; ~ (y; x; ) = e i(y(y) ỹ(y)) % 1 n (x; y)% 1 n (x; y) J ; ~ (x; y) ; (y; x; ) 2 V ;

where y = (y j ) and ỹ = (ỹ k ) are two n.c.s. with origin x such that @y j =@ ỹk (x) = jk (of course (y j ) and (ỹ k ) are normal with respect to and ~ respectively). We can notice easily that J

; ~ = J 1 ~ ; 2 C 1 (V ) and ; ~ = 1 ~ ; 2 C 1 (V )
. By returning to the de…nition of the normal coordinate systems, one can show that there exist a neighborhood W V of M and a map (x; y) 2 W 7 ! (x; L(x; y)) 2 Iso(T M; T M ) of class C 1 having the two following properties:

-For all x 2 M , L(x; x) is equal to the identity of T x M ; -If (x i ) is a coordinate system de…ned on U M , then

(x ỹ(y)) = t M L (x; y) 1 (x y(y)); 8(x; y) 2 U M \ W; (6.5)
where M L (x; y) is the matrix of L(x; y) with respect to (x i ) and y = (y j ) and ỹ = (ỹ k ) are the two normal coordinate systems with origin x associated to (x i ).

I can now give this theorem.

Theorem 6.6. Let A 2 m ; 1 (M ; ; ; )\ m ; 2 (M ; ; ; ~ ). Under the condition max( 1 ; 2 ; 1=2) < , if ~ A (x; ) is a symbol of A with respect to ~ , then each symbol A (x; ) of A with respect to veri…es the following asymptotic expansion

A (x; ) X 1 ! r y ; ~ (y; x; ) =y=x D ~ A (x; ): Proof. Let ~ A (x; ) be a symbol of A with respect to ~ . Then K A (x; y) = %1 (x; y) Z e i' 0 (x; ;y) ~ A (x; ) † + f (x; y); (x; y) 2 V ~ with f (x; y) 2 C 1 (V ~ ; ;1
). Now by using (6.5), we get iii) Let M and N be two smooth manifolds of same dimension n and let G : M ! N be a di¤eomorphism. Let and ~ be two connections de…ned on M and N respectively. I put V = (x; y) 2 V ~ : (G 1 (x); G 1 (y)) 2 V and G; ; ~ = G ; ~ . One veri…es that the function G; ; ~ is given by G; ; ~ (y; x; ) = e ify(G 1 (y)) ỹ(y)g % 1 n (G 1 (x); G 1 (y))% 1 n (x; y) det @(y j G 1 ) @ ỹk (y) ; for (y; x; ) 2 V , where ỹ = (ỹ j ) is a n.c.s. with origin x and y = (y j ) is the n.c.s. with origin G 1 (x) which veri…es @(y j G 1 )=@ ỹk (x) = jk . According to Remark 6.4, Proposition 6.5 and Theorem 6.6 we have the following theorem which is considered the natural generalization of the theorem 4.2 in [Sh]. 4. Transposed Operators and Adjoint Operators. Let be a connection on M . Remarks (5.4) and (6.2) immediately give the following theorem.

K A (x; y) = % 1 (x; y) Z e i' 0 (
Theorem 6.8. Let A 2 m ; (M ; ; ; ) ( > 0, < 1). First we assume that the operator A is given by Au = au; u 2 C 1 (M ; ) with a 2 C 1 (M ; ) (m 1 = 0). So we have K BA (x; y) = a(y)K B (x; y) = % 1 (x; y) Z e i' 0 (x; ;y) a(y) B (x; ) † + f (x; y); (x; y) 2 V where f 2 C 1 (V ; ;1 ). From here we deduct that BA 2 m2 ; (M ; ; ; ), furthermore, Proposition 5.1 shows that we have To treat the general case, we need some notation. We will put % ; (x; y; z) = % 2 (x; z)% 1 (z; y)% 1 (y; x); % (x; y; z) = % ; (x; y; z) = % 2 (x; z)% 1 (z; y)% 1 (y; x); (x; ; y; z) = ' 0 (x; ; y) + ' 0 (x; ; z) + ' 0 (z; (x; z) ; y); with f 2 C 1 (V ; ;1 ) and a(y; x; ) = Z e i' 0 (x; ;z) (x; z) (z; y) B (x; + )

A (z; (x; z) )% ; (x; y; z)e i (x; ;y;z) dz † :

After that we continue exactly as in the proof of the theorem 8.3 in [Sa] for arriving to the wanted.

Remark 6.11. The result of Theorem 6.10 remains valid if we replace one of its conditions by the following condition: A 2 m1 1;0 (M ; ; ; ) or B 2 m2 1;0 (M ; ; ; ).

  ::;jq (x) = D X (y) % (x; y)T i1;:::;ip j1;:::;jq (y) =y=x = D X (x) % (z; ::;jq (x) = D X (y)T

D

  k1 (y) D kr (y) % (x; y)T i1;:::;ip j1;:::;jq (y) =y=x , D k1 (y) D kr (y)T i1;:::;ip j1;:::;jq (y) =y=x

  ip a(y; ) =(y; )=(x; ) o

P j a j .

 j Proposition 4.1. Let a 2 S m ; (T M ; ; ). For all p 2 N the two density tensors with the components r x; y)a(y; )) =(y; )=(x; ) belong to the space S m+p ;

r

  X1 (y; ) r Xp (y; )a(y; ) =(y; )=(x; ) 2 S m+p ; (T M; ; ); r X1 (y; ) r Xp (y; )(% (x; y)a(y; )) =(y; )=(x; ) 2 S m+p ; (T M; ; ): Remark 4.3. Let V = f(y; x; ) : (x; y) 2 V ; 2 T x M g, (t; s) 2 [0; 1] 2 and (p; q) 2 N N. If a 2 S m ; (V ; ; ; ), then by modifying the equality (4y; ỹ)% (x; x)a(ỹ; x; ~ )) =(ỹ;x; ~ )=(zs;zt; (x;zt) )

  x; ;y) a(y; x; ) † + f (x; y); (x; y) y) J ; ~ (x; y)1 jdet L(x; y)j ~ A (x; L(x; y) ): Since 0 1 = max(1 1 ; ) < , Proposition 5.1 shows that if A (x; ) is a symbol of A with respect to , then A (x; ) X 1 ! D r y a(y; x; ) =y=x :We can also write this asymptotic expansion as follows A (x; )X P (x; )D ~ A (x; )where P (x; ) is a polynomials in independent of ~ A (x; ) and its degree is strictly lower of j j . Therefore if A is a di¤erential operator of order m on M , thenA (x; ) = X j j m P (x; )D ~ A (x; ); (x; ) 2 T M:But a direct use of (6.1) and (6.2) gives y; x; ) =y=x D ~ A (x; ); (x; ) 2 T M: So P (x; ) = r y ; ~ (y; x; ) =y=x ; 8 because the polynomials P (x; ) are independent of A.

  Theorem 6.7. Let A 2 m ; (N ; ; ); 1 < 1. If A (x;) is a symbol of A with respect to ~ , then each symbol B (x; ) of B = G AG 1 with respect to veri…es the following asymptotic expansion ; ~ (y; G(x); t DG(x) 1 ) =y=G(x) D A (G(x); t DG(x) 1 ):

  be a connection on M . We can see easily that S m

	means that the classes S m ; ( ) doesn't depend on when the properties of the classes S m ; ( ) are analogous to those of the classes S m 1 . In general, ; ,
	and for example, from the de…nition we immediately get the following:
	-a 2 S m1 ; ( ; ; ) ^b 2 S m2 ; ( ; ; ) ) a + b 2 S	max(m1;m2) ;	( ; ; );
	-a 2 S m1 ; (N T M; ; ; ) ^b 2 S m1 ; (N T M; S m1+m2 ; (N T M ; + 0 ; + 0 ; );	0 ; 0 ; ) ) a b 2
	-a 2 S m ; ( ; ; ) ) @ a 2 S	m j j ;	( ; ; );
	-a 2 S m ; (N T M ; ;0 ; ) ) r X1	r Xp a 2 S m+p ;
	where 0 = max(1	; ). Therefore if		1	; ; = S m S m ; ( ) S m ; 0 then S m ; ( ), and this

  Every density A;t (x; ) veri…es this condition is called a t-symbol of the operator A, and according to Corollary 5.3 we deduce that the map A 7 !

	1 ); V of M and a density a(y; x; ) 2 e i' 0 (x; ;y) a(y; x; ) † ; (x; y) 2 W . Remark 6.1. From (5.2) it follows that m (ii) There exist a neighborhood W S m ; (W ; 0; ; ) such that K A (x; y) = % 1 (x; y) Z ; (M ; ; ; ) m ; 0 (M ; ; ) where 0 = max( ; 1 ). Therefore m ; (M ; ; ; ) = m ; (M ; ; ) when 1 . Remark 6.2. Let t 2 [0; 1]. Based on Proposition 5.1 we conclude that if 0 < 1, then the condition (ii) is equivalent to the following condition: (iii) t There exists a density A;t (x; ) 2 S m ; (T M ; ; ) such that K A (x; y) = % (x; z t )% 1 (z t ; y) Z e i' t (x; ;y) A;t (x; ) † + f t (x; y) in V ; (M ; ; ; )= 1 ; (M ; ; ) to the space S m ; (T M ; ; )=S 1 (T M ; ). Also, from Proposition 5.1 it follows that if A;t (x; ) is t-symbol of A and s 2 [0; 1], then A;s (x; ) X (t s) j j ! D d x A;t (x; ): where f A;t (x; ) is an isomorphism from the space m ; (M ; ; ; ) with 0 < 1 and > 1=2. Remark 6.3. Let A 2 m From the de…nition and Remark 6.2 we deduce that all t-symbol of A (t 2 [0; 1]) is a principal symbol of A 2 m ; 0 (M ;

t (x; y) 2 C 1 (V ; ;1 ). ; ).

  Then t A; A 2 Remark 6.9. A operator A 2 m ; (M ; ;1 ; ) is said to be formally selfadjoint if A = A and nearly formally self-adjoint if A = A + R with R 2 1 (M ; ;1 ). Under the condition 0 < 1, the previous theorem shows that A is nearly formally self-adjoint if and only if the values of the symbol W A (x; ) = A;1=2 (x; ) are real. Then a di¤erential operator A 2 Di¤(M ; ;1 ) is formally self-adjoint if and only if the values of its Weyl symbol are real. 5. Composition formula of intrinsic symbols. Let be a connection on M and let A 2 m1 ; (M ; ; ; ), B 2 m2 ; (M ; ; ; ); 0 < 1, from which at least one is properly supported. In this case, does the operator BA belong to m1+m2 ; (M ; ; ; )? if yes, what the relation that exists between its symbols and those of A and B?

	m ; (M ; 1 ;1 ; ). Moreover, if 1	<	1, then t A;s (x; ) =
	A;1 s (x; ) and A ;s (x; ) = A;1 s (x; ) for all s 2 [0; 1]. So we have
	t A;s (x; )	X (2s 1) j j !	D d x A;s (x; );
	A ;s (x; )	X (1 2s) j j !	D d x A;s (x; ):

I now give some examples. a) Let a 2 C 1 (M ; ). Based on (6.2) and Remark 6.2 we conclude that all symbols of the operator u 2 C 1 (M ; ) 7 ! au 2 C 1 (M ; + ) are equal to a.

b) Let X = P j X j @ @x j be a vector …eld on M . We de…ne two di¤erential operators of order 1 A X and B X by

By using (3.2) and ( 6.2) we get

Now the Remarks 3.6 and 6.2 show that if t 2 [0:1], then

c) Let's assume that M is a pseudo-Riemannian manifold and is the Levi-Civita connection on M . We put ; = g g where is the usual Laplace operator on M , g is the canonical density of M and ( ; ) 2 R 2 . From (2.5), (3.5) and (6.2) it follows that

where S(x) is the scalar curvature of M at x and j j 2 x = P jk g jk (x) j k . Then since D g jk = 0 for all 2 N n 0, the Remarks 3.6 and 6.2 give the following

3. Formulae for transformations of symbols. I now discuss the action of the transformations on the symbols. For this let M and N be two smooth manifolds of same dimension n and let G :

) is a linear continuous operator, then the distribution kernel of B = G AG 1 is given by

Concerning the action on the intrinsic symbols, we will divide our discussion on three steps.

i) Let and ~ be two connections de…ned on M and N respectively and let A 2 m ; (N ; ; ; ~ ). In general, G AG 1 doesn't belong to m ; (M ; ; ; ) when < 1

, but if G is a¢ ne transformation, this belonging is always true. Let's recall that G : M ! N is said to be an a¢ ne transformation of (M; ) into (N; ~ ) if it veri…es

Under this condition, we verify easily that we have 8 > > < > > :

~ G(x);G(y) (s) = G( x;y (s)); 's (G(x); t DG(z s ) 1 ; G(y)) = ' s (x; ; y); (x; y) = t DG(y) ~ (G(x); G(y)) t DG(x) 1 ; %(x; y) = jJ G (x; G(x))j

1 %(G(x); G(y)) jJ G (y; G(y))j (6.4) for all (x; y) 2 V , s 2 [0; 1] and 2 T zs M . All the objects corresponding to ~ are marked by s in order that we distinguish them of those corresponding to .

Remark 6.4. Let ~ be a connection on N . If G : M ! N is a di¤eomorphism, then G is an a¢ ne transformation of (M; ) into (N; ~ ) where = G ~ .

The relations (6.3) and (6.4) immediately give the following proposition.

Proposition 6.5. Let G : (M; ) ! (N; ~ ) be an a¢ ne transformation and let A 2 m ; (N ; ; ; ~ ); < 1, > 0. Then G AG 1 2 m ; (M ; ; ; ). Moreover, under the condition 0

ii) Let's assume that M = N and let A 2 m ; 1 (M ; ; ; )\ m ; 2 (M ; ; ; ~ ) (max( 1 ; 2 ) < ). In this case, what the relation that exists between the symbols of the operator A with respect to and its symbols with respect to ~ ? To give a partial answer to this question, I need some notations.

where

) is a coordinate system de…ned on U M , we put (as Safarov did) P ( ; ) ;

(x; ) = (@ y + @ z ) @ y 0 @ X j 0 j j j

; (x; ) = (@ y + @ z ) @ y 0 @ X j 0 j j j

where (x; ) 2 T U (here the operation of the derivation is made in the normal coordinate systems with origin x associated to (x i )). It is clear that P ( ; ) ;

(x; ) and P ( )

; (x; ) are polynomials in , and we can verify (see [Sa]) that the degree of each one of these two polynomials is lower or equal to min(j j ; j j), furthermore, if is symmetric, this degree is lower or equal to min(j j ; j j ; (j j + j j)=3). In the case where is ‡at, we have P ; (x; ) 0 when j j + j j 1.

Theorem 6.10. Let A 2 m1 ; (M ; ; ; ), B 2 m2 ; (M ; ; ; ); 0 < 1, from which at least one is properly supported. Let's suppose that at least one of the following conditions is ful…lled:

(ii) is symmetric and > 1=3; Proof. The idea of the proof is as follows: We choose a function 2 C 1 (M M ) such that supp V , 1 in a small neighborhood of M and the two projections 1 ; 2 :supp ! M are proper maps. Next we use the de…nition to get K BA (x; y) = % 1 (x; y) Z e i' 0 (x; ;y) a(y; x; ) † + f (x; y); (x; y) 2 V