Aoustin

M Langard
email: morgan.langard@outlook.fr

Ph Lucidarme

N Delanoue

R Guyonneau

F Mercier

C Chevallereau

Ph Wenger

Y Aoustin

Design and Optimization of a Planar Biped Leg Based on Direct Drive Linear Actuators

Design and Optimization of a Planar Biped Leg Based on Direct Drive Linear Actuators

Introduction

One shortcoming of generic robots is their capability of evolving in their environment, which results in either tracked or wheeled mechanisms. However, it limits their use to nearly 50% of Earth's landmass, which is unsuitable for such movement schemes [START_REF] Raibert | Legged Robots that Balance[END_REF]. is limitation motivates the design of bipedal walking robots as the human locomotion system allows one to perform a wide range of motions in uneven environments. However, biped motions such as squatting, walking, or running are complex to perform. ey require the actuation system to produce forces with important variations and to be able to cope with impacts on the ground. Moreover, the forces required from the motors will be greatly in uenced by the robot's parameters. erefore, choosing a relevant leg architecture and motors for the actuation is of utmost importance when designing a bipedal robot. Concerning the motor type, there are two main solutions: rotary and linear motors. To take into account the impacts involved in classic biped motions, one can consider the use of backdrivable motors as they o er inherent lowimpedance and impact mitigation [START_REF] Kaminaga | Development of Backdrivable Hydraulic Joint Mechanism for Knee Joint of Humanoid Robots[END_REF][START_REF] Chevallereau | Leg design for biped locomotion with mono-articular and bi-articular linear actuation[END_REF][START_REF] Li | Design, modeling and testing of a compact variable stiffness mechanism for exoskeletons[END_REF][START_REF] Raibert | BigDog, the rough-terrain quadruped robot[END_REF][START_REF] Rosendo | Pneupard: a biomimetic musculoskeletal approach for a feline-inspired quadruped robot[END_REF][START_REF] Spielberg | Functional co-optimization of articulated robots[END_REF][START_REF] Wensing | Proprioceptive actuator design in the MIT cheetah: impact mitigation and high-bandwidth physical interaction for dynamic legged robots[END_REF].

Rotary motors are a popular option for current bipedal robots. eir characteristics are well known from being used for multiple decades in industrial robots such as the PUMA in the 80s and later on for humanoid robots like the Honda [START_REF] Hirai | e development of Honda humanoid robot[END_REF]. is type of motor allows unlimited joint rotation and can be connected to a reduction gear to increase the output torque. Such motors have some drawbacks. e output axis must be aligned with the articulated bodies to avoid dynamic imbalance. ey also have limited backdrivability whenever a reduction gear is used. Research was done to cope with the low backdrivability by using a variable sti ness mechanism [START_REF] Li | Design, modeling and testing of a compact variable stiffness mechanism for exoskeletons[END_REF] or selective elastic actuators [START_REF] Sharbafi | A new biarticular actuator design facilitates control of leg function in Bio-Biped3[END_REF]. e latter adds control complexity and limitations to the actuation bandwidth. An alternative is the use of Direct-Drive (DD) actuators, but DD rotative motors have limited torque and might require the addition of a low-ratio gearbox. is forces one to compromise between motor size, backdrivability, bandwidth, and output torque [START_REF] Wensing | Proprioceptive actuator design in the MIT cheetah: impact mitigation and high-bandwidth physical interaction for dynamic legged robots[END_REF][START_REF] Kalouche | GOAT: a legged robot with 3D agility and virtual com4pliance[END_REF].

Linear motors can also be considered (see Figure 1). ey allow for more freedom when designing a leg architecture, and a lot of systems are naturally backdrivable. Some examples are pneumatic and hydraulic motors [START_REF] Raibert | BigDog, the rough-terrain quadruped robot[END_REF][START_REF] Rosendo | Pneupard: a biomimetic musculoskeletal approach for a feline-inspired quadruped robot[END_REF]. Unfortunately, compressors are an issue for embedded systems and hydraulic motors are costly, heavy, and noisy. We believe DD electric motors have better potential in these aspects as they have good dynamic properties, allowing direct control over the output force [START_REF] Jakubik | An autonomous biped: concept and design[END_REF]. Such motors, having high backdrivability, can produce high forces at high speeds.

ey also mimic the human muscular system, making them good candidates to design a human-like bipedal robot.

Another aspect is the use of motors in bi-articular position. A motor in bi-articular position actuates two consecutive joints.

is is contrary to a monoarticular one, which affects only one joint. Doorenbosch et al. [START_REF] Doorenbosch | e role of mono-and bi-articular muscles during contact control leg tasks in man[END_REF] showed that mono and bi-articular muscles are present in the human leg structure, making such motors an option to consider. A design methodology for bipedal robots using a combination of monoarticular and bi-articular DD linear motors is proposed in [START_REF] Chevallereau | Leg design for biped locomotion with mono-articular and bi-articular linear actuation[END_REF], as well as some architectures without redundancy selected to perform a computed trajectory. As most bipedal robots limit themselves to nonredundant architectures, redundancy while using linear actuators is still a subject to be explored.

is paper proposes to go further by studying different architectures involving monoarticular and bi-articular DD linear motors with and without redundancy, as well as evaluating their efficiency when performing a movement to ultimately build a bipedal robot using such motors. Another originality in our approach is to base our work on motioncaptured motions in order to reinforce the biomimetic aspect of the robot. To do so, a simulator was built to evaluate potential designs. A numerical optimization was also implemented to obtain optimal placement of the motors for each architecture, as shown in Figure 2. is paper will be organized as follows: first, the problem and methodology will be presented. We will then explain the simulator's behavior before presenting the obtained results and discussing them.

Problem Statement

2.1. Selected Architectures and Geometries. Even though it was decided to use linear motors, their number as well as their placement on a robotic leg still needs to be fixed. e rotation of the joint using linear motors is done through the elongation of the actuator. In this aspect, some of the followed guidelines are as follows:

(i) e motor's positions should be similar to their equivalent muscles in the human body when possible. (ii) Each motor should be unique in the considered architecture to keep the design simple (iii) Even when considering redundancy, the number of motors should still be limited to conserve the human-like aspect and reduce its weight is resulted in considering a monoarticular motor for each joint (hip, knee, and ankle) and two bi-articular actuators spanning the hip-knee joints and knee-ankle joints, respectively, for a maximum of five motors. From these choices, 14 architectures, i.e., motor's type and number, are deduced, taking as a base the nonredundant architectures presented in [START_REF] Chevallereau | Leg design for biped locomotion with mono-articular and bi-articular linear actuation[END_REF], and adding six architectures with redundancy (5 four-motor architectures and one five-motor architecture, shown in Figure 3). e motors used in each architecture are detailed in Figure 4.

Once the studied architectures are decided, what remains is the choice of the position of the fixation points on the robot limbs for all the motors. is is what we call the geometry of a robot. Per the previous guidelines, the geometry should conserve the human-like aspect of the biped while ensuring its capacity to perform a given motion. Indeed, as long as linear motors are used, a singularity will occur whenever the motor's axis intersects with the joint axis. In this singular configuration, no force can be transmitted to the joint. is states a problem where the motor's position must avoid singularities during the motion but should still be close enough to the limbs as to keep the robot size within human-like proportions.

e latter leads to defining a bounded area around each limb where the motor attachments should be placed.

Direct Drive (DD) Electric Linear

Motor. DD-electric linear motors have a structure similar to rotative ones. Both have a static stator, but while a rotative motor has a movable rotor to generate rotation, a linear motor has a slider to produce translation. e stator is composed of multiple coils that will create an electromagnetic field when circulating an electric current. is will in turn fix the position of the slider. An interesting feature of this mechanism is that it allows one to tune the resistance to perturbations with the input current. is creates a virtual spring to absorb shocks. e geometry required for such a robot comes directly from the use of linear motors.

With a linear motor, the torque is not generated directly like a rotative one but comes from the available leverage between the motor axis and the affected joint axis. A singularity problem can occur and must be taken into account. Moreover, the maximal available force of a linear motor is dependent on its elongation, as seen in Figure 5, which reinforces the importance of geometry.

Formal Problem Statement.

Our goal is to find the best architecture among the candidates along with its optimal geometry in order to perform a given motion. As the motor singularities and torques required from them depend on the configuration of the biped, the results will depend on the studied motions. A set of motions comprised squats, walking, and running at different speeds can be selected to have multiple kinds of motions at different intensities. A simulator was implemented in order to evaluate the performance of an architecture A (with A ∈ 1, . . . , 14 { }, see Figure 3) and one of its geometries P (see Figure 6). In other words, for a fixed architecture, multiple geometries are evaluated to obtain the best suited for the chosen motions.

e evaluated criterion is the feasibility M A (P) of the motion using the geometry P, defined as

M A (P) � 1 T 􏽚 T 0 b A (P, t)dt, (1)
where [0, T] is the time interval of interest. We define the Boolean b A (P, t) as true if and only if all the required forces to actuate the joints can be produced by the linear motors of the given architecture A. e criterion of feasibility represents the doable percentage for the considered motion with the given architecture and the given geometry.

Simulator

3.1.

Workflow. e simulator's behavior is shown on Figure 7, with the input being the following:

(x i , y i ) ∈ [a i , b i ] × [c i , d i
] the higher 2D anchor point, (x i , y i ) the lower 2D anchor point, and o i ∈ [e i , f i ] the offset of each motor. For each motor, the five-dimensional vector p i � (x i , y i , x i , y i , o i ) combines the 2D positions of its anchor points and the offset. is brings either 15, 20, or 25 optimization variables depending on the considered architecture. For instance, with architecture 1, the geometry P is of the following form: (p 1 , p 2 , p 3 ). q(t) the joint configuration at time t of the studied motion, obtained from the HuMoD database [START_REF] Wojtusch | HuMoD -a versatile and open database for the investigation, modeling and simulation of human motion dynamics on actuation level[END_REF]. Γ(t) the joint torques at time t of the studied motion, computed from t ↦q(t) with the Newton-Euler formalism.

It is relevant to remark that even though q(t) and Γ(t) are coherent between one another. But in reality, the latter is dependent on P. In our approach, we assume that the position of the linear motors should not affect neither the mass repartition nor the inertia matrices of the different limbs.

e Boolean b A (P, t) is computed by comparing at each instant t, the forces F(t) required from the motors with the maximum forces available from them F(P, q(t)); formally, b A (P, t) ⇔ F(t) ≤ F(P, q(t)).

(

) 2
e latter is then integrated over time and outputs the feasibility denoted by M A (P).

Forces Computation.

As a linear motor cannot produce directly a torque but instead generate a force along the motor's axis, some transformation is needed to convert this forces into a torque. e relationship between the two also depends on the number and positioning (i.e., the affected joints) of the motors. As explained in [START_REF] Chevallereau | Leg design for biped locomotion with mono-articular and bi-articular linear actuation[END_REF], when considering all the motors, the relation between the joint torques and their corresponding motor forces can be written as

Γ h Γ k Γ a ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ � j h 0 0 j h,hk 0 0 j k 0 j k,hk j k,ka 0 0 j a 0 j a,ka ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ • F h F k F a F hk F ka ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ( 3
)
with Γ h , Γ k , and Γ a being the hip, knee, and ankle joint torque, respectively, F h , F k , F a , F hk , and F ka the forces provided by the actuators (see Figure 8). In Eq. ( 2), the matrix containing all the j α,β components will be referred simply as J. Something worth noting is that this 3 × 5 matrix which represents a three-joint/five-motor architecture will be modified when removing a motor by suppressing its corresponding column in the matrix. is highlights the nonredundant architectures limited to three motors (thus a 3 × 3 matrix) and the redundant ones with up to 5 actuators (thus 3 × 4 or 3 × 5 depending on the architecture).

As linear actuators are considered, generically, the real coefficient j α,β is defined as the leverage between the actuator's axis m β �→ and the affected joint position P α , defined as

j α,β � P α M β ����� �→ × m β �→ � � � � � � � � � � m β �→ � � � � � � � � � � , ( 4
)
with M β being a point belonging to the motor's line. In our case, the point M β is chosen as a fixation point. Even though J is known, there might be some issues to be addressed depending on the architecture. In fact, for redundant architectures, there exist many admissible couples of forces. is situation is handled by performing an optimization of the motor's power. To do so, the motor's velocities are computed from the following:

p 1 = (x 1 , y 1 , x 1 , y 1 , o 1 ) p 2 = (x 2 , y 2 , x 2 , y 2 , o 2 ) p 3 = (x 3 , y 3 , x 3 , y 3 , o 3 ) (x 1 , y 1 ) (x 2 , y 2 ) (x 3 , y 3 ) (x 2 , y 2 ) (x 3 , y 3 )
Figure 6: e geometry parameters for a given architecture.

q(t) Γ(t) Jacobian F (t) F (P, q(t)) P = (p i ) b A (P, q(t))
M A (P)

Figure 7: System inputs and outputs.

V motors � J T • dq dt , ( 5
)
with J T the transpose of the matrix J linking the vector of torques Γ and the vector of forces F in [START_REF] Kaminaga | Development of Backdrivable Hydraulic Joint Mechanism for Knee Joint of Humanoid Robots[END_REF]. en, an optimization is performed in order to find the forces distribution with minimal power while satisfying the torque requirement constraints. e latter optimization problem is defined as a least squares linear problem of the following form:

min F 1 2 V motors • F � � � � � � � �such that Γ � J • F 􏼚 􏼛. (6)
If the required forces cannot be performed by the actuators, the optimization is redone without boundaries in order to still obtain a force distribution, even though the Boolean b(p, t) will not be true.

e method to obtain the maximum available force of a motor F(p, q(t)) is composed of two steps. First, the motor's elongation which corresponds to the distance between its two fixation points is computed. en, the maximum force available for this length is retrieved from its datasheet (see Figure 5).

Numerical Optimization

e previously shown simulator is used within an optimization loop in order to obtain for each studied architecture an optimal placement of the motors. is optimization was implemented in the MATLAB ® environment with the fminsearchbnd function, which uses the Nelder-Mead simplex method with boundaries to find a local minimum for a nonlinear problem. e following parts will present the practical choices made within the team to perform this optimization and the results are obtained.

Experimental Setup.

To realize a numerical optimization, some concerns such as the robot's dimension and the studied motions must be addressed. In [START_REF] Spielberg | Functional co-optimization of articulated robots[END_REF][START_REF] Ha | Computational co-optimization of design parameters and motion trajectories for robotic systems[END_REF], these elements were included as optimization variables instead of using preexisting or simulated data. For the sake of simplicity, and as the choice of the motor's type and number as well as their naive placement comes from mimicking the human leg structure, the robot's parameters and motions were chosen to be identical to those of a human subject. To do so, the geometric parameters and joint trajectories were picked from the HuMoD database [START_REF] Wojtusch | HuMoD -a versatile and open database for the investigation, modeling and simulation of human motion dynamics on actuation level[END_REF]. We obtained them from a subject (A in the database) performing multiple motions on an instrumented treadmill while recorded by a motion capture system.

is study focuses on squats, walking, and running motions available in the database.

ese motions are considered to be performed alone and consecutively. is leads to motion-specialized geometries as well as generic geometries for each architecture and evaluates their overall efficiency. A preemptive study was performed to check for motions considered impossible regarding the torques required and future bounds of the optimization algorithm.

Indeed, the selected motors can produce a force up to 585 N (P01-48 × 240/30 × 240 from LinMot ® ), so some available motions could be too demanding for the motors and should not be considered. is resulted in selecting the following motions:

Squats

Walking at 1, 1.5 and 2 m/s

Running at 2 m/s Figures 9-11 show the joint torques required for a 3 m/s running as an example, which is not a part of the set of studied motions due to the motion being too demanding. Different leverages (from 5 to 20 centimeters) were then assumed with the maximum motor force available (585 N), and their corresponding torques were computed to draw feasible zones depending on the maximum leverage allowed. For instance, the knee joint would require 20 centimeters of leverage to generate the required torque. From this, faster running was considered too ambitious as it would require a leverage of at least 20 centimeters in order to perform with the selected motors.

Concerning the bounds for each motor's position (i.e., their fixation points), they are limited to a rectangle spanning the length of the limb attached to and up to 85 millimeters from the limb's axis. Last but not least, it is assumed that the robot will have masses and inertia matrices identical to the HuMoD subsect. With all these assumptions in mind, a cluster was put up to perform multiple optimizations in parallel.

Γ a Γ k F a F ka F hk F k Γ h F h
is allows us to optimize all the considered architectures simultaneously for a given set of motions. An experiment (i.e., an optimization of all the architectures for one set of motions) takes up to one week, as for each instant (30,000 to 50,000 time-stamped instants per motion) the actuator's positions, maximal and required forces are computed. An evaluation takes up to 5 minutes to be performed in the case that all the motions are taken into account.

For the following results, a threshold of 95% was fixed to consider a motion feasible by a given architecture. e convergence time and feasibility values were recorded during the optimization process and are shown in Figure 12. Furthermore, the motor forces, once the optimal geometries were obtained, were computed (see Figures 1314151617). e source code for the simulator is available online at https:// laris.univ-angers.fr/fr/projets/projets-actuels/robibio.html.

e following sections will present our results when considering a single motion or multiple motions at once.

Specialized Geometries.

is section concerns the optimization results obtained when considering each motion separately. Each value shown in Figure 18 represents the best feasibility obtained by an architecture for a given motion. A noticeable gap can be seen between the nonredundant and redundant architectures. In fact, only redundant architectures can perform up to the running motion, with architecture 11 being an exception as only one motor is actuating the knee joint. Moreover, there is only one architecture with an existing geometry to perform each motion entirely, namely, the five-motor architecture number 14. However, the four-motor architecture number 12 can still pass the threshold for every studied motion.

e nonredundant architecture cannot perform beyond walking at 1 m/s. In this aspect, architecture 8 (Figure 19) can be highlighted as the only three-motor architectures with geometries able to perform the squat or the walking motion. It is worth noting that, for the squat motion, it is mandatory to have two motors affecting the hip joint (the monoarticular hip and the bi-articular hip-knee), hence the need of bi-articular actuators.

Versatile Geometries.

With multiple motions, the first experiment takes two of them simultaneously: the squat and the walking motion at 1 m/s. We then add one motion after the other for each new experiment until all five motions are considered. In practice, the additional motions are added to the joint trajectory vectors. Here, only redundant architectures seem to have the potential to perform multiple motions, as shown in Figure 20. As in the previous case, only architecture 14 (i.e., the 5-actuator architecture) has a solution satisfying all the motion requirements. It can be noted that when disregarding the running motion, architectures 12 (Figure 21) and 13 are valid alternatives even though they use only 4 actuators.

Discussion.

In this paper, all the optimizations are performed considering the complete timeline of the HuMoD motions. Indeed, we could have restricted our attention to a cycle considered representative of the whole trajectory. However, it led to specialized optimal solutions for the given "representative" cycle. From the previously shown results, electric DD linear motors appear as a valid actuation scheme when operating a biped robot. A bi-articular motor and a larger scale redundancy can greatly improve the capabilities of a biped robot compared to a classic 3 joints/3 monoarticular actuator architecture. e optimality of the results is an issue as our system possesses a high number of local minima. Furthermore, even though architecture 14 is able to perform nearly 100% of the studied motions, Figure 22, which exposes the optimal geometry for each architecture, shows that possible collisions between the actuators and/or body segments might be an issue that should be addressed.

Future work involves the validation of the architectures with real actuator masses in a 3D modeling software as well as performing a new set of optimizations with more powerful actuators. Finally, we expect to start building a robot with such actuation in the upcoming months to verify these theoretical results. is work could also be reproduced with input torques obtained from an optimized joint trajectory for a specific motion to confirm these results. Mathematical Problems in Engineering 13

Conclusion and Perspectives

is study investigates the potential of electric DD linear motors for the actuation scheme of a planar biped robot. We also considered the use of bi-articular motors and redundant architectures. Such motors are an interesting alternative to rotative actuators and allow more freedom in their positioning. Furthermore, relying on bi-articular actuators can increase the available torque on a specific joint to perform a motion that would be impossible with only monoarticular actuators. Redundancy seems like a prerequisite in order to perform high-dynamic motions such as fast walking and running. Many development choices were made, such as the actuators chosen (new models seem to be able to deliver over 1000 N), their number and position, as well as the hypothesis concerning the actuator masses. (Table 1).

Figure 1 :

 1 Figure 1: An example of a DD electric linear actuator.

Figure 2 :Figure 4 :

 24 Figure 2: Computer-aided design of the obtained optimal monoarticular architecture.

Figure 3 :

 3 Figure 3: Architectures (blue: nonredundant/orange: redundant).

Figure 5 :

 5 Figure 5: Standard DD electric linear motor characteristics (P01-48 × 240/30 × 240 from LinMot ® ).

Figure 8 :

 8 Figure 8: Actuator's forces and joint torques.

Figure 9 :

 9 Figure 9: Hip torques for a 3 m/s run and leverage required.

Force

Figure 13 :Figure 12 :

 1312 Figure 13: Kip motor force for the optimal geometry of architecture 14 (red: (a), blue: (b)).

Figure 15 :Figure 14 :Figure 17 :Figure 16 :Figure 19 :

 1514171619 Figure 15: Ankle motor force for the optimal geometry of architecture 14 (red: (a), blue: (b)).

Figure 20 :

 20 Figure 20: Feasibilities after optimization for cumulative motions.

Figure

  Figure Optimal versatile geometry for architecture 12.

Figure 22 :

 22 Figure 22: Optimal geometries for each architecture.

Data Availability

e source code of the simulator is available online at https:// laris.univ-angers.fr/fr/projets/projets-actuels/robibio.html.

Conflicts of Interest

e authors declare that they have no conflicts of interest.


Jean-Philippe Anker

Stefano Meda

Vittoria Pierfelice

Maria Vallarino

Hong-Wei Zhang

Schrödinger equation on noncompact symmetric spaces

Keywords: 2020 Mathematics Subject Classification. 22E30, 35J10, 35P25, 43A85, 43A90 non-compact symmetric space, semilinear Schrödinger equation, pointwise kernel estimates, dispersive property, Strichartz inequality, global well-posedness

published or not. The documents may come   L'archive ouverte pluridisciplinaire

1. Introduction 1.1. Strichartz inequality. The Strichartz inequality, which has proved to be an efficient tool in the study of the nonlinear Schrödinger equation, has been extensively investigated over the past five decades. Let us start by recalling some well-known results in the Euclidean setting. Consider the free Schrödinger equation

i∂ t u(t, x) + ∆ x u(t, x) = 0, t ∈ R, x ∈ R d , u(0, x) = f (x), (1.1)
whose solution is given by the convolution of the initial data with the kernel of the Schrödinger propagator:

u(t, . ) = e it∆ f = f * (4πit) -d 2 e i | . | 2 4t
.

(1.2)

The Strichartz inequality refers to an a priori estimate of the solution to (1.1): in dimension d ≥ 2, if a couple (p, q) is admissible, in the sense that

2 p + d q = d 2
with p, q ≥ 2 and (p, q, d) = (2, ∞, 2), (1.3) then there exists a constant C > 0 such that global solutions to (1.1) satisfy

u L p (R; L q (R d )) ≤ C f L 2 (R d ) .
(1.4)

Such an inequality means that, endowed with suitable space-time Lebesgue norms, the solution to the Schrödinger equation can be estimated in terms of the L 2 norm of the initial data. This type of estimate appeared in the pioneering works [Tom75; Seg76; Str77] around 1976 and was fully developed after Keel-Tao completed the proof for the endpoint in 1998, see [START_REF] Keel | Endpoint Strichartz estimates[END_REF]. Over these twenty years, there have been many remarkable contributions to this problem, and we refer to [GiVe95; Caz03; Tao06] for more details.

A related fundamental problem is to figure out how the underlying geometry affects the Strichartz inequality. This helps us to understand the behavior of solutions to the partial differential equations in the non-Euclidean background. Let M be a Riemannian manifold of dimension d ≥ 2 and ∆ be the Laplace-Beltrami operator on M. The question is, for which (p, q) and s ≥ 0, the following inequality holds:

e it∆ f L p (I; L q (M)) ≤ C f H s (M) .
(1.5)

We say that (1.5) holds global-in-time if I = R. Here H s denotes the standard L 2 Sobolev spaces. If (1.5) is available only for some s > 0, we say that the Strichartz inequality holds with loss of s derivatives. The inequality (1.4) shows that, when M = R d , the global-in-time Strichartz inequality holds without loss for all (p, q) satisfying the admissible condition (1.3). However, this is not always the case on manifolds. Apart from some non-trapping manifolds (see, for instance, [StTa02; HTW05; BoTz07]), the Strichartz inequality is more or less understood on compact manifolds. In general, (1.5) cannot hold global-in-time in the compact case. Bourgain proved that, on the flat torus M = T d , if a couple (p, q) satisfies the admissible condition (1.3), then (1.5) holds for I = T and s > d 4 -1 2 , see [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations[END_REF][START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation[END_REF]. Later, Burq, Gérard and Tzvetkov [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] showed that, on arbitrary compact Riemannian manifolds without boundary, (1.5) holds with loss of s = 1/p derivatives for any finite time interval I and (p, q) satisfying (1.3). This result is sharp in some particular cases, such as on a three-dimensional sphere, where a 1/2 loss must occur in (1.5).

In this paper we consider non-compact Riemannian symmetric spaces of any rank, which form an important class of non-positively curved Riemannian manifolds. Due to their exponential volume growth at infinity and the validity of the Kunze-Stein phenomenon, one expects stronger dispersive phenomena than in the Euclidean setting, hence better Strichartz inequality, well-posedness and scattering results. This was indeed brought to light in the study of the Schrödinger equation on real hyperbolic spaces, which are the simplest symmetric spaces of non-compact type and rank one, see for instance [Pie06; Ban07; BCS08; AnPi09; IoSt09]. See also [START_REF] Anker | Schrödinger equations on Damek-Ricci spaces[END_REF] for similar results on Damek-Ricci spaces (a class of harmonic manifolds which includes all non-compact symmetric spaces of rank one). Extending these rank one results to higher rank is a natural but challenging problem since the Schrödinger kernel is sensitive to the geometry of the underlying manifolds, and the spherical Fourier analysis is well-known to be much more intricate on the higher rank symmetric spaces. In the next subsections, we will explain in detail the difficulties involved and share some new ideas to overcome them. Roughly saying, we establish in the present paper a stronger global-in-time Strichartz inequality, in the sense that the related family of admissible pairs is significantly larger than the best possible in the Euclidean setting. This is achieved by a subtle analysis of the Schrödinger kernel. Once the Schrödinger equation is understood on globally symmetric spaces, studying it on locally symmetric spaces is a further natural problem. See [BGH10; FMM18; Wan19] for some first results in rank one. 1.2. Statements of main results. We adopt the standard T T * duality argument to establish the Strichartz inequality, see [START_REF] Kato | On nonlinear Schrödinger equations[END_REF][START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF]. This argument relies on the dispersive estimates of the Schrödinger propagator, which can be easily obtained in R d since the corresponding convolution kernel is explicitly defined. However, such a fundamental piece of information is lacking on a general manifold. Our primary task is to find sharp pointwise estimates of the Schrödinger kernel.

Consider a non-compact symmetric space X = G/K of rank , where G and K are suitable Lie groups. Let d ≥ 2 and D ≥ 3 be its manifold dimension and dimension at infinity (see Sect. 2 for more details about these notations). We denote by ∆ the Laplace-Beltrami operator on X and consider the free Schrödinger equation

i∂ t u(t, x) + ∆ x u(t, x) = 0, t ∈ R, x ∈ X, u(0, x) = f (x), (1.6)
whose solution is given by

u(t, x) = e it∆ f (x) = f * s t (x) = X dy f (y) s t (y -1 x).
Here s t is the bi-K-invariant convolution kernel of the Schrödinger propagator e it∆ . As mentioned above, except for a few particular cases, for instance when G is complex (see [START_REF] Gangolli | Asymptotic behavior of spectra of compact quotients of certain symmetric spaces[END_REF]), the kernel s t has no explicit expression as in (1.2). Moreover, the expression obtained by using the inverse Abel transform on real hyperbolic spaces (see [AnPi09, p.1859]) is also unavailable in higher rank. Our main result is the the following sharp-in-time kernel estimates on non-compact symmetric spaces of any rank.

Theorem 1.1 (Pointwise kernel estimates). There exist C > 0 and N > 0 such that the following estimates hold, for all t ∈ R * and x ∈ X :

|s t (x)| ≤ C (1 + |x + |) N e -ρ, x +    |t| -d 2 if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1, (1.7)
where x + denotes the radial component of x in the Cartan decomposition and ρ is the half sum of all positive roots.

Remark 1.2. Notice that ρ, x + > 0 as ρ and x + are two -dimensional vectors in the socalled positive Weyl chamber, see Sect. 2 for more details. Hence the large polynomial factor (1 + |x + |) N is harmless for proving the dispersive estimates because of the exponential factor e -ρ,x + .

In contrast to the Euclidean setting, the kernel s t here (expressed as (3.1) according to the inverse spherical Fourier transform) behaves differently for small and large times and satisfies no rescaling. Let us emphasize that the existing methods in rank one fail to produce desired estimates since the Plancherel density |c(λ)| -2 occurring in (3.1) is not always a differential symbol in general, hence the standard stationary phase method fails. This is in fact a well-known difficulty in the study of higher rank spherical harmonic analysis, see for instance [START_REF] Branson | Huyghens' principle in Riemannian symmetric spaces[END_REF]. As for the recent works about the wave equation on symmetric spaces [AnZh20; Zha21], the Schrödinger propagator does not enjoy the finite propagation speed as the wave propagator, and a more involved analysis is required. However, by borrowing some ideas from previous works and by combining them cleverly, we can obtain Theorem 1.1 on general symmetric spaces of non-compact type, see Sect. 3.

Once we establish these key pointwise kernel estimates, the dispersive properties of e it∆ follow from the Kunze-Stein phenomenon (instead of Young's convolution inequality) and interpolation. Then we deduce the Strichartz inequality by using the T T * argument. The proofs are not so different from the ones on real hyperbolic spaces, but we still include the details in Sect. 4 for the reader's convenience.

Theorem 1.3 (Dispersive properties). Let 2 ≤ q, q < +∞. Then there exists a constant C > 0 such that following estimates hold for all t ∈ R * :

e it∆ L q (X) → L q (X) ≤ C    |t| -max( 1 2 -1 q , 1 2 -1 q ) d if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1, (1.8)
where q and q are dual indices in the sense that 1 q + 1 q = 1. Remark 1.4. Theorem 1.3 covers previously known results in rank 1, where the dimension at infinity is D = 3. Notice that the large time decay in (1.8) becomes faster in higher rank, see (2.1). Notice also that the estimate (1.8) is quite different from the one in the Euclidean setting where

e it∆ R d L q (R d ) → L q (R d ) ≤ C |t| -( 1 2 -1 q )d ∀ t ∈ R * .
In particular, the large time decay in (1.8) depends no more on the indices q or q. Such a particular phenomenon on non-compact symmetric spaces yields a stronger Strichartz inequality.

Theorem 1.5 (Strichartz inequality). Let (p, q) and ( p, q) be two admissible pairs corresponding to the triangle

1 p , 1 q ∈ 0, 1 2 × 0, 1 2 2 p + d q ≥ d 2 0, 1 2 .
(1.9)

Then there exists a constant C > 0 such that, for any bounded or unbounded I ⊆ R, the solution to the inhomogeneous Schrödinger equation

i∂ t u(t, x) + ∆ x u(t, x) = F (t, x), u(0, x) = f (x), (1.10) satisfies u L p (I;L q (X)) ≤ C f L 2 (X) + F L p (I;L q (X)) .
(1.11)

1 p 1 q 1 2 1 2 -1 d 0 1 2 1 p = d 2 1 2 -1 q Figure 1. Admissibility in dimension d ≥ 3.
Remark 1.6. The inequality (1.11) itself is analogous to the one in the Euclidean setting, but the admissible set (1.9) is much larger than (1.3). The latter one corresponds only to the lower edge of the admissible triangle (1.9), see Fig. 1. This is due to the large-scale geometry of symmetric spaces, which yields better dispersive properties.

1.3. Well-posedness and scattering for the semilinear Schrödinger equation. The fixed point argument, which consists in finding a suitable contraction mapping defined on an appropriate Banach space, is nowadays a standard method to prove the well-posedness of nonlinear partial differential equations. Due to the stronger Strichartz inequality proved in Theorem 1.5, we are able to obtain the following better results on non-compact symmetric spaces. The proofs are adapted straightforwardly from the rank one case considered in [AnPi09; APV11] and are therefore omitted.

Consider the semilinear Schrödinger equation:

i∂ t u(t, x) + ∆ x u(t, x) = F (u(t, x)), t ∈ R, x ∈ X, u(0, x) = f (x). (NLS)
where F is a power-like nonlinearity of order γ > 1 in the sense that

|F (u)| ≤ C |u| γ and |F (u) -F (v)| ≤ C (|u| γ-1 + |v| γ-1 ) |u -v|.
Let H 1 (X) be the Sobolev space defined as the image of L 2 (X) under the operator (-∆) -1 2 . We have the following well-posedness results for the semilinear Schrödinger equation (NLS):

• If 1 < γ ≤ 1 + 4 d , the Cauchy problem (NLS) is globally well-posed for small L 2 data. • If 1 < γ < 1 + 4
d , the Cauchy problem (NLS) is locally well-posed for arbitrary L 2 data. Moreover, if F is in addition gauge invariant, namely, if Im{F (u)ū} = 0, then the L 2 conservation of mass implies the global well-posedness for arbitrary L 2 data in this subcritical case.

• If 1 < γ ≤ 1 + 4 d-2
, the Cauchy problem (NLS) is globally well-posed for small H 1 data.

• If 1 < γ < 1 + 4 d-2
, the Cauchy problem (NLS) is locally well-posed for arbitrary H 1 data. Moreover, if F is in addition defocusing, namely, if there exists a nonnegative C 1 function G such that F (u) = G (|u| 2 )u, then the H 1 conservation of energy implies the global well-posedness for arbitrary H 1 data in this subcritical case.

Notice that these results are better than the known ones on Euclidean spaces. For instance, global well-posedness for small L 2 initial data holds for any exponent 1 < γ ≤ 1 + 4 d on X, while on R d one must assume in addition gauge invariance. However, under this condition, one can handle arbitrary L 2 data by using conservation laws.

The Strichartz inequality can also be used to prove scattering results, which means that global solutions to the nonlinear Schrödinger equation behave asymptotically as solutions to the linear equation. Specifically, Theorem 1.5 implies the following scattering results.

• If 1 < γ ≤ 1+ 4
d , then the global solution u(t, x) to the Cauchy problem (NLS) corresponding to small L 2 data satisfies the following scattering property: there exists

u ± ∈ L 2 (X) such that u(t, •) -e it∆ u ± L 2 (X) -→ 0 as t -→ ±∞. • If 1 < γ ≤ 1+ 4 d-2
, then the global solution u(t, x) to the Cauchy problem (NLS) corresponding to small H 1 data satisfies the following scattering property: there exists

u ± ∈ H 1 (X) such that u(t, •) -e it∆ u ± H 1 (X) -→ 0 as t -→ ±∞.
This scattering property remains valid for any H 1 initial data in the subcritical case 1 < γ < 1 + 4 d-2 if the nonlinearity F is assumed to be defocusing. Notice that, on R d , these scattering properties are known to fail for small powers γ ∈ (1, 1+ 2 d ]. However, our stronger Strichartz inequality on non-compact symmetric spaces makes these properties possible in the full range. 1.4. Layout. After reviewing spherical Fourier analysis on non-compact symmetric spaces in Sect. 2, we prove in Sect. 3 our main result, namely the pointwise kernel estimates. More precisely, we start with the large time estimate, which is surprisingly straightforward, see Sect. 3.1. In small time, we get the desired estimates by combining the stationary phase method based on the barycentric decomposition in Theorem 2.1, the subordination formula (3.11), and the improved Hadamard parametrix (3.13), see Sect. 3.2. By adapting the method carried out in rank one, we deduce in Sect. 4 our stronger dispersive and Strichartz inequalities.

Preliminaries

In this section we review briefly harmonic analysis on Riemannian symmetric spaces of noncompact type. We adopt the standard notation and refer to [Hel78; Hel00; GaVa88] for more details. Throughout this paper, the symbol A B between two positive expressions means that there is a constant C > 0 such that A ≤ CB. The symbol A B means that A B and B A.

2.1. Non-compact symmetric spaces. Let G be a semisimple Lie group, connected, noncompact, with finite center, and K be a maximal compact subgroup of G. The homogeneous space X = G/K is a Riemannian symmetric space of non-compact type. Let g = k ⊕ p be the Cartan decomposition of the Lie algebra of G, the Killing form of g induces a K-invariant inner product . , . on p, hence a G-invariant Riemannian metric on X. Fix a maximal abelian subspace a in p. The rank of X is the dimension of a. We identify a with its dual a * by means of the inner product inherited from p. Let Σ ⊂ a be the root system of (g, a) and denote by W the Weyl group associated to Σ. Once a positive Weyl chamber a + ⊂ a has been selected, Σ + (resp. Σ + r or Σ + s ) denotes the corresponding set of positive roots (resp. positive reduced roots or simple roots). Let d be the dimension of X and let D be the so-called dimension at infinity or pseudo-dimension of X:

d = + α∈Σ + m α and D = + 2|Σ + r |, (2.1)
where m α is the dimension of the positive root subspace g α . Notice that these two dimensions behave quite differently. For example,

D = 3 while d ≥ 2 is arbitrary in rank one, D = d if G is complex, and D > d (actually D = 2d -) if G is split.
Let n be the nilpotent Lie subalgebra of g associated to Σ + and let N = exp n be the corresponding Lie subgroup of G. We have the decompositions

G = N (exp a) K (Iwasawa), G = K (exp a + ) K (Cartan).
In the Cartan decomposition, the Haar measure on G writes

G dx f (x) = const. K dk 1 a + dx + δ(x + ) K dk 2 f (k 1 (exp x + )k 2 ) , with density δ(x + ) = α∈Σ + (sinh α, x + ) mα α∈Σ + α, x + 1 + α, x + mα e 2 ρ,x + ∀ x + ∈ a + .
(2.2)

Here ρ ∈ a + denotes the half sum of all positive roots α ∈ Σ + counted with their multiplicities m α :

ρ = 1 2 α∈Σ + m α α.
2.2. Spherical Fourier analysis on symmetric spaces. Let S(K\G/K) be the Schwartz space of bi-K-invariant functions on G. The spherical Fourier transform H is defined by

Hf (λ) = G dx ϕ -λ (x) f (x) ∀ λ ∈ a, ∀ f ∈ S(K\G/K),
where ϕ λ ∈ C ∞ (K\G/K) denotes the spherical function of index λ ∈ a, which is a smooth bi-K-invariant eigenfunction of all invariant differential operators on X, in particular of the Laplace-Beltrami operator:

-∆ϕ λ (x) = (|λ| 2 + |ρ| 2 ) ϕ λ (x).
The spherical functions have the following integral representation

ϕ λ (x) = K dk e iλ+ρ,A(kx) ∀ λ ∈ a, (2.3)
where A(kx) denotes the a-component in the Iwasawa decomposition of kx. They satisfy the basic estimate

|ϕ λ (x)| ≤ ϕ 0 (x) ∀ λ ∈ a, ∀ x ∈ G,
where

ϕ 0 (exp x + ) α∈Σ + r (1 + α, x + ) e -ρ,x + ∀ x ∈ G. (2.4)
Denote by S(a) W the subspace of W -invariant functions in the Schwartz space S(a). Then H is an isomorphism between S(K\G/K) and S(a) W . The inverse spherical Fourier transform is given by

f (x) = C 0 a dλ |c(λ)| -2 ϕ λ (x) Hf (λ) ∀ x ∈ G, ∀ f ∈ S(a) W , (2.5)
where C 0 > 0 is a constant depending only on the geometry of X. By using the Gindikin-Karpelevič formula of the Harish-Chandra c-function (see [START_REF] Helgason | Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions[END_REF] or [START_REF] Gangolli | Harmonic analysis of spherical functions on real reductive groups[END_REF]), we can write the Plancherel density as

|c(λ)| -2 = α∈Σ + r |c α ( α, λ )| -2 , (2.6) with c α (v) = Cα Γ( α,ρ α,α + 1 2 mα) Γ( α,ρ α,α ) Γ( 1 2 α,ρ α,α + 1 4 mα+ 1 2 m 2α ) Γ( 1 2 α,ρ α,α + 1 4 mα) Γ(iv) Γ(iv+ 1 2 mα) Γ( i 2 v+ 1 4 mα) Γ( i 2 v+ 1 4 mα+ 1 2 m 2α ) . Notice that |c α | -2 is an inhomogeneous differential symbol on R of order m α + m 2α , for every α ∈ Σ + r . Hence |c(λ)| -2
is a product of one-dimensional symbols, but not a symbol on a in general. It satisfies indeed

|c(λ)| -2 |λ| D- if |λ| ≤ 1, |∇ k a c(λ)| -2 |λ| d-if |λ| ≥ 1, (2.7)
for all k ∈ N. The estimate away from the origin means that one cannot obtain any additional decay from general derivatives, hence the standard stationary phase method fails. To overcome this difficulty, we introduce the following barycentric decomposition.

Theorem 2.1. There exists a smooth partition of unity w∈W 1≤j≤ χ w.S j = 1 on a {0}

(2.8) consisting of homogeneous symbols χ w.S j of order 0, and some nonzero vectors w.Λ j ∈ supp χ w.S j such that for all w ∈ W and 1 ≤ j ≤ ,

| w.Λ j , λ | |λ| ∀ λ ∈ supp χ w.S j (2.9)
and the following dichotomy holds: for every α ∈ Σ,

• either α, w.Λ j = 0,

• or | α, λ | |λ| for all λ ∈ supp χ w.S j .
This theorem allows us to split up the Cartan subspace a into several subcones supp χ w.S j by using some appropriate cut-off functions χ w.S j . In each subcone, we can choose a vector w.Λ j such that if a root α is not orthogonal to w.Λ j , then the inner product between α and any vector λ in this subcone is comparable to |λ|, see [AnZh20, Subsection 2.2] for more details. Thanks to this decomposition, the Plancherel density can be handled as if it were a differential symbol, provided that we differentiate it in supp χ w.S j along the well-chosen vector w.Λ j .

Corollary 2.2. For every w ∈ W , 1 ≤ j ≤ and k ∈ N, we have

∂ k w.Λ j |c(λ)| -2 = O(|λ| d--k ) ∀λ ∈ supp χ w.S j .
(2.10)

Proof. Let α be an arbitrary root in Σ. If α, w.Λ j = 0, then all terms

∂ k w.Λ j |c α ( α, λ )| -2 ∀ k ∈ N *
vanish. Otherwise, we deduce from the previous dichotomy that

∂ k w.Λ j |c α ( α, λ )| -2 (1 + | α, λ |) mα+m 2α -k (1 + |λ|) mα+m 2α -k
for all λ ∈ supp χ w.S j , since |c α | -2 is an inhomogeneous differential symbol of order m α + m 2α . We conclude by using (2.1) and (2.6).

Pointwise estimates of the Schrödinger kernel

For simplicity, we consider in this section the shifted Schrödinger propagator e -itD 2 with D = -∆ -|ρ| 2 , and denote by s t its bi-K-invariant convolution kernel. By using the inverse formula of the spherical Fourier transform, we have

s t (x) = C 0 a dλ |c(λ)| -2 ϕ λ (x) e -it|λ| 2 ∀ t ∈ R * , ∀ x ∈ X. (3.1)
As usual, such an oscillatory integral makes sense by applying standard procedures (as a limit of convergent integrals and/or after performing several integrals by parts). We will study (3.1) differently, depending whether |t| is large or small. Let us begin with the easier case where |t| is large.

3.1. Large time kernel estimate. Assume that |t| ≥ 1. In this case, we establish the following pointwise kernel estimate, by using the standard stationary phase method based on the elementary estimate (2.7).

Theorem 3.1. There exist an integer N > max{d, D} and a constant C > 0 such that

|s t (x)| ≤ C |t| -D 2 1 + |x| N ϕ 0 (x) ∀ |t| ≥ 1, ∀ x ∈ X.
Proof. By using the integral expression (2.3) of the spherical function, we write

s t (x) = C 0 K dk e ρ,A(kx) a dλ |c(λ)| -2 e -it|λ| 2 e i λ,A(kx) I(t,A(kx)) (3.2)
where A(kx) denotes the a-component in the Iwasawa decomposition of kx, which satisfies |A(kx)| ≤ |x| and which we abbreviate by A in the sequel. Theorem 3.1 will follow from the estimate

|I(t, A)| |t| -D 2 1 + |A| N . (3.3)
Let us split up

I(t, A) = I 0 (t, A) + I ∞ (t, A) = a dλ χ 0 t (λ) . . . + a dλ χ ∞ t (λ) . . . where χ 0 t (λ) = χ( |t||λ|) is a radial cut-off function such that supp χ 0 t ⊂ B(0, 2|t| -1 2 ), χ 0 t = 1 on B(0, |t| -1
2 ) and χ ∞ t = 1 -χ 0 t . On the one hand, by using (2.7), we easily estimate

|I 0 (t, A)| |λ| |t| -1 2 dλ |c(λ)| -2 |t| -D 2 . (3.4)
On the other hand, after performing N integrations by parts based on

e -it|λ| 2 = -1 2it j=1 λ j |λ| 2 ∂ ∂λ j e -it|λ| 2 , (3.5)
we obtain

I ∞ (t, A) = (2it) -N a dλ e -it|λ| 2 j=1 ∂ ∂λ j • λ j |λ| 2 N χ ∞ t (λ) |c(λ)| -2 e i λ,A . (3.6)
Assume that

• N 0 derivatives are applied to the cut-off function χ 0 t (λ), which produces O(|t|

N 0 2 ),
• N 1 derivatives are applied to the factors

λ j
|λ| 2 , which produces O(|λ| -N -N 1 ), • N 2 derivatives are applied to the Plancherel density |c(λ)| -2 , which is not a symbol in general and which produces

O(|λ| D-) if |λ| ≤ 1, O(|λ| d-) if |λ| ≥ 1,
• N 3 derivatives are applied to the exponential factor e i λ,A , which produces O(|A| N 3 ),

with N 0 + N 1 + N 2 + N 3 = N .
If some derivatives hit the cut-off function χ 0 t (λ), i.e., if N 0 ≥ 1, then the integral reduces to a spherical shell where |λ| |t| -1 2 , and the contribution to (3.6) is estimated by

|t| -N 2 |t| N 0 2 |t| N 1 2 |t| -D 2 |A| N 3 |t| -D 2 1 + |A| N , (3.7) since |t| ≥ 1. If N 0 = 0, then |I ∞ (t, A)| |t| -N |λ| |t| -1 2 dλ ∇ N 2 λ |c(λ)| -2 |λ| -N -N 1 |A| N 3 |t| -N |A| N 3 |t| -1 2 |λ|≤1 dλ |λ| D--N -N 1 + |λ|≥1 dλ |λ| d--N -N 1 |t| -D 2 (1 + |A|) N + |t| -N (1 + |A|) N |t| -D 2 (1 + |A|) N (3.8)
provided that N > d and N ≥ D 2 . In conclusion, (3.3) follows from (3.4), (3.7) and (3.8).

Remark 3.2. The analysis carried out in the proof of Theorem 3.1 yields at best the following small time estimate

|s t (x)| |t| -d (1 + |x|) d ϕ 0 (x) ∀ 0 < |t| < 1, ∀ x ∈ X.
(3.9) 3.2. Small time kernel estimate. Assume that 0 < |t| < 1. Our aim is to reduce the negative power |t| -d in (3.9) to |t| - Theorem 3.3. The following estimate holds, for 0 < |t| < 1 and |x| ≤ |t|:

|s t (x)| |t| -d 2 ϕ 0 (x).
Proof. By resuming the notation in the proof of Theorem 3.1, we have

s t (x) = C 0 K dk e ρ,A(kx) I 0 (t, A) + I ∞ (t, A) .
Clearly,

|I 0 (t, A)| = a dλ χ 0 t (λ) |c(λ)| -2 e -it|λ| 2 e i λ,A |λ| |t| -1 2 dλ |c(λ)| -2 |t| -d 2 .
(3.10)

In order to estimate

I ∞ (t, A) = a dλ χ ∞ t (λ) |c(λ)| -2 e -it|λ| 2 e i λ,A ,
we split up

I ∞ (t, A) = w∈W 1≤j≤ a dλ χ w.S j (λ) χ ∞ t (λ) |c(λ)| -2 e -it|λ| 2 e i λ,A I w.S j (t,x)
.

according to the barycentric decomposition (2.8). Next, we study I w.S j (t, x) by performing N integrations by parts based on e -it|λ| 2 = -1 2it 1 w.Λ j ,λ ∂ w.Λ j e -it|λ| 2 , which yields

I w.S j (t, x) = (2it) -N a dλ e -it|λ| 2 ∂ w.Λ j • 1 w.Λ j ,λ N χ w.S j (λ) χ ∞ t (λ) |c(λ)| -2 e i λ,A .
As in the the proof of Theorem 3.1, we assume that • N 0 derivatives are applied to the cut-off function χ ∞ t (λ):

∂ N 0 w.Λ j χ ∞ t (λ) = O(|t| N 0
2 ),

• N 1 derivatives are applied to the factors 1 w.Λ j ,λ , which produces O(|λ| -N -N 1 ), • N 2 derivatives are applied to the factor χ w.S j (λ), which is a homogeneous symbol of order 0:

∂ N 2 w.Λ j χ w.S j (λ) = O(|λ| -N 2 ),
• N 3 derivatives are applied to the factor e i λ,A :

∂ N 3 w.Λ j e i λ,A = O(|A| N 3 ),
• N 4 derivatives are applied to the Plancherel density |c(λ)| -2 (see Corollary 2.2):

∂ N 4 w.Λ j |c(λ)| -2 = O(|λ| d--N 4 ), with N 0 + N 1 + N 2 + N 3 + N 4 = N . Therefore, if no derivative hits the cut-off function χ ∞ t (λ), i.e., if N 0 = 0, then |I w.S j (t, x)| |t| -N |λ| |t| -1 2 dλ |λ| -N -N 1 -N 2 +d--N 4 |A| N 3 |t| -N |t| -d 2 + N 2 + N 1 2 + N 2 2 + N 4 2 |t| N 3 2 ≤ |t| -d 2 provided that N > d.
If N 0 ≥ 1, then the integral is reduced to a spherical shell where |λ| |t| -1 2 , and hence

|I w.S j (t, x)| |t| -N |t| N 0 2 |t| -d 2 + N 2 + N 1 2 + N 2 2 + N 4 2 |A| N 3 ≤ |t| -d 2 ,
since |A| ≤ |x| ≤ |t|. Together with (3.10), we conclude that |I(t, A)| |t| -d 2 and

|s t (x)| |t| -d 2 ϕ 0 (x),
for all 0 < |t| < 1 and x ∈ X such that |x| ≤ |t|.

The above proof shows that, for every λ ∈ (supp χ w.S j ) ∩ (supp χ ∞ t ), the Plancherel density |c(λ)| -2 behaves like an inhomogeneous symbol of order d -if we differentiate it along the direction w.Λ j . When |x| > |t|, we write the Schrödinger propagator in terms of the wave propagator according to the subordination principle, and use an improved Hadamard parametrix. Let us express the Schrödinger propagator

e -itD 2 = π -1 2 e -i π 4 sign (t) C 1 |t| -1 2 +∞ 0 ds e i 4t s 2 cos(sD)
in terms of the wave propagator and correspondingly

s t (x) = C 1 |t| -1 2 +∞ 0 ds e i 4t s 2 Φ s (x) (3.11)
for their bi-K-invariant convolution kernels. On the one hand, by finite propagation speed,

Φ s (x) = 0 if |x| > |s|.
(3.12)

On the other hand, recall the Hadamard parametrix

Φ s (exp H) = J(H) -1 2 |s| +∞ k=0 4 -k U k (H) R k-d-1 2 + (s 2 -|H| 2 ) ∀ s ∈ R * , ∀ H ∈ p, (3.13)
where J denotes the Jacobian of the exponential map p → G/K, which is given by

J(H) = α∈Σ + sinh α,H α,H mα ∀ H ∈ a + ,
and {R z + | z ∈ C} denotes the analytic family of Riesz distributions on R, which is defined by

R z + (r) = Γ(z) -1 r z-1 if r > 0 0 if r ≤ 0 ∀ Re z > 0.
This parametrix was constructed and used in various settings, see for instance [START_REF] Bérard | On the wave equation on a compact Riemannian manifold without conjugate points[END_REF][START_REF] Hörmander | The analysis of linear partial differential operators III[END_REF][START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on non-compact symmetric spaces[END_REF]. We refer to [AnZh20, Appendix B] for the details about the wave propagator cos (t √ -∆) associated to the unshifted Laplacian ∆ on non-compact symmetric spaces. Notice that the same results hold for cos (tD). Specifically (3.13) is an asymptotic expansion

Φ s (exp H) = J(H) -1 2 |s| [d/2] k=0 4 -k U k (H) R k-d-1 2 + (s 2 -|H| 2 ) + E(s, H) (3.14)
where the coefficients U k are Ad K-invariant smooth functions on p, which are bounded together with their derivatives, while the remainder satisfies

|E(s, H)| (1 + |s|) 3( d 2 +1) e -ρ,H ∀s ∈ R * , ∀H ∈ a + . (3.15) Let us split up +∞ 0 ds = +∞ 0 ds χ 0 ( s |x| ) + +∞ 0 ds χ 1 ( s |x| ) + +∞ 0 ds χ ∞ ( s |x| )
in (3.11) by means of a smooth partition of unity

1 = χ 0 + χ 1 + χ ∞ on R such that        supp χ 0 ⊂ (-1, 1), supp χ 1 ⊂ (-2C 2 , -1 2 ) ∪ ( 1 2 , 2C 2 ), supp χ ∞ ⊂ (-∞, -C 2 ) ∪ (C 2 , +∞)
where the choice of C 2 > 1 will be specified later. Then the contribution of the first integral vanishes according to (3.12) and we are left with

s t (x) = C 1 |t| -1 2 +∞ 0 ds χ 1 ( s |x| ) e i 4t s 2 Φ s (x) s 1 t (x) + C 1 |t| -1 2 +∞ 0 ds χ ∞ ( s |x| ) e i 4t s 2 Φ s (x) s ∞ t (x)
where s 1 t (x) and s ∞ t (x) are bi-K-invariant. Let us first study s ∞ t (x) by using again the barycentric decomposition. In comparison with the proof of Theorem 3.3, we have now |x| > |t| and there is an additional integral over s ∈ (1, ∞) to control. Let us state the theorem.

Theorem 3.4. The following estimate holds, for all 0 < |t| < 1 and |x| > |t| :

|s ∞ t (x)| |t| -d 2 ϕ 0 (x).
Proof. We express

s ∞ t (x) = 1 2 C 0 C 1 |t| -1 2 +∞ -∞ ds χ ∞ ( s |x| ) e i 4t s 2 a dλ |c(λ)| -2 ϕ λ (x) e -is|λ|
by evenness and by expressing the wave kernel Φ s by means of the inverse spherical Fourier transform. Let us split up

s ∞ t = 1 2 C 0 C 1 (s ∞,0 t + s ∞,∞ t ), where s ∞,0 t (x) = |t| -1 2 +∞ -∞ ds χ ∞ ( s |x| ) e i 4t s 2 a dλ χ 0 t (λ) |c(λ)| -2 ϕ λ (x) e -is|λ| I 0 (s,t,x) and s ∞,∞ t (x) = |t| -1 2 +∞ -∞ ds χ ∞ ( s |x| ) e i 4t s 2 a dλ χ ∞ t (λ) |c(λ)| -2 ϕ λ (x) e -is|λ|

I∞(s,t,x)

.

Recall that χ 0 t (λ) = χ( |t||λ|) is a radial cut-off function such that supp χ 0 t ⊂ B(0, 2|t| -1 2 ), χ 0 t = 1 on B(0, |t| -1 2 ) and χ ∞ t = 1 -χ 0 t .
Estimate of s ∞,0 t . Notice that the obvious estimate |I 0 (s, t, x)| |t| -d 2 |ϕ 0 (x) is not enough for our purpose. We need indeed to compensate on the one hand the factor |t| -1 2 and to get on the other hand enough decay in |s| to ensure the convergence of the external integral. To this end, we perform two integrations by parts based on (3.17)

Notice that

( ∂ ∂s ) k I 0 (s, t, x) |t| -d+k 2 ϕ 0 (x) ∀ k ∈ N.
If any derivative hits χ ∞ ( s |x| ) in (3.17), the integral reduces to two intervals where |s| |x|, and the corresponding contribution is estimated by

|t| -d-3 2 ϕ 0 (x) |s| |x| ds {s -2 |x| -2 + s -3 |x| -1 + s -2 |x| -1 |t| -1 2 } |t| -d 2 ϕ 0 (x), since |t| 1 2 < |x|.
Otherwise we end up with the estimate

|t| -d-3 2 ϕ 0 (x) |s| |x| ds {s -4 + s -3 |t| -1 2 + s -2 |t| -1 } |t| -d 2 ϕ 0 (x).
In conclusion,

|s ∞,0 t (x)| |t| -d 2 ϕ 0 (x),
for all 0 < |t| < 1 and x ∈ X such that |x| > |t|.

Estimate of s ∞,∞ t . Let us turn to

s ∞,∞ t (x) = +∞ -∞ ds χ ∞ ( s |x| ) e i 4t s 2 I ∞ (s, t, x).
We will prove the following estimate, for any integer N > d,

|I ∞ (s, t, x)| |s| -N |t| -d 2 + N 2 ϕ 0 (x) ∀ |s| ≥ C 2 |x|. (3.18)
Then, as |x| > |t| , we conclude easily that

|s ∞,∞ t (x)| |t| -d 2 + N 2 -1 2 ϕ 0 (x) |s| |x| ds |s| -N |t| -d 2 √ |t| |x| N -1 ϕ 0 (x) |t| -d 2 ϕ 0 (x). (3.19)
In order to establish (3.18), we express

I ∞ (s, t, x) = a dλ χ ∞ t (λ) |c(λ)| -2 ϕ λ (x) e -is|λ| = K dk e -ρ,A
w∈W 1≤j≤ a dλ χ w.S j (λ) χ ∞ t (λ) |c(λ)| -2 e -i(s|λ|-λ,A ) I w.S j (s,t,A) by using again the integral formula (2.3) and the barycentric decomposition (2.8). According to (2.9), we can choose

C 2 > 0 such that, if |s| ≥ C 2 |x|, then |∂ w.Λ j (s|λ| -λ, A )| = s w.Λ j ,λ |λ| -w.Λ j , A ≥ |s| | w.Λ j ,λ | |λ| 1 -| w.Λ j , A | |x| |s|,
for every λ ∈ (supp χ ∞ t ) ∩ (supp χ w.S j ). Under these assumptions, the phase function λ → s|λ| -λ, A has no critical point along the direction w.Λ j . By performing N integrations by parts based on e -i(s|λ|-λ,A ) = i ∂ w.Λ j (s|λ|-λ,A ) ∂ w.Λ j e -i(s|λ|-λ,A ) , we write

I w.S j (s, t, A) = (is) -N a dλ e -i(s|λ|-λ,A ) ∂ w.Λ j • s ∂ w.Λ j (s|λ|-λ,A ) N χ w.S j (λ) χ ∞ t (λ) |c(λ)| -2 .
Assume that

• N 0 derivatives are applied to the cut-off function χ ∞ t (λ), which produces O(|t|

N 0
2 ), • N 1 derivatives are applied to the factors s ∂ w.Λ j (s|λ|-λ,A ) , which produces O(|λ| -N 1 ),

• N 2 derivatives are applied to the cut-off functions χ w.S j (λ), which produces O(|λ| -N 2 ),

• N 3 derivatives are applied to the Plancherel density |c(λ)| -2 , which produces O(|λ| d--N 3 ), with N 0 + N 1 + N 2 + N 3 = N . Again, if some derivatives hit χ ∞ t (λ), i.e., if N 0 ≥ 1, then the integral reduces to a spherical shell where |λ| |t| -1 2 , and its contribution is estimated by

|s| -N |t| -2 |t| N 0 2 |t| N 1 2 |t| N 2 2 |t| -d 2 + 2 + N 3 2 = |s| -N |t| -d 2 + N 2 .
If N 0 = 0, then

|I w.S j (s, t, A)| |s| -N |λ| |t| -1 2 dλ |λ| -N 1 |λ| -N 2 |λ| d--N 3 |s| -N |t| -d 2 + N 2
provided that N > d. This proves (3.18) and hence (3.19).

Theorem 3.5. The following estimate holds, for all 0 < |t| < 1 and |x| > |t|:

|s 1 t (x)| |t| -d 2 (1 + |x|) 3 2 d+4 e -ρ,x + . Proof. Since s 1 t is bi-K-invariant, we have s 1 t (x) = C 1 2 |t| -1 2 J(x + ) -1 2 [d/2] k=0 4 -k U k (x + ) I k (t,|x|) +∞ 0 d(s 2 ) χ 1 ( s |x| ) e i 4t s 2 R k-d-1 2 + (s 2 -|x| 2 ) + C 1 2 |t| -1 2 +∞ 0 ds χ 1 ( s |x| ) e i 4t s 2 E(s, x + ) E(t,
k-d-1 2 + (s-1) = ∂ ∂s d-1 2 -k R 0 + (s-1), we obtain |I k (t, |x|)| = |x| 2k-d+1 +∞ 0 ds χ 1 ( √ s) e i|x| 2 4t s R k-d-1 2 + (s -1) = |x| 2k-d+1 +∞ 0 ds R 0 + (s-1) -∂ ∂s d-1 2 -k {χ 1 ( √ s) e i|x| 2 4t s .
As R 0 + (s-1) is the Dirac measure at s = 1, we conclude that

I k (t, |x|) = |x| 2k-d+1 -i 4 |x| 2 t d-1 2 -k = O(|t| k-d-1 2 ).
When d is even, we obtain similarly

|I k (t, |x|)| = π -1 2 |x| 2k-d+1 +∞ 1 ds √ s-1 -∂ ∂s d 2 -k χ 1 ( √ s) e i|x| 2 4t s ,
which is a linear combination of expressions

t j+k-d 2 |x| 1-2j +∞ 1 ds √ s-1 θ j (s) e i|x| 2 4t s J j (t,|x|) where 0 ≤ j ≤ d 2 -k and θ j ∈ C ∞ c (R) with supp θ j ⊂ (-4C 2 2 , 4C 2
1+ |t| |x| 2 1 ds ds √ s-1 = 2 √ s-1 s=1+ |t| |x| 2 s=1 = 2 √ |t| |x| .
After performing an integration by parts based on

|s 1 t (x)| |t| -d 2 J(x + ) -1 2 + |t| -1 2 |x| (1 + |x|) 3( d 2 +1) e -ρ,x + |t| -d 2 (1 + |x|) 3 2 d+4 e -ρ,x + when 0 < |t| < 1 and x ∈ X satisfies |x| > |t| .
In summary, we have divided our kernel analysis into three parts and deduced Theorem 1.1 from Theorem 3.1, Theorem 3.3 and Theorem 3.5. Notice that the method used to prove small time kernel estimates can be also used for large time.

Dispersive estimates and Strichartz inequalities on symmetric spaces

Once pointwise kernel estimates are available on symmetric spaces, one can deduce dispersive properties for the corresponding propagator, by using an interpolation argument based on the Kunze-Stein phenomenon. The following bi-K-invariant version is a straightforward generalization of [APV11, Theorem 4.2].

Lemma 4.1. Let κ be a reasonable bi-K-invariant function on G. Then • * κ L q (X)→L q (X) ≤ G dx ϕ 0 (x) |κ(x)| q 2 2 q for every q ∈ [2, +∞). In the limit case, we have

• * κ L 1 (X)→L ∞ (X) = sup x∈G |κ(x)|.
Proof of Theorem 1.3. At the endpoint q = q = 2, t → e it∆ is a one-parameter group of unitary operators on L 2 (X): e it∆ L 2 (X)→L 2 (X) = 1. For 2 < q < ∞, we deduce from estimates (1.7) and (2.2) that s t L q (X) ω(t) a dx + (1 + |x + |) qN e -(q-2) ρ,x + 1 q ω(t)

where

ω(t) =    |t| -d 2 if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1.
Moreover, since s t is the bi-K-invariant convolution kernel of the propagator e it∆ , we deduce, by combining the above lemma with (1.7), (2.2), and (2.4) that e it∆ q 2

L q (X)→L q (X) ≤ a dx + δ(x + ) ϕ 0 (exp x + ) |s t (exp x + )| q 2 ω(t)

q 2 a dx + (1 + |x + |) qN +D- 2 e -( q 2 -1) ρ,x + = O(1)
.

We conclude by interpolation between

         e it∆ L 1 (X)→L q (X) ≤ s t L q (X) |t| -d 2 ,
e it∆ L q (X)→L ∞ (X) ≤ s t L q (X) |t| -d 2 , e it∆ L 2 (X)→L 2 (X) = 1,

when |t| ≤ 1 and         
e it∆ L 1 (X)→L q (X) ≤ s t L q (X) |t| -D 2 , e it∆ L q (X)→L ∞ (X) ≤ s t L q (X) |t| -D 2 , e it∆ L q (X)→L q (X)

|t| -D 2 , when |t| ≥ 1.

Finally, we establish our Strichartz inequality (1.11) by using the T T * argument. Notice that the solution to the Schrödinger equation (1.10) is given by the Duhamel formula: u(t, x) = e it∆ f (x) -i t 0 ds e i(t-s)∆ F (s, x).

Consider the operator T and its formal adjoint T * : T f (t, x) = e it∆ f (x) and T * F (x) = R ds e -is∆ F (s, x).

By duality, T is bounded from L 2 x to L p t L q x if and only if T * is bounded from L p t L q x to L 2 x . Equivalently, the operator T T * F (t, x) = R ds e i(t-s)∆ F (s, x) (4.1) is bounded from L p t L q x to L p t L q x . We prove the latter for all (p, q) in the admissible triangle (1.9). The Strichartz inequality (1.11) follows from the Christ-Kiselev lemma, see [START_REF] Christ | Maximal functions associated to filtrations[END_REF] or [Tao06, Section 2.3].

Proof. The endpoint ( 1 p , 1 q ) = (0, 1 2 ) is settled by L 2 conservation and ( 1 p , 1 q ) = ( 1 2 , 1 2 -1 d ) can be handled by using the method carried out in [START_REF] Keel | Endpoint Strichartz estimates[END_REF]. Let us focus on the non-endpoint cases where ( 1 2 -1 q ) n 2 ≤ 1 p ≤ 1 2 and 1 2 -1 n < 1 q < 1 2 . According to the dispersive estimates (1.8), we have T T * F L p (I;L q (X)) ≤ R ds e i(t-s)∆ F (s, . ) L q (X) L p .

On the one hand, the convolution kernel |t-s| -D 2 1 {|t-s|≥1} on R defines a bounded operator from L p to L p for all 2 ≤ p ≤ ∞. On the other hand, the convolution kernel |t -s|

-( 1 2 -1 q )d 1 {|t-s|≤1}
is bounded from L p to L p for all 2 ≤ p < ∞ such that 0 ≤ 1 p -1 p ≤ 1 -( 1 2 -1 q )d, or in other words, when p satisfies ( 1 2 -1 q ) n 2 ≤ 1 p ≤ 1 2 .

Remark 4.2. We do not take full advantage of our strong large time decay |t| -D 2 in the dispersive estimates (1.8). Indeed, the convolution kernel |t -s| -1-ε 1 {|t-s|≥1} defines a bounded operator from L p to L p for any ε > 0 and 2 ≤ p ≤ ∞.

d 2 .

 2 We shall use different tools, depending on the size of |x| √ |t| . If |x| √ |t| is small, we decompose the Weyl chamber into several subcones according to the barycentric decompositions described in Theorem 2.1, and perform in each subcone several integrations by parts along a well chosen direction. If |x| √ |t| is large, we express in addition the Schrödinger propagator in terms of the wave propagator and use the Hadamard parametrix.

  ( s |x| ) I 0 (s, t, x) .

  (s-1) -1 2 + (s-1) -3 2 √ |t| |x|under the assumption |x| > |t| . Thus (3.21) holds as well when d is even. In conclusion,

qF

  )d F (s, . ) L q (X) L p t (I) + |t-s|≥1 ds |t -s| -D 2 (s, . ) L q (X) L p t (I)

  Let us first prove (3.21) when d is odd. By a change of variables and by using the fact that R

	follows from (3.15). On the other hand, we claim that
	|I k (t, |x|)|	|t| k-d-1 2	(3.21)
	if |x| > |t| > 0.
			(3.20)

|x|)

according to (3.14). On the one hand, the remainder estimate

| E(t, |x|)| |x| (1 + |x|) 3( d 2 +1) e -ρ,x +

  2 ). Notice that the elementary estimate J j (t, |x|) = O(1), together with the assumption |x| > |t| implies that

			|I k (t, |x|)| |x| |t| -d 2 +k ,
	which might be enough for our purpose as long as k > 0. The case k = 0 requires actually a
	more careful analysis. Let us show that
			J j (t, |x|)	√ |x| |t|
	by splitting up
	1	+∞	ds =	1	1+	|t| |x| 2	ds +	+∞ |x| 2 1+ |t|	ds
	in the definition of J

j (t, |x|). The contribution of the first integral is easily estimated by

Acknowledgments. The last author acknowledges financial support from the Methusalem Programme Analysis and Partial Differential Equations (Grant number 01M01021) during his postdoc stay at Ghent University.