Rémy Juvénal
email: remy.juvenal@institutoptique.fr

Caroline Kulcsár
email: caroline.kulcsar@institutoptique.fr

Henri-François Raynaud

Jean-Marc Conan

Gaetano Sivo

Performance assessment for the linear control of adaptive optics systems: noise propagation and temporal errors

Keywords: Adaptive Optics, Error budget, Linear Control, Performance analyses

We propose a detailed study of the temporal and noise propagation errors entering the error budget of Single Conjugate Adaptive Optics (SCAO) systems. A transfer-function oriented method is developed for the computation of these errors, in a general linear control context. We apply this formalism to performance comparison between Linear Quadratic Gaussian (LQG) and Integral action controllers. Simulation results are presented, corresponding to a typical SCAO system, in order to validate some approximations on the perturbation power spectral density. We then apply the formalism to evaluate the tip-tilt error budget on GeMS.

Introduction

Adaptive optics (AO) systems have taken a key role in diffraction limited astronomical instrumentation for ground based telescopes. However, performance of adaptive optics systems will have to be improved significantly in order to reach the scientific goals of future instruments under development for the VLTs and ELTs. In this context, error budgets are critical tools. Several papers have dealt with the problem of establishing error budgets in the design phase of new AO systems such as the TMT AO system, [START_REF] Gilles | Wavefront Error Budget Development for the Thirty Meter Telescope Laser Guide star Adaptive Optics System[END_REF] or the SAXO module of SPHERE. [START_REF] Fusco | High-order adaptive optics requirements for direct detection of extrasolar planets: Application to the SPHERE instrument[END_REF] Quirós-Pacheco et al. [START_REF] Pacheco | General aliasing and its implications in modal gain optimization for multi-conjugated adaptive optics[END_REF] studied the effect of spatial aliasing for integral control with optimized gain, but their method was limited to this particular type of controller. Moreover, error budgets are rarely computed from on-sky data to analyse the performance, but some examples can be found such as CANARY. [START_REF] Vidal | Analysis on on-sky MOAO performance of CANARY using natural guide stars[END_REF] We propose here, in the special case of a single-conjugate AO (SCAO) configuration, a method which enables to compute the control-dependent terms in the error budget. In the present article, we assume that all the phases can be represented by Zernike modes, which is not restrictive. This method is applicable to any linear controller, and thus can be used to compare the performance of different control laws.

The error budget for SCAO systems is classically expressed as a decomposition of the residual phase variance in a sum of independent terms:

We choose to separate these errors into two groups:

(i) The control-dependent errors, which include the temporal error σ 2 temp , resulting from the delays in the control loop, the noise propagation error σ 2 noise,filt , which corresponds to the impact of the wavefront sensor (WFS) measurement noise, and the aliasing error σ 2 alias resulting from WFS (spatial) aliasing.

(ii) The control-free errors, which include the fitting error σ 2 fitting , related to the finite number of actuators of the deformable mirror (DM), the non common path aberration (NCPA) error σ 2 NCPA , due to the differential optical aberrations between the scientific and WFS paths.

In Section 1, we introduce a general transfer function based formalism to describe the control dependent errors, with a focus on the temporal and noise propagation errors. A key ingredient in this calculation is the disturbance (turbulence + vibrations) power spectral density (PSD), which needs to be estimated. In Section 2, we show, with a simplified SCAO simulation, that classical approximations on the structure of the disturbance PSD give acceptable results for SCAO systems. We then illustrate the consistency of this formalism with a performance analysis of tip-tilt on-sky data recorded on the Gemini Multi-Conjugate AO instrument GeMS, evaluating the restrained error budget with the tool defined in previous sections. Finally, we discuss the results presented in this paper, and perspectives for future works.

THEORETICAL PERFORMANCE ASSESSMENT

A precise control oriented description of adaptive optics systems has been given by Kulcsár et al. [START_REF] Kulcsár | Minimum variance prediction and control for adaptive optics[END_REF] For our purpose, we just need to consider the classical adaptive optics block diagram represented in Fig. 1.

Figure 1: Generic block diagram of an adaptive optics system

Without any limitation to what follows, we can assume a two-frame delay in the system: one frame for WFS integration and another for read-out, centroiding, control computation and data transfer. The system in Fig. 1 has two independent inputs: the disturbance phase φ and the measurement noise w ; and one output: the residual phase φ res .

Adaptive optics transfer functions

From an input-output point of view, the diagram in Fig. 1 is completely equivalent to the concise diagram presented in Fig. 2. The two inputs and the output remain the same, but the AO system is completely described by two transfer functions:

-the rejection transfer function T rej , from φ to φ res , -the noise propagation transfer function T noise , from w to φ res .

If we consider a linear controller with transfer function G, we can derive from Fig. 1 the expressions of the AO loop transfer functions:

T rej (z) = I -z -2 N G(z)D -1 T noise (z) = -z -1 T rej (z)N G(z) ( 2
)
where N is the deformable mirror influence matrix, D is the WFS matrix and z is the z-transform variable. Using a transfer function, G, is a convenient way to characterize any linear controller, as shown in section 2.1 for LQG and integral controllers.

σ 2 φ res = lim T →∞ 1 T T 0 φ res (t) 2 2 dt (3) = trace       1 π π 0
T rej (e iω ) S φ (e iω ) T rej * (e iω )dω

σ 2 temp +σ 2 alias +σ 2 fitting + 1 π π 0 T noise (e iω ) Σ w T noise * (e iω )dω σ 2 noise,filt       (4)
where S φ is the PSD of φ, and Σ w is the measurement noise covariance matrix. The second term in the right hand side of equation ( 4) is an exact expression of the noise propagation error variance. As the measurement noise is generally assumed to be uncorrelated between the WFS subapertures, Σ w is a diagonal matrix, which is estimated on all operational systems. The first integral in (4) corresponds to the temporal, aliasing and fitting errors.

Perturbation power spectral density approximation

The first step to evaluate σ 2 φ res using equation ( 4) is the determination of the disturbance PSD S φ . We use Zernike modes in the following, but any other basis could be used as well. As we do not have access to the perturbation phase on a real system but to WFS measurements, we have to estimate each diagonal component of S φ by reconstructing the associated Zernike modes from the WFS measurements. The least-square reconstructor used for this operation is denoted by R φ and defined as:

φ = R φ y, R φ = D T D -1 D T , (5)
φ being the reconstructed phase vector, and y the WFS measurement vector. However, replacing directly the empirical S φ by the PSD S φ of φ leads to the propagation of the measurement noise in the rejection transfer function. Thus, in order to compute the true value of the residual phase variance, we have to remove this effect from the residual phase computation and to replace it by the true noise propagation in the AO loop:

σ 2 φ res = trace 1 π π 0 T rej (e iω ) S R φ y (e iω ) T rej * (e iω )dω -trace 1 π π 0 T rej (e iω ) R φ Σ w R T φ T rej * (e iω )dω +trace 1 π π 0 T noise (e iω ) Σ w T noise * (e iω )dω . ( 6
)
As S φ , as well as S φ , is a non diagonal matrix-valued function of the frequency ω, its estimation can be computationally demanding. So we also approximate S φ by a diagonal matrix, neglecting all the correlations between Zernike modes.

INTEGRAL/LQG CONTROLLERS COMPARISON IN SIMULATION

In this section, we particularize the error budget formalism developed in Section 1 by comparing integral and Linear Quadratic Gaussian (LQG) controllers with two simulations. First, we simulate in Section 2.2 a simplified SCAO system (without aliasing) in order to demonstrate the consistency of the approximations proposed in Section 1.3. In Section 2.3, we use this method to analyse the performance of the GeMS tip-tilt loop.

Controllers transfer functions 2.1.1 Integral controller

The integral controller is defined with its recurrence equation:

u k = u k-1 + gM com y k , ( 7
)
where u is the commands vector, g is the integrator gain and M com is the command matrix. Taking the ztransform of equation (7) gives directly the integral controller transfer function:

G int (z) = gM com 1 -z -1 . (8)

LQG controller

The LQG control relies on an underlying state space model:

x k+1 = Ax k + Γv k , (9)
φ k = C φ x k , (10)
y k = Cx k -DN u k-2 + w k . (11)
The equations ( 9) and ( 10) correspond to the perturbation model as the output of a shaping filter, with a white noise v as input. Equation ( 11) models the WFS measurement process, where C describes the mapping from the state vector x to the WFS measurement vector y and w is the measurement noise. v and w are assumed to be white, Gaussian, and mutually independent, with respectively Σ v and Σ w as covariance matrix. LQG control uses a Kalman filter in order to predict the future value of the perturbation through the estimation of the state vector. The structure of an LQG controller, in AO, is given by:

x k+1|k = A x k|k-1 + L y x -y k|k-1 (12)
y k|k-1 = C x k|k-1 -DN u k-2 (13)
u k = P u C φ x k+1|k (14)
where ( 12) and ( 13) define a Kalman filter with gain L, where x is the estimated state vector and y is the estimated WFS measurement vector. Equation ( 14) is the projection of the predicted perturbation on the deformable mirror, P u being defined as:

P u = N + = N T N -1 N T . ( 15
)
The LQG controller transfer function is then obtained by taking the z-transform from equations ( 12) to ( 14):

G(z) = P u C φ I -(A -LC)z -1 -LDN P u C φ z -2 L. ( 16
)

Multi-mode SCAO simulation

We perform a simplified SCAO simulation in order to validate the approximations suggested in Section 1.3.

The disturbance phase has been generated using Auto-Regressive models of order 2 (AR2) 10 , 14 using standard observation conditions: observation seeing = 0.7 arcsec, wind velocity norm V wind = 15 m.s -1 , damping coefficient ξ = 0.9. We simulate an 8 meter VLT telescope. We choose a 10x10 sub-apertures Shack-Hartmann WFS, with a 0.2 rad 2 measurement noise variance (in phase difference at the edge of the sub-apertures), which is representative of high flux observations. The perturbation phase is generated with the 27 first Zernikes modes without piston, according to a Kolmogorov statistics. We limit the number of Zernike modes on purpose in order to avoid any aliasing effect in the simulation. We focus the analysis on the control-dependent error budget. For this reason, we consider a mirror compensating directly all the 27 Zernike modes in order to avoid any fitting error. This way, the total error budget for the simulation is simply given by:

σ 2 φ res , simu = σ 2 temp + σ 2 noise,filt . (17)
We compare here an integral controller, with a gain of 0.5, and an LQG controller, based on the same AR2 model used to generate the turbulence. In that case, the LQG controller is the optimal (minimum variance) controller. The theoretical performance can be directly computed from equation ( 4), because all physical quantities are available, particularly the disturbance PSD (including non diagonal elements). Using the integral controller leads to a 0.50 rad 2 theoretical residual phase variance. Using the diagonal approximation proposed in Section 1.3 and estimating the diagonal PSD with WFS measurements, we obtain a 0.51 rad 2 approximated residual phase variance, which corresponds to an error of 2% on the theoretical performance. The LQG controller leads to a 0.25 rad 2 theoretical performance, and to an approximated residual phase variance of 0.27 rad 2 , which corresponds to a 8% error. We can push the analysis further by considering the modal decomposition of the theoretical and approximated residual phase variances, with both the approximation and the estimation, represented for the two controllers in Fig. 3. Approximated (red broken line) and theoretical (cyan line) residual phase variance decompositions over the Zernike modes are practically identical. LQG curves are close to each other, but less than for the integral controller ones. Indeed, in the case of the integral controller, we use a diagonal gainconstant here, but an optimized modal gain would not have changed the conclusion -to compute the correction phase. So the diagonal approximation is reflected by the structure of the integral controller, which does not account for the interactions between the modes of the perturbation phase. In the case of the LQG controller, the computation of the Kalman gain accounts for these interactions, so neglecting them leads to a visible error, which is here anyway low as illustrated in Fig. 3 The LQG controller appears to be twice better in terms of residual variance than the integral controller, which performance would be slightly imroved by increasing the gain g. The pie charts in Fig. 4 represent the error budget decomposition (17) of both controllers for this simulation. We find results close to those presented in Conan et al., 1 using similar simulation conditions. This includes a more balanced compromise between temporal and noise propagation errors for LQG controller. Indeed, the error budget of the integral controller is dominated by the temporal error (90 %). The LQG controller uses a Kalman filter which predicts the perturbation phase one step ahead, with a higher relative impact of the measurement noise than the integral controller.

Replay simulation with GeMS tip-tilt on-sky data

As the approximations suggested in Section 1.3 have given satisfactory results in simulations, we apply this tool to the performance analysis of on-sky data. We use tip/tilt data from the multi-conjugate AO system GeMS, [START_REF] Neichel | Gemini multi-conjugate adaptative optics system review II: Commissioning, operation and overall performance[END_REF][START_REF] Rigaut | Gemini multiconjugate adaptive optics system review -I. design, trade-offs and integration[END_REF] based at Gemini South Observatory, in Chile. The tip/tilt loop of GeMS is actually working exactly like a standard SCAO system. Indeed, the tip/tilt measurements from up to 3 off-axis natural guide stars WFSs are averaged, and the command is computed from this virtual on axis WFS measurements by an integral controller.

GeMS performance, as with most AO systems operating on VLTs, is affected by vibrations. The control law, which is a simple integral controller, cannot efficiently cope with these vibrations. We have collaborated with the GeMS AO team in order to implement an LQG controller with vibration filtering for the tip-tilt loop, the same type of control which was successfully tested on CANARY [START_REF] Sivo | First on-sky SCAO validation of full LQG control with vibration mitigation on the CANARY pathfinder[END_REF] and which is in operation on SPHERE. [START_REF] Petit | Sphere eXtreme AO control scheme: final performance assessment and on sky validation of the first auto-tuned LQG based operational system[END_REF] As noted in Section 2.1, LQG control relies on an underlying model of the perturbations (turbulence, vibrations, ...). A fine study of these vibrations, and of what would be brought by using an LQG controller using Meimon et al. [START_REF] Meimon | Tip-tilt disturbance model identification for kalman-based control scheme: application to XAO and ELT systems[END_REF] model identification method is presented in Leboulleux et al. [START_REF] Leboulleux | Vibration analysis on gems tip-tilt data and performance evaluation using an LQG controller[END_REF] For example, Fig. 5 represents the PSD of a common buffer from GeMS (black curve), acquired in April 2015. We can see a particularly intense vibration around 37Hz. We have studied several model identification methods 4 in order to find the most suitable technique for the GeMS tip/tilt loop. The identification method used to compute the results presented below is called cPEM 5 (for cascaded Prediction Error Minimization) and has provided the best results during the study. The green curve on Fig. 5 shows the PSD of the model, identified from half of the buffer. Before discussing the results, note that when using pseudo open-loop data to simulate an AO loop (replay mode), the residual phase variance evaluation is different from that of section 2.3. Indeed, we have access here neither to the real perturbation phase, nor to the measurement noise values to run the simulation, but only to the WFS measurements. The residual phase obtained is therefore in this case:

Φ res (z) = T rej (z)R φ Y (z) = T rej (z)R φ (DΦ(z) + W (z)) = T rej (z)R φ DΦ(z) + T rej (z)R φ W (z), (18)
where Φ res , Y , Φ and W are the z-transforms of respectively the residual phase φ res , the WFS measurements y, the perturbation phase φ and the measurement noise w. Therefore, in order to compute the correct residual phase variance (see eq. ( 4)), we have to remove from the residual phase variance evaluated in replay-mode the measurement noise filtered by the rejection transfer function and to add the impact of the noise propagation:

σ 2 φ res = σ 2 φ res -σ 2 noise T rej R φ 2 + σ 2 noise, filt . (19)
From (19), we evaluate the error budget of the integral controller, illustrated in Fig. 6(a), and of the LQG controller obtained from the cPEM identified model, illustrated in Fig. 6(b). The integral controller overall performance (here in on-sky angle, i.e. in residual phase standard deviation) is 19.4 mas and 15.4 mas for the LQG controller. The LQG performance is for this buffer around 25 % lower than for the integral controller (see Juvénal et al. [START_REF] Juvénal | Tiptilt modeling and control for GeMS: a performance comparison of identification techniques[END_REF] for more details). Looking at the error budget decomposition, we note first that the integral controller is completely dominated by the temporal error. The LQG controller achieves, again, a better RMS compromise between the two errors, compromise that is essential here in presence of vibrations. We can note that the temporal error is, however, largely dominant in the LQG error budget (95%). This is due to 2 phenomena:

• first, the measurement noise level was very low for this observation, i.e. the natural guide stars were bright (the operating point is a classical high-flux observation),

• secondly, the identified model, due to real-time control constraints, is limited to 15 cells of order 2. The perturbation model is consequently limited to the representation of 14 vibration peaks. Therefore, some small unfiltered vibrations can remain uncorrected, and impact the error budget of the LQG controller.

CONCLUSION

We have presented in this paper a general formalism for the control-dependent error budget evaluation, applicable for any SCAO system and any linear controller. We aim at designing a tool that allows accurate comparisons between controllers. The critical element of this error budget evaluation process is the estimation of the disturbance PSD, due to the lack of knowledge about the disturbance in a real system. We show that we can use reasonable approximations to evaluate the performance of a SCAO system. The first simulation verifies the validity of both the approximation and the estimation for the residual phase estimation. We can also use this formalism to study how any linear controller would perform on a specific system, by estimating its overall performance and evaluating all the terms in its error budget. This allows to determine on which aspects a controller should be improved.

We still need to model finely the impact of WFS aliasing for SCAO systems, which is an on-going work. A clue to achieve this is to study the modal structure of both the rejection and noise propagation transfer functions. The next step is to evaluate the control-dependent error budget terms for Wide-field Adaptive Optics systems, which will require the description of other errors like the tomographic error, together with special attention to the disturbance PSD estimation.

Figure 2 :

 2 Figure 2: Equivalent SCAO system block diagram 1.2 Theoretical residual phase variance evaluation Using Parseval's theorem and Fig. 2, we can express the residual phase variance in function of the AO transfer functions:

Figure 4 :

 4 Figure 4: Error budget repartition in simulation restricted to temporal and noise propagation errors comparison.

Figure 5 :

 5 Figure 5: PSD of tip from buffer ngs15108223248, acquired in April 2015 with GeMS. Black curve: sample empirical PSD. Green curve: PSD of the model identified with cPEM.

Figure 6 :

 6 Figure 6: Error budget repartition in replay mode with GeMS on-sky tip-tilt data. Buffer ngs15108223248 (tip), April 2015, evaluation made on the last 4000 samples, 350 Hz, 3 natural guide stars.

  .

						Integrator overall	LQG overall
							performance	performance
	10 -1		<	2 ?	res , int, th	= 0.50 rad	2	<	2 ?	res , lqg, th	= 0.25 rad	2
			<	2 ?	res , int, est	= 0.51 rad	2	<	2 ?	res , lqg, est	= 0.27 rad	2
	in rad 2
	res
	< 2 ? 10 -2
			Theoretical < Estimated < 2 ?	2 ? res , int res , int
	10 -3	0	5 Theoretical < Estimated < 2 ?	2 ? res , lqg res , lqg	10			15	20	25
												Zernike's mode number
	Figure 3: Modal decomposition of the residual phase variances obtained in simulation for LQG and integral
	controllers. Cyan curve: theoretical integral controller performance; red (dashed) curve: approximated integral
	controller performance; blue curve: theoretical LQG performance; yellow (dashed) curve: approximated LQG
	performance.

Acknowledgements

Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).


Long Meng
email: long.meng@enpc.fr

ON THE MIXED REGULARITY OF N -BODY COULOMBIC WAVEFUNCTIONS

On the mixed regularity of N -body Coulombic wavefunctions

Introduction

In most applications of molecular simulation, a molecule is described by an assembly of M static nuclei equipped with N electrons, with M , N in N `. We assume that the nuclei are fixed, according to the Born-Oppenheimer approximation, while the electrons are modeled quantum mechanically through a wavefunction and the N -body Hamiltonian operator:

H " ´1 2 N ÿ i"1 △ i ´Vne `Vee (1.1)
with

V ne :" N ÿ i"1 M ÿ ν"1 Z ν |x i ´aν | , and
V ee :" 1 2 N ÿ i,j"1 i‰j 1 |x i ´xj | ,
where a 1 ¨¨¨, a M P R 3 are the positions of nuclei with respective charges Z 1 , ¨¨¨, Z M P N `(in atomic units), and x 1 , ¨¨¨, x N P R 3 are the coordinates of given N electrons. We denote Z :" ř M ν"1 Z ν the total nuclear charge. The right-hand side terms in (1.1) model the kinetic energy, the Coulomb attraction between nuclei and electrons V ne and the Coulomb repulsion between electrons V ee , respectively.

Mathematically, the electronic ground -or excited -state problem can be expressed by the Euler-Lagrange equation of the eigenvalue problem of the operator (1.1):

Hu " λu, (1.2)
where u P H 1 ppR 3 q N q and }u} L 2 ppR 3 q N q " 1.

Remark 1.1. It is shown in [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] that any eigenvalue of (1.2) is negative.

The eigenvalue problem (1.2) is well-explored mathematically (see for example [START_REF] Hunziker | The quantum N -body problem[END_REF], as well as the regularity properties of eigenfunctions of problem (1.2) [3-5, 8-10, 12]).

In quantum mechanics, in addition to the spatial coordinates, a particle may have internal degrees of freedom, the most important one being the spin. Electrons, for example, have two kinds 2020 Mathematics Subject Classification. 35J10, 35B65, 41A25, 41A63 .

of spins r σ with value 1, 2. If a particle has q kinds of spins, we shall say that the particle has q spin states and we label them by the integer r σ P t1, ¨¨¨, qu.

From the mathematical point of view, it is interesting to consider an arbitrary q spin states in our system. For this reason, in this article, we will study the wavefunctions of N identical particles with q spin states instead of the electronic wavefunctions.

A wavefunction Ψ of identical N particles with q spin states can be written as

Ψ : pR 3 q N ˆt1, ¨¨¨, qu N Ñ C, px, σq Þ Ñ Ψpx, σq, (1.3)
where x :" px 1 , ¨¨¨, x N q and σ :" pσ 1 , ¨¨¨, σ N q with x i P R 3 and σ i P t1, ¨¨¨, qu.

There are two kinds of particles: fermions and bosons. Fermions, among them electrons, satisfy the Pauli exclusion principle: the sign of the wavefunction Ψ changes sign under an exchange of the space coordinates x i , x j , and the spins σ i , σ j of two identical fermions i, j. More precisely, Pauli exclusion principle writes : ΨpP pxq i,j x, P pσq i,j σq " ´Ψpx, σq (1.4) where P pxq i,j and P pσq i,j the permutation operators which exchange the space coordinates x i , x j and the spins σ i , σ j respectively, i.e. P pxq i,j p¨¨¨, x i , ¨¨¨, x j , ¨¨¨q :" p¨¨¨, x j , ¨¨¨, x i , ¨¨¨q,

and P pσq i,j p¨¨¨, σ i , ¨¨¨, σ j , ¨¨¨q :" p¨¨¨, σ j , ¨¨¨, σ i , ¨¨¨q.

(

On the other hand, bosons satisfy the Bose-Einstein statistics in which the particles occupy symmetric quantum states. Thus the bosonic wavefunctions Ψ satisfies (1.4) when the sign ´is replaced by `. Now we are going to fix the spin σ, and only consider the antisymmetry of the fermionic wavefunction with respect to x. As the eigenvalue problem (1.2) does not act upon the spin variables, for every fixed spin σ, the wavefunction Ψpx, σq in (1.3) can be represented by the wavefunction upxq which is defined by u : pR 3 q N Ñ C, x Þ Ñ Ψpx, σq.

(1.7)

Furthermore, for every fixed spin σ, the particles can be categorized into q subsets according to their spin states: I l :" ti P t1, ¨¨¨, N u; σ i " lu, l " 1, ¨¨¨, q, and I σ :" tI 1 , ¨¨¨, I q u.

(1.8)

In particular, if σ i ‰ l for any i " 1, ¨¨¨, N , we set I l " H and |I l | " 0. If i, j P I with I P I σ and |I| ą 1, then σ i " σ j . Thus P pσq i,j σ " σ. Therefore, the permutation operator P pσq i,j keeps the spin σ invariant if the i-th and j-th electrons have the same spin. Hence for every fixed σ and for any i, j P I with I P I σ and |I| ą 1, (1.4) implies the fermionic wavefunction u is antisymmetric with respect to x i , x j , i.e., upP pxq i,j xq " ´upxq.

(1.9)

In particular, if x i " x j then upxq " 0: thanks to the antisymmetry, the fermionic wavefunctions can counterbalance the singularity of the interaction potential 1 |x i ´xj | . Relying on this observation, a new regularity result about eigenfunctions of problem (1.2) has been proven in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] which can help to break the complexity barriers in computational quantum mechanics. More precisely, it is shown in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] that, for every fixed spin σ, any eigenfunction u of problem (1.2) satisfies ż

pR 3 q N ˜1 `N ÿ i"1 |2πξ i | 2 ¸˜ÿ IPIσ ź kPI p1 `|2πξ k | 2 q ¸|p u ˚pξq| 2 dξ ă `8, (1.10)
where p u ˚pξq :" F x 1 ,¨¨¨,x N pu ˚qpξq " ş pR 3 q N u ˚pxqe ´2πiξ¨x dx is the Fourier transform of u with ξ :" pξ 1 , ¨¨¨, ξ N q and ξ i P R 3 , i " 1, ¨¨¨, N . The proof is based on a Hardy-type inequality for the Coulomb system in the scalar product. Then based on this Hardy-type inequality, a hyperbolic cross space approximation of any eigenfunction of (1.2) has been studied. The convergence of this approximation is proven in [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF]. The hyperbolic cross space approximation is defined by (2.10) below.

Later, by using r12-methods and interpolation of Sobolev spaces, H.C. Kreusler and H. Yserentant [START_REF] Kreusler | The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces[END_REF] proved that any eigenfunction u ˚of problem (1.2) satisfies ż

pR 3 q N ˜1 `N ÿ i"1 |2πξ i | 2 ¸s ˜N ź k"1 p1 `|2πξ k | 2 q ¸t |p u ˚pξq| 2 dξ ă `8, (1.11)
for s " 0 and t " 1 or s " 1 and t ă 3{4. Notice that this regularity is independent of the choice of σ. It is shown in [START_REF] Kreusler | The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces[END_REF] that the bound 3{4 is the best possible: it can neither be reached nor surpassed except for the totally antisymmetric eigenfunctions. However, lacking Hardy-type inequalities associated with the new type of mixed regularity, they could not prove the convergence of the corresponding hyperbolic cross space approximation of eigenfunctions.

In this article, we are going to improve the results of [START_REF] Kreusler | The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces[END_REF] and [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] in two directions: a) we prove the convergence of the hyperbolic cross space approximation of eigenfunctions associated with the regularity (1.11); b) due to the Pauli exclusion principle, taking the antisymmetry of the wavefunctions into account, we prove a better mixed regularity of eigenfunctions and prove the convergence of the corresponding hyperbolic cross space approximation.

We generalize the concept of the antisymmetric functions such that, under this new definition, non-antisymmetric functions can also be regarded as special antisymmetric functions. Definition 1.2 (Generalized antisymmetric function). Let I Ă t1, ¨¨¨, N u. When |I| ą 1, a wavefunction u is antisymmetric with respect to I if and only if, for any i, j P I, upP pxq i,j xq " ´upxq, where P pxq i,j is defined by (1.5). When |I| " 1, every wavefunction u is antisymmetric with respect to I. Remark 1.3. According to (1.8), (1.9) and above definition, wavefunction u defined by (1.7) with q spin states and the fixed spin σ is antisymmetric with respect to I for any I P I σ .

Let u ˚given by (1.7) with q spin states and the fixed spin σ be an eigenfunction of (1.2). The main results of this paper (Theorem 2.3 and Corollary 2.4) then state that

ż pR 3 q N ˜1 `N ÿ i"1 |2πξ i | 2 ¸ÿ IPIσ ˜ź jPI p1 `|2πξ j | 2 q ¸αI ˜ź kPI c p1 `|2πξ k | 2 q ¸βI |p u ˚pξq| 2 dξ ă `8.
(1.12)

Here and below I c " t1, ¨¨¨, N uzI, α I P r0, 5{4q, β I P r0, 3{4q and α I `βI ă 3{2.

As mentioned above, this result improves the results in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF]. Actually, if we take α I " 1 and β I " 0, then (1.12) becomes (1.10). Thus the regularity (1.10) is a special case of (1.12). Furthermore, we can choose α I « 5 4 and β I « 1 4 in (1.12) which are much larger than the ones in (1.10).

If we assume in particular that, for the fixed spin σ, there exists l such that I l " t1, ¨¨¨, N u with I l given by (1.8), then u ˚is totally antisymmetric (i.e., u ˚is antisymmetric w.r.t. t1, ¨¨¨, N u) and (1.12) becomes ż

pR 3 q N ˜1 `N ÿ i"1 |2πξ i | 2 ¸˜N ź j"1 p1 `|2πξ j | 2 q ¸α |p u ˚pξq| 2 dξ ă `8
with any 0 ď α ă 5 4 . Then (1.12) is better than (1.10) and (1.11) for the totally antisymmetric case. Now we choose α I " β I . Then if u ˚is not totally antisymmetric, the condition on α I and β I shows that 0 ď α I " β I ă 3 4 , and (1.12) becomes ż

pR 3 q N ˜1 `N ÿ i"1 |2πξ i | 2 ¸˜N ź j"1 p1 `|2πξ j | 2 q ¸β |p u ˚pξq| 2 dξ ă `8 (1.13)
with any 0 ď β ă 3 4 . This is exactly (1.11) with s " 1 and t ă 3 4 , and this regularity is independent of the choice of the spin σ and of the antisymmetry of the eigenfunctions. Thus we provide an alternative proof for (1.11). As mentioned above and shown in [START_REF] Yserentant | The mixed regularity of electronic wave functions multiplied by explicit correlation factors[END_REF], our regularity is optimal in this case except for the totally antisymmetric eigenfunctions.

The proof of this new mixed regularity is based on a generalization of the Hardy-type inequality for Coulomb system in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF] (i.e., Theorem 2.2) for any α I and β I as in (1.12). As in [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF], from this new Hardy-type inequality, we can obtain the corresponding hyperbolic cross space approximation (i.e., Theorem 2.5). In particular, concerning the case α I " β I ă 3 4 , we prove the convergence of the hyperbolic cross space approximation of eigenfunctions associated with the regularity (1.11).

Set-up and main results

In this section, we introduce first the operators and functional spaces used in this paper, then we present our main results and give the main ideas of the proof.

2.1. Operators and functional spaces. For every set I Ă t1, ¨¨¨, N u, we define the Hilbert spaces L 2 I ppR 3 q N q and H 1 I ppR 3 q N q of the wavefunctions which are antisymmetric with respect to I by L 2 I ppR 3 q N q :" tu P L 2 ppR 3 q N q; u is antisymmetric with respect to Iu,

and H 1 I ppR 3 q N q :" tu P H 1 ppR 3 q N q; u is antisymmetric with respect to Iu,

respectively. It is easy to see that, when |I| " 1, H 1 I ppR 3 q N q " H 1 ppR 3 q N q and L 2 I ppR 3 q N q " L 2 ppR 3 q N q. However when |I| ą 1, we have L 2 I ppR 3 q N q Ĺ L 2 ppR 3 q N q and H 1 I ppR 3 q N q Ĺ H 1 ppR 3 q N q. Now we are going to define the new mixed Sobolev space like (1.11) in consideration of the antisymmetry with respect to I. Before going further, we define some fractional Laplacian-type operators associated with I. Define the operator L I,α,β by

L I,α,β :" ˜ź jPI p1 `|∇ j | 2 q α{2 ¸˜ź iPI c p1 `|∇ i | 2 q β{2 ¸, (2.3)
where I c " t1, ¨¨¨, N uzI and ∇ i is the gradient with respect to the coordinate x i P R 3 . This operator is defined with the help of the Fourier transform (see Section 3 for details). In particular, when I " t1, ¨¨¨, N u, then

L I,α,β " N ź j"1 p1 `|∇ j | 2 q α{2
is indeed independent of the choice of β.

In addition to the operator L I,α,β , the following operators will be useful,

L piq I,α,β :" ¨ź jPIztiu p1 `|∇ j | 2 q α{2 '¨ź iPI c ztiu p1 `|∇ i | 2 q β{2 ', and
L pi,jq I,α,β :" ¨ź jPIzti,ju p1 `|∇ j | 2 q α{2 '¨ź iPI c zti,ju p1 `|∇ i | 2 q β{2 '.
Thus,

L I,α,β " p1 `|∇ i | 2 q γ i {2 L piq I,α,β , L I,α,β " p1 `|∇ i | 2 q γ i {2 p1 `|∇ j | 2 q γ j {2 L pi,jq I,α,β , (2.4)
where γ k " α if k P I, and γ k " β if k P I c . We next introduce the corresponding functional spaces X I,α,β defined by

X I,α,β :" tu P H 1 I ; }L I,α,β u} H 1 ppR 3 q N q ă `8u, (2.5)
endowed with the norm }u} I,α,β :" }L I,α,β u} H 1 ppR 3 q N q .

(2.6)

We also define the following norm and semi-norm respectively, }u} 0,I,α,β :" }L I,α,β u} L 2 ppR 3 q N q , }u} 1,I,α,β :" }∇L I,α,β u} L 2 ppR 3 q N q .

(2.7)

Here ∇ :" p∇ 1 , ¨¨¨, ∇ N q is the gradient with respect to x P pR 3 q N . Obviously, }u} 2 I,α,β " }u} 2 0,I,α,β }u} 2 1,I,α,β . 2.2. Main results. Before going further, we need some assumptions on α and β. Assumption 2.1. We assume that α P r0, 5{4q, β P r0, 3{4q and α `β ă 3{2.

The key tool to prove the regularity of eigenfunctions is the following. Theorem 2.2 (Hardy-type inequality for Coulomb system in the scalar product). For every I Ă t1, ¨¨¨, N u, and under Assumption 2.1 on α, β, there is a constant C mix,α,β independent of N , Z such that for any u, v P X I,α,β ,

| L I,α,β pV ne `Vee qu, L I,α,β v | ď C mix,α,β ? N maxtZ, N u}u} 0,I,α,β }v} 1,I,α,β . (2.8)
It is shown in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF] that (2.8) holds for α " 1 and β " 0. Based on this inequality, the regularity of eigenfunction u ˚and the corresponding hyperbolic cross space approximation are also proven therein.

Our main result on the new mixed regularity of the wavefunction is the following.

Theorem 2.3 (Mixed regularity of eigenfunctions). Let I Ă t1, ¨¨¨, N u, and let u ˚P H 1 I ppR 3 q N q be a solution to the eigenvalue problem (1.2). Then, under Assumption 2.1 on α and β, u ˚P X I,α,β .

This proof is postponed until Section 5. According to Remark 1.3, we know that u given by (1.7) is situated in Ş IPIσ H 1 I ppR 3 q N q. Then we have Corollary 2.4. Let u given by (1.7) be an eigenfunction of the eigenvalue problem (1.2) with q spin states and a fixed spin σ. Then u ˚P Ş IPIσ X I,α I ,β I where α I and β I satisfy Assumption 2.1. We first recall the definition of the hyperbolic cross space approximation. Let Ω be a scaling parameter which will be given in Theorem 2.5. Let H I,α,β pR, Ωq be a region defined by

H I,α,β pR, Ωq :" # pω 1 , ¨¨¨, ω N q P pR 3 q N ; ź iPI ˆ1 `ˇˇωi Ω ˇˇ2 ˙α ź jPI c ˆ1 `ˇˇωj Ω ˇˇ2 ˙β ď R 2 + . (2.9)
Note that this region can be considered as cartesian product of hyperboloid-like regions, from which the notion hyperbolic cross space approximation originates. Then we define the projector pP R,Ω I,α,β uqpxq :"

ż pR 3 q N χ R,Ω I,α,β pξqp upξq exp p2πiξ ¨xqdξ (2.10)
where χ R,Ω I,α,β is the characteristic function of the domain H I,α,β pR, Ωq. The approximation (2.10) of u is called the hyperbolic cross space approximation.

In [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF], based on the mixed regularity (1.10), the convergence of hyperbolic cross space approximation of eigenfunctions is proven. Now we are going to prove the convergence of the hyperbolic cross space approximation of eigenfunctions associated with the regularity proven in Theorem 2.3.

Based on Theorem 2.2, we get the following.

Theorem 2.5 (Hyperbolic cross space approximation). Let I Ă t1, ¨¨¨, N u. For any eigenfunction u ˚P H 1 I of (1.2), and every Ω ě 2 π C mix,α,β ? N maxtZ, N u, under Assumption 2.1 on α, β, we have

}u ˚´P R,Ω I,α,β u ˚}L 2 ppR 3 q N q ď ? 2πe 5{8 R }u ˚}L 2 ppR 3 q N q , and
}∇pu ˚´P R,Ω I,α,β u ˚q} L 2 ppR 3 q N q ď 2 ? 2πe 5{8 R Ω}u ˚}L 2 ppR 3 q N q .
Here the constant C mix,α,β is defined in Theorem 2.2.

This proof is provided in Section 6.

2.3.

Main ideas of the proof of Theorem 2.2. As mentioned in Introduction, the extension of the results in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF][START_REF] Yserentant | The mixed regularity of electronic wave functions multiplied by explicit correlation factors[END_REF] is based on Theorem 2.2. Once Theorem 2.2 is proven, following the proofs in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] line by line, we can prove Theorem 2.3 and Theorem 2.5, respectively. Thus this paper is devoted mainly to the proof of Theorem 2.2. As the proof of Theorem 2.2 is quite technical, before entering the details, let us try to explain the main ideas and the main improvements with respect to the existing results. Let

L I :" â jPI ∇ j , L piq I :" â jPIztiu ∇ j and L pi,jq I :" â kPIzti,ju ∇ k .
The Hardy-type inequality used in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] can be expressed in the following way.

Theorem 2.6. [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF][START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] For every I Ă t1, ¨¨¨, N u, there is a constant C I independent of N , Z such that for any u, v P X I,1,0 ,

| L I pV ne `Vee qu, L I v | ď C I ? N maxtZ, N u}u} 0,I,α,β }v} 1,I,α,β . (2.11)
One can essentially recover Theorem 2.2 with our Theorem 2.6 by setting α " 1 and β " 0.

Remark 2.7. Actually, Theorem 2.2 is optimal at least for the case α " β if the wavefunction u is not totally antisymmetric. Otherwise, if (2.8) holds for some α " β ě 3 4 , then following the proofs in [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF] and [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF] as in Section 5 in this paper, Theorem 2.3 and Corollary 2.4 will also hold for these α and β. This means that for any eigenfunction u ˚which is not totally antisymmetric, u ˚P X I,α,β with some α " β ě 3 4 . However, [START_REF] Yserentant | The mixed regularity of electronic wave functions multiplied by explicit correlation factors[END_REF] (see also (1.11) and (1.13) in this paper) shows that α " β ă 3 4 is optimal. Then we reach a contradiction. Thus (2.8) is optimal for the case α " β if the wavefunction u is not totally antisymmetric. Now we compare these two inequalities technically. We first give a glimpse into the formulae on the left-hand side of (2.8) and (2.11):

L I,α,β V ne u " N ÿ i"1 M ÿ ν"1 p1 `|∇ i | 2 q γ i {2 ˆZν |x i ´aν | L piq I,α,β u ˙, (2.12
)

L I V ne u " N ÿ i"1 M ÿ ν"1 ∇ i ˆZν |x i ´aν | L piq I u ˙, (2.13)
L I,α,β V ee u " 1 2 ÿ i‰j p1 `|∇ i | 2 q γ i {2 p1 `|∇ j | 2 q γ j {2 ˆ1 |x i ´xj | L pi,jq I,α,β u ˙, (2.14)
and

L I V ee u " 1 2 ÿ i‰j ∇ γ 1 i i â ∇ γ 1 j j ˆ1 |x i ´xj | L pi,jq I u ˙, (2.15)
where γ k " α, γ We can not prove Theorem 2.2 as for Theorem 2.6 since the Leibniz rule fails for fractional Laplacian operators. In addition, the optimality of Theorem 2.2 for the case α " β and the singularity of the Coulomb potential make the proof of (2.8) in Theorem 2.2 much more delicate than the one of (2.11) in Theorem 2.6.

To prove Theorem 2.2, we need first to study the relationship between the fractional operator p1 `|∇ i | 2 q γ i {2 and the Coulomb type potentials. In this paper, this is equivalent to the study of the relationship between |∇ i | γ i and the Coulomb type potentials by introducing a bounded operator K s,y :" p1 `|∇ y | 2 q s{2 p1 `|∇ y | s q ´1 in Section 3.1.

The main tool of this paper is the following Hardy-type inequality (i.e., Theorem 3.3):

}|y| ´s|∇ y | ´sf } L 2 pR 3 q À s }f } L 2 pR 3 q , 0 ď s ă 3 2 ,
and its dual form:

}|∇ y | ´s|y| ´sf } L 2 pR 3 q À s }f } L 2 pR 3 q , 0 ď s ă 3 2 .
This is the most important inequality used in this paper and this gives the tool to study the electron-nucleus term V ne immediately (see (2.12)).

Concerning the electron-electron term V ee in (2.14), we need a corresponding version of Hardytype inequality for two particles. From the above Hardy-type inequality, one can deduce the following (see Lemma 3.6):

}|y ´z| ´s´t |∇ y | ´s|∇ z | ´tf } L 2 pR 3 ˆR3 q À s,t }f } L 2 pR 3 ˆR3 q , s, t ě 0, s `t ă 3 2 .
This gives the condition that β ă 3 4 and α `β ă 3 2 in Assumption 2.1. Thanks to antisymmetry, we can also generalize the standard Hardy inequality for two particles since the antisymmetry will counterbalance the singularity of the potential 1 |x i ´xj | . More precisely, for any function g P C 8 0 pR 3 ˆR3 q satisfying gpx, yq " ´gpy, xq, it is easy to see that gpx, yq " 0 for x " y. Thus |gpx, yq| ď C|x ´y| in any compact neighborhood of the set tx " yu. As a result,

› › › › g |y ´z| 2s › › › › L 2 pR 3 ˆR3 q ă `8, 0 ď s ă 5 4 .
Then arguing as for the standard Hardy inequality, we show in Corollary 3.8 and Lemma 3.9 the following inequalities with antisymmetry:

› › › › g |y ´z| 2s › › › › L 2 pR 3 ˆR3 q À s › › › › ∇ y ∇ z g |y ´z| 2s´2 › › › › L 2 pR 3 ˆR3 q À s }|∇ y | s |∇ z | s g} L 2 pR 3 ˆR3 q , 1 ď s ă 5 4 .
Returning back to (2.8) and (2.14), we will use the following extension of the Hardy-type inequality for two particles with antisymmetry in Lemma 4.5:

}|∇ y | s´1{2 |∇ y | s´1{2 |y ´z| ´1g} L 2 pR 3 ˆR3 q À s }|∇ y | s |∇ z | s g} L 2 pR 3 ˆR3 q , 0 ď s ă 5 4 .
This gives the condition α ă 5 4 in Assumption 2.1. In addition, this estimate shows that spatial antisymmetry implies regularity.

Once the fractional Laplacian operator is defined and the above inequalities and their extensions are established, we can obtain Theorem 2.2 immediately. This paper is organised as follows. As mentioned above, we will use the fractional Laplacian operator. Thus in Section 3, we will first study the fractional Laplacian. Then based on Hardy-type inequalities, we will deduce the above inequalities. In Section 4, we will use the above inequalities to prove Theorem 2.2. Then in Section 5 and Section 6, following the proofs in [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF], we will prove Theorem 2.3 and Theorem 2.5, respectively.

Fractional Laplacian operators and related inequalities

In the following, we denote the gradients ∇ y and ∇ z corresponding to the variables y and z in R 3 , respectively.

In the next subsection, we study the fractional Laplacian operator p1`|∇ y | 2 q s{2 on R 3 . Actually, we rather study the fractional Laplacian operator |∇ y | s . The relationship between |∇ y | s and p1 `|∇ y | 2 q s{2 will be studied equally. Then, for some s, t ą 0, we will study some Hardy-type inequalities associated with the operators |∇ y | ´s|y| ´s and |∇ y | ´s|∇ z | ´t|y ´z| ´s´t .

3.1. Fractional Laplacian. First of all, we define our convention for the Fourier transform. Let f P L 2 pR 3 q, g P L 2 pR 3 ˆR3 q and u P L 2 ppR 3 q N q, then the Fourier transforms of f , g and u are respectively

F y pf qpξ y q :" ż R 3
f pyqe ´2πiξy¨y dy, F y,z pgqpξ y , ξ z q :" F z ˝Fy pgqpξ y , ξ z q, and

F x 1 ,¨¨¨,x N puqpξq :" F x N ˝¨¨¨˝F x 1 puqpξq, ξ :" pξ 1 , ¨¨¨, ξ N q with ξ k P R 3 , k " 1, ¨¨¨, N.
For s ą 0, a function f P L 2 pR 3 q is said to be in H s pR 3 q if and only if

}f } 2 H s pR 3 q :" ż R 3
p1 `|ξ y | 2 q s |F y pf qpξ y q| 2 dξ y ă `8.

The fractional Laplacian |∇ y | s (or p´△ y q s{2 ) is defined on functions f P H s pR 3 q by the Fourier representation: F y p|∇ y | s f qpξ y q :" |2πξ y | s F y pf qpξ y q.

Similarly, p1 `|∇ y | 2 q s{2 is defined by F y pp1 `|∇ y | 2 q s{2 f qpξ y q :" p1 `|2πξ y | 2 q s{2 F y pf qpξ y q.

The operator L I,α,β which is defined on functions u P X I,α,β can be regarded as a composition of fractional Laplacian operators on R 3 in the following manner:

L I,α,β u :" p1 `|∇ 1 | 2 q γ 1 {2 ˝¨¨¨˝p1 `|∇ N | 2 q γ N {2 u,
where γ k " α if k P I, and γ k " β if k P I c . Applying the Fourier transform to solve the Poisson equation

|∇ y | s f 1 pyq " f 2 pyq in R 3 ,
we find that |2πξ y | s F y pf 1 qpξ y q " F y pf 2 qpξ y q. The inverse of the fractional Laplacian, or negative power of the Laplacian |∇ y | ´s with s ą 0 is defined on functions f in the Schwartz space SpR 3 q by F y p|∇ y | ´sf qpξ y q " |2πξ y | ´sF y pf qpξ y q for ξ y ‰ 0.

In principle, we need the restriction 0 ă s ă 3 because when s ě 3 the multiplier |ξ y | ´s does not define a tempered distribution (for more details, see e.g. [START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF]).

On the other hand, the term 1 |y| s is a tempered distribution for 0 ă s ă 3 with Fourier transform b s F y p| ¨|´s qpξ y q " b 3´s |ξ y | ´3`s , b s " π ´s{2 Γps{2q,

where Γ is the Gamma function. For the detail, see e.g., [6, Eqn. (3.

3)] (the difference of the definition of b s therein is because of the different definition of the Fourier transform) or [14, Theorem 5.9] by using the fact that F y pf gq " F y pf q ˚Fy pgq. Hence, if 0 ă s ă 3, the operator |∇ y | ´s can be rewritten as

|∇ y | ´sf pyq " b 3´s p2πq s b s ż R 3 |z ´y| ´3`s f pzqdz, f P SpR 3 q. (3.2)
Suppose that 0 ă s ă 3, then |∇ y | s |y| ´t is an L 1 loc pR 3 q-function for 0 ă t ă 3 ´s and, using (3.1), We end this subsection by studying the relationship between p1 `|∇ y | 2 q s{2 and |∇ y | s , then in the next subsections we will study Hardy-type inequalities associated with the fractional Laplacian operator |∇ y | s . Let

|∇ y | s |y| ´t " F ´1 y p|2πξ y | s F y p| ¨|´t qq " p2πq s b 3´t b t F ´1 y p| ¨|´3`t`s q " p2πq s b s`t b 3´t b 3´s´t b t |y| ´s´t . ( 3
K s,y :" p1 `|∇ y | 2 q s{2 p1 `|∇ y | s q ´1, (3.5)
which is defined by the Fourier transform: F y pK s,y f qpξ y q :" p1 `|2πξ| 2 q s{2 1 `|2πξ| s F y pf qpξ y q. By the Fourier transform, it is easy to see that K s,y " K s,y .

(3.6)

Then we have the following.

Lemma 3.1. For any 0 ď s ď 2, }K s,y } L 2 pR 3 qÑL 2 pR 3 q ď 1.

Proof. For any 0 ď s ď 2, f pyq P L 2 pR 3 q, }K s,y f } L 2 pR 3 q " }p1 `|2πξ y | 2 q s{2 p1 `|2πξ y | s q ´1F y pf q} L 2 pR 3 q .

As 0 ď s{2 ď 1, p1 `|2πξ y | 2 q s{2 ď p1 `|2πξ y | s q. Thus, }K s,y f } L 2 pR 3 q ď }F y pf q} L 2 pR 3 q " }f } L 2 pR 3 q .

Hence the lemma.

Remark 3.2. Actually, the operator K s,y is also bounded from below. One can use the inequalities pa p `bp q ď pa `bq p ď 2 p pa p `bp q for a, b ě 0. (We thank one of the referees for this remark.)

3.2. Hardy-type inequalities for a single particle. We now consider the term |∇ y | ´s|y| ´sf pyq with f P L 2 pR 3 q. Actually, |∇ y | ´s|y| ´s is the adjoint of the operator |y| ´s|∇ y | ´s which has been well studied in [START_REF] Herbst | Spectral theory of the operator pp 2 `m2 q 1{2 ´Ze 2 {r[END_REF] (see equally [2, Theorem 1.7.1]). The following holds.

Theorem 3.3 (Hardy-type inequality [START_REF] Herbst | Spectral theory of the operator pp 2 `m2 q 1{2 ´Ze 2 {r[END_REF]). Let p ´1 `q´1 " 1. Suppose s ą 0 and 3s ´1 ą p ą 1. Then }|y| ´s|∇ y | ´sf } L p pR 3 q ď C s,p }f } L p pR 3 q (3.7) where C s,p :" 2 ´s Γp 1 2 p3p ´1 ´sqqΓp 3 2 q ´1q Γp 3 2 pq ´1 `sqqΓp

}|y| ´sg} L p pR 3 q ď C s,p }|∇ y | s g} L p pR 3 q . (3.8)
When p " 2 and s " 1, (3.7) is indeed the Hardy inequality with the optimal constant C 1,2 " 2. When p " 2 and s " 1 2 , (3.7) is the Kato inequality with the optimal constant C 1 2 ,2 " ? π ?

2 (see e.g. [START_REF] Balinsky | Spectral analysis of relativistic operators[END_REF]Formula (1.7

.7)]).

In this paper, we only use the case p " 2. By duality, we also have the following: for 0 ă s ă 3 2

}|∇ y | ´s|y| ´sf } L 2 pR 3 q ď C s,2 }f } L 2 pR 3 q .
Notice that |∇ y | s rf py `aqs " r|∇ y | s f spy `aq. Then from Theorem 3.3, for any a P R 3 and 0 ă s ă 3 2 , }| ¨´a| ´sf } L 2 pR 3 q ď C s,2 }|∇ y | s f } L 2 pR 3 q .

(3.9) By Fubini's Theorem and for any hpy, zq P L 2 pR 3 ˆR3 q, we have }|y ´z| ´shpy, zq} L 2 pR 3 ˆR3 q ď C s,2 }|∇ y | s h} L 2 pR 3 ˆR3 q .

(3.10)

The dual version of (3.9) and (3.10) can be represented respectively as follows: for a P R 3 and 0 ă s ă 3{2,

}|∇ y | ´s| ¨´a| ´sf } L 2 pR 3 q ď C s,2 }f } L 2 pR 3 q (3.11)
and

}|∇ y | ´s|y ´z| ´sf } L 2 pR 3 ˆR3 q ď C s,2 }f } L 2 pR 3 ˆR3 q .
(3.12)

3.3. Hardy-type inequalities for two particles. The rest of this section is devoted to Hardytype inequalities for |∇ y | ´s|∇ z | ´t|z ´y| ´s´t terms for some s, t ě 0. By using Theorem 3.3, we have the following.

Lemma 3.6. For s, t ě 0, s `t ă 3{2, and f P L 2 pR 3 ˆR3 q, we have

}|∇ y | ´s|∇ z | ´t|y ´z| ´s´t f } L 2 pR 3 ˆR3 q ď 2c s`t }f } L 2 pR 3 ˆR3 q (3.13)
and

}|y ´z| ´s´t |∇ y | ´s|∇ z | ´tf } L 2 pR 3 ˆR3 q ď 2c s`t }f } L 2 pR 3 ˆR3 q , (3.14)
with c 0 :" 1 and, for 0 ă s ă 3{2, c s :" C s,2 where C s,2 is defined in Theorem 3.3.

Proof. For simplicity, we use the shorthand }f } L 2 for }f } L 2 pR 3 ˆR3 q . We prove first

}|∇ y | ´s|∇ z | ´t|y ´z| ´s´t f } L 2 ď 2c s`t }f } L 2 .
If s " t " 0, then }f } L 2 pR 3 ˆR3 q ď 2c 0 }f } L 2 pR 3 ˆR3 q where c 0 " 1. Now we assume s " 0 and t ‰ 0. Thanks to (3.12), we have

}|∇ z | ´t|y ´z| ´tf } L 2 ď c t }f } L 2 ď 2c t }f } L 2 .
The case t " 0 and s ‰ 0 can be treated in the same manner. Now we only need to consider the case s, t ą 0. For any function upy, zq P L 2 pR 3 ˆR3 q, by Fourier transform on R 3 ˆR3 , we have

}|∇ y | ´s|∇ z | ´t|y ´z| ´s´t f } L 2 " p2πq ´s´t }|ξ y | ´s|ξ z | ´tF y,z p|y ´z| ´s´t f qpξ y , ξ z q} L 2 .
Notice that |τ | t ď |τ | s`t `1 for τ P R. Then for τ " |ξ y |{|ξ z |, we have

|ξ y | ´s|ξ z | ´t ď |ξ y | ´s´t `|ξ z | ´s´t . Thus, }|ξ y | ´s|ξ z | ´tF y,z p|y ´z| ´s´t f qpξ x , ξ y q} L 2 ď }|ξ y | ´s´t F y,z p|y ´z| ´s´t f qpξ y , ξ z q} L 2 `}|ξ z | ´s´t F y,z p|y ´z| ´s´t f qpξ y , ξ z q} L 2 " p2πq s`t }|∇ y | ´s´t |y ´z| ´s´t f } L 2 `p2πq s`t }|∇ z | ´s´t |y ´z| ´s´t f } L 2 .
As 0 ă s `t ă 3{2, by (3.12), we have

}|∇ y | ´s´t |y ´z| ´s´t f } L 2 ď c s`t }f } L 2 ,
and

}|∇ z | ´s´t |y ´z| ´s´t f } L 2 ď c s`t }f } L 2 .

Consequently, we deduce

}|∇ y | ´s|∇ z | ´t|x ´y| ´s´t f } L 2 ď 2c s`t }f } L 2 .
By duality, Eqn. (3.14) follows.

For the wavefunction u, the antisymmetry with respect to I will counterbalance the singularities of the potential between electrons. Based on this observation, Lemma 3.6 can be extended in consideration of the antisymmetry. In Theorem 2.2, we only focus on the case t " s. The extension is based on the following. Lemma 3.7. Let a P R 3 and k P r1, 3{2q Y p3{2, 5{2q. If f P C 8 0 pR 3 q for k P r1, 3{2q, or if f P C 8 0 pR 3 ztauq for k P p3{2, 5{2q, we have

› › › › f | ¨´a| k › › › › L 2 pR 3 q ď 2 |2k ´3| › › › › ∇ y f | ¨´a| k´1 › › › › L 2 pR 3 q .
The proof is inspired by [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF]Lemma 2].

Proof. We have the relationship:

p2k
|2k ´3| 2 4 ż R 3 |f | 2 |y ´a| 2k dy ď ż R 3 |∇f | 2 |y ´a| 2k´2 dy.
This ends the proof. Using Fubini's theorem and Lemma 3.7, the following holds for antisymmetric functions.

Corollary 3.8. For s P r1, 5{4q and f P C 8 0 pR 3 ˆR3 q with f py, zq " ´f pz, yq, we have

› › › › f |y ´z| 2s › › › › L 2 pR 3 ˆR3 q ď 4 |4s ´5||4s ´3| › › › › ∇ y ∇ z f |y ´z| 2s´2 › › › › L 2 pR 3 ˆR3 q .
Proof. This is a generalization of [START_REF] Yserentant | On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives[END_REF]Eqn. (3.9)]. We first fix z P R 3 , and let g z pyq " f py, zq. As f py, zq " ´f pz, yq and f P C 8 0 pR 3 ˆR3 q, we know that g z P C 8 0 pR 3 ztzuq. Thus Lemma 3.7 shows that for any

z P R 3 › › › › g z | ¨´z| 2s › › › › L 2 pR 3 y q ď 2 |4s ´3| › › › › ∇ y g z | ¨´z| 2s´1 › › › › L 2 pR 3 y q .
Taking L 2 -norm with respect to z in the above inequality, we get

› › › › f py, zq |y ´z| 2s › › › › L 2 pR 3 ˆR3 q ď 2 |4s ´3| › › › › ∇ y f py, zq |y ´z| 2s´1 › › › › L 2 pR 3 ˆR3 q . Now 1 ď 2s ´1 ă 3
2 and ∇ y f py, ¨q P C 8 pR 3 q for any fixed y P R 3 . Arguing as above, fixing y P R 3 , and using Lemma 3.7 again, we infer

› › › › f py, zq |y ´z| 2s › › › › L 2 pR 3 ˆR3 q ď 4 |4s ´5||4s ´3| › › › › ∇ y ∇ z f py, zq |y ´z| 2s´2 › › › › L 2 pR 3 ˆR3 q .
This ends the proof.

Combining Lemma 3.6 with Corollary 3.8, we have the following.

Lemma 3.9. For s P r1, 5{4q and f P C 8 0 pR 3 ˆR3 q with f py, zq " ´f pz, yq, we have

› › › › f |y ´z| 2s › › › › L 2 pR 3 ˆR3 q ď c 2s }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q (3.15)
where c 2s :" 8c 2s´2 p5´4sqp4s´3q and c 2s´2 " C 2s´2,2 .

Remark 3.10. Denote the functional space Y anti,s pR 3 ˆR3 q by Y anti,s pR 3 ˆR3 q :" tf P L 2 pR 3 ˆR3 q; f py, zq " ´f pz, yq,

|∇ y | s |∇ z | s f P L 2 pR 3 ˆR3 qu.
Then Y anti,s pR 3 ˆR3 q is a completion of C 8 0 pR 3 ˆR3 q with the antisymmetry f py, zq " ´f pz, yq. Thus by density, (3.15) still holds for f P Y anti,s pR 3 ˆR3 q.

Hardy-type inequalities for the Coulomb system

Now, we are going to use these inequalities to prove Theorem 2.2. We will study separately the contribution of the potential V ne in Section 4.1 and the potential V ee in Section 4. Before going further, we recall that Z " ř M ν"1 Z ν and that the constant c k is defined by : c 0 " 1, c k " C k,2 for k P p0, 3{2q with C k,p defined in Theorem 3.3, and c k " 8c k´2 p5´2kqp2k´3q for k P r2, 5{2q.

Contribution of the electrons-nuclei interaction.

In this subsection, we are going to prove the following. Lemma 4.1 (Contribution of V ne ). Let I Ă t1, ¨¨¨, N u. For any α, β P r0, 3{2q, there is a constant C 1,mix,α,β independent of N, Z such that for any u, v P X I,α,β ,

| L I,α,β V ne u, L I,α,β v | ď C 1,mix,α,β ? N Z}u} 0,I,α,β }v} 1,I,α,β . (4.1)
It is an immediate result of the following.

Lemma 4.2. For s P r´1, 1{2q, a P R 3 , and f P H 1`s pR 3 q, we have

› › |∇ y | s | ¨´a| ´1f › › L 2 pR 3 q ď C s }|∇ y | 1`s f } L 2 pR 3 q , where C s " pc 1`s `cs qc 1´s if s ą 0, C 0 " c 1 " 2, C s " c ´sc 1`s if ´1 ď s ă 0.
Before proving Lemma 4.2, we use it to prove Lemma 4.1 first.

Proof of Lemma 4.1.

Recall that K s,y " p1 `|∇ y | 2 q s{2 p1 `|∇ y | s q ´1.
When 0 ă s ă 1{2, we use the formal identity |∇| s " |∇||∇| s´1 . Thus,

› › |∇ y | s |y ´a| ´1f pyq › › L 2 pR 3 q " › › |∇ y | s´1 ∇ y |y ´a| ´1f pyq › › L 2 pR 3 q ď }|∇ y | s´1 |y ´a| ´1∇ y f pyq} L 2 pR 3 q `}|∇ y | s´1 |y ´a| ´3py ´aqf pyq} L 2 pR 3 q .
Using (3.11) again, we get

}|∇ y | s´1 |y ´a| ´1∇ y f pyq} L 2 pR 3 q ď c 1´s }|y ´a| ´s∇ y f pyq} L 2 pR 3 q ď c 1´s c s }|∇ y | 1`s f } L 2 pR 3 q .
Analogously, as 0 ă s ă 1{2, we have

}|∇ y | s´1 |y ´a| ´3py ´aqf pyq} L 2 pR 3 q ď c 1´s }|y ´a| ´1´s f pyq} L 2 pR 3 q ď c 1´s c 1`s }|∇ y | 1`s f } L 2 pR 3 q .
We conclude now that, for s P p0, 1{2q, Lemma 4.2 holds with C s " c 1´s pc s `c1`s q, i.e.,

› › |∇ y | s | ¨´a| ´1f › › L 2 pR 3 q ď C s }|∇ y | 1`s f } L 2 pR 3 q
, This ends the proof. 4.2. Contribution of the electron-electron interaction. In this subsection, we are going to prove the following.

Lemma 4.3 (Contribution of V ee ). Let I Ă t1, ¨¨¨, N u. Under Assumption 2.1 on α, β, there is a constant C 2,mix,α,β independent of N, Z such that for any u, v P X I,α,β , | L I,α,β V ee u, L I,α,β v | ď C 2,mix,α,β N 3{2 }u} 0,I,α,β }v} 1,I,α,β . (4.3)
Recall that b s " π ´s{2 Γps{2q where Γp¨q is the Gamma function and recall that Y anti,s pR 3 ˆR3 q :" tf P L 2 pR 3 ˆR3 q; f py, zq " ´f pz, yq,

|∇ y | s |∇ z | s f P L 2 pR 3 ˆR3 qu.
To prove Lemma 4.3, we need the followings.

Lemma 4.4. Define Y s,t pR 3 ˆR3 q :" tg P L 2 pR 3 ˆR3 q; |∇ y | s |∇ z | t g P L 2 pR 3 ˆR3 qu. Let 0 ď t ď s, s `t ă 3{2 and f P Y s,t pR 3 ˆR3 q. Then,

}|∇ y | s`t´1 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď C 1 s,t }|∇ y | s |∇ z | t f } L 2 pR 3 ˆR3 q ,
where C 1 s,t :"

´2s`t π s`t´2 b s`t c s`t b 3´s´t
`2c 1 ¯if 1 ă s `t ă 3{2, and C 1 s,t :" 2c 1´s´t c s`t if 0 ď s `t ď 1.

Lemma 4.5. Let 0 ď s ă 5{4 and f P Y anti,s pR 3 ˆR3 q. Then,

}|∇ y | s´1{2 |∇ z | s´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď C anti,s }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q ,
where C anti,0 :" 2c 1 , C anti,s :" 4c 3´2s c 2s´2 `4c 3´2s c 2s´1 `4c 3´2s c 2s if 1 ď s ă 5{4, and C anti,s :" Proof of Lemma 4.3. Recall that K s,y :" p1 `|∇ y | 2 q s{2 p1 `|∇ y | s q ´1. Following (2.14) and (3.6), we have

C 1´s anti,0 C s anti,1 if 0 ă s ă 1. We prove Lemma 4.
L I,α,β V ee u, L I,α,β v " 1 2 N ÿ i,j"1 i‰j p1 `|∇ i | 2 q γ i {2 p1 `|∇ j | 2 q γ j {2 |x i ´xj | ´1L pi,jq I,α,β u, L I,α,β v " 1 2 ÿ i‰j p1 `|∇ i | γ i qp1 `|∇ j | γ j q|x i ´xj | ´1L pi,jq I,α,β u, K γ i ,x i K γ j ,x j L I,α,β v ,
We first consider the case ti, ju Ć I. Then as γ j ď γ i , we have γ j " β P r0, 3{4q. Thus by using Lemma 3.1 and the formal identity

|∇ i | γ i " |∇ i | γ i `γj ´1|∇ i | 1´γ j , ˇˇ |∇ i | γ i |∇ j | γ j |x i ´xj | ´1L pi,jq I,α,β u, K γ i ,x i K γ j ,x j L I,α,β v ˇ" ˇˇ |∇ i | γ i `γj ´1|x i ´xj | ´1L pi,jq I,α,β u, |∇ j | γ j |∇ i | 1´γ j K γ i ,x i K γ j ,x j L I,α,β v ˇď }|∇ i | γ i `γj ´1|x i ´xj | ´1L pi,jq I,α,β u} L 2 ppR 3 q N q }|∇ j | γ j |∇ i | 1´γ j v} 0,I,α,β .
Notice that |τ | γ j ď 1 `|τ | for any τ P R and 0 ď γ j ă 3{4. For τ "

|ξ j | |ξ i | , we have |ξ j | γ j |ξ i | 1´γ j ď |ξ j | `|ξ i |. Thus, }|∇ j | γ j |∇ i | 1´γ j v} 0,I,α,β ď }|∇ i |v} 0,I,α,β `}|∇ j |v} 0,I,α,β .
On the other hand, thanks to Lemma 4.4, Assumption 2.1 and the fact

|2πξ k | γ k ď p1 `|2πξ k | 2 q γ k {2
with k " i or k " j, we have

}|∇ i | γ i `γj ´1|x i ´xj | ´1L pi,jq I,α,β u} L 2 ppR 3 q N q ď C 1 γ i ,γ j }u} 0,I,α,β . (4.8)
We conclude that, if ti, ju Ć I, then

ˇˇ |∇ i | γ i |∇ j | γ j |x i ´xj | ´1L pi,jq I,α,β u, K γ i ,x i K γ j ,x j L I,α,β v ˇď C 1
γ i ,γ j }u} 0,I,α,β p}|∇ i |v} 0,I,α,β `}|∇ j |v} 0,I,α,β q . Now we consider the case ti, ju Ă I. Then γ i " γ j " α with α P r0, 5{4q. Besides, the function u is antisymmetric with respect to ti, ju. We fix the variables px k q kPt1,¨¨¨,N uzti,ju , and let f px i , x j q " upxq with x " px 1 , ¨¨¨, x N q. Thus f px i , x j q P Y anti,α pR 3 x i

ˆR3

x j q, and by Lemma 4.5,

}|∇ i | α´1{2 |∇ j | α´1{2 |x i ´xj | ´1f } L 2 pR 3 x i ˆR3 x j q ď C anti,α }|∇ i | α |∇ j | α f } L 2 pR 3 x i

ˆR3

x j q . Finally, for i, j P I, by the formal identity

|∇ i | α |∇ j | α " |∇ i | α´1{2 |∇ j | α´1{2 |∇ i | 1{2 |∇ j | 1{2 , we get ˇˇ |∇ i | α |∇ j | α |x i ´xj | ´1L pi,jq I,α,β u, K α,x i K α,x j L I,α,β v ˇ" ˇˇ |∇ i | α´1{2 |∇ j | α´1{2 |x i ´xj | ´1L pi,jq I,α,β u, |∇ j | 1{2 |∇ i | 1{2 K γ i ,x i K γ j ,x j L I,α,β v ˇď }|∇ i | α´1{2 |∇ j | α´1{2 |x i ´xj | ´1L pi,jq I,α,β u} L 2 ppR 3 q N q }|∇ j | 1{2 |∇ i | 1{2 v} 0,I,α,β ď C anti,α 2 }|∇ i | α |∇ j | α L i,j
I,α,β u} L 2 ppR 3 q N q p}|∇ i |v} 0,I,α,β `}|∇ j |v} 0,I,α,β q ď C anti,α 2 }u} 0,I,α,β p}|∇ i |u} 0,I,α,β `}|∇ j |v} 0,I,α,β q .

(4.9)

Let C 2 α,β :" maxt

C anti,α 2
, C 1 α,β , C 1 β,β u. For the last term on the right-hand side of (4.4), under Assumption 2.1 on α, β, we conclude ˇˇˇˇÿ We split the condition 0 ď s `t ă 3{2 into two cases: 0 ď s `t ď 1 and 1 ă s `t ă 3{2. For the case 0 ď s `t ď 1, by Lemma 3.6, we have

i‰j |∇ i | γ i |∇ j | γ j |x i ´xj | ´1L pi,jq I,α,β u, K γ i ,x i K γ j ,x j L I,α,β v ˇˇˇˇď C 2 α,β N 3{2
}|∇ y | s`t´1 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď c 1´s´t }|y ´z| ´s´t f } L 2 pR 3 ˆR3 q ď 2c 1´s´t c s`t }|∇ y | s |∇ z | t f } L 2 pR 3 ˆR3 q .
Thus, C 1 s,t :" 2c 1´s´t c s`t if 0 ď s `t ď 1. Now, we turn to prove the case 1 ă s `t ă 3{2. It is based on the study of the fractional Laplacian in Section 3.1.

We consider first the Fourier transform of f py,zq |y´z| τ . For any function gpy, zq P L 2 pR 3 ˆR3 q, we have F y,z pgqpξ y , ξ z q " F z ˝Fy pgqpξ y , ξ z q. Besides, thanks to (3.1), we have F y p|¨´z| ´τ qpξ y q " b 3´τ bτ e ´2iπz¨ξy |ξy| 3´τ . Thus,

F y,z ˆf |y ´z| τ ˙pξ y , ξ z q " b 3´τ b τ ż R 3 1 |l| 3´τ F z pe ´2iπz¨l F y pf qpξ y ´l, zqqdl " b 3´τ b τ ż R 3 1 |l| 3´τ F y,z pf qpξ y ´l, ξ z `lqdl. (4.11)
Thus for s `t ą 1, by Plancherel's Theorem,

}|∇ y | s`t´1 |y ´z| ´1f } L 2 pR 3 ˆR3 q " 2 s`t´1 π s`t´2 › › › › ż R 3 |ξ y | s`t´1 F y,z pf qpξ y ´l, ξ z `lq |l| 2 dl › › › › L 2 pR 3 ˆR3 q .
For any k P R 3 and 1 ă s `t ă 3{2, we have |ξ y | s`t´1 ď |ξ y ´k| s`t´1 `|k| s`t´1 . Let k " l, thus

}|∇ y | s`t´1 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď p2πq s`t´1 π ´1 › › › › ż R 3 |F y,z pf qpξ y ´l, ξ z `lq| |l| 3´s´t dl › › › › L 2 pR 3 ˆR3 q `p2πq s`t´1 π ´1 › › › › ż R 3 |ξ y ´l| s`t´1 |F y,z pf qpξ y ´l, ξ z `lq| |l| 2 dl › › › › L 2 pR 3 ˆR3 q .
(4.12)

Using (4.11) and Plancherel's Theorem again, we get

› › › › ż R 3 |F y,z pf qpξ y ´l, ξ z `lq| |l| 3´s´t dl › › › › L 2 pR 3 ˆR3 q " b s`t b 3´s´t
}|y ´z| ´s´t F ´1 y,z p|F y,z pf q|q} L 2 pR 3 ˆR3 q .

By Lemma 3.6, we finally deduce

}|y ´z| ´s´t F ´1 y,z p|F y,z pf q|q} L 2 pR 3 ˆR3 q ď 2c s`t }|∇ y | s |∇ z | t F ´1 y,z p|F y,z pf q|q} L 2 pR 3 ˆR3 q " 2c s`t p2πq s`t }|ξ y | s |ξ z | t F y,z pf q} L 2 pR 3 ˆR3 q " 2c s`t }|∇ y | s |∇ z | t f } L 2 pR 3 ˆR3 q .
On the other hand, for the second term on the right-hand side of (4.12), we have

› › › › ż R 3 |ξ y ´l| s`t´1 |F y,z pf qpξ y ´l, ξ z `lq| |l| 2 dl › › › › L 2 pR 3 ˆR3 q " p2πq 1´s´t › › › › ż R 3 |F y,z p|∇ y | s`t´1 f qpξ y ´l, ξ z `lq| |l| 2 dl › › › › L 2 pR 3 ˆR3 q " 2 1´s´t π 2´s´t }|y ´z| ´1F ´1 y,z p|F y,z p|∇ y | s`t´1 f q|q} L 2 pR 3 ˆR3 q .
As 0 ď t ď s and t `s ă 3{2, we have t ď 3{4. Thus by Lemma 3.6,

}|y ´z| ´1F ´1 y,z p|F y,z p|∇ y | s`t´1 f q|q} L 2 pR 3 ˆR3 q ď 2c 1 }|∇ y | 1´t |∇ z | t F ´1 y,z p|F y,z p|∇ y | s`t´1 f q|q} L 2 pR 3 ˆR3 q " 4πc 1 }|ξ y | 1´t |ξ z | t F y,z p|∇ y | s`t´1 f q} L 2 pR 3 ˆR3 q " 2c 1 }|∇ y | s |∇ z | t f } L 2 pR 3 ˆR3 q .
Consequently, for 0 ď t ď s and 1 ă s `t ă 3{2, we deduce

}|∇ y | s`t´1 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď ˆ2s`t π s`t´2 b s`t c s`t b 3´s´t `2c 1 ˙}|∇ y | s |∇ z | t f } L 2 pR 3 ˆR3 q .
Thus C 1 s,t :"

´2s`t π s`t´2 b s`t c s`t b 3´s´t
`2c 1 ¯if 1 ă s `t ă 3{2 and 0 ă t ď s. This ends the proof.

4.2.2.

Proof of Lemma 4.5. When s " 0, by Lemma 3.6, we have

}|∇ y | ´1{2 |∇ z | ´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď 2c 1 }f } L 2 pR 3 ˆR3 q .
Thus, C anti,0 :" 2c 1 Now, we assume 1 ď s ă 5{4. By the formal identity

|∇ y | s´1{2 |∇ z | s´1{2 " |∇ y ||∇ y | s´3{2 |∇ z ||∇ z | s´3{2 , we have }|∇ y | s´1{2 |∇ z | s´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q " }∇ y b ∇ z |∇ y | s´3{2 |∇ z | s´3{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď }|∇ y | s´3{2 |∇ z | s´3{2 |y ´z| ´1∇ y b ∇ z f } L 2 pR 3 ˆR3 q `}|∇ y | s´3{2 |∇ z | s´3{2 |y ´z| ´3py ´zq b ∇ z f } L 2 pR 3 ˆR3 q `}|∇ y | s´3{2 |∇ z | s´3{2 |y ´z| ´3pz ´yq b ∇ y f } L 2 pR 3 ˆR3 q `}|∇ y | s´3{2 |∇ z | s´3{2 p∇ y b ∇ z |y ´z| ´1qf } L 2 pR 3 ˆR3 q . (4.13)
As ´1{2 ď s ´3{2 ă ´1{4, by Lemma 3.6, we have

}|∇ y | s´3{2 |∇ z | s´3{2 g} L 2 pR 3 ˆR3 q ď 2 c 2s´3 }|y ´z| 3´2s g} L 2 pR 3 ˆR3 q .
Besides, it is not difficult to see that

|∇ y b ∇ z |y ´z| ´1| ď 6 |y ´z| ´3.
Thus,

}|∇ y | s´1{2 |∇ z | s´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď 2c 3´2s }|y ´z| 2´2s ∇ y b ∇ z f } L 2 pR 3 ˆR3 q `2c 3´2s }|y ´z| 1´2s ∇ z f } L 2 pR 3 ˆR3 q `2c 3´2s }|y ´z| 1´2s ∇ y f } L 2 pR 3 ˆR3 q `12c 3´2s }|y ´z| ´2s f } L 2 pR 3 ˆR3 q .
(4.14) By Lemma 3.6, we have

}|y ´z| 2´2s ∇ y b ∇ z f } L 2 pR 3 ˆR3 q ď 2c 2s´2 }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q ,
and, as 1 ď 2s ´1 ă 3{2,

}|y ´z| 1´2s ∇ z f } L 2 pR 3 ˆR3 q ď 2c 2s´1 }|∇ y | s |∇ z | s´1 ∇ z f } L 2 pR 3 ˆR3 q " 2c 2s´1 }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q .
Analogously, we have

}|y ´z| 1´2s ∇ y f } L 2 pR 3 ˆR3 q ď 2c 2s´1 }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q .
Finally, by Lemma 3.9 and Remark 3.10, we get

}|y ´z| ´2s f } L 2 pR 3 ˆR3 q ď c 2s }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q .
Thus, when 1 ď s ă 5{4, we have

}|∇ y | s´1{2 |∇ z | s´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď C anti,s }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q
with C anti,s :" 4c 3´2s c 2s´2 `4c 3´2s c 2s´1 `24c 3´2s c 2s . Finally, by using the interpolation theory between the cases s " 0 and s " 1, we immediately obtain the conclusion: for 0 ă s ă 1,

}|∇ y | s´1{2 |∇ z | s´1{2 |y ´z| ´1f } L 2 pR 3 ˆR3 q ď C anti,s }|∇ y | s |∇ z | s f } L 2 pR 3 ˆR3 q ,
where C anti,s " C 1´s anti,0 C s anti,1 . This ends the proof.

Application to mixed regularity of eigenfunctions

This section is devoted to the proof of Theorem 2.3. Let I Ă t1, ¨¨¨, N u. Imitating the proof in [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF], we split the eigenfunction u ˚P H 1 I ppR 3 q N q of (1.2) into the high-frequency part and the low-frequency part. Denote the projector P Ω to the high-frequency part by

F x 1 ,¨¨¨,x N pP Ω uqpξq :" ½ |¨|ěΩ pξqF x 1 ,¨¨¨,x N puqpξq, u P L 2 I ppR 3 q N q, ξ " pξ 1 , ¨¨¨, ξ N q, (5.1)
where Ω is a constant such that

Ω ě 2 π C mix,α,β ? N maxtN, Zu. (5.2)
For any function u P L 2 I ppR 3 q N q, let u H :" P Ω u, u L :" p1 ´PΩ qu.

Given a functional space Y , the subspaces P Ω Y and p1 ´PΩ qY are formally defined by P Ω Y :" tu H ; u P Y u and p1 ´PΩ qY :" tu L ; u P Y u.

Then, under Assumption 2.1 on α, β, the low-frequency part u L P X I,α,β for any u P H 1 I ppR 3 q N q. Hence u ˚,L :" p1 ´PΩ qu ˚P X I,α,β . Thus to prove u ˚P X I,α,β , it suffices to prove u ˚,H :" P Ω u ˚P X I,α,β .

To prove u ˚,H P X I,α,β , we consider the following variational problem in u for the eigenvalue problem (1.2):

L I,α,β Hu, L I,α,β v H ´λ L I,α,β u, L I,α,β v H " 0 for any v H P P Ω X I,α,β , (5.3)
from which, using the fact that u " u H `uL , we deduce

L I,α,β pH ´λqu H , L I,α,β v H " ´ L I,α,β pV ne `Vee qu L , L I,α,β v H for any v H P P Ω X I,α,β . (5.4)
Obviously, when α " β " 0, L I,α,β " 1 and X I,α,β " H 1 I . Thus for α " β " 0, u ˚solves (5.3), then u ˚,H solves (5.4) with u L " u ˚,L .

Before going further, we study the properties of the variational problem (5.4).

Lemma 5.1. Under Assumption 2.1 on α, β, for any given u L P p1 ´PΩ qX I,α,β , the variational problem (5.4) admits a unique solution ψ H,α,β pu L q P P Ω X I,α,β .

Application to the hyperbolic cross space approximation

Finally, in this section, we study the hyperbolic cross space approximation and prove Theorem 2.5. We need to replace in Theorem 2.2 the operator L I,α,β by L I,α,β,τ which is defined by

L I,α,β,τ :" ź jPI ˆ1 `|∇ j | 2 τ 2 ˙α{2 ź iPI c ˆ1 `|∇ i | 2 τ 2
˙β{2 .

We consider equally the following norms }u} 0,I,α,β,τ :" }L I,α,β,τ u} L 2 ppR 3 q N q , }u} 1,I,α,β,τ :" }∇L I,α,β,τ u} L 2 ppR 3 q N q .

It is easy to see that for τ ě 1 τ ´αN }u} 0,I,α,β ď }u} 0,I,α,β,τ ď }u} 0,I,α,β , τ ´αN }u} 1,I,α,β ď }u} 1,I,α,β,τ ď }u} 1,I,α,β , while for 0 ă τ ă 1 }u} 0,I,α,β ď }u} 0,I,α,β,τ ď τ ´αN }u} 0,I,α,β , }u} 1,I,α,β ď }u} 1,I,α,β,τ ď τ ´αN }u} 1,I,α,β .

Before going further, we need the following. where C mix,α,β is defined in Theorem 2.2.

Proof. Let u τ pxq :" τ ´3N {2 upτ ´1xq and v τ pxq :" τ ´3N {2 vpτ ´1xq where x " px 1 , ¨¨¨, x N q P pR 3 q N . Let On the other hand, }u τ } 0,I,α,β " }L I,α,β u τ } L 2 I ppR 3 q N q " }L I,α,β,τ u} L 2 I ppR 3 q N q " }u} 0,I,α,β,τ (6.4) and }v τ } 1,I,α,β " }∇L I,α,β v τ } L 2 I ppR 3 q N q " τ ´1}∇L I,α,β,τ v} L 2 I ppR 3 q N q " τ ´1}v} 1,I,α,β,τ . (6.5)

Gathering together (6.2) to (6.5), Eqn. (6.1) follows.

Let I Ă t1, ¨¨¨, N u. Let u ˚P H 1 I pR 3 q N be an eigenfunction of (1.2), and let u ˚,H " P Ω u ˚, u ˚,L " p1 ´PΩ qu ˚with P Ω defined by (5.1). We consider the following variational problem: for any v H P P Ω X I,α,β , L I,α,β,τ pH ´λqu H , L I,α,β,τ v H " ´ L I,α,β,τ pV ne `Vee qu ˚,L , L I,α,β,τ v H .

( ? 2 e 5{8 }u ˚}L 2 ppR 3 q N q , }u ˚}1,I,α,β,2πΩ ď 2 ? 2 πe 5{8 Ω }u ˚}L 2 ppR 3 q N q .

Proof. The proof is in the spirit of [START_REF] Yserentant | The hyperbolic cross space approximation of electronic wavefunctions[END_REF]Theorem 9]. Under Assumption 2.1 on α, β, we have 0 ď β ď α ă 5{4 and

ź jPI ˆ1 `|ξ j | 2 |Ω| 2 ˙α ź iPI c ˆ1 `|ξ i | 2 |Ω| 2 ˙β ď exp ˆ5|ξ| 2 4|Ω| 2 ˙,
where ξ :" pξ 1 , ¨¨¨, ξ N q P pR 3 q N . Thus, by (5.1) and using the fact that u ˚,L " p1 ´PΩ qu ˚, we have 4|Ω| 2 ˙|F x 1 ,¨¨¨,x N pu ˚qpξq| 2 dξ ď e 5{4 }u ˚}2 L 2 ppR 3 q N q .

Let now τ " 2πΩ. Then Eqn. (6.10) implies }u ˚}0,I,α,β,2πΩ ď ? 2e 5{8 }u ˚}L 2 ppR 3 q N q , }u ˚}1,I,α,β,2πΩ ď 2 ? 2πe 5{8 Ω}u ˚}L 2 ppR 3 q N q .

This ends the proof.

Finally, we turn to the proof of Theorem 2.5.

Proof of Theorem 2.5. For any ξ P H I,α,β pR, Ωq c where H I,α,β pR, Ωq is defined by (2.9), we have

1 ď 1 R 2 ź iPI ˜1 `ˇˇˇ2 πξ i 2πΩ ˇˇˇ2 ¸α ź jPI c ˜1 `ˇˇˇ2 πξ i 2πΩ ˇˇˇ2 ¸β .
Thus, by Lemma 6.2, it is easy to see that }p1 ´PR,Ω I,α,β qu ˚}L 2 ppR 3 q N q ď 1 R }p1 ´PR,Ω I,α,β qu ˚}0,I,α,β,2πΩ ď 1 R }u ˚}0,I,α,β,2πΩ ď ? 2e 5{8 R }u ˚}L 2 ppR 3 q N q .

Analogously, }∇pu ˚´P R,Ω I,α,β u ˚q} L 2 ppR 3 q N q ď 1 R }u ˚}1,I,α,β,2πΩ ď 2 ? 2πe 5{8 R Ω}u ˚}L 2 ppR 3 q N q .

This ends the proof.

. 3 )

 3 This equation can also be found in[START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF] Eqns. (3.4)-(3.5)]. Also, |∇ y | ´s|y| ´t is a L 1 loc pR 3 q-function for 0 ă s ă t ă 3 and |∇ y | ´s|y| ´t " F ´1 y p|2πξ y | ´sF y p| ¨|´t qq " b t´s b 3´t p2πq s b 3`s´t b t |y| s´t . (3.4)

2 .

 2 Theorem 2.2 follows immediately from Lemma 4.1 and Lemma 4.3 with C mix,α,β " C 1,mix,α,β `C2,mix,α,β where C 1,mix,α,β and C 2,mix,α,β are given in Lemma 4.1 and Lemma 4.3 respectively.

  3 first. The proofs of Lemma 4.4 and Lemma 4.5 are postponed to Subsection 4.2.1 and Subsection 4.2.2 respectively.

4 . 2 . 1 .

 421 }u} 0,I,α,β }v} 1,I,α,β . (4.10) Combining (4.5),(4.6),(4.7) and (4.10), we finally get| L I,α,β V ee u, L I,α,β v | ď C 2,mix,α,β N 3{2 }u} 0,I,α,β }v} 1,I,α,β , with C 2,mix,α,β :" pc 1 `2C α,β`C2 α,β q{2 independent of Z, N . This ends the proof. Proof ofLemma 4.4.

Lemma 6 . 1 .

 61 Under Assumption 2.1 on α, β, we have for any u, v P X I,α,β ,| L I,α,β,τ pV ne `Vee qu, L I,α,β,τ v | ď C mix,α,β ? N maxtZ, N u}u} 0,I,α,β,τ }v} 1,I,α,β,τ ,(6.1)

  i ´τ a ν | . It is easy to see u τ , v τ P X I,α,β since u, v P X I,α,β . Then, by Theorem 2.2, we have | L I,α,β pV τ ne `Vee qu τ , L I,α,β v τ | ď C mix,α,β ? N maxtZ, N u}u τ } 0,I,α,β }v τ } 1,I,α,β . (6.2) The scaling definition yields | L I,α,β pV τ ne `Vee qu τ , L I,α,β v τ | " τ ´1 | L I,α,β,τ pV ne `Vee qu, L I,α,β,τ v | . (6.3)

ˆ1 `|ξ i | 2 |Ω| 2 dξ ď ż |ξ|ďΩ exp ˆ5|ξ| 2

 222 ˙β |F x 1 ,¨¨¨,x N pu ˚qpξq| 2

  6.6) Following the proof of Theorem 2.3 in Section 5 and under Assumption 2.1 on α, β, we know that u ˚,H is the unique solution to the variational problem(6.6).Recall that Ω ě 2 I,α,β,τ pV ne `Vee qu ˚,L , L I,α,β,τ u ˚,H | ď πΩ 2 }u ˚,L } 0,I,α,β,τ }u ˚,H } 1,I,α,β,τ . (6.8) By (6.7) and (6.8), we get πΩ 2 }u ˚,H } 0,I,α,β,τ ď 1 4 }u ˚,H } 1,I,α,β,τ ď πΩ 2 }u ˚,L } 0,I,α,β,τ . (6.9) It follows from (6.9) and the identity }u ˚}2 0,I,α,β,τ " }u ˚,L } 2 0,I,α,β,τ `}u ˚,H } 2 0,I,α,β,τ that }u ˚,H } 0,I,α,β,τ ď ? 2}u ˚}0,I,α,β,τ , }u ˚,H } 1,I,α,β,τ ď 2 ? 2πΩ}u ˚}0,I,α,β,τ . (6.10) Lemma 6.2. Let Ω ě 2 π C mix,α,β ? N maxtN, Zu be large enough. Under Assumption 2.1 on α, β, we have }u ˚}0,I,α,β,2πΩ ď

π C mix,α,β ? N maxtN, Zu and let v H " u ˚,H . Then we have | L I,α,β,τ pH ´λqu ˚,H , L I,α,β,τ u ˚,H | ě 1 4 }u ˚,H } 2 1,I,α,β,τ (6.7) and | L

Acknowledgement

The author thanks Eric Séré and Isabelle Catto for their fruitful discussions on the full work of the paper. The author also would like to thank the reviewers for their useful comments and efforts towards improving this work. This work was supported by grants from Région Ile-de-France.

Following (2.12), (3.6) and the formal identity |∇ i ||∇ i | ´1 " 1, we have

where γ k " α if k P I, and γ k " β if k P I c . By definition (2.7), }u} 2 1,I,α,β "

Thus by Lemma 3.1,

By Lemma 4.2, we have

, where s " 0 or s " γ i ě 0 and the last inequality holds since |2πξ i | s ď p1 `|2πξ i | 2 q γ i {2 . Using (4.2) again, we finally get

with C 1,mix,α,β :" C 0 `maxtC α´1 , C β´1 u independent of N, Z. This ends the proof. Now, we turn back to prove Lemma 4.2.

Proof of Lemma 4.2. When s " 0, it is just the Hardy inequality and C 0 " c 1 " 2.

When s P r´1, 0q, by (3.11), we have

where γ k " α if k P I, and γ k " β if k P I c . Then by using the formal identity

For the first term on the right-hand side of (4.4), it follows from (3.7), Lemma 3.1 and the fact

Thus, according to (4.2), we get ˇˇˇˇÿ i‰j L pi,jq

For the second term, it follows from Lemma 3.1, Lemma 4.2 and the fact γ j P r0, 3{2q that

where the last inequality holds since |2πξ j | γ j ď p1 `|2πξ j | 2 q γ j {2 . Thus,

with C α,β :" maxtC α , C β u. Analogously, for the third term, ˇˇˇˇÿ

Finally, we deal with the last term on the right-hand side of (4.4). It is the most delicate term in (4.4) and Assumption 2.1 on α, β is necessary. Before going further, we assume that γ j ď γ i . The case γ i ď γ j can be treated in the same manner.

Proof. We will prove this lemma by using Lions-Lax-Milgram's Theorem (see e.g., [15, Theorem 2.1, Chpt. III.2]). Thanks to Theorem 2.2 and (5.2), for any u H , v H P P Ω X I,α,β , we have

since }u H } L 2 ppR 3 q N q ď p2πΩq ´1}∇u H } L 2 ppR 3 q N q . Then, according to Remark 1.1, λ ă 0, and we have

Thus we get the weak coercivity: for u H , v

the continuity:

and the continuity of the right-hand side term in (5.4):

Thus by Lions-Lax-Milgram's Theorem, under Assumption 2.1 on α, β, for any given u L P p1 ṔΩ qX I,α,β , (5.4) admits a unique solution ψ H,α,β pu L q P P Ω X I,α,β . Theorem 2.3 can be immediately obtained by the following. Lemma 5.2. For any α, β satisfying Assumption 2.1 and for u L " u ˚,L , ψ H,α,β pu ˚,L q " u ˚,H is the unique solution to the variational problem (5.4). Thus u ˚,H P X I,α,β .

Proof of Lemma 5.2. Let u L " u ˚,L in (5.4). When α " β " 0, by Lemma 5.1, ψ H,0,0 pu ˚,L q is the unique solution to (5.4). On the other hand, for α " β " 0, u ˚,H solves equally (5.4) with u L " u ˚,L . Thus by the uniqueness of solution to (5.4), ψ H,0,0 pu ˚,L q " u ˚,H .

To end the proof, it suffices to prove ψ H,α,β pu ˚,L q " u ˚,H for any α, β satisfying Assumption 2.1. As the operator L I,α,β is invertible, we denote the functional space X I,´α,´β by X I,´α,´β :" tu; L ´1 I,α,β u P H 1 I ppR 3 q N qu. Thus for any v H P P Ω X I,α,β , we have L 2 I,α,β v H P P Ω X I,´α,´β . On the other hand, for any φ H P P Ω X I,´α,´β , L ´2 I,α,β φ H P P Ω X I,α,β . Let v H " L ´2 I,α,β φ H , then (5.4) can be rewritten as pH ´λqu H , φ H " ´ pV ne `Vee qu L , φ H for any φ H P P Ω X I,´α,´β .

(5.8)

Now, for any α 1 , β 1 satisfying Assumption 2.1, let ψ H,α 1 ,β 1 pu L q P P Ω X I,α 1 ,β 1 be the unique solution to (5.4) for α " α 1 and β " β 1 . Obviously, H 1 I ppR 3 q N q Ă X I,´α 1 ,´β 1 . Then P Ω H 1 I ppR 3 q N q Ă P Ω X I,´α 1 ,´β 1 . Thus thanks to (5.8), for any φ H P P Ω H 1 I ppR 3 q N q, we have pH ´λqψ H,α 1 ,β 1 pu L q, φ H " ´ pV ne `Vee qu L , φ H , which implies that ψ H,α 1 ,β 1 pu L q also solves (5.4) for α " β " 0. Then, by Lemma 5.1, for any α 1 , β 1 satisfying Assumption 2.1, ψ H,α 1 ,β 1 pu L q " ψ H,0,0 pu L q. As ψ H,0,0 pu ˚,L q " u ˚,H , we finally get ψ H,α 1 ,β 1 pu ˚,L q " u ˚,H . By Lemma 5.1, u ˚,H " ψ H,α 1 ,β 1 pu ˚,L q P X I,α 1 ,β 1 . This ends the proof.