IET Control Theory & Applications -Special issue on

Continuous-time model identification Guest editors Hugues Garnier, Torsten Söderström and Juan Yuz April 19, 2011

1 General background and motivation for the special issue

The use of system models in different aspects of control engineering and in different application areas has been increasing in recent years. This growth in the use of models to accomplish different objectives in the design of industrial control systems has been accompanied by a similar growth in the science of system identification. A mainstay of the control system modelling paradigm are continuous-time models because they arise naturally when describing the physical phenomena of systems and processes. These models of physical systems usually involve differential equations that stem from the application of physical and chemical laws. However, the best known system identification schemes have been based on discrete-time models, without much concern for the merits of natural continuous-time model descriptions and their associated identification methods.

In fact, early research on system identification focused on identification of continuoustime models from continuous-time data. Subsequently however, developments in digital data acquisition and computing technology and the concomitant sampled data led to an emphasis on the use of discrete-time system models, discrete-time control designs and discrete-time-based system identification algorithms from the mid 1960s onward. The last decade has, however, witnessed a renewed interest in the techniques for the identification of continuous-time models from sampled data. This approach to the modelling of deterministic/stochastic dynamic systems from discrete-time data has many advantages in scientific terms. It provides differential equation models that conform with models used in most scientific research. Continuous-time models are defined by a unique set of parameter values that do not depend on the sampling interval. As a consequence, there is no need for conversion from discrete to continuous-time domain, which is an essential element of indirect approaches to estimation based on discrete-time models. These continuous-time model identification methods have proven to be particularly well suited in a number of cases. As examples we can mention the case of mildly non-uniformly sampled data, dominant system modes with widely different natural frequencies (stiff systems), fast sampled data, or when the input does not respect the usual zero-order hold assumption. The first significant survey in the field of continuous-time model identification appeared in 1981 [START_REF] Young | Parameter estimation for continuous-time models -a survey[END_REF]. Subsequently, further developments in the field were tracked by Un-1 behauen and Rao [START_REF] Unbehauen | Continuous-time approaches to system identification -a survey[END_REF][START_REF] Unbehauen | A review of identification in continuous-time systems[END_REF] and Garnier et al. [2] while the latest survey was published only a few years ago in IET CTA [START_REF] Rao | Identification of continuous-time systems[END_REF]. Several books have also been dedicated to the subject of continuous-time model identification from sampled data [START_REF] Unbehauen | Identification of Continuous Systems. Systems and control series[END_REF][START_REF] Sinha | Identification of Continuous-time Systems[END_REF]. In fact, the lastest book on this topic appeared very recently in 2008 [START_REF]Identification of Continuous-time Models from Sampled Data[END_REF]. Continuous-time model has clearly now matured and several of the methods are now incorporated in the continuous-time system identification CONTSID (www.cran.uhp-nancy.fr/contsid) toolbox for use with Matlab. Clearly, continuous-time model identification has been one of the dominant directions in system identification over the last decade, as shown by the numerous special sessions organised for the IFAC world congresses in Barcelona (2002), Prague (2005) and Seoul (2008), but also for the IFAC Symposia on System Identification (SYSID) in Rotterdam (2003), Newcastle (2006) and Saint-Malo (2009). The large number of publications in this research area reflects the intensive effort devoted to the developments of the theory and applications of these techniques.

The aim of this special issue is to give a broad perspective of the state-of-the-art on the subject and to provide the community with an up-to-date account of recent advances in the continuous-time model identification field, such as for identification from non-uniformly fast sampled data, identification in closed loop, linear parameter varying (LPV) system identification, fractional order modelling, time-delay estimation and timevarying system identification.

The first two papers consider the identification of state-space models. The paper by Juan Yuz, Jared Alfaro, Juan Carlos Agüero and Graham Goodwin investigates the use of the Expectation-Maximization (EM) algorithm to identify δ-operator state-space models from non-uniformly fast sampled data.

Then, Marco Bergamasco and Marco Lovera present two subspace-based methods for the identification of continuous-time state-space models. The time-derivative problem is handled by using Laguerre filters and the methods are shown to be useful to systems operating in closed loop.

The next two papers aim at filling up gaps and focus on new developments for the refined optimal instrumental variable (RIVC) method, which has been used successfully for years in many application areas. The first contribution by Xian'en Liu, Jiandong Wang and Wei Xing Zheng establishes the convergence property of the RIVC estimates. The paper by Vincent Laurain, Roland Toth, Marion Gilson and Hugues Garnier presents the use of the RIVC method to identify input/output LPV continuous-time models when the additive measurement noise is assumed to be coloured.

Fractional order models are the main subject of the next two papers. Jean-Denis Gabano, Thierry Poinot and Houcem Kanoun consider the identification of LPV fractional order models. The proposed approach is successfully applied to the modelling of nonlinear diffusion systems. Then, the time-delay estimation issue of fractional order models is investigated in the paper by Anuj Narang, Sirish Shah and Tongwen Chen. Their method simultaneously estimates time-delays along with other model parameters in an iterative manner by solving simple linear regression equations. The proposed approach is successfully applied to identify a simulated thermal diffusion in a wall.

Identification in the frequency domain can constitute an elegant alternative to timedomain identification. The next two papers provide some new results for identifying linear continuous-time time-varying dynamical systems. First, Ai Hui Tan and Chin Leei Cham describe a frequency-domain identification technique combined with an adaptive estimation of the variable time-delay through a simple gridding method. The proposed method is successfully applied to a cooling system. Then John Lataire and Rik Pintelon present a frequency-domain least squares-based estimator for identifying linear ordinary differential equations whose coefficients vary as polynomials in time. Several consistent parameter estimation methods based on the total least squares, generalised total least squares, and weighted nonlinear least squares estimators are constructed.

In the last contribution, Céline Casenave and Gérard Montseny presents the use of the so-called diffusive representation to identify non-rational input-output convolution models.

A special issue is the result of hard work by many people. We are first very grateful to the authors who have worked hard to take into account the comments from the reviewers and to respect the time schedule. We would also like to thank the reviewers who have provided careful and detailed reviews of the submitted papers. We are convinced that all these efforts have resulted in a special issue which will constitute a useful presentation of the current trends in continuous-time model identification, and will be helpful to delineate new challenging research directions in the whole area of system identification. We wish you a pleasant and interesting reading.

  to Aug. 2004, he visited the Centre for Complex Dynamic Systems and Control, University of Newcastle, Australia. In 2006 and 2007, he hold visiting positions at different universities in Australia including the University of Newcastle, the Royal Melbourne Institute of Technology and the University of Technology in Sydney. He is a member of the IFAC Technical Committee "Modelling, Identification and Signal Processing" and of the IEEE Technical Committee "System Identification and Adaptive Control". He has also been a member of the International Program Committee for the IFAC Symposia on System Identification (SYSID) since 2005. Hugues Garnier's main research interest is related to the modelling of stochastic dynamical systems. This includes time series analysis and prediction, parameter estimation and system identification. He has been very active in the last decade to promote and develop new techniques for continuous-time model identification. He is also behind CONTSID, a Matlab toolbox for direct continuous-time model identification from sampled data. He is a member of the Editorial Board of International Journal of Control. He is the (co)editor of two books (with L. Wang): 'Identification of Continuous-time Models from Sampled Data', Springer Verlag, 2008 and 'System Identification, Environmetric Modelling and Control System Design', Springer Verlag, 2011.

•

  Torsten Söderström received the MSc degree ('civilingenjör') in engineering physics in 1969 and the PhD degree in automatic control in 1973, both from Lund Institute of Technology, Lund, Sweden. He is a Life Fellow of IEEE, and an IFAC Fellow. During 1967-1974 he held various teaching positions at the Lund Institute of Technology. Since 1974, he has been with the Department of Systems and Control, Uppsala University, Uppsala, Sweden, where he is a professor of automatic control. Dr Söderström is the author or coauthor of many technical papers. His main research interests are in the fields of system identification, signal processing, and control. He is the (co)author of four books: 'Theory and Practice of Recur-Executive Board member and Awards Committee Chair. He has been the Automatica editor for the area of System Parameter Estimation since 1992. • Juan I. Yuz received his professional title of Ingeniero Civil Electronico and Master degree in Electronics Engineering from Universidad Tecnica Federico Santa Maria (UTFSM), Chile, in 2001, obtaining the Best Electronic Engineering Student Award. He received his Ph.D. in Electrical Engineering from The University of Newcastle, Australia, in 2006. He currently holds a research position at the Department of Electronic Engineering, UTFSM where he is also involved in teaching at undergraduate and postgraduate levels. His main research areas include control and identification of linear and nonlinear sampled-data systems. He is (co)author of the book 'Linear System Analysis' (in Spanish), Pearson, Spain, and several book chapters on systems theory, robust identification, and sampled-data models.

	sive Identification', MIT Press, 1983 (with L Ljung), 'The Instrumental Variable
	Methods for System Identification', Springer-Verlag, 1983 (with P. Stoica), 'System
	Identification', Prentice-Hall, 1989 (with P. Stoica) and 'Discrete-Time Stochastic
	Systems', Prentice-Hall, 1994; second edition, Springer-Verlag, 2002. In 1981 he
	was, with coauthors, given an Automatica Paper Prize Award. Within IFAC (In-
	ternational Federation of Control) he has served in several capacities including
	Council member,


Etienne Huot

Isabelle Herlin

Nicolas Mercier

Gennady Korotaev

Evgeny Plotnikov

Observation and Modeling for Environmental Sciences

Keywords: data assimilation, motion estimation, shallow-water assimilation de données, estimation du mouvement, shallowwater

This report concerns the validation of surface velocity estimated from satellite images. The estimation is obtained with a dynamic model based on shallow-water equations. We first compare the stationary assumption to the shallow-water heuristics to justify our choice. Second, we quantify the quality of the estimation by measuring the misfit between the model output and the altimetry measures. Experiments are achieved on Sea Surface Temperature data acquired by the NOAA/AVHRR satellites over the Black Sea. The altimetry measures are obtained by two radar sensors: Envisat and GFO. The good adequacy between the shallow-water output and the altimetry data validates our motion estimation approach.

Introduction

The issue of surface velocity estimation from satellite images has been extensively studied in the literature [1,[START_REF] Nagel | Displacement vectors derived from second-order intensity variations in image sequences[END_REF][START_REF] Cohen | Optical flow and phase portrait methods for environmental satellite image sequences[END_REF][START_REF] Mémin | Optical flow estimation and object-based segmentation with robust techniques[END_REF][START_REF] Béréziat | A generalized optical flow constraint and its physical interpretation[END_REF][START_REF] Corpetti | Dense estimation of fluid flows[END_REF][START_REF] Isambert | A multiscale vector spline method for estimating the fluids motion on satellite images[END_REF]. Data Assimilation (DA) techniques have been applied in the last five years and gain importance in the scientific community [START_REF] Herlin | Estimation of a motion field on satellite images from a simplified ocean circulation model[END_REF][START_REF] Papadakis | Image assimilation for motion estimation of atmospheric layers with shallow-water model[END_REF][START_REF] Béréziat | Solving ill-posed Image Processing problems using Data Assimilation[END_REF][START_REF] Titaud | Assimilation of image sequences in numerical models[END_REF]. The key points of the DA approach are: availability of heuristics on the dynamics of a satellite sequence, knowledge on links between velocity and image data.

This paper proposes an analysis and a validation of the DA approach for motion estimation from ocean satellite images. Two Image Models were proposed in [START_REF] Korotaev | Retrieving ocean surface current by 4-D variational assimilation of sea surface temperature images[END_REF][START_REF] Huot | Assimilation of SST Satellite Images for Estimation of Ocean Circulation Velocity[END_REF][START_REF] Huot | Estimating apparent motion on satellite acquisitions with a physical dynamic model[END_REF]. They express heuristics on the dynamic of the motion field. The comparison of the estimation using the two models allows us to analyze the impact of these heuristics. The main issue of the paper is then to validate the estimation approach by evaluating the quality of the result, compared to real data.

The motion estimation is performed with NOAA/AVHRR Sea Surface Temperature (SST) data acquired over the Black Sea. The analysis is conducted by comparing the stationary and the shallow-water heuristics. The validation is obtained by quantifying the discrepancy of the water layer thickness, estimated with the shallow-water image model, and the one computed from altimetry data. The altimetry measures, used in this study, come from the Envisat and GFO sensors.

The paper is organized as follows. Section 2 summarizes the principles of variational data assimilation. The definition of the Stationary Image Model (SIM) and Shallow Water Image Model (SWIM) is given in Section 3. Section 4 describes the application of DA to perform motion estimation. Section 5 describes the SST images (5.1), displays and analyzes the estimated motion result (5.2), describes the altimetry data (5.3), and validates the approach (5.4).

Variational Data Assimilation

Mathematical setting

Let X being the state vector depending on the spatial coordinate x (x = (x, y) for image data) and time t. X is defined on A = Ω × [0, τ ], Ω being the bounded spatial domain and [0, τ ] the temporal domain.

We assume X is evolving in time according to:

∂X ∂t (x, t) + ▼(X)(x,t) = 0 (1)
▼, named the evolution model, is supposed differentiable.

Observations Y(x, t), for instance satellite image acquisitions, are available at location x and date t and linked to the state vector through an observation equation:

Y(x, t) = ❍(X)(x,t) + E O (x, t) (2)
In this paper, we assume that one component of X is directly comparable to Y.

Consequently, ❍ reduces to a projection operator. The observation error E O simultaneously represents the imperfection of the observation operator ❍ and the measurement errors.

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We consider having some knowledge on the initial condition of the state vector at t = 0:

X(x, 0) = X b (x) + E b (x) (3)
X b is named background value of the initial condition and E b the background error. E b and E O are assumed to be Gaussian and fully characterized by their covariance matrices B and R.

Variational formulation

In order to solve the system (1), ( 2), (3) with respect to X having a maximal a posteriori probability given the observations, a functional E(X) is defined and minimized:

E(X) = A [Y(x, t) -❍(X)(x,t)] T R -1 (x, t)[Y(x, t) -❍(X)(x,t)])dxdt + Ω [X(x, 0) -X b (x)] T B -1 (x)[X(x, 0) -X b (x)]dx + Reg (4)
In this formulation, we consider no correlation of the errors between two spacetime positions. Reg is a regularization term used to obtain a convex function and allow the minimization process to converge to a global minimum. The minimization of E(X) is carried out with an iterative method based on the one described in [START_REF] Béréziat | Solving ill-posed Image Processing problems using Data Assimilation[END_REF] and summarized in the following.

At each iteration k, the analysis X k a is obtained from the background X k b by computing the increment δX at t = 0.

X k a (x, 0) = X k b (x, 0) + δX(x) (5)
1.

Initialization (a) k = 0 (b) Compute X 0 b (x, t) from the initial condition X b (x) of the state vector at t = 0 in (3): X 0 b (x, 0) = X b (x) (6)
∂X 0 b ∂t (x, t) + ▼(X 0 b )(x, t) = 0, for t = 0 to τ (7) (c) Initialize the analysis X 0 a (x, t): X 0 a (x, t) = X 0 b (x, t) ∀t ∈ [0, τ ] (8)
2. Repeat (a) Compute the adjoint variable λ from t = τ to t = 0:

λ(x, τ ) = 0 (9) - ∂λ ∂t (t)+ ∂▼ ∂X * λ(t) = ❍ T R -1 [Y(t)-❍X k a ], for t = τ to 0 (10) RR n°7658
(b) Update the value of the background variable:

X k+1 b = X k a (11)
(c) Compute the incremental variable δX at t = 0:

δX(x) = B(x)λ(x, 0) (12)
(d) Update the value of the analysis variable:

X k+1 a (x, 0) = X k+1 b (x, 0) + δX(x) (13)
(e) Compute X k+1 a (x, t) from the initial condition:

∂X k+1 a ∂t (x, t) + ▼(X k+1 a )(x, t) = 0, for t = 0 to τ (14) (f) k = k + 1 Until ||δX|| 2 ≤ ε 3. Final result is X k a .
Equation ( 10) makes use of the adjoint model ∂▼ ∂X *

. In our study, the discrete adjoint model is automatically obtained by the Tapenade software 1 .

Image models

The two Image Models used in the paper are based on the assumption that a pixel value is a passive tracer transported by the surface velocity field. The state vector X includes the motion vector W and a tracer q that can be directly compared to the image observations. The evolution of q is given by the advection-diffusion equation:

∂q ∂t + W • ∇q = ν q ∆q (15)
with ν q standing for the diffusion coefficient. The Stationary Image Model (SIM) is based on the restrictive assumption that, at each position, the velocity is constant over time. The underlying hypothesis is that the surface velocity field evolves much slower than the temperature field. This heuristic is acceptable for a large range of marine processes. If a vortex, whose spatial scale is more than 10 -50km, is transported with a velocity less than 0.1 to 0.5m/s, then the temporal scale of that phenomenon will be more than one day. It means that the surface velocity field can be considered as stationary during one day. Defining X = (u, v, q) T , with u and v the two components of the 2D motion vector W, SIM is defined as:

               ∂u ∂t = 0 ∂v ∂t = 0 ∂q ∂t = -u ∂q ∂x -v ∂q ∂y + ν q ∆q (16)
1 http://www-sop.inria.fr/tropics/ RR n°7658

However, the stationary hypothesis makes this image model only applicable on a short temporal window. The shallow-water equations, derived from the Navier-Stokes equations, link the 2D velocity (u, v) of the layer to its thickness h and take into account the gravity and Coriolis forces. The state vector X is (u, v, h, q) T and the Shallow Water Image Model (SWIM) is defined as:

                       ∂u ∂t = - ∂B ∂x + v(f + ξ) + ν∆u ∂v ∂t = - ∂B ∂y -u(f + ξ) + ν∆v ∂h ∂t = - ∂hu ∂x - ∂hv ∂y ∂q ∂t = -u ∂q ∂x -v ∂q ∂y + ν q ∆q (17)
with B = gh + 1 2 (u 2 + v 2 ), g the reduced gravity, f the Coriolis parameter (depending on the latitude), ξ the vorticity (ξ = ∂v ∂x -∂u ∂y ).

Application of Data Assimilation

Data Assimilation is applied to perform motion estimation. The sequence of SST images T (x, t) is assimilated in the two models SIM or SWIM, using the incremental method described in Section 2.2.

As said in Section 2.1, the pixel value T (x, t) is directly comparable to the component q(x, t) of the state vector. The observation operator ❍ reduces to a projection operator, ❍(X(x,t)) = q(x, t). The regularization term is based on the L 2 -norm of the motion gradient (to obtain a smooth vector field) and on the motion divergency (incompressibilty assumption). Its impact is analyzed in [START_REF] Huot | Estimating apparent motion on satellite acquisitions with a physical dynamic model[END_REF]. As we consider perfect models, the value of X(t) is obtained from the initial conditions X(0) by integrating in time. Hence, the cost function (4) only depends on the initial conditions and is rewritten as:

E(X(0)) = A (T -q) T R -1 (x, t)(T -q)dxdt + Ω X(0) -X b T B -1 (x) X(0) -X b dx Ω α(|∇u| 2 + |∇v| 2 )dx + Ω β|div v| 2 dx (18)
The choice of the covariance matrix R is crucial for the quality of results. As the satellite images are provided with meta-data information (see section 5.1), the quality of the acquisitions is approximately known. R -1 (x, t) is then given a small value when the acquisition is noisy at (x, t) (because of cloud occlusion for instance). The choice of the initial background conditions has also a strong impact on the quality of the result. It has been discussed in [START_REF] Huot | Estimating apparent motion on satellite acquisitions with a physical dynamic model[END_REF] that the best results are obtained with the first observation as background for q, null value for W and a constant value h m for h, with h m being the thickness value at rest state. As the background value of q is reliable, B q is given a small value.

Results

Image data

A huge amount of images are acquired over the ocean by space remote sensors. Those obtained by optical instruments, such as Sea Surface Temperature (SST) data, display a high space-time coherence. The images, used in the paper, are acquired on-board NOAA-AVHRR satellites. Their spatial resolution is 1.1 km 2 at nadir and the temporal revisit is at best one day. However, several acquisitions over the same area are usually acquired on the same day by different satellites. Some of these data are contaminated by clouds or corrupted by noise. Figure 1

Analysis

In this paper, motion estimation is tested on a sequence of four images, displayed on Figure 2. The cyan areas, on the third and fourth frames, correspond to missing data.

The two Image Models are used to estimate the surface velocity on these data. Figure 3 compares the motion fields estimated with SIM and SWIM, at t = 0. The results obtained with SWIM visualize a cyclonic vortex on the western part of the Black Sea. SWIM, due to its physical assumptions on the dynamic, permits a more realistic motion estimation and characterizes structures occuring on the sea surface. In comparison, the potential of SIM highly depends on the size of the temporal window compared to the dynamics involved during that period. That makes SIM no more relevant for data such as those displayed in Figure 2. In conclusion, the DA approach for motion estimation permits to retrieve the major currents of the Black Sea basin. Moreover, the high resolution of NOAA/AVHRR images allows to better evaluate the size of some well known mesoscale structures [START_REF] Korotaev | Retrieving ocean surface current by 4-D variational assimilation of sea surface temperature images[END_REF].

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Altimetry data

Satellite altimeters provide an accurate measure of the Sea Level Anomaly (SLA) that corresponds to the sea surface deviation from its rest state (see the black curve on Figure 4). The altimeters are nadir-pointing instruments providing an along-track acquisition. The coverage of Envisat1 over the Black Sea is for instance displayed on Figure 5. In this paper, we use altimetry measures provided by Envisat2 with a 35 days cycle and by GFO3 with a 17 days cycle.

Validation

The outputs of SWIM are W, the surface velocity, and h the thickness of the surface layer. The thickness anomaly, denoted h SW IM , is estimated from h as its deviation from the value at rest. On another hand, the altimeters are 1dimensional instruments measuring the Sea Level Anomaly, denoted h alt , along their tracks. We then compare h SW IM and h alt at the same positions. The physical formula linking these two quantities is:

ρ × h alt = ∆ρ × h SW IM (19)
with ρ being the density of the upper layer, ∆ρ the difference of density between the upper and the lower layer, h alt the sea level anomaly measured by the satellite, h SW IM the thickness anomaly (h -h m ) of the shallow-water model. The number of altimetry measures available on the same space-time period than the SST data is rather small. However, we apply the conversion given in (19) and perform a quantitative comparison of h alt and h SW IM along a track. Figure 7 displays these curves for the two tracks displayed on Figure 6: on the left with Envisat and on the right with GFO. Black crosses locate the altimeter measures. The shapes and values of h alt and h SW IM curves are very similar. There is no error in the slope directions. The extrema are well localized. It is almost perfect in the case of Envisat. As the velocity field is strongly related to the shape of the thickness image, these promising results on thickness estimation validate the estimation of the motion. Figure 8 illustrates the link between velocity and thickness: a bump correspond to an anticyclonic velocity field and a bowl to a cyclonic one.

Conclusion

In this paper, we propose an analysis and validation of the data assimilation approach for motion estimation from satellite image sequences. We compared two dynamic assumptions, i.e. we assimilated the same data in two image models, SIM and SWIM, and analyzed motion results. Moreover, we used altimetry data to quantify the quality of the estimation. The comparison between the surface anomaly estimated by SWIM and measured by altimeters validates our approach.

  displays a SST image acquired over the Black Sea in October 2005, with the cyan color corresponding to clouds or noise.

Figure 1 :

 1 Figure 1: Cyan area corresponds to clouds or noise.

Figure 2 :

 2 Figure 2: SST data acquired from October 23th to October 24th, 2005.

Figure 3 :

 3 Figure 3: Motion estimation. Up: SIM; down: SWIM.

Figure 4 :

 4 Figure 4: Sea Level Anomaly.

Figure 5 :

 5 Figure 5: 35 days cycle of Envisat1.

Figure 6 :Figure 6

 66 Figure 6: Two altimeter tracks displayed over the average of h SW IM .

Figure 7 :

 7 Figure 7: Sea Level Anomaly, given by the altimeters, compared with the estimation with SWIM.

Figure 8 :

 8 Figure 8: From left to right. 1. 3D water layer thickness. 2. Its 2D projection. The magenta line figures the track of an altimeter. 3. SLA along this track. 4. Velocity field.

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http://envisat.esa.int

http://ilrs.gsfc.nasa.gov/satellite_missions/list_of_satellites/gfo1_general.html RR n°7658


Hua Chen
email: chenhua@whu.edu.cn

Wei-Xi Li
email: wei-xi.li@whu.edu.cn

And Chao-Jiang Xu
email: chao-jiang.xu@univ-rouen.fr

GEVREY REGULARITY OF SUBELLIPTIC MONGE-AMP ÈRE EQUATIONS IN THE PLANE

Keywords: 2000 Mathematics Subject Classification. 35B65, 35H20 Monge-Ampère equations, Gevrey regularity, subellipticity

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge-Ampère equations in the plane, under the assumption that one principle entry of the Hessian is strictly positive and an appropriately finite type degeneracy degeneracy, then we prove that the solution of the degenerate Monge-Amp'ere equation will be smooth in Gevrey classes.

Introduction

In this paper, we study the regularity problem for the real Monge-Ampère equation (1.1) det

D 2 u = k(x), x ∈ Ω ⊂ R d ,
where Ω is an open domain of R d , d ≥ 2. We consider the convex solution u of equation (1.1), then k is a nonnegative function. In the case when k > 0, the equation (1.2) is elliptic and the theory is well developed. For instance, it's shown in [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation[END_REF] that there exists a unique solution u to the Dirichlet problem for (1.1), smooth up to the boundary of Ω, provided that k is smooth and the boundary ∂Ω of Ω is strictly convex. In the degenerate case, i.e., Σ k = {x ∈ Ω; k(x) = 0, ∇k(x) = 0} = ∅.

The equation (1.1) is then a full nonlinear degenerate elliptic equation. The existence and uniqueness of the solution for the Dirichlet problem of the equation (1.1) have already been studied in [START_REF] Guan | On the Dirichlet problem for degenerate Monge-Ampère equations[END_REF]. Also in [START_REF] Hong | Existence of C ∞ local solutions for the Monge-Ampère equation[END_REF], they proved that the Monge-Ampère equation has a C ∞ convex local solution if the order of degenerate point for the smooth coefficient k is finite.

As far as the regularity problem is concerned, a result in [START_REF] Zuily | Sur la régularité des solutions non strictement convexes de l'équations de Monge-Ampère réelle[END_REF] proved that, for the degenerate Monge-Ampère equation, if the solution u ∈ C ρ for ρ > 4 (so that it is a classical solution), then u will be C ∞ smooth.

However, in general, the convex solution u to (1.1) is at most in C 1,1 if k is only smooth and nonnegative (see [START_REF] Guan | C 2 a priori estimates for degenerate Monge-Ampère equations[END_REF] for example). To get a higher regularity, some extra assumptions are needed to impose on k. This problem has been studied by P. Guan [START_REF] Guan | Regularity of a class of quasilinear degenerate elliptic equations[END_REF] in two dimension case, in which the smoothness of C ∞ for a C 1,1 solution u of the equation (1.1) is obtained, if k vanishes in finite order, i.e. k ≈ x 2ℓ + Ay 2n with ℓ ≤ n, A ≥ 0, and one principal curvature of u is strictly positive. In a recent paper [START_REF] Guan | Sawyer Regularity of subelliptic Monge-Ampère equations in the plane[END_REF], the last assumption is relaxed to the bounding of trace of Hessian from below, i.e., △u ≥ c 0 > 0. For such C ∞ regularity problem, see also earlier work of C.-J. Xu [START_REF] Xu | Opérateurs sous-elliptiques et régularité des solutions d'équations aux dérivées partielles non linéaires du second ordre en deux variables[END_REF] which is concerned with the C ∞ regularity for general two-dimensional degenerate elliptic equation. In a recent work [START_REF] Rios | A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations[END_REF], the authors extended Guan's two-dimensional result of [START_REF] Guan | Regularity of a class of quasilinear degenerate elliptic equations[END_REF] to higher dimensional case.

It is natural to ask that, in the degenerate case, would it be the best possible for the regularity of solution here to be C ∞ smooth? One may expect that, in case of coefficient k with higher regularity, the solution u would have better regularity than C ∞ smooth. we will introduce the Gevrey class, an intermediate space between the spaces of the analytic functions and the C ∞ functions. There is well-developed theory on the Gevrey regularity (see the definition later) for nonlinear elliptic equations of any order, see [START_REF] Friedman | On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations[END_REF] for instance. For the linear degenerate elliptic problem, there have been many works on the Gevrey hypoellipticity of linear subelliptic operators of second order (e.g. [START_REF] Derridj | Zuily Sur la régularité Gevrey des opérateurs de Hörmander[END_REF][START_REF] Durand | Régularité Gevrey d'une classe d'opérateurs hypo-elliptiques[END_REF] and the reference therein). The difficulty concerned with equation (1.1) lies on the mixture of degeneracy and nonlinearity.

In this paper, we attempt to explore the regularity of solutions of equation (1.1) in the frame of Gevrey class. We study the problem in two dimension case (1.2) u xx u yyu 2 xy = k(x, y), (x, y) ∈ Ω, and assume that u yy > 0, then we can apply the classic partial Legendre transformation (see [START_REF] Schulz | Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions[END_REF] for instance), to translate the equation (1.2) to the following divergence form quasi-linear equation (1.3) ∂ ss w(s, t) + ∂ t {k(s, w(s, t))∂ t w(s, t)} = 0.

This quasi-linearity allows us to adopt the idea used in [START_REF] Chen | Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations[END_REF], to obtain the Gevrey regularity for the above divergence form equation. In order to go back to the original problem, i.e., the Gevrey regularity for the equation (1.2), a key point would be to show that the Gevrey regularity is invariant under the partial Legendre transformation, which will be proved in Section 3. Now let us recall the definition of the space of Gevrey class functions, which is denoted by G σ (U ), for σ ≥ 1, with U an open subset of R d and σ being called Gevrey index. We say that f ∈ G σ (U ) if f ∈ C ∞ (U ) and for any compact subset K of U , there exists a constant C K , depending only on K, such that for all multi-

indices α ∈ Z d + , ∂ α f L ∞ (K) ≤ C |α|+1 K (|α|!) σ .
The constant C K here is called the Gevrey constant of f. We remark that the above inequality is equivalent to the following condition:

∂ α f L 2 (K) ≤ C |α|+1 K (|α|!) σ .
In this paper, both estimates above will be used. Observe that G 1 (U ) is the space of real analytic functions in U .

We state now our main result as follows, where Ω is an open neighborhood of origin in R 2 .

Theorem 1.1. Let u be a C 1,1 weak convex solution to the Monge-Ampère equation (1.2). Suppose that u yy ≥ c 0 > 0 in Ω and that k(x, y) is a smooth function defined in Ω, satisfying

(1.4) c -1 (x 2ℓ + A y 2n ) ≤ k(x, y) ≤ c(x 2ℓ + A y 2n ), (x, y) ∈ Ω where c > 0, A ≥ 0 and ℓ ≤ n are two nonnegative integers. Then u ∈ G ℓ+1 (Ω), provided k ∈ G ℓ+1 (Ω). Remark 1.2. If k is C ∞ smooth
and satisfies the condition (1.4), and u yy > 0, P. Guan [START_REF] Guan | Regularity of a class of quasilinear degenerate elliptic equations[END_REF] has proved that a C 1,1 solution of the equation (1.2) will be C ∞ smooth.

In [START_REF] Guan | Sawyer Regularity of subelliptic Monge-Ampère equations in the plane[END_REF], the assumption that u yy > 0 is relaxed to the bounding of trace of Hessian from below, i.e., △u ≥ c 0 > 0, but the assumption (1.4) is changed to A > 0 and ℓ = n. Our main contribution here is to obtain the Gevrey regularity G ℓ+1 .

Remark 1.3. The regularity result of main theorem seems the best possible, since in the particular case of ℓ = 0, we have u ∈ G 1 (Ω) (i.e., the solution is analytic in Ω), and in this case the equation (1.2) is elliptic, thus our result coincides with the well-known analytic regularity result for nonlinear elliptic equations. We can also justify that if k is independent of second variable y (then A = 0), the equation (1.3) is linear, it is known that, see [START_REF] Derridj | Zuily Sur la régularité Gevrey des opérateurs de Hörmander[END_REF], the optimal regularity result is that the solution lies in G ℓ+1 .

Remark 1.4. The extension of above result to higher dimensional cases and more general models of the Monge-Amère equations with k = k(x, u, Du) is our coming work. By using the results of [START_REF] Rios | A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations[END_REF], the idea is the same.

The paper is organized as follows: the section 2 is devoted to proving the Gevrey regularity for the quasi-linear equation (1.3). In Section 3 we prove our main result by virtue of the classic partial Legendre transformation. We prove the technical lemmas in Section 4.

Gevrey regularity of quasi-linear subelliptic equations

In this section we study the Gevrey regularity of solutions for the following quasi-linear equation near the origin of R 2

(2.1)

∂ ss w + ∂ t (k(s, w)∂ t w) = 0.
We assume that k(s, w) satisfies the condition

(2.2) c -1 (s 2ℓ + Aw 2n ) ≤ k(s, w) ≤ c(s 2ℓ + Aw 2n ),
where c > 1, A ≥ 0 are two constants and ℓ ≤ n are two positive integers. Since Gevrey regularity is a local property, we study the problem on the unit ball in R 2 , B = (s, t) s 2 + t 2 < 1 ,

and denote W = [-1, 1] × [-w L ∞ ( B) , w L ∞ ( B)
]. We prove the the following result in this section.

Theorem 2.1. Suppose that w(s, t) ∈ C ∞ ( B) is a solution to the quasi-linear equation (2.1), and that k ∈ G ℓ+1 (W ). Then w ∈ G ℓ+1 (B).

We recall some notations and elementary results for the Sobolev space and pseudo-differential operators. Let H κ (R 2 ), κ ∈ R, be the classical Sobolev space equipped with the norm

• κ . Observe • 0 = • L 2 (R 2 ) . Recall that H κ (R 2 ) is an algebra if κ > 1.
We need also the interpolation inequality for Sobolev space: for any ε > 0 and any r 1 < r 2 < r 3 ,

(2.3) h r2 ≤ ε h r3 + ε -(r2-r1)/(r3-r2) h r1 , ∀ h ∈ H r3 (R 2 ).
Let U be an open subset of R 2 and S a (U ), a ∈ R, be the symbol space of classical pseudo-differential operators. We say P = P (s, t, D s , D t ) ∈ Op(S a (U )), a pseudodifferential operator of order a, if its symbol σ(P )(s, t; ζ, η) ∈ S a (U ) with (ζ, η) the dual variable of (s, t). If P ∈ Op(S a (U )), then P is a continuous operator from H κ c (U ) to H κ-a loc (U ). Here H κ c (U ) is the subspace of H κ (R 2 ) consisting of the distributions having their compact support in U , and H κ-a loc (U ) consists of the distributions h such that φh ∈ H κ-a (R 2 ) for any φ ∈ C ∞ 0 (U ). For more detail on the pseudo-differential operator, we refer to the book [START_REF] Trèves | Introduction to pseudodifferential and Fourier integral operators[END_REF]. Remark that if P 1 ∈ Op(S a1 ), P 2 ∈ Op(S a2 (U )), then [P 1 , P 2 ] ∈ Op(S a1+a2-1 (U )). In this paper, we shall use the pseudo-differential operator Λ r = 1

+ |D s | 2 + |D t | 2 r 2 of order r, r ∈ R, whose symbol is given by σ(Λ r ) = 1 + ζ 2 + η 2 r 2 .
In the following discussions, we denote, for P ∈ Op(S a ),

P ∂ m v κ = |α|=m P ∂ α v κ and v j,U = |γ|=j ∂ γ v L ∞ (U) .
We consider the following linearized operator corresponding to (2.1) and the solution w,

L = ∂ ss + ∂ t k(s, t)∂ t • ,
where k(s, t) = k(s, w(s, t)). To simplify the notation, we extended smoothly the function k to R 2 by constant outside of B, similar for k. We have firstly the following subelliptic estimate.

Lemma 2.2. Under the assumption (2.2), for any r ∈ R, there exists C r > 0 such that

(2.4) v 2 r+ 1 ℓ+1 + ∂ s Λ r v 2 0 + k 1 2 ∂ t Λ r v 2 0 ≤ C r Lv 2 r-1 ℓ+1 + v 2 0 ,
for any v ∈ C ∞ 0 (B), where C r depends only on k j,

B , 0 ≤ j ≤ 2.
Remark 2.3. By using Faà di Bruno's formula, k j,

B is bounded by a polynomial of k i,W , w i, B , 0 ≤ i ≤ j.

Proof. Firstly, we study the case of r = 0. Observe (2.5)

∂ s v 2 0 + k 1 2 ∂ t v 2 0 = ∂ s v 2 0 + R 2 k(s, t)|∂ t v(s, t)| 2 dsdt = -(Lv, v) .
Then the assumption (2.2) implies

∂ s v 2 0 + s ℓ ∂ t v 2 0 ≤ c ∂ s v 2 0 + k 1 2 ∂ t v 2 0 = -c (Lv, v) .
Since the vector fields {∂ s , s ℓ ∂ t } satisfies the Hörmander's condition of order ℓ, we get (see [START_REF] Fefferman | Subelliptic eigenvalue problems[END_REF][START_REF] Rotschild | Hypoelliptic differential operators and nilpotent groups[END_REF])

(2.6) v 2 1 ℓ+1 ≤ C 0 ∂ s v 2 0 + k 1 2 ∂ t v 2 0 + v 2 0 = -C 0 (Lv, v) + C 0 v 2 0 .
By Cauchy-Schwarz inequality, we have proved (2.4) with r = 0. Since we have extended k to R 2 , (2.5) (2.6) also hold for any v ∈ S(R 2 ). Now for the general case, we have

∂ s Λ r v 2 0 + k 1 2 ∂ t Λ r v 2 0 = -(Λ r Lv, Λ r v) -k, Λ r ∂ t v, ∂ t Λ r v ≤ Lv 2 r-1 ℓ+1 + v 2 r+ 1 ℓ+1 -k, Λ r ∂ t v, ∂ t Λ r v . Since for v ∈ C ∞ 0 (B), we have Λ r v ∈ S(R 2 ). Then (2.6) implies v 2 r+ 1 ℓ+1 + ∂ s Λ r v 2 0 + k 1 2 ∂ t Λ r v 2 0 (2.7) ≤ C 0 Lv 2 r-1 ℓ+1 + v 2 0 -k, Λ r ∂ t v, ∂ t Λ r v .
We consider now the commutator k, Λ r , the pseudo-differential calculus give

σ k, Λ r = |α|=1 ∂ α s,t k(s, t)∂ α ζ,η σ Λ r (ζ, η) + σ(R 2 )(s, t, ζ, η), with σ(R 2 ) ∈ S r-2 (R 2 ) and |(R 2 ∂ t v, ∂ t Λ r v)| ≤ C 2 v 2 r ,
where C 2 depends only on k j,

B , 0 ≤ j ≤ 2. Thus k, Λ r ∂ t v, ∂ t Λ r v ≤ C 0 v r ∂ s k ∂ t Λ r v 0 + ∂ t k ∂ t Λ r v 0 + C v 2 r .
Moreover, note that k is nonnegative, and hence we have the following well-known inequality

|∂ s k(s, t)| 2 + |∂ t k(s, t)| 2 ≤ 4[ k ] 2,R 2 k(s, t). (2.8)
For the sake of completeness, we will present the proof of the above inequality later. By Cauchy-Schwarz inequality and interpolation inequality (2.3), one has

k, Λ r ∂ t v, ∂ t Λ r v ≤ 1 2C 0 k 1 2 ∂ t Λ r v 2 0 + v 2 r+ 1 ℓ+1 + C r v 2 0 .
Thus Lemma 2.2 follows. Now it remains to show (2.8). For any h ∈ R, the following formula holds

k(s + h, t) = k(s, t) + ∂ s k(s, t)h + 1 2 ∂ ss k(s 0 , t)h 2 , s 0 ∈ R. Observe k ≥ 0, then for all h ∈ R we get 0 ≤ k(s, t) + ∂ s k(s, t)h + 1 2 [ k ] 2,R 2 h 2 . So the the discriminant of this polynomial is nonpositive; that is, |∂ s k(s, t)| 2 ≤ 2[ k ] 2,R 2 k(s, t). Similarly |∂ t k(s, t)| 2 ≤ 2[ k ] 2,R 2 k(s, t). This gives (2.8).
Remark 2.4. With same proof, we can also prove the following estimate

v 2 r+m+ 1 ℓ+1 + |α|≤m ∂ s Λ r ∂ α v 2 0 + k 1 2 ∂ t Λ r ∂ α v 2 0 ≤ C r,m Lv 2 r+m-1 ℓ+1 + v 2 0 for any v ∈ C ∞ 0 (B).
A key technical step in the proof of Gverey regularity is to choose a adapted family of cutoff functions. For 0 < ρ < 1, set

B ρ = (s, t) s 2 + t 2 < 1 -ρ .
For each integer m ≥ 2 and each number 0 < ρ < 1, we choose the cutoff function ϕ ρ,m satisfying the following properties:

(2.9)

     supp ϕ ρ,m ⊂ B (m-1)ρ m , and ϕ ρ,m (s, t) = 1 in B ρ , sup (s,t)∈B ∂ k ϕ ρ,m ≤ C k m ρ k .
For such cut-off functions, we have the following Lemma 2.5 (Corollary 0.2.2 of [START_REF] Durand | Régularité Gevrey d'une classe d'opérateurs hypo-elliptiques[END_REF]). There exists a constant C, such that for any 0 ≤ κ ≤ 4, and any f ∈ S(R 2 ), (2.10)

∂ j ϕ ρ,m f κ ≤ C m ρ j f κ + m ρ j+κ f 0 , 0 ≤ j ≤ 2.
We prove now Theorem 2.1 by the following Proposition.

Proposition 2.6. Let w ∈ C ∞ ( B) be a smooth solution of the quasi-linear equation (2.1). Suppose k ∈ G ℓ+1 (R 2 ). Then there exists a constant L, such that for any integer m ≥ 5, we have the following estimate

ϕ ρ,m ∂ m w 2+ j ℓ+1 + ∂ s Λ 2+ j-1 ℓ+1 ϕ ρ,m ∂ m w 0 + k 1 2 ∂ t Λ 2+ j-1 ℓ+1 ϕ ρ,m ∂ m w 0 ≤ L m-2 ρ (ℓ+1)(m-3) m ρ j (m -3)! ℓ+1 , 0 ≤ j ≤ ℓ + 1, 0 < ρ < 1. (2.11)
Remark 2.7. The constant L in Proposition 2.6 depends on ℓ, w 8, B , the Gevrey constant of k, and is independent of m.

As an immediate consequence, for each compact subset K ⊂ B, if we choose ρ 0 = 1 2 dist (K, ∂B). Then ϕ ρ0,m = 1 on K for any m, and (2.11) for j = 0 yields,

∂ m w L 2 (K) ≤ L ρ ℓ+1 0 m+1 (m!) ℓ+1 , ∀ m ∈ N.
This gives u ∈ G ℓ+1 (B). The proof of Theorem 2.1 is thus completed.

The proof of Proposition 2.6 is by induction on m. We state now the following two Lemmas, and postpone their proof to the last section.

Lemma 2.8. Let k ∈ G ℓ+1 (R 2 ) and w ∈ C ∞ ( B) be a solution of equation (2.

1).

Suppose that for some N > 5, (2.11) is satisfied for any 5 ≤ m ≤ N -1, and that for some 0 ≤ j 0 ≤ ℓ, we have

ϕ ρ,N ∂ N w 2+ j 0 ℓ+1 + ∂ s Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 + k 1 2 ∂ t Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 ≤ C 0 L N -3 ρ (ℓ+1)(N -3) N ρ j0 (N -3)! ℓ+1 , ∀ 0 < ρ < 1, (2.12)
where C 0 is a constant independent of L, N . Then there exists a constant C 1 independent of L, N, such that for any 0 < ρ < 1,

(2.13) Lϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 1 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 . Lemma 2.9. Let k ∈ G ℓ+1 (R 2
) and w ∈ C ∞ ( B) be a solution of equation (2.1). Suppose that for some N > 5, (2.11) is satisfied for any 5 ≤ m ≤ N -1. Then there exists a constants C 2 independent of L, N, such that

(2.14) Lϕ ρ,N ∂ N -1 w 2+ j-1 ℓ+1 ≤ C 2 L N -3 ρ (ℓ+1)(N -4) N -1 ρ j+1 (N -4)! ℓ+1 for all 0 < ρ < 1, 0 ≤ j ≤ ℓ + 1.
Here and throughout the proof, C and C j are used to denote suitable constants which depend on ℓ, k 0,B , w 8, B and the Gevrey constant of k, but it is independent of m and L.

Proof of Proposition 2.6. The proof is by induction on m. Firstly by using (2.10), the direct calculus implies, for m = 5, all 0 < ρ < 1 and all integers j with 0

≤ j ≤ ℓ + 1, ϕ ρ,m ∂ m w 2+ j ℓ+1 + ∂ s Λ 2+ j-1 ℓ+1 ϕ ρ,m ∂ m w 0 + k 1 2 ∂ t Λ 2+ j-1 ℓ+1 ϕ ρ,m ∂ m w 0 ≤ M 1 ρ 4
with M 1 a constant depending only on k 0,B , w

m = N if it is true for all 3 ≤ m ≤ N -1.
We prove this claim again by induction on j, for 0 ≤ j ≤ ℓ + 1. Case of j = 0: We apply Remark 2.4 with

r = 2 -1 ℓ+1 , m = 1 and v = ϕ ρ,N ∂ N -1 w ∈ C ∞ 0 (B), ϕ ρ,N ∂ N w 2 2 + ∂ s Λ 2-1 ℓ+1 ϕ ρ,N ∂ N w 2 0 + k 1 2 ∂ t Λ 2-1 ℓ+1 ϕ ρ,N ∂ N w 2 0 ≤ ϕ ρ,N ∂ N -1 w 2 2+1 + ∂ s Λ 2-1 ℓ+1 ∂ 1 ϕ ρ,N ∂ N -1 w 2 0 + k 1 2 ∂ t Λ 2-1 ℓ+1 ∂ 1 ϕ ρ,N ∂ N -1 w 2 0 + C ∂ 1 ϕ ρ,N ∂ N -1 w 2 3-1 ℓ+1 ≤ C 3 Lϕ ρ,N ∂ N -1 w 2 3-2 ℓ+1 + ϕ ρ,N ∂ N -1 w 2 0 + ∂ 1 ϕ ρ,N ∂ N -1 w 2 3-1 ℓ+1
.

By the induction assumption, we use now Lemma 2.9, to get

Lϕ ρ,N ∂ N -1 w 3-2 ℓ+1 = Lϕ ρ,N ∂ N -1 w 2+ ℓ-1 ℓ+1 ≤ C 2 L N -3 ρ (ℓ+1)(N -4) N -1 ρ ℓ+1 (N -4)! ℓ+1 ≤ 2 ℓ+1 C 2 L N -3 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 .
Hence the proof will be complete if we can show that (the term ϕ ρ,N ∂ N -1 w 0 is easier to treat)

(2.15) ∂ 1 ϕ ρ,N ∂ N -1 w 3-1 ℓ+1 ≤ C 4 L N -3 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 .
Setting ρ 1 = (N -1)ρ N , then for any k ≥ 2,

ϕ ρ1,k = 1, on B ρ1 ,
which implies that ϕ ρ1,k = 1 on the Supp ϕ ρ,N ⊂ B ρ1 for any k ≥ 2. From (2.10), we have

∂ 1 ϕ ρ,N ∂ N -1 w 3-1 ℓ+1 = ∂ 1 ϕ ρ,N ϕ ρ1,N -1 ∂ N -1 w 2+ ℓ ℓ+1 ≤ C 5 N ρ ϕ ρ1,N -1 ∂ N -1 w 2+ ℓ ℓ+1 + N ρ 4-1 ℓ+1 ϕ ρ1,N -1 ∂ N -1 w 0 .
On the other hand, the induction assumption with

m = N -1, j = ℓ, 0 ≤ ρ 1 ≤ 1, yields N ρ ϕ ρ1,N -1 ∂ N -1 w 2+ ℓ ℓ+1 ≤ N ρ L N -3 ρ 1 (ℓ+1)(N -4) N -1 ρ 1 ℓ (N -4)! ℓ+1 ≤ (2e) ℓ+1 L N -3 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 . Setting now ρ1 = (N -2)ρ1 N -1 , then for any k ≥ 2, ϕ ρ1,k = 1, on B ρ1 ,
which implies that ϕ ρ1,k = 1 on the Supp ϕ ρ1,N ⊂ B ρ1 for any k ≥ 2. The induction assumption with m = N -3, j = 0, 0 ≤ ρ1 ≤ 1, yields

N ρ 4-1 ℓ+1 ϕ ρ1,N -1 ∂ N -1 w 0 = N ρ 4-1 ℓ+1 ϕ ρ1,N -1 ∂ 2 ϕ ρ1,N -3 ∂ N -3 w 0 ≤ N ρ 4-1 ℓ+1 ϕ ρ1,N -3 ∂ N -3 w 2 ≤ N ρ 4-1 ℓ+1 L N -5 ρ(ℓ+1)(N-6) 1 (N -6)! ℓ+1 ≤ C ℓ L N -5 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 ,
where we have used the fact that

3(ℓ + 1) -4 + 1 ℓ + 1 ≥ 0, ∀ ℓ ≥ 0.
Therefore, we get (2.15) with C 4 = C 5 ((2e) ℓ+1 + 2C ℓ , and finally for all 0 < ρ < 1,

ϕ ρ,N ∂ N w 2 + ∂ s Λ 2-1 ℓ+1 ϕ ρ,N ∂ N w 0 + k 1 2 ∂ t Λ 2-1 ℓ+1 ϕ ρ,N ∂ N w 0 ≤ L N -2 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 , (2.16) if we choose L ≥ 2C 1/2 3 2 ℓ+1 C 2 + C 4 .
We prove now that (2.11) is true for m = N and j = j 0 + 1 if it is true for m = N and 0 ≤ j ≤ j 0 . We apply (2.4) with r = 2 + j0 ℓ+1 and

v = ϕ ρ,N ∂ N w ∈ C ∞ 0 (B), ϕ ρ,N ∂ N w 2 2+ j 0 +1 ℓ+1 + ∂ s Λ 2+ j 0 ℓ+1 ϕ ρ,N ∂ N w 2 0 + k 1 2 ∂ t Λ 2+ j 0 ℓ+1 ϕ ρ,N ∂ N w 2 0 ≤ C 3 Lϕ ρ,N ∂ N w 2 2+ j 0 -1 ℓ+1 + ϕ ρ,N ∂ N w 2 0 .
Firstly,

ϕ ρ,N ∂ N w 2 0 ≤ ϕ ρ1,N -2 ∂ N -2 w 2 ≤ L N -4 ρ 1 (ℓ+1)(N -5) (N -5)! ℓ+1 ≤ e 2(ℓ+1) L N -4 ρ (ℓ+1)(N -5) (N -5)! ℓ+1 . Now for the term Lϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1
, we are exactly in the hypothesis of Lemma 2.8, (2.13) implies that

ϕ ρ,N ∂ N w 2+ j 0 +1 ℓ+1 + ∂ s Λ 2+ j 0 ℓ+1 ϕ ρ,N ∂ N w 0 + k 1 2 ∂ t Λ 2+ j 0 ℓ+1 ϕ ρ,N ∂ N w 0 ≤ C 1/2 3 (C 1 + e ℓ+1 )L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 .
Finally, if we choose

L ≥ max M 1 , 2C 1/2 3 2 ℓ+1 C 2 + C 4 , C 1/2 3 (C 1 + e ℓ+1 ) ,
we get the validity of (2.11) for j = j 0 + 1, and hence for all 0 ≤ j ≤ ℓ + 1. Thus the proof of Proposition 2.6 is completed.

Gevrey regularity of solutions for Monge-Ampère equations

In this section we prove Theorem 1.1. In the following discussions, we always assume u(x, y) is a smooth solution of the Monge-Ampère equation (1.2) and u yy > 0 in Ω, a neighborhood of the origin.

We first introduce the classic partial Legendre transformation (e.g. [START_REF] Schulz | Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions[END_REF]) to translate the Gevrey regularity problem to the divergence form quasi-linear equation (2.1). Define the transformation T : (x, y) -→ (s, t) by setting

(3.1) s = x, t = u y .
It is easy to verify that

J T = s x s y t x t y = 1 0 u xy u yy , and
J -1 T = x s x t y s y t = 1 0 - uxy uyy 1 uyy .
Thus if u ∈ C ∞ and u yy > 0 in Ω, then the transformations

T : Ω -→ T (Ω), T -1 : T (Ω) -→ Ω are C ∞ diffeomorphism.
In [START_REF] Guan | Regularity of a class of quasilinear degenerate elliptic equations[END_REF], P. Guan proved that if u(x, y) ∈ C 1,1 (Ω) is a weak solution of the Monge-Ampère equation (1.2) and u yy > 0 in Ω, then y(s, t) ∈ C 0,1 (T (Ω)) is a weak solution of equation

(3.2) ∂ ss y + ∂ t k(s, y(s, t))∂ t y = 0.
He proved also the smoothness of y(s, t) ∈ C ∞ (T (Ω)) and u ∈ C ∞ (Ω).

We prove now the following theorem which, together with Theorem 2.1, implies immediately Theorem 1.1.

Theorem 3.1. Let y(s, t) ∈ G ℓ+1 (T (Ω)) be a solution of equation (3.2). Assume that k(x, y) ∈ G ℓ+1 (Ω). Then u(x, y) ∈ G ℓ+1 (Ω) .
We begin with the following results, which can be found in Rodino's book [START_REF] Rodino | Linear partial differential operators in Gevrey spaces[END_REF] (page 21).

Lemma 3.2. If g(z), h(z) ∈ G ℓ+1 (U ) , then (g h)(z) ∈ G ℓ+1 (U )
, and moreover

1 g(z) ∈ G ℓ+1 (U ) if g(z) = 0. If H ∈ G ℓ+1 (Ω) and the mapping v : U -→ Ω is G ℓ+1 (U ), then H(v(•)) ∈ G ℓ+1 (U ).
We study now the stability of Gevrey regularity by non linear composition. The following result is due to Friedman [START_REF] Friedman | On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations[END_REF].

Lemma 3.3 (Lemma 1 of [START_REF] Friedman | On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations[END_REF]). Let M j be a sequence of positive numbers satisfying the following monotonicity condition:

(3.3) j! i!(j -i)! M i M j-i ≤ C * M j , (i = 1, 2, • • • , j; j = 1, 2, • • • ) with C * a constant. Let F (z,
∂ γ z ∂ i p F (z, p) ≤ C |γ|+i M |γ|-2 M i-2 ,
for all γ ∈ Z 2 + , i ∈ Z + with |γ| , i ≥ 2. Then there exist two constants C, C * , depending only on the above constants C * and C, such that for every H 0 , H 1 > 1 with H 1 ≥ CH 0 , if the smooth function ξ(z) satisfies that max z∈Ω |ξ(z)| < b and that

max z∈B ∂ β z ξ(z) ≤ H 0 , for β with |β| ≤ 1, (3.4) max z∈B ∂ β z ξ(z) ≤ H 0 H |β|-2 1 M |β|-2 , for all β ∈ Z 2 + with 2 ≤ |β| ≤ N, (3.5)
where N ≥ 2 is a given integer, then for all α ∈ Z 2 + with |α| = N, max

z∈B |∂ α z (F (z, ξ(z)))| ≤ C * H 0 H N -2 1 M N -2 .
Remark 3.4. Under the same assumptions as the above lemma, if we replace (3.4) and (3.5), respectively, by

max z∈Ω ∂ m zi ξ(z) ≤ H 0 , for m ≤ 1, max z∈Ω ∂ m zi ξ(z) ≤ H 0 H m-2 1 M m-2 , for all m ∈ Z + with 2 ≤ m ≤ N,
with 1 ≤ i ≤ 2 some fixed integer and N ≥ 2 a given integer, then

max z∈Ω ∂ N zi (F (z, ξ(z))) ≤ C * H 0 H N -2 1 M N -2 .
We prepare firstly two propositions. In the follows, let K be any fixed compact subset of Ω. Proposition 3.5. Assume that y(s, t) ∈ G ℓ+1 (T (Ω)) and k(x, y) ∈ G ℓ+1 (Ω), then the functions F m (s, t) ∈ G ℓ+1 (T (Ω)) , m = 1, 2, 3, where

F 1 (s, t) = (u xy • T -1 )(s, t) = u xy (x(s, t), y(s, t)), F 2 (s, t) = (u xx • T -1 )(s, t) = u xx (x(s, t), y(s, t)),
and

F 3 (s, t) = (u yy • T -1 )(s, t) = u yy (x(s, t), y(s, t)).
Proof. Indeed, since y(s, t) ∈ G ℓ+1 (T (Ω)) , then we conclude y s (s, t), y t (s.t) ∈ G ℓ+1 (T (Ω)) ; that is u xy (x(s, t), y(s, t)) u yy (x(s, t), y(s, t)) , 1 u yy (x(s, t), y(s, t))

∈ G ℓ+1 (T (Ω)) .

Lemma 3.2 yields that F 3 (s, t), F 1 (s, t) ∈ G ℓ+1 (T (Ω)). Moreover, the fact that k(x, y) ∈ G ℓ+1 (Ω) and x(s, t) = s, y(s, t) G ℓ+1 (T (Ω)) , implies k(s, y(s, t)) ∈ G ℓ+1 (T (Ω)), we have, in view of the equation (1.2),

F 2 (s, t) = u xx (x(s, t), y(s, t)) ∈ G ℓ+1 (T (Ω)) .
This gives the conclusion.

As a consequence of the above proposition, there exists a constant M * , depending only on the Gevrey constants of k(x, y) and y(s, t), such that for all i, j ∈ Z + with i, j ≥ 2, (3.6) max (s,t)∈T (K)

∂ i s ∂ j t F m (s, t) ≤ M i+j * (i -2)! ℓ+1 (j -2)! ℓ+1 , m = 1, 2, 3.
Proposition 3.6. Assume that y(s, t) ∈ G ℓ+1 (T (Ω)) and k(x, y) ∈ G ℓ+1 (Ω). There exists a constant M, depending only on the Gevrey constants of the functions y(s, t) and k(x, y), such that for all i ≥ 2, (3.7) max

(x,y)∈K ∂ i x u y (x, y) + max (x,y)∈K ∂ i x u x (x, y) ≤ 2 u 3,K M i-2 (i -2)! ℓ+1 .
Proof. We first use induction on integer i to show that

(3.8) max (x,y)∈K ∂ i x u y (x, y) ≤ u 3,K M i-2 (i -2)! ℓ+1 , i ≥ 2.
Obviously, (3.8) is valid for i = 2. Now assuming (3.9) max

(x,y)∈K ∂ i x u y (x, y) ≤ u 3,K M i-2 ((i -2)!) ℓ+1 , for all 2 ≤ i ≤ N
with N ≥ 2 an integer, we need to show that (3.10) max

(x,y)∈K ∂ N +1 x u y (x, y) = max (x,y)∈K ∂ N x u xy (x, y) ≤ u 3,K M N -1 (N -1)! ℓ+1 .
Observe that

F 1 = u xy • T -1 which implies u xy (x, y) = (F 1 • T )(x, y) = F 1 (x, u y (x, y)).
Thus the desired estimate (3.10) will follow if we can prove (3.11) max

(x,y)∈K ∂ N x F 1 (x, u y (x, y)) ≤ u 3,K M N -1 (N -1)! ℓ+1 .
In the following we shall apply Remark 3.4 to deduce the above estimate. Define

M j = (j!) ℓ+1 , H 0 = u 3,K , H 1 = M.
Clearly {M j } satisfies the monotonicity condition (3.3). Furthermore, (3.9) and (3.6) yield max

(x,y)∈K ∂ i x u y (x, y) ≤ H 0 , for i ≤ 1, max (x,y)∈K ∂ i x u y (x, y) ≤ H 0 H i-2 1 M i-2
, for all i with 2 ≤ i ≤ N, and max (s,t)∈T (K)

∂ i s ∂ j t F 1 (s, t) ≤ M i+j * M i-2 M j-2 , for all i, j ∈ N with i, j ≥ 2.
Then it follows from the above three inequalities that the conditions in Remark 3.4 are satisfied, with z i = x, ξ(z) = u y (x, y) and F (z, ξ(z)) = F 1 (x, u y (x, y)). This yields max

(x,y)∈K ∂ N x F 1 (x, u y (x, y)) ≤ C * H 0 H N -2 1 M N -2 = C * u 3,K M N -2 (N -2)! ℓ+1
with C * a constant depending only on M * and hence on the Gevrey constants of y(s, t) and k(x, y). Then estimate (3.11) follows if we choose M large enough such that M ≥ C * . This completes the proof of (3.8). Now it remains to prove max (x,y)∈K

∂ i x u x (x, y) ≤ u 3,K M i-2 (i -2)! ℓ+1 , i ≥ 2.
This can be deduce similarly as above. In view of (3.8) and (3.6), we can use Remark 3.4, with z = (x, y), z i = x, ξ(z) = u y (x, y) and F (z, ξ(z)) = F 2 (x, u y (x, y)), to obtain the above estimate.

End of the proof of Theorem 3.1: Now we can show u ∈ G ℓ+1 (Ω), i.e., (3.12) max

(x,y)∈K |∂ α u(x, y)| ≤ 2[u] 3,K M m-3 (m -3)! ℓ+1 , ∀ |α| = m ≥ 3,
where M is the constant given in (3.7). We use induction on m. The validity of (3.12) for m = 3 is obvious. Assuming, for some positive integer m 0 ≥ 4, (3.13) max

(x,y)∈K |∂ γ u(x, y)| ≤ 2[u] 3,K M m-3 (m-3)! ℓ+1 , ∀ 3 ≤ |γ| = m ≤ m 0 -1.
we need to show the validity of (3.12) for m = m 0 . In the following discussions, let α be any fixed multi-index with |α| = m 0 . In view of (3.7), we only need to consider the case when

∂ α = ∂ α∂ 2 y with α a multi-index satisfying |α| = m 0 -2. Observe F 3 = u yy • T -1 which implies u yy (x, y) = (F 3 • T ) (x, y) = F 3 (x, u y (x, y)). Hence ∂ α u = ∂ αu yy = ∂ α F 3 (x, u y (x, y)) , | α| = m 0 -2.
So the validity of (3.12) for m = m 0 will follow if we show that, for any | α| = m 0 -2,

(3.14) max (x,y)∈K ∂ α F 3 (x, u y (x, y)) ≤ 2[u] 3,K M m0-3 (m 0 -3)! ℓ+1 .
To obtain the above estimate, we take M j , H 0 , H 1 as in the proof of Proposition 3.6; that is

M j = (j!) ℓ+1 , H 0 = u 3,K , H 1 = M.
Then from (3.6) and the induction assumption (3.13), one has max

(x,y)∈K |∂ γ u y (x, y)| ≤ 2H 0 , for |γ| = m ≤ 1, max (x,y)∈K |∂ γ u y (x, y)| ≤ 2H 0 H m-2 1 M m-2 , for all 2 ≤ |γ| = m ≤ m 0 -2,
∂ i s ∂ j t F 3 (s, t) ≤ M i+j * M i-2 M j-2 , for all i, j ∈ N with i, j ≥ 2.
Consequently, Lemma 3.3, with z = (x, y), ξ(z) = u y (x, y), N = m 0 -2 and

F (z, ξ(z)) = F 3 (x, u y (x, y)), yields for any |α| = m 0 -2 max (x,y)∈K ∂ α F 3 (x, u y (x, y)) ≤ 2 CH 0 H m0-4 1 M m0-4 = 2 C u 3,K M m0-4 (m 0 -4)! ℓ+1 ,
where C is a constant depending only the Gevrey constants of k(x, y) and y(s, t). Thus (3.14) follows if we choose M large enough such that M ≥ 2 C. This gives validity of (3.12) for m = m 0 and hence for all m ≥ 3, completing the proof of Theorem 3.1.

Technical lemmas

In this section, we prove the technical Lemmas( Lemma 2.8 and Lemma 2.9) used in the section 2. Firstly as an analogue of Lemma 3.3, we have Lemma 4.1. Let N > 4 and 0 < ρ < 1 be given. Let {M j } be a positive sequence satisfying the monotonicity condition (3.3) and that M j ≥ ρ -j , j ≥ 0. Suppose F (s, t, p), g(s, t) are two smooth functions satisfying the following two conditions:

1) There exists a constant C such that for any j, l ≥ 2,

∂ γ s,t ∂ l p F C 4 ( B×[-b,b]) ≤ C j+l M j-2 M l-2 , ∀ |γ| = j,
where b = g 0, B and

• C 4 ( B×[-b,b]
) is the standard Hörder norm.

2) There exist two constants H 0 , H 1 ≥ 1, satisfying H 1 ≥ CH 0 with C a constant depending only on the above constant C, such that g 6, B ≤ H 0 and for any 0 < ρ * < 1 with ρ * ≈ ρ and any j, 2 ≤ j ≤ N,

ϕ ρ * ,j ∂ j g ν ≤ H 0 H j-2 1 M j-2 ,
where 1 < ν < 4 is a real number.

Then there exists a constant C * depending only on C, such that

ϕ ρ,N ∂ N F •, g(•) ν ≤ C * H 0 H N -2 1 M N -2 .
Proof. The proof is similar to Lemma 5.3 of [START_REF] Chen | The Gevrey hypoellipticity for a class of kinetic equations[END_REF], so we give only main idea of the proof here. In the proof, we use C n to denote constants which depend only on n and may be different in different contexts. By Faà di Bruno' formula,

ϕ ρ,N D α [F (•, g(•))]
is the linear combination of terms of the form

ϕ ρ,N ∂ β s,t ∂ l p F (•, g(•)) • l j=1 ∂ γj g, (4.1)
where |β| + l ≤ |α| and γ 1 + γ 2 + • • • + γ l = αβ, and if γ i = 0, D γi g doesn't appear in (4.1). Since H ν (R 2 ) for ν > 1 is an algebra, then we have

ϕ ρ,N ∂ β s,t ∂ l p F (•, g(•)) • l j=1 ∂ γj g ν ≤ ψ ∂ β s,t ∂ l p F (•, g(•)) ν • l j=1 ϕ ρ,|γj | ∂ γj g ν ,
where ψ ∈ C ∞ 0 (R 2 ) and ψ = 1 on supp ϕ ρ,N . The above inequality allows us to adopt the approach used by Friedman to prove Lemma 3.3, to get the desired estimate. Instead of the L ∞ norm in Lemma 3.3, we use H ν -norm here. But there is no additional difficulty since H ν (R 2 ) is an algebra. We refer to [START_REF] Friedman | On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations[END_REF] for more detail.

Applying the above result to the functions k(s, t) def = k(s, w(s, t)) and kw (s, t) def = k y (s, w(s, t)), we have Corollary 4.2. Let N 0 > 4 and j 0 ∈ [0, ℓ + 1] be any given integers. Suppose k(x, y) ∈ G ℓ+1 (R 2 ) and w(s, t) ∈ C ∞ ( B) satisfying that for all 5 ≤ m ≤ N 0 and all ρ with 0 < ρ < 1,

(4.2) ϕ ρ,m ∂ m w 2+ j 0 -1 ℓ+1 ≤ c * L m-2 ρ (ℓ+1)(m-3) m ρ j0 (m -3)! ℓ+1 ,
where L, c * are two constants with c * independent of L. Then there exists a constant c, depending only on the Gevrey constants of k, w, and the above constant c * , such that for all 5 ≤ m ≤ N 0 and all ρ with 0 < ρ < 1,

ϕ ρ,m ∂ m k 2+ j 0 -1 ℓ+1 + ϕ ρ,m ∂ m kw 2+ j 0 -1 ℓ+1 (4.3)
≤ cL m-2 ρ (ℓ+1)(m-3) m ρ j0 (m -3)! ℓ+1 .
Proof. We set

H 0 = c * w 8, B + 1 , H 1 = L and M 0 = 1 ρ 3 , M j = (j -1)! ℓ+1 ρ (ℓ+1)(j-1) j + 2 ρ j0 , j ≥ 1.
Then by (4.2), we have

ϕ ρ,m ∂ m w 2+ j 0 -1 ℓ+1 ≤ H 0 H m-2 1 M m-2 , 2 ≤ m ≤ N 0 .
On the other hand, the fact that

k ∈ G ℓ+1 (R 2 ), k y ∈ G ℓ+1 (R 2 ) and M j ≥ (j - 1)! ℓ+1 implies ∂ i x ∂ j y k(x, y) C 4 (Ω) + ∂ i x ∂ j y k y (x, y) C 4 (Ω) ≤ C i+j M i-2 M j-2 , ∀ i, j ≥ 2,
where C is the Gevrey constant of k. Then by Lemma 4.1, the desired inequality (4.3) will follow if we show that {M j } satisfies the monotonicity condition (3.3). For every 0 < i < j, we compute

j i M i M j-i = j! i!(j -i)! (i -1)! ℓ+1 ρ (ℓ+1)(i-1) i + 2 ρ j0 (j -i -1)! ℓ+1 ρ (ℓ+1)(j-i-1) j -i + 2 ρ j0 = 1 ρ (ℓ+1)(j-2) j! (i -1)! ℓ (j -i -1)! ℓ i(j -i) i + 2 ρ j0 j -i + 2 ρ j0 ≤ 9 j0 ρ (ℓ+1)(j-2) j! (j -2)! ℓ i(j -i) i ρ j0 j -i ρ j0 ≤ 9 ℓ+1 ρ (ℓ+1)-j0 j 2 i j0-1 (j -i) j0-1 (j -1) ℓ+1 (j + 2) j0 (j -1)! ℓ+1 ρ (ℓ+1)(j-1) j + 2 ρ j0 ≤ 9 ℓ+1 ρ (ℓ+1)-j0 j 2 j 2(j0-1) (j -1) ℓ+1 (j + 2) j0 (j -1)! ℓ+1 ρ (ℓ+1)(j-1) j + 2 ρ j0 ≤ C ℓ M j ,
where C ℓ is a constant depending only on ℓ. In the last inequality we used the fact that ℓ + 1j 0 ≥ 0. This completes the proof of Corollary 4.2.

We prove now the technical Lemmas of section 2. We present a complete proof of Lemma 2.8, but omit the proof of Lemma 2.9 since it is similar.

Proof of Lemma 2.8. We recall the hypothesis of Lemma 2.8; that is, one has (1) k ∈ G ℓ+1 (R 2 ) and Lw = 0;

(2) for some N > 5, (2.11) is satisfied for any 5 ≤ m ≤ N -1;

(3) for some 0 ≤ j 0 ≤ ℓ,

ϕ ρ,N ∂ N w 2+ j 0 ℓ+1 + ∂ s Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 + k 1 2 ∂ t Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 ≤ C 0 L N -3 ρ (ℓ+1)(N -3) N ρ j0 (N -3)! ℓ+1 . (4.4)
We want to prove

(4.5) Lϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 1 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 for all 0 < ρ < 1.
It follows from Lw = 0 that

Lϕ ρ,N ∂ α w = L, ϕ ρ,N ∂ α w + ϕ ρ,N L, ∂ α w, |α| = N.
Hence the desired estimate (4.5) will follow if we can prove that

L, ϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 1 L N -3 2ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 , (4.6) and |α|=N ϕ ρ,N L, ∂ α w 2+ j 0 -1 ℓ+1 ≤ C 1 L N -3 2ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 . (4.7)
We shall proceed to show the above two estimates by the following steps. As a convention, in the sequel we use C j to denote different constants independent of L, N.

Step 1. We claim

(4.8) ϕ ρ,m ∂ m w 0 ≤ C 1 L N -3 ρ (ℓ+1)(N -3) N ρ -2(ℓ+1) (N -3)! ℓ+1 , ∀ 3 ≤ m ≤ N.
To confirm this, we set ρ = (m-1)ρ m . Then

ϕ ρ,m ∂ m w 0 = ϕ ρ,m ∂ 2 ϕ ρ,m-2 ∂ m-2 u 0 ≤ ϕ ρ,m-2 ∂ m-2 u 2 ,
we can use (2.11) with j = 0 to compute

ϕ ρ,m-2 ∂ m-2 u 2 ≤ L (m-2)-2 ρ(ℓ+1)((m-2)-3) ((m -2) -3)! ℓ+1 ≤ C 0 L N -4 ρ (ℓ+1)(m-5) (m -5)! ℓ+1 ≤ N ρ -2(ℓ+1) C 0 L N -3 ρ (ℓ+1)(N -3) (N -3)! ℓ+1 ,
which implies (4.8) at once.

Step 2. In this step, we shall prove the following two inequalities:

(4.9) (∂ t ϕ ρ,N ) k∂ t ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 2 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 and (4.10) (∂ s ϕ ρ,N ) ∂ s ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 3 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 .
To prove the first inequality (4.9), we use (2.10) to get

(∂ t ϕ ρ,N ) k∂ t ∂ N w 2+ j 0 -1 ℓ+1 = (∂ t ϕ ρ,N ) k∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 4 N ρ k∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + N ρ 3+ j 0 -1 ℓ+1 k∂ t ϕ ρ1,N ∂ N w 0 .
Furthermore, the interpolation inequality (2.3) gives N ρ

3+ j 0 -1 ℓ+1 k∂ t ϕ ρ1,N ∂ N w 0 ≤ N ρ k∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + N ρ 4+ j 0 -1 ℓ+1 k∂ t ϕ ρ1,N ∂ N w -1 ≤ N ρ k∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + C 5 N ρ 4+ j 0 -1 ℓ+1 ϕ ρ1,N ∂ N w 0 ≤ N ρ k∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + C 6 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 ,
where we have used (4.8) and Λ -1 k∂ t is bounded in L 2 . On the other hand, note that

k∂ t ϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ k∂ t Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 + k, Λ 2+ j 0 -1 ℓ+1 ∂ t ϕ ρ,N ∂ N w 0 ≤ C 7 k 1 2 ∂ t Λ 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N w 0 + ϕ ρ,N ∂ N w 2+ j 0 ℓ+1
, which together with (4.4) yields:

k∂ t ϕ ρ,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 8 L N -3 ρ (ℓ+1)(N -3) N ρ j0 (N -3)! ℓ+1 ,
and hence we obtain the desired inequality (4.9), combining the above inequalities. Similar arguments can be applied to prove (4.10). This completes the proof.

Step 3. We now claim that

(∂ ss ϕ ρ,N ) ∂ N w 2+ j 0 -1 ℓ+1 + (∂ tt ϕ ρ,N ) k ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 9 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3) ℓ+1 . (4.11)
To confirm this, we use (2.10) to get

(∂ ss ϕ ρ,N ) ∂ N w 2+ j 0 -1 ℓ+1 + (∂ tt ϕ ρ,N ) k ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 10 N ρ 2 ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + N ρ 4+ j 0 -1 ℓ+1 ϕ ρ1,N ∂ N w 0 .
The interpolation inequality (2.3) yields

N ρ 2 ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ N ρ ϕ ρ1,N ∂ N w 2+ j 0 ℓ+1 + N ρ 2(ℓ+1)+j0+1 ϕ ρ1,N ∂ N w 0 .
The above two inequalities, together with (4.4) and (4.8), give the desired estimate (4.11) at once.

Step 4. Now we are ready to prove (4.6), the estimate on the commutator of L with the cut-off function ϕ ρ,N . Firstly, one has

L, ϕ ρ,N =2 (∂ s ϕ ρ,N ) ∂ s + (∂ ss ϕ ρ,N ) + 2 (∂ t ϕ ρ,N ) k∂ t + (∂ tt ϕ ρ,N ) k + (∂ t ϕ ρ,N ) ∂ t k . Observe that (∂ t ϕ ρ,N ) ∂ t k ∂ N w 2+ j 0 -1 ℓ+1 ≤C 11 N ρ ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 + N ρ 3+ j 0 -1 ℓ+1 ϕ ρ1,N ∂ N w 0 ,
hence from (4.4) and (4.8), we have

(∂ t ϕ ρ,N ) ∂ t k ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 12 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3) ℓ+1 .
Together with (4.9), (4.10) and (4.11), this yields the desired estimate (4.6) at once.

Step 5. In this step we shall deal with the non linear terms, and prove (4.12)

ϕ ρ,N ∂ t ∂ N k 2+ j 0 -1 ℓ+1 ≤ C 13 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 . Recall ϕ ρ,N ∂ t ∂ N k 2+ j 0 -1 ℓ+1 = |α|=N ϕ ρ,N ∂ t ∂ α k 2+ j 0 -1 ℓ+1
. Leibniz's formula gives, for any α with |α| = N,

ϕ ρ,N ∂ t ∂ α k = 1≤|β|≤|α| α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w + ϕ ρ,N kw ∂ t ∂ α w = 5≤|β|≤|α|-4 α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w + ϕ ρ,N kw ∂ t ∂ α w + R α with kw (s, t) = k w (s, w(s, t))) and R α = 1≤|β|≤4 α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w + |α|-3≤|β|≤|α| α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w . Since H κ (R 2 ), κ > 1 is an algebra, we have |α|=N 5≤|β|≤|α|-4 α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 ≤ |α|=N 5≤|β|≤|α|-4 α β ϕ ρ1,|β| ∂ β kw 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 ≤ N -4 i=5 N ! i!(N -i)! ϕ ρ1,i ∂ i kw 2+ j 0 -1 ℓ+1 ϕ ρ,N ∂ N -i+1 w 2+ j 0 -1 ℓ+1 .
We can use (4.3) in Corollary 4.2, to get for each i with 5

≤ i ≤ m ϕ ρ1,i ∂ i kw 2+ j 0 -1 ℓ+1 ≤ C 14 L i-2 ρ 1 (ℓ+1)(i-3) i ρ 1 j0 (i -3)! ℓ+1 ≤ C 15 L i-2 ρ (ℓ+1)(i-3) i ρ j0 (i -3)! ℓ+1 .
Observing Ni + 1 ≤ N for each i ≥ 1, we use (4.8) and the induction assumptions (2.11) and (4.4), to compute

ϕ ρ,N ∂ N -i+1 w 2+ j 0 -1 ℓ+1 ≤ C ϕ ρ1,N -i+1 ∂ N -i+1 w 2+ j 0 -1 ℓ+1 + N ρ 2+ j 0 -1 ℓ+1 ϕ ρ1,N -i+1 ∂ N -i+1 w 0 ≤ ϕ ρ1,N -i+1 ∂ N -i+1 w 2+ j 0 -1 ℓ+1 + C 16 L N -i-1 ρ (ℓ+1)(N -i-2) N -i + 1 ρ j0 (N -i -2)! ℓ+1 ≤ ϕ ρ1,N -i+1 ∂ N -i+1 w 2+ j 0 ℓ+1 + C 16 L N -i-1 ρ (ℓ+1)(N -i-2) N -i + 1 ρ j0 (N -i -2)! ℓ+1 ≤ C 17 L N -i-1 ρ (ℓ+1)(N -i-2) N -i + 1 ρ j0 (N -i -2)! ℓ+1 . Then |α|=N 5≤|β|≤|α|-4 α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 ≤ 5≤i≤N -4 N ! i! (N -i)! C 15 L i-2 ρ (ℓ+1)(i-3) i ρ j0 (i -3)! ℓ+1 × C 16 L N -i-1 ρ (ℓ+1)(N -i-2) N -i + 1 ρ j0 (N -i -2)! ℓ+1 ≤ C 18 L N -3 ρ (ℓ+1)(N -5) N ρ 2j0 5≤i≤N -3 N ! i 3 (N -i) 2 (i -3)! ℓ (N -i -2)! ℓ ≤ C 18 L N -3 ρ (ℓ+1)(N -4) N ρ j0+1 5≤i≤N -4 (N -5)!N 5+(j0-1) i 3 (N -i) 2 (N -5)! ℓ ≤ C 19 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 5≤i≤N -4 N 4+j0+1 N 2(ℓ+1) i 3 (N -i) 2 ≤ C 19 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 5≤i≤N -4 N 2 i 3 (N -i) 2 .
Here the last inequality holds since 4 + j 0 -2(ℓ + 1) ≤ 2. Moreover, observing that the series 5≤i≤N -4 In the last inequality we used the interpolation inequality (2. the last inequality following from (4.4) and (4.8). The proof is thus completed.

N 2 i 3 (N -i) 2
Step 6. Now we prepare to prove the inequality (4.7), the estimate on the commutator of L with the differential operator ∂ α . Direct verification yields .

L, ∂ α w = - 0<β≤α α β ∂ t ∂ β k ∂ t ∂ α-β w - 0<β≤α α β ∂ β k ∂ tt ∂ α-β w .

So

For S 1 , we have treated the term of β = α by (4.12), and the terms of 0 < β < α can be deduced similarly to (4.13); this gives

S 1 ≤ C 28 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! (ℓ+1) .
For S 2 , we have treated the term of |β| = 1 by (4.14), and the terms of 2 ≤ |β| ≤ |α| can be deduced similarly to (4.13); this gives also

S 2 ≤ C 29 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! (ℓ+1) .
This complete the proof of Lemma 2.8.

  p) be a smooth function defined on Ω×(-b, b) ⊂ R 2 ×R satisfying that, for some constant C, max (z,p)∈Ω×(-b,b)

  and max (s,t)∈T (K)

. 1 ℓ+1= 1 ℓ+1∂ϕϕ

 11 is dominated from above by a constant independent of N , then we get|α|=N 5≤|β|≤|α|-4 α β ϕ ρ,N ∂ β kw ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 ≤ C 20 L N -3 ρ (ℓ+1)(N -3) N ∂ β kw ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 (4.13) ≤ C 22 L N -3 ρ (ℓ+1)(N -3) N ρ j0+1 (N -3)! ℓ+1 . Observe ϕ ρ,N ∂ t ∂ N k N ∂ β kw ∂ t ∂ α-β w 2+ j 0 -1 ℓ+1 + ϕ ρ,N kw ∂ t ∂ N w 2+ j 0 -1 ℓ+1So to get the desired estimates (4.12) it remains to estimate the last term above. Direct calculations yield thatϕ ρ,N kw ∂ t ∂ N w 2+ j 0 -ϕ ρ,N kw ∂ t ϕ ρ1,N ∂ N w 2+ j 0 -1 ℓ+1 ≤ C 23 kw Λ 2+ j 0 -1 ℓ+1 ∂ t ϕ ρ1,N ∂ N w 0 + kw , Λ 2+ j 0t ϕ ρ1,N ∂ N w 0 + N ρ 2+ j 0 -1 ℓ+1 kw ∂ t ϕ ρ1,N ∂ N w 0 ≤ C 24 kw Λ 2+ j 0 -1 ℓ+1 ∂ t ϕ ρ1,N ∂ N w 0 + ϕ ρ1,N ∂ N w 2+ j 0 -1 ρ1,N ∂ N w 1 . ≤ C 25 kw Λ 2+ j 0 -1 ℓ+1 ∂ t ϕ ρ1,N ∂ N w 0 + ϕ ρ1,N ∂ N w 2+ j 0 -1 ρ1,N ∂ N w 0 .

1 2 2 , 1 ℓ+1 ≤ C 26 k 1 2 Λ

 1212 3). Combining the fact that |k w (s, w)| ≤ C sup w∈R |k ww (s, w)| (k(s, w)) 1which can be deduced from the nonnegativity of k(s, w), we obtainϕ ρ,N kw (s, w)∂ t ∂ N w 2+ j 0 -2+ j 0 -1 ℓ+1 ∂ t ϕ ρ1,N ∂ N w 0 + ϕ ρ1,N ∂ N w 2+ j 0

(4. 15 ) 1 ℓ+1≤ S 1 + S 2 with S 1 ,

 15121 |α|=N ϕ ρ,N L, ∂ α w 2+ j 0 -S 2 given by S 1 = |α|=N 0<β≤α α β ϕ ρ,N ∂ t ∂ β k ∂ t ∂ α-β w N ∂ β k ∂ tt ∂ α-β w 2+ j 0 -1 ℓ+1