Mario Bermudez Guzman

Oleg Gomozov
email: oleg.gomozov@ensam.eu

Xavier Kestelyn
email: xavier.kestelyn@ensam.eu

Federico Barrero
email: fbarrero@us.es

Ky Ngac

Eric Nguyen

Semail

M Bermudez

N K Nguyen

E Semail
email: eric.semail@ensam.eu

Model predictive

Keywords: Multiphase drives, Model predictive control, Current and voltage limits, Optimal reference currents, Real-time simulation environments

come

Introduction

The interest in recent times in multiphase drives relies on their fault-tolerance inherent capabilities and on the ability to manage the power with lower torque pulsation and lower current harmonic content than conventional threephase ones [START_REF] Barrero | Recent advances in the design, modeling and control of multiphase machines -Part 1[END_REF][START_REF] Duran | Recent advances in the design, modeling and control of multiphase machines -Part 2[END_REF][START_REF] Levi | Advances in converter control and innovative exploitation of additional degrees of freedom for multiphase machines[END_REF], which usually reduce the electrical stresses on the machine and the power electronic components. Such advantages make them ideal candidates for applications where electrical limits are normally reached, becoming the reliability a key control issue for economical and safety reasons when multiphase drives are considered. For example, high-speed operation in traction and power generation applications using multiphase drives are typically required, while it is also desirable to extract the maximum torque capability over the entire speed range. These control requirements will depend on the voltage and current limits of the multiphase machine and the voltage source inverter (VSI), making essential the application of optimal control methods that take into account such limits or constraints.

While optimal controllers are mature control techniques in conventional three-phase drives, this is not the case and the situation becomes much more complicated when multiphase systems are considered. One dq reference frame appears in three-phase drives, which simplifies the definition of the analytical expressions of the optimal stator current reference while respecting the imposed constraints. The machine flux is usually weakened (the d-current stator component is reduced) to respect the imposed voltage limit, adjusting at the same time the q-current stator component with the aim of not exceeding the current limit. Many algorithms have been proposed in the scientific literature based on this flux-weakening control idea for induction [START_REF] Harnefors | Torque-maximizing field-weakening control: design, analysis, and parameter selection[END_REF][START_REF] Xu | Selection of the flux reference for induction machine drives in the field weakening region[END_REF] and permanent magnet [START_REF] Kim | Speed control of interior permanent magnet synchronous motor drive for the flux weakening operation[END_REF] machines. However, when multiphase drives are considered, permanent magnet and concentrated windings induction machines have been analyzed in [START_REF] Parsa | Five-phase permanent-magnet motor drives[END_REF][START_REF] Wang | Dual-rotor multiphase permanent magnet machine with harmonic injection to enhance torque density[END_REF][START_REF] Xu | Five-phase induction motor drives with DSP-based control system[END_REF], where the torque density is increased adding a third spatial harmonic in the magnetic field, but the operation in the field weakening region under electrical limits has not been studied. This absence of scientific analyses for the multiphase case relies on the appearance of multiple orthogonal dq spaces, where the optimization problem becomes quite complex and makes difficult to find analytical expressions for the reference stator currents in the different dq reference sets. Indeed, the phase peak value of the electrical magnitude depends on the peak value of each harmonic but also on their respective phase shifts. Therefore, some simplifications are normally assumed to get analytical expressions for the reference values when limits are considered. This is the case in [START_REF] Levi | Analytical determination of DC-bus utilization limits in multiphase VSI supplied AC drives[END_REF][START_REF] Mengoni | High-torque-density control of multiphase induction motor drives operating over a wide speed range[END_REF], where it is assumed that all harmonics components reach their peak values at the same time instant and suboptimal results are obtained. A different alternative appears in recent research works, where the optimal current references are based on offline procedures that generate look-up tables [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Lu | Computation of optimal current references for fluxweakening of multi-phase synchronous machines[END_REF]. These works also consider the steady state operation of the drive, and do not take into account the electrical and mechanical dynamics of the real system. Then, there is a need for new research works and methodologies to find optimal reference values when multiphase systems with current and voltage limits are considered.

A potential solution to the optimization problem can be the application of the Model Predictive Control (MPC) technique. This control strategy offers a high flexibility facing multi-input multi-output systems subject to constraints. The method is based on an accurate model of the system that it is used to predict the future behavior of the system through time, in order to select the optimal value of the control variables by minimizing a predefined cost function [START_REF] Cortes | Predictive control in power electronics and drives[END_REF]. MPC techniques can be categorized into two major types: Continuous-Control-Set MPC or CCS-MPC, where an average model of the system is defined and controlled with the purpose of generating continuous reference signals; and Finite-Control-Set MPC or FCS-MPC, which takes advantage of the limited number of switching states available in the VSI for solving the optimization problem using an iterative algorithm. It is noteworthy that MPC techniques have been widely utilized to solve control problems in electrical applications with power converters [START_REF] Cortes | Predictive control in power electronics and drives[END_REF]. Different control objectives and/or restrictions are easily included, and MPC has been employed for controlling multiphase drives giving a high flexibility [START_REF] Arahal | Multiphase current control using finite-state model-predictive control[END_REF][START_REF] Lim | Model predictive control of a two-motor drive with five-leg-inverter supply[END_REF][START_REF] Rodriguez | State of the art of finite control set model predictive control in power electronics[END_REF]. Nevertheless, none of these proposals takes into account electrical limits for the drive in the control strategy, up to the authors' knowledge. In [START_REF] Kestelyn | Investigation on model predictive control of a five-phase permanent magnet synchronous machine under voltage and current limits[END_REF], these limits have been barely considered to obtain optimal reference currents using classical PI regulators to guarantee the current tracking, while the present work goes beyond all mentioned proposals, stating that optimal reference currents can be obtained using model-based predictive methods. The main objective of the paper is then to introduce a new multiphase MPC method, named as two-stage Model-based Predictive optimal Control (2S-MPC). The proposed controller generates online optimal current references by means of a CCS-MPC stage that respects imposed voltage and current limits. Then, a control stage based on a FCS-MPC method is applied for the current regulation of the system. The achievements of the 2S-MPC technique compared to classical methods can be summarized as follows:  The proposed controller allows the consideration of electrical restrictions in the regulator strategy, including voltage and current limits imposed by the power converter and the electrical machine. An important industrial demand, in relation with obtaining the higher requirements in the peak torque and power density of modern motor drives, is then attended because an increment in the reliability levels of the drive is forced introducing stringent controllers with the ability of managing failure mechanisms and critical electrical limits.  The modulation stage using an inner current controller based on the FCS-MPC method can be suppressed, which improves the close-loop torque performance providing faster torque transient [START_REF] Lim | A comparative study of synchronous current control schemes based on FCS-MPC and PI-PWM for a two-motor three-phase drive[END_REF][START_REF] Lim | Model predictive control of a two-motor drive with five-leg-inverter supply[END_REF]. Note also that the tuning of the controller is practically effortless, in contrast to that of PI-PWM based methods.  Although optimal controllers have been previously stated in the research bibliography, only suboptimal solutions have been raised up in the multiphase drives' field. In this particular research area, some simplifications and/or assumptions were considered in the control strategy to take into account the considered electrical limits as stated before. However, the proposed method permits the online computation of the optimal current reference, ensuring the optimality condition of the problem.

A performance analysis of the controller is also done prior to the application to a real system. A real-time simulation environment based on the OPAL-RT technology is selected for this purpose, in order to accelerate the analysis, to reduce risks associated with conducting tests on a physical system and to simplify its future implementation [START_REF] Bélanger | The what, where and why of real-time simulation[END_REF]. The paper is organized as follows. First, Section 2 analyzes the five-phase PMSM drive as well as the limits that will be taken into account in the study. The proposed 2S-MPC technique is shown in Section 3. Section 4 details the implementation of the 2S-MPC method in the real-time system, providing the validation results that show the closed-loop performance of the multiphase drive working with the considered limits. Finally, conclusions are presented in Section 5.

Presentation of the studied system

Modeling of the five-phase PMSM drive

The system under study is based on a five-phase star-coupled PMSM, supplied by a five-phase two-level VSI. A simplified scheme of the drive is shown in Fig. 1. The voltage, flux and torque equations of the drive are normally obtained considering the following assumptions:  Some effects like magnetic saturation, hysteresis and iron losses are neglected.  Only first and third harmonics of periodical variables are considered.  Slot effects are assumed to be negligible.

Taking this into account, the stator voltage equation is given by: where R is the stator resistance, i and e are the stator current and back-EMF vectors, respectively, and L is the inductance matrix. The existing magnetic coupling between phase windings makes difficult the control of the five-phase machine in the phase frame. A coordinate transformation is normally applied to reduce this complexity, which converts the phase variables into two independent rotating reference frames, called dq1 and dq3. It is possible to link a two-phase fictitious machine to each dq subspace, as it is presented in [START_REF] Kestelyn | Vectorial multi-machine modeling for a five-phase machine[END_REF]. The first (second) fictitious machine is called the main (secondary) machine, modeled in the dq1 (dq3) subspace and associated with the fundamental (third harmonic) of the real machine variables. The coordinate transformation is done by the extended Park transformation matrix detailed in [START_REF] Barrero | Recent advances in the design, modeling and control of multiphase machines -Part 1[END_REF], producing new voltage equations presented in (3)- [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF]. Note that this transformation will generate constant dq1 and dq3 components in steady state. (2)

e t i L Ri v    d d (1) i a v a - + S b S a A B S a S b V DC 2 V DC 2 N N' i b v b - + i c v c - + S d S c C D S c S d i d v d - + S e E S e
                                                                                                                                                            2
                                    p p
1 1 1 1 1 1 d d q q d d d d i L p t i L Ri v     (3)              1 1 1 1 1 1 1 2 5 d d f d d q q q q i L p t i L Ri v  (4) 3 3 3 3 3 3 3 d d q q d d d d i L p t i L Ri v     (5)              3 3 3 3 3 3 3 2 5 3 d d f d d q q q q i L p t i L Ri v  (6)
where:  vd1, vq1, id1, iq1 and vd3, vq3, id3, iq3 are the projections of the phase voltages and currents in the subspaces dq1 and dq3, respectively.  Ld1, Lq1, Ld3, Lq3 are the inductances along the d and q axes associated with the first and third harmonics of the air gap flux.  p is the number of pole pairs.  Φf1 and Φf3 are the fluxes along the d axis created by the permanent magnets associated with the first and third harmonics of the air gap flux, respectively. Following this approach, the electromagnetic torque of the real machine is determined as the sum of the torques developed by both fictitious machines, as it is stated down below:

T T T   (7)               1 1 1 1 1 1 1 2 5 q f q d q d em i i i L L p T (8)               3 3 3 3 3 3 3 2 5 3 q f q d q d em i i i L L p T ( 9
)
being Tem1 and Tem3 the electromagnetic torque created by the first and third harmonics of the air gap flux, respectively.

Considered constraints

From the physical point of view, maximizing the torque capability of the multiphase drive depends on the voltage and current constraints of the machine and the VSI. The voltage limit comes from the maximum DC-link voltage that the VSI can apply to the machine (maximum peak phase-to-phase voltage, VDC), and it is obtained in the fluxweakening region, where the available torque decreases when the machine operates above the base speed. On the other hand, current limits are imposed by: the power converter and its electronic switches, which limit a maximum peak phase current value (IVSI); and the copper losses in the machine, which generate a maximum RMS phase current (IRMS). For the sake of simplicity, it is considered in this work that the RMS phase current never exceeds the maximum available value and safety margins are included for controllability reasons for the maximum peak phase current value and the maximum peak phase-to-phase voltage (Imax and Vmax, respectively). Then, the electrical constraints that will be taken into account in what follows are summarized as:

  VSI max phase I I t i   (10)   DC max phase to phase V V t v     (11)

Proposed 2S-MPC technique

The scheme of the proposed 2S-MPC method is shown in Fig. 2, where the two main stages are identified. The optimization problem (first control stage) is a reference current generator based on CCS-MPC that finds the best way to divide currents between d and q axes. It can be considered as an extra part of the inner current controller (second control stage), where the FCS-MPC method is used for the stator current regulation. An outer PI-based speed control loop should be also considered to implement a variable speed drive based on the utilization of a multiphase electric machine. However, the study is focused on the performance of the multiphase drive when online optimal reference currents are applied to get the reference torque, while minimizing the copper losses and respecting the defined maximum peaks values of currents and voltages. Then, the analysis is done in the open-loop torque regulation configuration to avoid any interference of the outer speed controller in the electrical performance of the system. Notice that a machine simulator stage is plotted (see next section) to identify that we are not using an experimental test rig but a real-time simulation environment based on OPAL-RT technologies for modeling the multiphase drive.

S opt ω r T * em V max I max Current Controller (FCS-MPC)
Extended Park Transformation Eq. ( 2) The optimization problem to be solved in the first stage is presented down below:

i q3 i d3 , i q1 i d1 , i * d1 i * q1 , i * d3 i *
        (9)
2 em em T q d q d i V u u u u peak I i i i i i peak to subject T T i i i i f min                 (12)
where two weighting factors ωi and ωT are introduced in the objective function f to give more or less importance to the minimization of the copper losses with respect to the reference torque tracking. In this case, the multiphase drive has been analyzed using a steady state model of the system and a simulation environment based on Matlab© tools. It is concluded that ωi = 1 and ωT = 10000 are appropriate values to guarantee a good tracking of the reference torque when the drive operates inside the feasible reference region (i.e. below the maximum torque versus speed characteristic of the electrical machine) and the production of the maximum possible torque when the reference current is not feasible and the operating point is outside the feasible region. The procedure to solve the optimization problem is summarized in Fig. 3 (left plot), where the model of the multiphase drive described in equations ( 3)-( 9) is required. The model is discretized using a forward Euler method and used with a possible dq current optimal reference to obtain reference dq voltage and torque values that can be applied to the machine. Then, phase currents and phaseto-phase voltages are calculated from previous dq current and voltage values, using the inverse of the Park transformation matrix detailed in [START_REF] Barrero | Recent advances in the design, modeling and control of multiphase machines -Part 1[END_REF]. After that, the peak values of these variables are extracted to verify if the constraints are respected, and if this is the case, the objective function is also evaluated for the proposed solution (dq current references) and the generated torque. If the constraints are not respected, a new possible dq current optimal reference is analyzed in the same way, and the process starts over. This optimization problem is rewritten in the standard form of a quadratic programming problem and solved in an iterative manner by the algorithm presented in [START_REF] Gomozov | Adaptive energy management system based on a real-time model predictive control with non-uniform sampling time for multiple energy storage electric vehicle[END_REF].

On the other hand, the FCS-MPC method is utilized as an inner predictive stator current controller. It is based on the discretization of the five-phase PMSM model, which is used along with the measured speed ωr k and the stator currents idq k in the time instant k to obtain the future stator current values in the next sampling period k+1 (idq k+1 ). The predictive model is based on dq transformation and it is discretized employing the forward Euler integration method. The control objective of the predictive current controller lies on defining a cost function J and finding the switch configuration to be applied in the VSI (Sopt). This optimum value Sopt is obtained computing the predictive model for every available switching state (2 5 = 32 for our five-phase machine) to predict the future stator current and find the one that minimizes J. The cost function considered in this work consists in tracking the optimal current references that were calculated in the previous stage, as it is shown in [START_REF] Levi | Advances in converter control and innovative exploitation of additional degrees of freedom for multiphase machines[END_REF]. This second optimization process is detailed in the flow diagram shown in the right plot of Fig. 3.

2 k q k q 2 k d k d 2 k q k q 2 k d k d i i i i i i i i J                                         1 3 1 * 3 1 3 1 * 3 1 1 1 * 1 1 1 1 * 1 (13)
To summarize, the optimization problem that leads to the optimal current references is convex and thus it has only one optimal solution without local minima, checking the optimality condition and providing the optimal reference to the FCS-MPC controller. Then, the FCS-MPC stage uses a finite set of switching states, checking all of them to apply the optimal one that minimizes the designed cost function. Notice that a one-step prediction horizon is used in this study. Although better results are expected with longer prediction horizons, a bigger online calculation effort would be required and a prediction horizon equal to one should be enough to state the interest of the proposal. Note also that the optimization problem is adapted to a continuous physical system with a time constant higher than 1 ms, while the second stage is adjusted to the finite state and switching nature of the power converter.

Validation of the 2S-MPC technique in a real-time system based on OPAL-RT technologies

To validate the 2S-MPC proposed method, an experimental set-up system is programed in a real-time system based on the OP5600 real-time simulator target platform complemented with the OP5607 extension module from OPAL-RT technologies. The Xilinx Virtex 7 FPGA of the OP5607 module is used for building the machine simulator and the FCS-MPC current controller, using an internal clock frequency of 200 MHz. On the other hand, the Intel Xeon CPU of the OP5600 platform is used to communicate with a host PC and to solve the optimal references generation, sending the obtained results to the FPGA as well as the reference inputs (such as the reference torque and speed). The CPU performance allows up to 3 ms of sampling time for the optimal reference currents generator for single core mode, although further improvements could be obtained by parallelizing the solver. Notice however that the switching frequency is limited in real applications by the switching losses in the IGBT modules of the power converter. Then, the frequency of the FCS-MPC is set to 20 kHz due to this limitation, although the obtained frequency is higher. The obtained margin is used in our case to reduce the resource use in the FPGA by further pipelining the mathematical operations in the predictor of the FCS-MPC controller.

Focusing on the machine simulator, it is running in the FPGA at 1 MHz with 1 µs of sampling time. The model of the machine is discretized using the forward Euler method since it is the easiest discretization process to be implemented. It also provides sufficient stability and precision for the used simulation time step (1 µs), which is much smaller than the smallest time constant of the machine. The inputs of the model are the mechanical load torque, phase voltages (or inverter switch configuration) and the DC-link voltage. The outputs are stator voltage and current values in both dq and phase frames, along with the electrical torque and the speed of the machine. They are connected to four analog outputs of the OPAL-RT simulator and can be configured from the CPU part of the simulator.

The simulation model is split into multiple interconnected computation cores, which are responsible for various parts of the model. The general architecture of the FPGA implementation is shown in Fig. 4. The model is managed using a main controller that it is configured and supervised by the CPU part of the OPAL-RT simulator. Its main function is to generate synchronization signals, controlling the data flows between computation cores and insuring simulation synchronization with its time step. Other computation cores are: park transformation, fictitious machines, torque estimator and mechanical model of the machine's rotor. The states of the simulation model are stored inside the corresponding cores and include the Park matrix, currents in dq frame, the electrical torque, and the speed and position of the rotor, respectively.

The computations are done in parallel and controlled through the synchronization signals. The timeline is shown in Fig. 5. First, the input voltages are converted from phase to dq frame (vph -› vdq) and the previous state currents from dq to phase frame (idq -› iph), by using the Park matrix stored in the registers of its generator (P k ). Additionally, the new electrical torque (Tem k+1 ) and speed (ωr k+1 ) are computed based on previous states of the model. Once the new speed is available, the new rotor position can be found (θr k+1 ) and the Park matrix can be updated (P k+1 ). When both new speed and dq voltages are ready, new currents in main and secondary fictitious machines can be calculated in parallel (idq1 k+1 and idq3 k+1 , respectively), and a new simulation step can begin.

Since the Park matrix depends on the position of the rotor, each simulation step is generated using two look-up tables containing the values for sines and cosines, and it is stored in double registers. The Park matrix generator structure is shown in Fig. 6, where the sequencer is controlled by the external synchronization signal and iteratively populates the registers of the Park matrix using one of the look-up table and the corresponding electrical angle θe. Once the registers have been all written, they are stored in a second register array in such a way that the Park matrix changes instantly for the external cores. This approach allows to reduce the resource use in FPGA, taking advantage of the computation time of other cores of the model.

The main computation of the simulation occurs in the fictitious machines cores, based on dq frame. The equations for both fictitious machines are independent and can be simulated in parallel. For example, consider the equations ( 3) and (4), which represent the voltages of the main machine. These equations are rewritten in the ordinary first order differential equation form, as follows:

P k P k+1
  1 1 1 1 1 1 d d q q d d d d i L p Ri v L 1 t i     (14)
                     1 1 1 1 1 1 1 2 5 1 d d f d d q q q q i L p Ri v L t i  (15)
Then, new values for the currents are calculated at each simulation step using the following equations:

  ) ( ) ( ) ( ) ( ) ( ) 1 ( 1 1 1 1 1 1 1 k i L k p k Ri k v L h k i k i q q d d d d d       (16)
                       1 1 1 1 1 1 1 1 2 5 ) ( ) ( ) ( ) ( ) ( ) 1 ( f d d q q q q q k i L k p k Ri k v L h k i k i  (17)
being h a simulation step used for the modeling. The discretized equations for the secondary machine can be obtained in a similar way. Summarizing, the equations for both fictitious machines can be written as follows:

) ( ) ( ) ( ) ( ) 1 ( 3 2 1 k i k k k v k k i k k i q d d d      (18)
 

7 6 5 4 ) ( ) ( ) ( ) ( ) 1 ( k k i k k k v k k i k k i d q q q       (19)
where coefficients k1 to k7 are parameters of the model and vary for each fictitious machine. These calculations define the core of the simulation and are implemented as a pipelined architecture. The previous values for the currents in dq frame are stored in registers controlled by the external synchronization signal and updated every simulation step. Different tests have been carried out to validate the proposed 2S-MPC, using the machine parameters and limits detailed in Table 1 as well as the aforementioned real-time system. First, the ability of the 2S-MPC technique to generate optimal current references while respecting the limits over the entire speed range (from zero to the limit speed of the machine) is analyzed, summarizing Fig. 7 the obtained results. Figs. 8 and9 complement this analysis, showing the steady state operation of the system in different operating points that include the operation without electrical limits, considering one current or voltage limit and with the application of both limits. The dynamic behavior of 2S-MPC is also verified, and Fig. 10 summarizes the obtained results. In this case, the multiphase machine is operated outside the feasible region to focus on the operation of the system when the voltage and current limits are applied.

The first set of tests includes the steady state analysis of the controlled system in different operating points. A speed ramp is imposed to the machine from 0 rad/s to the limit speed of 240 rad/s, while the reference torque is always set higher than the maximum value (19.27 N-m, see Table 1), making that the peak value of phase currents is equal to the established limit (50 A) during the whole test. The maximum torque versus speed characteristic of the electrical machine is then obtained, once the steady state operation of the system is reached, as it is shown in Fig. 7a (blue ink). Figure 7a also illustrates the considered case studies, when the electrical limits are not reached (reference points in region 1 or case 1), if the current limit must be applied (reference points in region 2 or case 2), if the voltage limit is considered (reference points in region 3 or case 3), and taking into account the voltage and current constraints (reference points in region 4 or case 4). The evolution of dq1 and dq3 currents is also plotted (see Fig. 7b), showing the steady state performance of the system depending on the operating point. Notice that below the base speed (100 rad/s), the voltage limit is never reached, therefore obtained id1 and id3 values are equal to zero and the machine is not in the flux-weakening region. Furthermore, iq1 is always positive in order to produce a positive electrical torque when the fundamental of the flux is applied, but iq3 (which represents the third harmonic component of the stator current) is negative to guarantee that the maximum peak value for the phase current is not exceeded at the price of generating a negative electrical torque in the system. Once the base speed is reached, id1 and id3 currents become negative in order to comply with the voltage limit. It is important to note that the machine is now operating in the well-known flux-weakening region and the system is also regulated in our case optimizing the third harmonic current components, id3 and iq3 values, to be respectful with the established limits. The present analysis shows that the 2S-MPC method manages the system taking into account the limits and following the real-time established optimization problem (a desired torque is generated considering the copper losses produced in the system in order to minimize them).

The previous analysis is complemented in Figs. 8 and9, where the steady state performance of the system in four different case studies (corresponding with the cases illustrated in Fig. 7a) is analyzed. These reference points are detailed in Table 2, where the applied reference torque and speed are shown. The evolution of the phase current is plotted in Figs. 8a and 9a in order to see if the current limit is reached in the analyzed cases. Notice that only one phase (phase 'a') is shown for the sake of clarity, but similar results are obtained for the rest of the stator current phases, from 'b' to 'e'. Figs. 8b and 9b depict all filtered phase-to-phase voltages to find out that the voltage limits are regulated. When the machine operates and the established limits are not reached (case 1, Fig. 8a, left plot), it is observed that the obtained phase current using the optimal references from the 2S-MPC method (orange ink) is quite similar to the one corresponding with the analytical solution (green ink) that is found imposing the analytical optimal dq reference currents from the copper losses point of view [START_REF] Kestelyn | A vectorial approach for generation of optimal current references for multiphase permanent-magnet synchronous machines in real time[END_REF]: being εqk the speed-normalized back-EMF. This result verifies a good similarity between the proposed 2S-MPC technique and the analytical optimal solution if the constrains are not reached. In all the other cases, the current and voltage limits are considered and regulated by the proposed 2S-MPC technique, although the obtained current cannot be compared with any analytical optimal solution up to the authors' knowledge. Obtained results prove the reliability of the 2S-MPC method, which complies the voltage and current limitations in every case study. Finally, the dynamic operation of the system using the proposed controller is analyzed imposing a reference torque step from 0 to 20 N-m (at t = 0.01 s approximately) while the machine is regulated at 150 rad/s during the whole test (the system is working over the base speed operating point). It is important to note that this reference point (20 N-m and 150 rad/s) is clearly out of the feasible region of the machine. The obtained torque response is shown in Fig. 10a, where the generated torque cannot achieve the reference. This is an expected performance because the reference point is out of the maximum torque versus speed characteristic of the machine. The obtained torque is about 12 N-m, which corresponds with the maximum available torque that the machine can produce operating at 150 rad/s (see Fig. 7a). On the other hand, the evolution of dq1 and dq3 currents is depicted in Fig. 10b. The machine is in the flux-weakened region during the whole experiment: id1 and id3 currents are not zero, being optimally adapted in real-time to respect the voltage limit. Meanwhile, iq1 and iq3 currents are zero before t = 0.01 s because no generated torque is required. Once the reference torque step is applied, iq1 and iq3 currents become positive to maximize the produced torque while respecting the current limit. It is shown again that the obtained reference currents using the 2S-MPC technique are appropriate and the proposed controller is viable. It is interesting to highlight that the obtained results do not consider the parameter uncertainty of the system, which is out of the scope of this work. Parameter uncertainty is a complex research field in electromechanical systems when model-based predictive controllers are used [START_REF] Martín | Sensitivity of predictive controllers to parameter variation in fivephase induction motor drives[END_REF], but also when conventional PI-PWM control techniques are applied, where the estimation of the rotor flux position is mandatory.

0 *  dk i (20) * 3 , 1 2 * em i qi qk qk T i      (21)

Conclusion

This paper introduces a novel two-stage MPC technique, or simply 2S-MPC method, to generate online optimal reference currents and the maximum electrical torque in a close-loop controlled multiphase machine when current and voltage constrains are considered in the control strategy. It fills a gap in the scientific literature, where only suboptimal solutions have been raised, normally making some simplifications or assumptions to obtain the analytical expressions for the current references when current and voltage limits are considered. The implementation of the technique in a real-time simulation environment based on OPAL-RT technologies is used to validate the effectiveness of the proposal in steady and transient state operating conditions, where optimal reference currents are generated and the system is close-loop controlled.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the five-phase PMSM drive.

Fig. 2 .

 2 Fig. 2. Proposed 2S-MPC technique for a five-phase PMSM.

Fig. 3 .

 3 Fig. 3. Description of the optimization processes in CCS-MPC (left plot) and FCS-MPC (right plot) stages.

Fig. 4 .

 4 Fig. 4. General architecture of the five-phase PMSM real-time model simulator in FPGA.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Simulation model computation timeline.

Fig. 7 .

 7 Fig.7. Steady state analysis of the multiphase drive system, including a speed ramp test where the speed is varied in the machine from 0 rad/s to 240 rad/s and the reference torque is higher than the maximum value. (a) Obtained electrical torque versus speed characteristic of the system. (b)Evolution of dq1 and dq3 stator current values.

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. Steady state operation in cases 1 and 2: operation without considering voltage or current limits (left plots) and considering the current limit (right plots). (a) Stator phase current. (b) Filtered phase-to-phase stator voltages.

Table 1 .

 1 Machine parameters and limits.

	Parameter	Value
	Resistance R	37 mΩ
	Inductances Ld1 and Lq1	0.155 mH
	Inductances Ld3 and Lq3	0.051 mH
	Flux Φf1	19.4 mWb
	Flux Φf3	0.675 mWb
	Pole pairs p	7
	Voltage limit Vmax	35 V
	Current limit Imax	50 A
	Maximum torque Tem,max	19.27 N-m
	Base speed ωb	100 rad/s
	Maximum mechanical speed ωm,max	240 rad/s

Table 2 .

 2 Analyzed steady state reference points in Figs.8 and 9.

	Case	Operation	Reference torque	Speed
	1	Below the limits	10 N-m	50 rad/s
	2	Under current limit	25 N-m	50 rad/s
	3	Under voltage limit	5 N-m	150 rad/s
	4	Under current and voltage limit	20 N-m	150 rad/s

1 em em em

Acknowledgements

The authors would like to acknowledge the doctoral school of ENSAM and the "Ministerio de Economía y Competitividad" of the Spanish Government under reference DPI2016-76144-R for the financial support of this work.


Ivan Gentil
email: gentil@math.univ-lyon1.fr

Simon Zugmeyer
email: zugmeyer@math.univ-lyon1.fr

A family of Beckner inequalities under various curvature-dimension conditions

Keywords: Curvature-dimension condition, Poincaré inequality, Beckner inequalities, Entropy flows

In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincaré and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD(ρ, n) condition, either with ρ = 0 and n > 0, or with ρ > 0 and n ∈ R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization to what was proved in [BGS18, Ngu18].

Introduction

The family of the Beckner inequalities interpolate between the Poincaré and the logarithmic Sobolev inequalities. For instance, let dµ = e -V dx be a probability measure on R d , where V : R d → R is a smooth function satisfying ∇ 2 V ≥ ρ Id for some ρ > 0. Then, Bakry-Émery's curvature-dimension condition implies that, for any p ∈ (1, 2], and any nonnegative smooth function f , p p -1

f 2 dµ - f 2/p dµ p ≤ 2 ρ |∇f | 2 dµ, (1)
these results can be found in [BGL14, Sec. 7.6.2]. When p = 2, that is the usual Poincaré inequality for the measure µ and when p → 1 the inequality becomes the logarithmic Sobolev inequality,

f 2 log f 2 f 2 dµ dµ ≤ 2 ρ |∇f | 2 dµ,
both optimal when µ is the standard Gaussian measure. Similar inequalities have first been proved by Bidaut-Véron-Véron in [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] for the sphere, using the method proposed in [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF]. In this article, we refer to them as Beckner inequalities, in reference to [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF], where Beckner proves inequallity (1) for the Gaussian measure. They are sometimes called convex inequalities.

These inequalities play a major role among functional inequalities for probability measures, being useful to understand both asymptotic behaviour of parabolic equations, and also the geometry of measured spaces. It is interesting to notice that for the Gaussian measure, although the Poincaré and the logarithmic Sobolev inequalities are optimal and have nonconstant extremal functions, whenever p ∈ (1, 2), Beckner inequalities do not have extremal functions, and even admit various improvements (see for instance [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF][START_REF] Bolley | Phi-entropy inequalities for diffusion semigroups[END_REF][START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: new methods and consequences[END_REF]).

Other attractive examples appear for measures which satisfy a (weighted) Poincaré inequality but no logarithmic Sobolev inequality. More precisely, let ϕ : R d → R * + be a smooth and positive function such that for any β > d/2, ϕ -β dx < +∞. Now, define

dµ β = c β ϕ -β dx, (2)
where c β is a normalization constant such that µ β is a probability measure on R d . In [START_REF] Bakry | Sharp beckner-type inequalities for cauchy and spherical distributions[END_REF], the authors prove that, when ϕ = 1 + |x| 2 then for any β ≥ d + 1 and p ∈ [p * , 2].

p p -1 f 2 dµ β - f 2/p dµ β p ≤ 1 β -1 |∇f | 2 ϕdµ β , (3)
where p * = 1+1/(β -d). This inequality is rich enough to be equivalent to the Sobolev inequality on the sphere. This result has then been extended by N'Guyen in [START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF], where the author proves similar inequalities in R d with a function ϕ satisfying the convex assumption ∇ 2 ϕ ≥ c Id for some constant c > 0. The only difference with inequality (3) is that the constant (β -1) -1 , in front of the right hand side, becomes 2(c(β -1)) -1 , which is consistant with previous results, since c = 2 for ϕ : x → 1 + |x| 2 . In that regard, N'Guyen's result is more general, but there exists a limitation: the range of the parameter p for which the inequality remains valid is strictly smaller.

We would like to extend Beckner's inequality more precisely inequality (3) in the general context of curvature-dimension conditions. The goal is twofold. First, we extend some results of [START_REF] Bakry | Sharp beckner-type inequalities for cauchy and spherical distributions[END_REF][START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF] in the context of a Riemannian manifold. Then, under a curvature-dimension condition CD(ρ, n) with n negative, we also prove Beckner inequalities. We recover for instance the weighted Poincaré inequality for generalized Cauchy distributions.

More precisely, we prove functional inequalities under two kinds of assumptions, the curvaturedimension conditions CD(0, n) with n > 0 or CD(ρ, n) with ρ > 0 and n ∈ R. Let us give here a flavour of our results.

• In Theorem 3.1, we prove the following Poincaré inequality under CD(0, n), n > 0. Let (M, g) be a smooth d-dimensional Riemannian manifold with a nonnegative Ricci curvature and let ϕ be a positive function such that Hess(ϕ) ≥ cg, c > 0. Then for any function f and

β ≥ d + 1, Var µ ϕ,β (f ) ≤ 1 c(β -1) Γ(f )ϕdµ ϕ,β , (4)
where µ ϕ,β = Z ϕ,β ϕ -β dx is a probability measure. This inequality generalizes previous results on the subject.

• In Theorem 4.4, we prove a family of Beckner inequalities under CD(ρ, n) with ρ > 0 and n ∈ R. On a d-dimensional Riemannian manifold, for any nonnegative function f ,

p p -1 f 2 dµ - f 2/p dµ p ≤ 2 n -1 ρn Γ(f )dµ. -for all p ∈ (1, 2] if n ≥ d, -for all p ∈ [p * , 2] if n < -2, where p * = 1 + 1-4n 2n 2 +1 .
In this context, µ is the reversible measure and Γ is the carré du champ operator. The use of a negative dimension is new, up to our knowledge.

The article is organized as follows. In the next section, we state various definitions useful for the rest of the paper. In Section 3 we prove weighted Beckner inequalities, like inequality (4) under CD(0, n) conditions, n > 0. The Φ-entropy inequalities are also studied in this context. Section 4 is devoted to Beckner's inequality under the curvature-dimension condition CD(ρ, n) conditions with ρ > 0 and n ∈ R. Finally in Section 5, we apply our methods in the one dimensional case, giving another way to prove optimal weighted Poincaré inequality on R.

Acknowledgements: This research was supported by the French ANR-17-CE40-0030 EFI project. The authors warmly thank L. Dupaigne for discussing the problem and proofreading a draft version of this work.

Settings and definitions

Consider a connected, C ∞ Riemannian manifold (M, g) of dimension d and the Laplace-Beltrami operator ∆ g given by, in a local chart

∆ g f = 1 |g| ∂ i |g|g ij ∂ j f ,
where g ij are the components of the inverse metric tensor, g -1 . In this formula and in what follows, the Einstein notation was used, where the summation on indices is implied. On this manifold, define the symmetric diffusion operator L = ∆ g + Γ ∆g (V, .), where Γ ∆g is the carré du champ operator (the definition of which is recalled below) associated to ∆ g , and V is a C ∞ function. Most of the notions and results related to this operator Γ can be found in the quite thorough [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

Definition 2.1 (Carré du champ operator). Given a differential operator L on a smooth manifold M , the carré du champ operator Γ L is a symmetric bilinear map from

C ∞ (M ) × C ∞ (M ) onto C ∞ (M ). It is defined by Γ L (a, b) = 1 2 (L(ab) -aLb -bLa).
An important example to keep in mind is the manifold (R d , I d ), on which the Laplace-Beltrami operator is the usual Laplacian, and its carré du champ operator is, for any smooth functions a and b,

Γ ∆ (a, b) = ∇a • ∇b.
More generally, respectively using the Levici-Vita connection ∇, and in a local chart, operator Γ ∆g is given by

Γ ∆g (a, b) = g(∇a, ∇b) = g ij ∂ i a∂ j b.
We iterate this definition to get the second order operator, Γ 2 .

Definition 2.2 (Iterated carré du champ operator). Given a differential operator L on a smooth manifold M , the iterated carré du champ operator

Γ L 2 is a symmetric bilinear map from C ∞ (M )× C ∞ (M ) onto C ∞ (M ) defined by Γ L 2 (a, b) = 1 2 L(Γ L (a, b)) -Γ L (a, Lb) -Γ L (b, La) .
We point out that if L = ∆ g + Γ ∆g (V, .), then Γ L = Γ ∆g : the drift part of L, namely V , does not appear in this operator. For this reason, and for readability's sake, we shall simply use the notation Γ instead of Γ ∆g in what follows, as long as it is not ambiguous. We will also write Γ(a, a) = Γ(a) for brievety. We will do the same for Γ 2 , but one should keep in mind that the drift does play a role in Γ 2 , and Γ L 2 = Γ ∆g 2 . The idea behind the carré du champ operator is that it contains all the information about the geometry of the space (M, g), and it thus proves worthwhile to stick to its use in the (sometimes heavy) calculations. Generally, a formula that is valid for the standard Laplacian on R d involving only the Γ operator will remain valid on more general manifolds. For instance, if a, b, and c are smooth functions defined on R d , the Hessian of a may be expressed in the following fashion

Hess(a)(∇b, ∇c) = 1 2 (∇b • ∇(∇a • ∇c) + ∇c • ∇(∇a • ∇b) -∇a • ∇(∇b • ∇c)),
hence the following lemma Lemma 2.3 (Hessian of a function). Let a, b, c be smooth functions on (M, g). Then, the Hessian of a is given by

∇ 2 a(∇b, ∇c) = 1 2 (Γ(b, Γ(a, c)) + Γ(c, Γ(a, b)) -Γ(a, Γ(b, c))),
where Γ is the carré du champ operator associated to the Laplace-Beltrami operator ∆ g .

Proof. Since the Hessian is symmetric and bilinear, it is sufficient to prove that

∇ 2 a(∇b, ∇b) = Γ(b, Γ(a, b)) - 1 2 Γ(a, Γ(b)).

By definition,

Hess(a)(∇b, ∇b) = g(∇ ∇b ∇a, ∇b)

= ∇ ∇b g(∇a, ∇b) -g(∇a, ∇ ∇b ∇b)

= g(∇b, ∇g(∇a, ∇b)) -g(∇a, g(∇∇b, ∇b))

= g(∇b, ∇g(∇a, ∇b)) -1 2 g(∇a, ∇g(∇b, ∇b))

which is the claimed formula.

In the development of this article, we shall assume so-called curvature-dimension conditions on the different diffusion operators:

Definition 2.4 (Curvature-dimension conditions CD(ρ, n)). A diffusion operator L is said to satisfy a CD(ρ, n) condition for ρ ∈ R and n = 0, if for every smooth function f , Γ 2 (f ) ≥ ρΓ(f ) + 1 n (Lf ) 2 . ( 5
)
For example, the Bochner-Lichnerowicz formula implies that the Laplace-Beltrami operator ∆ g satisfies the CD(ρ, n) condition with n ≥ d and ρ ∈ R whenever the Ricci curvature is uniformly bounded form below by ρg, [BGL14, Sec. C.6].

Denoting by dx the Riemannian measure associated to (M, g), we define µ to be the reversible measure associated to L, i.e. dµ = Z V e -V dx, where Z V a normalizing constant such that µ is a probability measure if finite, and Z V = 1 otherwise. The triple (M, Γ, µ) is a Markov triple, as defined in [BGL14, Sec. 3.2]. Then, for any smooth functions a, b such that the integrals are well defined, we have the following integration by parts formula:

M Γ(a, b)dµ = - M aLbdµ = - M bLadµ, a notable consequence of which is that M Ladµ = 0.
Remark 2.5. We could have chosen to study a general symmetric Markov semigroup, for instance a generic Markovian triple (E, Γ, µ) as proposed in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. However, assuming L is a symmetric diffusion operator in a smooth Riemannian manifold, which is the case of interest for us, there exists a metric g and a function V such that the operator can be rewritten L = ∆ g + Γ ∆g (V, .), so we are not, in fact, losing any generality.

Weighted inequalities under nonnegative Ricci curvature

For a more pedestrian approach, we first explain the case of the Poincaré inequality, only to tackle more general inequalities later on.

Weighted Poincaré inequality

Let ϕ be a C 2 positive function on M such that

∇ 2 ϕ ≥ cg (6)
for some positive constant c. For β ∈ R such that ϕ -β is integrable with respect to µ, let µ ϕ,β = Z ϕ,β ϕ -β µ, with the constant Z ϕ,β such that this new measure is a probability measure. The main result is the following:

Theorem 3.1 (Weighted Poincaré inequality). Assume that the diffusion operator L satisfies a CD(0, n) condition with n ≥ d, and fix a real number β ≥ n + 1. Then for all smooth bounded functions f ,

Var µ ϕ,β (f ) = f 2 dµ ϕ,β - f dµ ϕ,β 2 ≤ 1 c(β -1) Γ(f )ϕdµ ϕ,β . (7)
Remark 3.2. As explained in Section 2, the main example to keep in mind is the Laplace-Beltrami operator on a Riemannian manifold with a nonnegative Ricci curvature, in which case

µ ϕ,β = Z ϕ,β ϕ -β dx.
This inequality happens to be optimal whenever (M, g) = (R d , Id), and ϕ(x) = 1+|x| 2 , where the optimal constant is reached for projectors x → x i , 1 ≤ i ≤ d. This optimal case has been proved in [BBD + 07] and also in [START_REF] Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF][START_REF] Arnaudon | Intertwinings and generalized brascamplieb inequalities[END_REF] with different methods.

Proof. Fix β ∈ R\{2}, and define

L := ϕL -(β -1)Γ(ϕ, .).
Note that the operator L is, in fact, of the form L = ∆ g + Γ ∆g (V , .), where g = ϕ -1 g and V = V + (d/2 -β) log ϕ, so the operator we are considering here is obtained from the first one through a conformal transformation. In what follows, everything written with an overline relates to objects associated to the operator L or the manifold (M, g). For instance, the carré du champ operator is given by Γ = ϕΓ, its reversible measure is μ = µ ϕ,β , and

Γ 2 (f ) = Γ L 2 (f ) = ϕ 2 Γ 2 (f ) + (β -1)ϕ∇ 2 ϕ(∇f, ∇f ) + Γ(f ) 2 (ϕLϕ -(β -1)Γ(ϕ)) + ϕΓ(ϕ, Γ(f )) -ϕLf Γ(f, ϕ), (8)
a proof of which can be stringed together with information from [BGL14, Sec. 6.9.2], for instance. Now, fix f , a smooth and bounded function on M , and consider the Markov semigroup (f t ) t≥0 , solution of the initial-value system

∂ t f t = Lf t on (0, +∞) × M, f 0 = f on M. ( 9
)
Consider the variance of f t along the flow:

Λ(t) := Var μ(f t ) = f 2 t dμ - f t dμ 2 = f 2 t dμ - f dμ 2 ,
because L is mass-preserving. Then, we use the following estimate:

Lemma 3.3. If β > 1 and β = 2, then for all t ≥ 0, Λ (t) ≥ -2c(β -1)Λ (t) + 4 β -2 ϕ 2 (β -1)Γ 2 (f t ) -(Lf t ) 2 ) dμ. (10)
Furthermore, there is equality in (10) for all t ≥ 0 whenever the Hessian of ϕ is a constant, i.e. when inequality (6) is an equality.

Assume that L satisfies the CD(0, n) condition for some n > 0, and that β -1 ≥ n, and β > 2. Then, we deduce from equation (10) that

Λ (t) ≥ -2c(β -1)Λ (t)
which we can integrate once betwen 0 and t to find that -Λ (t) ≤ -Λ (0)e -2c(β-1)t and then once again, between t = 0 and t = +∞,

Λ(0) -lim t→+∞ Λ(t) ≤ -1 2c(β -1) Λ (0).
The Markov semigroup studied is ergodic, in other words, it ensures the convergence of f t towards its mean in L 2 (μ), so that lim t→+∞ Λ(t) = 0, and the previous inequality is simply the stated result, for smooth and bounded functions, and for β > 2. The general case is established by approximation. The particular case β = 2 (also implying n = d = 1) is proved by letting β → 2 directly in inequality (7).

Proof of Lemma 3.3. By definition of μ, we may integrate by parts the derivative of Λ to find

Λ (t) = 2 f t Lf t dμ = -2 Γ(f t )dμ,
and, differentiating once again,

Λ (t) = -4 Γ(f t , Lf t )dμ = -4 Γ(f t , Lf t )dμ + 2 L(Γ(f t )) = 4 Γ 2 (f t )dμ.
We may now use formula (8) to find that

Λ (t) = 4 (β -1)ϕ∇ 2 ϕ(∇f t , ∇f t )dμ + 4 ϕ 2 Γ 2 (f t ) + Γ(f t ) 2 (ϕLϕ -(β -1)Γ(ϕ)) + ϕΓ(ϕ, Γ(f t )) -ϕLf t Γ(f t , ϕ) dμ. (11)
First, the convexity assumption (6) on ϕ yields

4 (β -1)ϕ∇ 2 ϕ(∇f t , ∇f t ) ≥ 4c(β -1) Γ(f t )dμ = -2c(β -1)Λ (t).
Now, in the second integral of equation ( 11), we may not directly use the integration by parts formula, because μ is the invariant measure for Γ, and not for Γ. We must thus rewrite it in terms of µ. First,

Γ(f t ) 2 ϕLϕdμ = Z ϕ,β Γ(f t ) 2 ϕ 1-β Lϕdµ = -Z ϕ,β Γ ϕ, ϕ 1-β Γ(f t ) 2 dµ = - ϕ 2 Γ(ϕ, Γ(f t ))dμ - 1 -β 2 Γ(f t )Γ(ϕ)dμ.
Then,

ϕΓ(ϕ, Γ(f t ))dμ = Z ϕ,β ϕ 1-β Γ(ϕ, Γ(f t ))dµ = Z ϕ,β 2 -β Γ(ϕ 2-β , Γ(f t ))dµ = 1 β -2 ϕ 2 LΓ(f t )dμ,
and likewise,

-ϕLf t Γ(f t , ϕ)dμ = Z ϕ,β β -2 Lf t Γ(f t , ϕ 2-β )dµ = Z ϕ,β β -2 Γ(f t , ϕ 2-β Lf t ) -ϕ 2-β Γ(f t , Lf t ) dµ = -1 β -2 ϕ 2 (Lf t ) 2 + Γ(f t , Lf t ) dμ.
We conclude putting these three identities together.

Φ-entropy and weighted Beckner inequalities

Instead of the variance, we may consider a generic Φ-entropy along the flow. Choose a strictly convex real function Φ ∈ C 4 (I), (where

I ⊂ R *
+ is an open interval) and consider the Φ-entropy of any function f : M → I such that the integrals below are well defined

Ent Φ µ ϕ,β (f ) := Φ(f )dµ ϕ,β -Φ f dµ ϕ,β ,
so that Var µ ϕ,β = Ent x →x 2 µ ϕ,β . Generalizations of inequalities like (7) to Φ-entropies have already been studied under CD(ρ, ∞) condition, ρ > 0 in [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF][START_REF] Bolley | Phi-entropy inequalities for diffusion semigroups[END_REF]. We find a generalized version of Theorem 3.1: Theorem 3.4 (Φ-entropy inequalities). Assume that the diffusion operator L satisfies a CD(0, n) condition with n ≥ 0, and fix a real number β ≥ n+1. Let Φ : I → R be a strictly convex function such that

Φ (4) Φ ≥ C n,β (Φ (3) ) 2 , ( 12
)
with

C n,β = 8(β -1 -n)(2β -1) + 9n 8(β -1 -n)(β -1) .
Then, for all smooth bounded functions f : M → I, there holds

Ent Φ μ (f ) ≤ 1 2c(β -1) Φ (f )Γ(f )ϕdμ. ( 13
)
Proof. Let us first assume that β > 2, the case β = 2 can be proved by passing to the limit in (13). Assume also that f is a smooth and bounded function, the general case can be proved by approximations. Consider, just like before, the function f t , solution of the initial-value system (9) starting from f , and define, for t ≥ 0,

Λ(t) = Ent Φ μ (f t ) = Φ(f t )dμ -Φ f t dμ = Φ(f t )dμ -Φ f dμ ,
where, again, μ = µ ϕ,β . Differentiating the entropy yields

Λ (t) = - Γ(Φ (f t )) Φ (f t ) dμ,
and, from [BG10, Lem. 4],

Λ (t) = 2 Γ 2 (Φ (f t )) Φ (f t ) + -1 Φ (f t ) Γ(Φ (f t )) Φ (f t ) 2 dμ.
We follow exactly the same steps as in Section 3.1. For brievety, we write h := Φ (f t ). First, expanding Γ 2 in terms of Γ 2 , Γ and L, and also using the convexity hypothesis on ϕ (6), we find,

Λ (t) ≥ -2c(β -1)Λ (t) + 2 Φ (f t ) ϕ 2 Γ 2 (h) + Γ(h) 2 (ϕLϕ -(β -1)Γ(ϕ)) + ϕΓ(ϕ, Γ(h)) -ϕLhΓ(h, ϕ) + ϕ 2 -1 Φ (f t ) Γ(h) Φ (f t ) 2 dμ.
We may now give these terms the same treatment as in the previous section: the goal is to remove all derivatives on ϕ. The calculations are not made explicit here; they involve the exact same ingredients we used before, only with more terms appearing. We finally find, using only integration by parts, that

Λ (t) ≥ -2c(β -1)Λ (t) + 1 β -2 ϕ 2 Φ (f t ) a 0 Γ 2 (h) + a 0 (Lh) 2 + a 2 Γ(h)Lh + a 3 Γ(h) 2 dμ, where                    a 0 = 2(β -1), a 0 = -2, a 2 = 3 Φ (f t ) (Φ (f t )) 2 , a 3 = (β -1) Φ (4) (f t ) (Φ (f t )) 3 + (1 -2β) (Φ (f t )) 2 (Φ (f t )) 4 .
Just like for the Poincaré inequality proof, this inequality becomes an equality when Hess(ϕ) = cg.

Invoke the CD(0, n) condition to find that

Λ (t) ≥ -2c(β -1)Λ (t) + 1 β -2 ϕ 2 Φ (f ) a 1 (Lh) 2 + a 2 Γ(h)Lh + a 3 Γ(h) 2 dμ, (14)
where a 1 = a 0 /n + a 0 = 2 n (β -1 -n). In the same way as before, we want this integrated quantity to be nonnegative. Since Φ is strictly convex, it is sufficient to require the polynomial function X → a 1 X 2 + a 2 X + a 3 to be nonnegative, which itself is equivalent to

a 1 ≥ 0 and a 2 2 -4a 1 a 3 ≤ 0.
Straightforward computation yields that this, in turn, is equivalent, whenever n ≥ 0, to

β ≥ n + 1 and Φ (4) Φ ≥ C β,n (Φ ) 2 ,
where

C β,n = 8(β -1 -n)(2β -1) + 9n 8(β -1 -n)(β -1) .
If this condition is satisfied, the integrand is pointwise nonnegative, thus

Λ (t) ≥ -2c(β -1)Λ (t),
and integrating this twice yields the theorem.

We can explicit this theorem in the particular case where Φ

(X) = X p , p > 1. Condition (12) is then equivalent to p ∈ [p * , 2],
where

p * = 1 + 8(β -1 -n) + 9n 8β(β -1 -n) + 9n ∈ (1, 2]. (15)
Thus, if this condition is satisfied,

f p dμ - f dμ p ≤ p(p -1) 2c(β -1) f p-2 ϕΓ(f )dμ.
Rewriting this inequality with f = f 2/p in place of f , we find Corollary 3.5 (Weighted Beckner inequalities). Under the same assumptions as Theorem 3.4, for all p ∈ [p * , 2], and all smooth bounded functions f , we have

p p -1 f 2 dµ ϕ,β - f 2/p dµ ϕ,β p ≤ 2 c(β -1) ϕΓ(f )dµ ϕ,β , (16)
where p * is given by (15).

Corollary 3.5 is optimal (and thus so is theorem 3.4) in the sense that there exists no constant 0 < C < (c(β -1)) -1 such that L satisfies a Beckner inequality B p (C) (see Definition 4.5 below), because B p (C) (for any p ∈ (1, 2)) implies the Poincaré inequality with constant C. Indeed, testing inequality (16) with the function 1 + εf , f bounded, we find that

p p -1 (1 + εf ) 2 dμ - (1 + εf ) 2/p dμ p ≤ 2ε 2 c(β -1) ϕΓ(f )dμ,
which, after being expanded when ε is small, turns into

f 2 dμ - f dμ 2 + o(1) ≤ 1 c(β -1) ϕΓ(f )dμ,
which is exactly the optimal weighted Poincaré inequality 3.1. Remark 3.6. In the case of interest, when Φ(X) = X p , we may explicit inequality ( 14)

Λ (t) ≥ -2c(β -1)Λ (t) + 1 β -2 ϕ 2 h 2-p p-1 a 1 (Lh) 2 + a 2 Γ(h) h Lh + a 3 Γ(h) 2 h 2 dμ,
where the a i , i ∈ {1, 2, 3} are real constants. The inequality can be then improved using the fact that

a 1 X 2 + a 2 X + a 3 = a 1 X + a 2 2a 1 2 - (a 2 ) 2 -4a 1 a 3 4a 1 ,
so that, whenever δ := -1 4a 1 ((a 2 ) 2 -4 a 1 a 3 ) ≥ 0, we find that

Λ (t) ≥ -2c(β -1)Λ (t) + δ β -2 ϕ 2 h 2-p p-1 Γ(h) 2 h 2 dμ ≥ -2c(β -1)Λ (t) + δ β -2 ϕh 2-p p-1 Γ(h)dμ 2 h p p-1 dμ = -2c(β -1)Λ (t) + C p Λ (t) 2 Λ(t) ,
by Jensen's inequality, where, for reference,

C p = δ β -2 (p -1) 2 p 2-p p-1 ≥ 0.
This leads, when integrated, to a refined version of Beckner's inequality, which we will come back to in section 4, and specifically, corollary 4.6. Remark 3.7. As explained in Section 1, we extend the result of Nguyen [START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF] in two aspects. First, Beckner inequalities (16) are proved in the more general context of Riemannian manifolds satisfying a CD(0, n) condition, and secondly, the range of parameter p ∈ [p * , 2] given by ( 15) contains strictly the one proposed in [START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF].

On the other hand, in [START_REF] Bakry | Sharp beckner-type inequalities for cauchy and spherical distributions[END_REF], corollary 3.5 is proved in the special case of M = R d and ϕ(x) = 1+|x|

2 . Interestingly, the range found in their article is greater than what we can manage here. Indeed, it is valid for all p ∈ [p * BGS , 2], where

1 < p * BGS = 1 + 1 β -d < p * .
It might be worth it to note that in our case, we used the fact that the second degree polynomial appearing in the integral is greater than 0, when this is in fact quite a gross lower bound. Indeed, the argument is that

ϕ 2 h 2-p p-1 hLh Γ(h) + a 2 2a 1 2 Γ(h) 2
h 2 dμ ≥ 0, but the squared term can probably be controlled in a way that leads to a wider range of p, since

hLh + a 2 2a 1 Γ(h) dµ = a 2 2a 1 -1 Γ(h)dµ < 0 whenever h ≡ 0.

Spaces with positive curvature and real dimension

The family of inequalities considered in this article, especially for sections 4 and 5, is the following interpolation between the Poincaré inequality and the logarithmic Sobolev inequality Definition 4.1 (Beckner inequalities). The Markov triple (M, Γ, µ) is said to satisfy a Beckner inequality B p (C) with parameter p ∈ (1, 2] and constant C > 0 if, for all nonnegative smooth bounded functions f ,

p p -1 f 2 dµ - f 2/p dµ p ≤ 2C Γ(f )dµ. ( 17
)
The constants in front of the integrals are chosen so that for p = 2, this is exactly the Poincaré inequality with constant C, and the limiting case p → 1 corresponds to the logarithmic Sobolev inequality, again with constant C. Indeed,

lim p→1 p p -1 f 2 dµ - f 2/p dµ p = f 2 log f 2 f 2 dµ dµ.
Remark 4.2. The weighted Beckner inequalities proved in the previous section can be seen as classical Beckner inequalities. Indeed, theorem 3.5 states that B p ((c(β -1)) -1 ) is valid for the triple (M, ϕΓ, µ ϕ,β ).

In this section, we consider a diffusion operator L defined on (M, g), juste like in Section 3, with µ being its reversible measure. We assume that L satisfies a curvature-dimension condition CD(ρ, n), with ρ > 0 and n ∈ R. That means, specifically but not exclusively, that we will consider negative n. Such spaces, sometimes referred to as having negative effective dimension, have been studied in the past, with articles dating as far back as 2003, up to our knowledge [START_REF] Scheffer | Local Poincaré inequalities in non-negative curvature and finite dimension[END_REF][START_REF] Milman | Beyond traditional curvature-dimension. I: New model spaces for isoperimetric and concentration inequalities in negative dimension[END_REF][START_REF] Milman | Harmonic measures on the sphere via curvature-dimension[END_REF][START_REF] Bakry | Sharp beckner-type inequalities for cauchy and spherical distributions[END_REF]. Furthermore, one can easily construct examples of such operators with [START_REF] Ohta | K, N )-convexity and the curvature-dimension condition for negative N[END_REF]Cor. 4.13]. Writing down the CD(ρ, n) inequality in good local coordinates like in [START_REF] Bakry | Sharp beckner-type inequalities for cauchy and spherical distributions[END_REF], one can check that a necessary criterion for the inequality to be true is that d/n ≤ 1, which means that we are actually restricted to n ∈ R\[0, d]. The main difference, compared to the previous section, is that the curvature is positive, which is a stronger assumption, but n can be negative, which is a weaker assumption.

Let us first recall the well known result for the Poincaré inequality, the proof of which is given in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]Thm. 4.8.4].

Theorem 4.3 (Poincaré inequality under CD(ρ, n)). Assume that the diffusion operator L satisfies a CD(ρ, n) condition, with ρ > 0 and n ∈ R\[0, d). Then the following Poincaré inequality holds,

Var µ (f ) ≤ n -1 ρn Γ(f )dµ. (18)
We extend this result to Beckner inequalities in the following theorem:

Theorem 4.4 (Beckner inequalities under CD(ρ, n)). Assume that the diffusion operator L satisfies a CD(ρ, n) condition, with ρ > 0 and n ∈ R\[-2, d). Then, B p n-1 ρn is satisfied • for all p ∈ (1, 2] if n ≥ d,
• for all p ∈ [p * , 2] if n < -2, where

p * = 1 + 1 -4n 2n 2 + 1 .
Interestingly, we find nothing for n ∈ [-2, 0), which corresponds to the weakest CD(ρ, n) conditions possible, even though the Poincaré inequality remains valid in that range. It seems like there is not enough structure in that case.

Proof. In the same fashion as the previous section, fix f , a bounded smooth nonnegative function on M , and consider the function f t , solution of the initial-value system

∂ t f t = Lf t on (0, +∞) × M, f 0 = f on M, (19)
and consider its Φ-entropy along the flow, where Φ is assumed to be strictly convex.

Λ(t) = Ent Φ µ (f t ) = Φ(f t )dµ -Φ f t dµ ,
Invoking once again,

Λ (t) = - Γ(Φ (f t )) Φ (f t ) dµ, Λ (t) = 2 Γ 2 (Φ (f t )) Φ (f t ) + -1 Φ (f t ) Γ(Φ (f t )) Φ (f t ) 2 dµ.
Classically, one can assume -1/Φ to be convex, whence the CD(ρ, n) condition yields

Λ (t) ≥ 2 Φ (f t ) ρΓ(Φ (f t )) + 1 n (LΦ (f t )) 2 dµ,
and, when n > 0, this leads to

Λ (t) ≥ -2ρΛ (t),
which, ultimately, proves Poincaré-type inequalities, using arguments like the ones in section 3. The problem is that the constant appearing in the inequality is, in fact, not optimal, and furthermore, the argument fails whenever n < 0. For the sake of simplicity, we will now assume that Φ(X) = X p , with p ∈ (1, 2], but the argument could very well be generalized to more general Φ.

The derivatives of the entropy then become

Λ (t) = - p p -1 f 2-p t Γ(f p-1 t )dµ, Λ (t) = p p -1 2f 2-p t Γ 2 (f p-1 t ) + 2 -p p -1 f 4-3p t Γ(f p-1 t ) 2 dµ.
Rewriting these quantities with respect to q := 2-p p-1 ∈ [0, +∞), and

h = f p-1 t , p -1 p Λ (t) = -h q Γ(h)dµ, p -1 p Λ (t) = 2h q Γ 2 (h) + qh q-2 Γ(h) 2 dµ.
To get the most information out of this, we shall apply the CD(ρ, n) condition not to the function h, but to η(h), where η is some function to be chosen later. Expanding every term in the CD(ρ, n) inequality (5)

Γ 2 (η(h)) ≥ ρΓ(η(h)) + 1 n (Lη(h)) 2 yields η 2 (h)Γ 2 (h) + η (h)η (h)Γ(h, Γ(h)) + (η (h)Γ(h)) 2 ≥ ρη 2 (h)Γ(h) + 1 n (η (h)Lh + η (h)Γ(h)) 2 .
This inequality is, in particular, true for the power function η(x) = x θ+1 , with θ ∈ R, so that

Γ 2 (h) ≥ ρΓ(h) i +2θ 1 n Γ(h)Lh h ii -θ Γ(h, Γ(h)) h iii +θ 2 1 -n n Γ(h) 2 h 2 iv + 1 n (Lh) 2 v .
Multiplying this inequality by h q and then integrating it, we are left with five terms to consider. The first term corresponds to the first derivative of the entropy, Λ (t), so we may leave it as it is. Terms ii and iii are trickier, because their sign is not known, so we must take care of them. Term iv is always negative since n / ∈ [0, 1], but it is compensated by another (positive) term of the same nature appearing naturally in Λ (t). The sign of the last term depends on the sign of n, so we must take care of it as well, at least for negative n.

Term ii: an integration by parts yields

h q-1 Γ(h)Lhdµ = - h q-1 Γ(h, Γ(h)) + (q -1)h q-2 Γ(h) 2 dµ.
Term iii: we do not do anything for now with this term, and will adjust θ later so that it disappears.

Term v: we use the fact that, by definition,

Γ 2 (h)dµ = 1 2 L(Γ(h)) -Γ(h, Lh) dµ = (Lh) 2 dµ
to prove that, for any real function η,

η 2 (h)(Lh) 2 dµ = η 2 (h)Γ 2 (h) + 3η(h)η (h)Γ(h, Γ(h)) + 2(η 2 (h) + η (h)η (h))Γ(h) 2 dµ.
In particular, h q (Lh) 2 dµ = h q Γ 2 (h) + q(q -1)h q-2 Γ(h) 2 + 3 2 qh q-1 Γ(h, Γ(h)) dµ.

Finally, we are left with an equality still involving the parameter θ:

h q Γ 2 (h)dµ ≥ ρn n -1 h q Γ(h)dµ + Ah q-1 Γ(h, Γ(h)) + Bh q-2 Γ(h) 2 dµ where        A = 1 n -1 3q 2 -θ(n + 2) , B = q(q -1) n -1 -θ 2 -2θ q -1 n -1 .
Choosing θ so that A = 0, i.e. θ = 3q 2(n+2) , we find that

2f q Γ 2 (h) + qh q-2 Γ(h) 2 dµ ≥ 2ρn n -1 h q Γ(h)dµ + α h q-2 Γ(h) 2 dµ, with α = 2B + q = q 2(n + 2) 2 (q(4n -1) + 2n(n + 2)) = 1 2(n + 2) 2 2 -p p -1 2 -p p -1 (4n -1) + 2n(n + 2)
Remembering that q ≥ 0, this constant α turns out to be nonnegative for the following range of parameters n ≥ d q ≥ 0 or n < -2 q ∈ [0, q * ] with q * = 2n(n + 2) 1 -4n , or equivalently, in terms of the exponent p,

n ≥ d p ∈ (1, 2] or n < -2 p ∈ [p * , 2] with p * = 1 + 1 -4n 2n 2 + 1 ,
Whenever α ≥ 0, we may, at last, compare Λ to Λ . Indeed, we find that

Λ (t) ≥ - 2ρn n -1 Λ (t),
which proves the claimed B p n-1 ρn inequality when integrated twice.

To prove this theorem, we used the nonnegativity of a specific term in a differential inequality, but in fact, we can do a little bit better and compare it to the other terms, in order to prove a refined version of the Beckner inequalities we are considering.

≥ 0 if, whenever θ = 1, p p -1 1 1 -θ f 2 dµ - f 2/p dµ (1-θ)p f 2 dµ θ ≤ 2C Γ(f )dµ
for all smooth functions f , and, when θ = 1,

p p -1 f 2 dµ log f 2 dµ f 2/p dµ p ≤ 2C Γ(f )dµ.
With this definition, B p (C) is the same as B * p (C, 0). The inequality given for θ = 1 is simply the limit of the other inequality when θ → 1. This indeed corresponds to an improved version of the Beckner inequality, because for all x, y > 0 and θ ∈ R + \{1},

x -y 1-θ x θ 1 -θ ≥ x -y,
= 1 2(n + 2) 2 p p -1 2 -p p -1 2 -p p -1 (4n -1) + 2n(n + 2) .
Proof. Coming back to the proof of theorem 4.4, we have that

2f q Γ 2 (h) + qh q-2 Γ(h) 2 dµ ≥ 2ρn n -1 h q Γ(h)dµ + α h q-2 Γ(h) 2 dµ, or, in terms of Λ, Λ (t) ≥ - 2ρn n -1 Λ (t) + α p p -1 h q-2 Γ(h) 2 dµ.
Invoking Jensen's inequality, we find that

h q-2 Γ(h) 2 dµ ≥ h q Γ(h)dµ 2 h q+2 dµ = p -1 p 2 Λ (t) 2 Λ(t) , so that Λ (t) ≥ - 2ρn n -1 Λ (t) + α p p -1 Λ (t) 2 Λ(t)
, just like we found in remark 3.6. Writing θ = α p p-1 , we may now integrate this inequality, to find that

- Λ (t) Λ(t) θ ≤ - Λ (0) Λ(0) θ exp - 2ρn n -1 t ,
which, integrated once more between 0 and +∞, leads to the claimed inequality.

Remark 4.7. As it turns out, the operator L from last section does not verify a good enough CD(ρ, n) condition. The best constant c(β -1) only arises using an integrated CD(ρ, n) criterion.

To see this, we can use the following result from [START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF]: the operator L = ∆ g + Γ(V, .), defined on (M, g), satisfies a CD(ρ, n) condition if, and only if,

n -d n Ric g -∇ 2 V -ρg ≥ 1 n ∇V ⊗ ∇V,
this tensorial reformulation being valid for any ρ ∈ R and n ∈ [0, d]. As stated in the proof to theorem 3.1, the operator Lf = ϕLf -(β -1)Γ(ϕ, f ) on (M, g) is related to a Laplace-Beltrami operator through the conformal transformation with conformal factor ϕ -1 . Thus, writing L = ∆ g + Γ ∆g (V , .), for some explicit function V , we find a somewhat easier to verify criterion for the CD(ρ, n) condition in (M, g):

Ric g +(β -1) ∇ 2 ϕ ϕ + 2 -d - (2β -1) 2 n -d ∇ϕ ⊗ ∇ϕ 4ϕ 2 + ∆ϕ 2ϕ -2β Γ(ϕ) 4ϕ 2 - ρ ϕ g ≥ 0. (20)
We leave it to the courageous to verify that indeed, even in the case where everything is nice and explicit, for instance for ϕ(x) = 1 + |x| 2 , there exists no couple (ρ, n) ∈ R * + × (R\[0, d]) such that inequality (20) is verified and ρn n -1 = c(β -1), making theorem 3.1 truly an integrated CD(ρ, n) criterion result.

For n = 2(1 -β), the first term in the above inequality disappears and the condition becomes

β - 1 2 ϕ -ρ ≥ 0,
which is clearly true for ρ = c β -1 2 , which proves that L satisfies the CD(c β -1 2 , 2(1 -β)) condition. We may now apply the theorem 4.3 to conclude that the Poincaré inequality is valid with constant n -

1 ρn = 1 -2β c(1 -β)(2β -1) = 1 c(β -1)
, which is the claimed result.

This proof proves more than just the Poincaré inequality, since theorem 4.4 provides a range of exponants for which the Beckner inequality holds, but this is only true when n < -2, which corresponds to β > 2. As far as Beckner inequalities go, this result is exactly the same as the one in section 3, and as such only constitutes an example of application of the results in section 4. The interest, however, lies in the fact that we extend the range of β for which the Poincaré inequality is valid.

As it turns out, this inequality is optimal for β ≥ 3/2, but it is not optimal anymore for β ∈ [1, 3/2), as proved in [START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF] for the function ϕ(x) = 1 + x 2 . In fact, they find that inequality (21) is valid for β ∈ (1/2, 3/2], and the optimal constant in that range changes from (2(β -1)) -1 to (β -1/2) -2 . It might be worth noting that the method presented in theorem 5.1 actually works for the full range of β, as summed up in the following proposition Proposition 5.2. Let ϕ : x → 1 + x 2 . Fix a real number β > 1 2 . Then for all smooth bounded functions f ,

f 2 dµ ϕ,β - f dµ ϕ,β 2 ≤ 1 C β (f ) 2 ϕdµ ϕ,β , (23)
where

C β =        2(β -1)if β ≥ 3 2 , β - 1 2 2 if β ∈ 1 2 , 3 2 . ( 24
)
Proof. The proof builds on the proof for theorem 5.1, using the explicit form of ϕ. The condition (22) becomes -(2β -1) 1 n -1 (2β -1) + 1 x 2 1 + x 2 + (2β -1) -ρ ≥ 0, for all x ∈ R, or, equivalently,

ρ ≤ 2β -1 and - (2β -1) 2 n -1 -ρ ≥ 0.
Restricting our study to nonnegative curvatures, we thus prove that the condition CD(ρ, n) (with ρ > 0) is satisfied if, and only if, 0 < ρ ≤ 2β -1 and 0 < ρ(1 -n) ≤ (2β -1) 2 (and n ∈ [0, 1]).

We are looking, for a fixed β > 1/2, for the parameters (ρ, n) satisfying those criteria that lead to the best possible value of ρn/(n -1).

  Definition 4.5. The Markov triple (M, Γ, µ) is said to satisfy a refined Beckner inequality B * p (C, θ) with parameter p ∈ (1, 2] and constants C > 0 and θ

  and more generally, B * p (C, θ) implies B * p (C, θ ) for all θ ∈ [0, θ]. Such improvements have been shown in [AD05, BG10] under the CD(ρ, ∞) condition. The limit case, that is, for the usual entropy, is proposed in [BGL14, Thm. 6.8.1]. Theorem 4.6 (Improved Beckner inequalities). Under the same assumptions of in Theorem 4.4, the inequality B * p n-1 ρn , θ is satisfied for the same range of parameter p, and for θ given by

  θ

Results on the real line

In dimension d = 1, simplifications happen, so that we are able to do calculations directly. All manifolds of dimension 1 are conformal to (R, 1), and even though the study of a generic manifold (R, g) does not reduce exactly to the study of (R, 1), the calculations are similar, and so we will only consider the Lebesgue measure λ for the reference measure. Following the work in section 3, assume ϕ is a C 2 , positive, convex function such that ϕ ≥ c for some constant c > 0. Fix β ∈ R such that ϕ -β is in L 1 (R, λ). Then, we find the following result version of the Poincaré inequality, for the probability measure µ ϕ,β = Z ϕ,β ϕ -β λ Theorem 5.1. Fix a real number β > 1. Then for all smooth bounded functions f ,

For β > 2, this is in fact theorem 3.1, so this theorem is an extension to smaller exponants β.

Proof. The idea is to apply Theorem 4.4 to the special case of the operator L defined by

after proving it satisfies a CD(ρ, n) condition with negative dimension. Since we are working in dimension 1, expliciting the CD(ρ, n) is not too hard, and that is just what we do. Indeed, straightforward computations yield

for all smooth functions f if, and only if,

since C ≥ 0, this is in turn equivalent to B 2 -4AC ≤ 0, which, after simplifications, boils down to the condition

Assuming (ρ, n) is such a couple, since n/(n -1) is decreasing on R * -, it is necessary that ρ(1 -n) = (2β -1) 2 , which then implies that ρ < 2(β -1) 2 . We thus reduce the problem to finding the maximum of the function

under the constraints 0 < ρ < max(2β -1, (2β -1) 2 ). An easy study yields that this maximum is C β as defined in equation (24). In other terms, the operator Lf

and this leads to the proposition.

Remark 5.3. This method is applicable to other functions than just x → 1 + x 2 . For instance, for the function ϕ : x → 1 + x 2 + x 4 , one finds that the condition CD(2β -1, 4β -1) is satisfied, and it is the one that leads to the best possible ρn/(n -1) constant for the corresponding operator.