Kidiyo Kpalma
email: kidiyo.kpalma@insa-rennes.fr

Joseph Ronsin

Multiscale contour description for pattern recognition

Keywords: pattern analysis and recognition, multiscale smoothing, scale-space analysis, curvature zero-crossing, intersection point map

In this paper, we present a new method and its preliminary results within the context of pattern analysis and recognition. This method is based on the multiscale analysis of a curve and deals with the contour of planar objects. Our method uses a low-pass Gaussian kernel to gradually smooth the contour by decreasing the filter bandwidth. Applying gain control to the smoothed contour stretches it to the same scale as the original one so that both contours intersect. By varying the bandwidth and marking all the intersection points between the smoothed contour and the original one we can generate the Intersection Points Map (IPM) function. The initial results obtained by applying this method to various contours appears to indicate that the IPM function has some very interesting properties within the context of pattern recognition. It is translation and rotation insensitive and also scale change resistant for a large range of scaling. The IPM function generated when applied to noisy contours shows that the method is resistant to noise for a range of noise energy. Applying the features extracted by this method to retrieve a pattern from a database confirms the efficiency of the method.

Introduction

Within the context of automatic pattern recognition, it is necessary to have a simple, fast and efficient algorithm especially when dealing with large databases. This algorithm must be resistant to rotation, translation, scale change and noise. Many techniques have been described in the literature related to this topic, however pattern feature extraction, which is the main problem in pattern recognition, is complex and an enormous amount of parameters are required to measure [START_REF] Zhang | Review of shape representation and description techniques[END_REF][START_REF] Sossa | Object Recognition, Summer School on Image and Robotics[END_REF]: shape area, elongation, moment invariants, number of holes, colour and texture… It is well known that objects, especially natural ones, consist of a more or less large range of scales; and that the aspect of the object can change from one scale to another. Faced with this situation, it is very difficult to significantly describe a pattern using only one meaningful scale. To overcome this problem, increasingly more pattern analysis (or pattern description) techniques are based on multiscale or multiresolution representation methods [START_REF] Lindeberg | Principles for Automatic Scale Selection[END_REF][START_REF] Lindeberg | Scale-Space Theory in Computer Vision[END_REF]. Within this context, methods based on the pattern itself [START_REF] Kadyrov | Object descriptors invariant to affine distortions[END_REF][START_REF] Torres-Méndez | Translation, Rotation, and Scale-Invariant Object Recognition[END_REF][START_REF] Grigorescu | Distance Sets for Shape Filters and Shape Recognition[END_REF][START_REF] Belongie | Shape matching and object recognition using shape contexts[END_REF] exist as well as methods based on pattern contour behaviour [START_REF] Matusiak | Planar Closed Contour Representation by Invariant Under a General Affine Transformation[END_REF][START_REF] Wang | Multiscale curvature-based shape representation using Bspline wavelets[END_REF][START_REF] Roh | 2-D object recognition using invariant contour descriptor and projective refinement[END_REF][START_REF] Bruckstein | Recognizing objects using scale space local invariants[END_REF][START_REF] Mokhtarian | Efficient and Robust Retrieval by Shape Content through Curvature Scale Space[END_REF][START_REF] Mokhtarian | A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves[END_REF][START_REF] Bruckstein | Similarity invariant signatures for partially occluded planar shapes[END_REF][START_REF] Latecki | Shape Dscriptors for Non-rigid Shapes with a Single Closed Contour[END_REF][START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF]. This study deals exclusively with pattern contour methods. In this paper, we present the initial results of a new method (MSGPR: A Multi-Scale curve smoothing for Generalised Pattern Recognition) [START_REF] Kpalma | A Multi-Scale curve smoothing for Generalised Pattern Recognition (MSGPR)[END_REF] within the context of pattern analysis for pattern recognition. This scale-space [START_REF] Matusiak | Planar Closed Contour Representation by Invariant Under a General Affine Transformation[END_REF][START_REF] Wang | Multiscale curvature-based shape representation using Bspline wavelets[END_REF][START_REF] Bruckstein | Recognizing objects using scale space local invariants[END_REF][START_REF] Mokhtarian | Efficient and Robust Retrieval by Shape Content through Curvature Scale Space[END_REF][START_REF] Lindeberg | Scale-Space Theory in Computer Vision[END_REF][START_REF] Mokhtarian | A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves[END_REF][START_REF] Bruckstein | Similarity invariant signatures for partially occluded planar shapes[END_REF] method is based on multiscale smoothing of a planar pattern contour. According to its framework, this method is totally translation insensitive. As we showed in the initial studies [START_REF] Kpalma | A Multi-Scale curve smoothing for Generalised Pattern Recognition (MSGPR)[END_REF], the MSGPR method is also rotation insensitive and robust against scale change for a large range of scaling. More sophisticated methods exist that deal with local invariant features, associated with planar curve smoothing: this enables to recognise occulted patterns by analysing the non-occulted part of their contour [START_REF] Bruckstein | Similarity invariant signatures for partially occluded planar shapes[END_REF]. The main weakness of the MSGPR approach is that it fails to handle occulted contours and contours having undergone a non-rigid deformation. Nevertheless, when concerned by non-occulted planar contours, MSGPR method is advantageous in that it is simple and easy to apply.

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The preliminary results indicate that the method is also resistant to additive noise. By comparing pattern retrieval results obtained with this method with those provided by an adapted-CSS method, we are able to say that our method brings a positive point in the description of planar forms. After this introduction, in the next section, we recall the method structure. In section 3, we present some of the main properties of the IPM function and we then go on to propose some interesting attributes that can be extracted from the IPM function for pattern characterisation. Section 4 concentrates on the application results within the context of pattern recognition by using the proposed attributes. Finally, in section 5, we present our conclusions together with some discussions.

Description of the MSGPR method

The framework of this method can be broken down into four main stages as follows (see Fig. 1):

1. The input contour is separated into two parameterised functions, 2. Both functions are low-pass filtered (smoothed), 3. Gain control is then applied to both filtered functions so that the corresponding smoothed contour has the same scale as the input one, 4. Finally, the intersection points map (IPM) is constructed by detecting the intersection points of the input contour and the smoothed gain-controlled one.

Pre-processing

Before applying the MSGPR method, the input contour is normalised in relation to the contour length. This pre-processing method consists of the following three tasks:

• Contour resampling. The contour is resampled so that there are 360 equally spaced points. This is achieved by applying linear interpolation to the original contour points: if the number of points is greater than 360, this corresponds to sub-sampling. Using 360 equally spaced points produces points with a space of 1° between two consecutives points, in the circle limit case: of course, any other number of points can be chosen. • Scale changing. Since the input contour points are pixels, their coordinates must be integer numbers.

We therefore apply a scale factor to the resampled contour to generate a contour with 360 equally spaced points with approximately one unit distance between two consecutive points. The coordinates of these points are rounded up to integer values. After this pre-processing, the subsequent operations are real-valued so the results are no more rounded up to integer values.

• Rotation compensation so that the major principal axis is aligned with the x-axis and the point furthest from the centre of gravity is on the right: the centre of this rotation is set to the centre of gravity of the contour. This defines the P 0 starting point for subsequent processing (see Fig. Fig. 2). NB: In the following sections, the expression "input contour" will be used to describe the pre-processed contour.

Coordinate separation

The input contour is represented by a series of points defined by their (x,y) coordinates. First, the input contour is separated into two functions x(u) and y(u) which are functions of the normalised curvilinear u parameter that varies from 0 to 2π relative to the curve length (see Fig. 3). Each point of the curve is then represented by its coordinates

)) u ( y ), u ( x ( .

For each contour, the origin of the u parameter is set to the P 0 starting point on the x-axis as defined previously. The points are then reorganised in a counter clockwise direction as illustrated in Fig. 3. Since the contour is a closed curve, the two functions are periodic with period T=2π relative to the u parameter.

Curve smoothing

Functions x(u) and y(u) are then gradually smoothed by decreasing the filter bandwidth. Similarly to the curvature scale space (CSS) method [START_REF] Matusiak | Planar Closed Contour Representation by Invariant Under a General Affine Transformation[END_REF][START_REF] Bruckstein | Recognizing objects using scale space local invariants[END_REF][START_REF] Mokhtarian | A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves[END_REF] or other scale-space methods and according to the arguments stated in [START_REF] Lindeberg | Scale-Space Theory in Computer Vision[END_REF], smoothing is based on the Gaussian filters g(σ,u):

3 of 33 2 2 2 u e 2 1 ) u , ( g σ - π σ = σ (1)
where σ is the standard deviation of the Gaussian kernel. The frequency response of the filter is then given by G(σ, f)

2 2 2 f 2 e ) f , ( G σ π - = σ (2)
where f denotes the frequency variable. Since the bandwidth is conversely proportional to σ, it is clear that the bandwidth decreases as σ increases.

Thus the filter cuts increasingly lower so that the output functions move towards their mean values when σ tends towards infinity. Fig. 4.a shows the filtered functions for σ=10 and σ=180 with their corresponding smoothed contours. Due to this filtering process, the X(σ,u) and Y(σ,u) functions are attenuated so that the reconstructed smoothed contour does not have the same scale as the original one (see Fig. 4.a).

Gain control

After low-pass filtering, the gain control system stretches the output contour so that it reaches the same scale as the input one and so that both curves intersect at certain points. The smoothed gain-controlled contour is then given by the coordinates

)) u , ( Y ), u , ( X ( GC GC σ σ defined by [ ] ) ( X ) ( X ) u , ( X A ) u , ( X G G x GC σ + σ - σ = σ (3)
[ ]

) ( Y ) ( Y ) u , ( Y A ) u , ( Y G G y GC σ + σ - σ = σ (4) where )
) ( Y ), ( X ( G G σ σ
are the coordinates of the centre of gravity of the smoothed contour. The (A x and A y )

gain control coefficients can be determined in different ways and we can look out for the optimal one. For the results presented in this paper, A x and A y coefficients are defined as follows:

∑ ∑ = = - - = N n G n N n G n x ) ( X ) u , ( X x ) u ( x A 1 1 σ σ and ∑ ∑ = = - - = N n G n N n G n y ) ( Y ) u , ( Y y ) u ( y A 1 1 σ σ (5)
where {u n } are the N (N=360) samples of the u parameter. Fig. 4.b shows the smoothed contour after the gain control system. The input and smoothed gain-controlled contours are now on the same scale so that they can intersect.

By increasing σ, the output gain-controlled contour moves towards a convex curve and has some intersection points with the input contour. By marking these intersection points for each σ, we obtain the intersection points map (IPM) function defined below which characterises the pattern.

Definition of the IPM function

After the gain control system, the IPM function is generated as follows. For each σ value, we define a function which is an image in the (u,σ) plane so that (see Fig. 5):

• 0 ) , u ( IPM = σ (black) if the )) u ( y ), u ( x (
point is an intersection point between the original curve and the filtered gain-controlled one,

• 1 ) , u ( IPM = σ (white) if point )) u ( y ), u ( x (
is not an intersection point.

Fig. 5.a shows the input and the smoothed gain-controlled one (that corresponds to σ=180). This figure also represents the IPM points for the corresponding σ. As σ increases, the number of IPM points decreases and we obtain "vertically" oriented lines or bands with variable widths. These remaining points constitute the points referred to as "limit IPM points". On Fig. 5.b, we represent the IPM function for σ, varying from 0 to 180, the IPM points indicated by ( 1) to ( 6) refer to the corresponding intersection points on Fig. 5.a. Notice that for small σ values corresponding to large bandwidth filters, most of the contour points are IPM points.

By analysing Fig. 5 and Fig. 6, it appears that low values of σ produce fine details on a contour and high values produce coarse structures. In this way, we can say that high frequency information lies in the lower part of the IPM function since low frequency behaviour is found in its higher part. In other words, we can say that the IPM function contains all the information concerning the pattern from finer scales (details) to coarser scales. As σ increases, details are eliminated from the contour which becomes smoother and smoother so that the remaining intersection points can reasonably be expected to be significantly representative of the pattern being studied.

We also notice that the concave and convex parts of a contour are characteristic of the IPM function as illustrated on Fig. 6. In this figure, the convex parts labelled (a), (b) and (c) on the contour lead to the structures labelled (A), (B) and (C), respectively, on the IPM function.

Pattern characterisation

To show that this function is characteristic of the contour being studied, Fig. 7 presents two other patterns and their IPM functions. Fig. 7.c represents the IPM function of the letter "M" (Fig. 7.a) and Fig. 7.d shows the IPM function of the letter "W" (Fig. 7.b). As can be seen, these IPM functions are, somewhat, topologically similar. However, in terms of the distance between two consecutive IPM points at scale σ, we notice that these IPM functions are different from each another and also different from those presented in Fig. 5. This shows that the IPM function is a good candidate to generate features for pattern recognition.

In the following sections, we will observe the behaviour of the IPM function regarding transformations such as scale change and rotation. We will then propose some features that can be derived from the IPM function.

The IPM function and scaling

In this section, we present the behaviour of the IPM function in relation of the scaling. The scaling transformation is obtained by applying a multiplicative coefficient to the contour coordinates. Fig. 8 shows that the IPM function is scale insensitive for a large range of scales. In this figure, the multiplicative coefficient next to the IPM function represents the scaling factor applied to the corresponding contour.

According to the initial results obtained by applying the MSGPR method to the contour, the amount of limit IPM points remains constant and the IPM function does not change. For scale factors higher than the unit, the (entire) IPM function remains unchanged.

The IPM function and rotation

Fig. 9 shows a pattern (see Fig. 9.a) and its rotated copies (see Fig. 9.b through Fig. 9.f). The rotation angles are 60°, 120°, 180°, 240° and 300° counter-clockwise. As can be seen in these figures, all IPM functions are topologically identical, except for a "circular" delay. In this way, rotating the contour around its centre of gravity is equivalent to a circular delay in the IPM function. By analysing the IPM functions obtained from a contour and its rotated copy, we can directly read the rotation angle between both patterns. This is illustrated in Fig. 10. Since the u parameter is normalised to 2π, we can locate an IPM point A in one IPM function, determine its u 0 abscissa and then locate the corresponding IPM point A' in the second IPM function and also determine its abscissa u 1 . The γ rotation angle between both contours is then determined by the difference between both abscissa:

0 1 u u - = γ
. Since the pre-processing method aligned the input contour with its major principal axis, the information related to the orientation is not available for subsequent processing. Therefore, to be able to directly read the rotation angle, we need to keep the compensated angle in mind during the preprocessing stage.

Feature definition and selection

With this IPM function, we can predetermine a value for σ and use the number N(σ) of IPM points as characterisation features. Since the number of IPM points is circular-delay insensitive, the two patterns in Fig. 7 are likely to be identical. To differentiate them, we need to use certain other features such as those suggested below. By analysing the IPM functions obtained earlier, we notice that certain measurements remain invariant regardless of the transformation. These measurements can be used to characterise a pattern for recognition purposes. At present, we can suggest some heuristic features such as:

• The number of IPM limits points (those corresponding to the highest value of σ),

• The set of distances between two consecutive IPM points at a given value of σ,

• The minimum and maximum distances between two consecutive IPM limit points,

• Measurements related to convex/concave parts of irregular objects. As mentioned earlier (see Fig. 6), the IPM function corresponding to irregular objects produces certain particular noticeable structures that can produce some interesting features such as their height, area and width at a fixed value of σ, • The ratio of arclength of arcs between two consecutive IPM points on the original contour over the Cartesian distance between both points for a given value of σ.

Application of the MSGPR method to pattern recognition

To evaluate the IPM function within the context of pattern recognition, we apply it to a pattern from a database. This database consists of 1,167 different contours plus 100 transformed copies of the same contour (kk707) and 25 transformed copies of 25 other contours. Transformations applied to the contours are affine transformations which are a combination of a rotation around the centre of gravity and scaling. Table I shows a set of ten (R i=0,1,…,9 ) rotation angles and a set of ten (H i=0,1,…,9 ) scaling factors. With these references, RmHnXYZ is obtained after applying an R m rotation angle and an H n scaling factor to the XYZ pattern. By way of example (see Table I), the R7H3kk707 pattern is generated by applying a 270° rotation angle and a 0.75 scaling factor to the kk707 contour of the database. Fig. 11 provides a sample of the contours contained in our test database. Note that during the transformation procedure, the output point coordinates are all integer values. This enables us to deal with points as if they were pixels produced by edge detection in a bitmap image. Of course, as we will see below, this is rather unfavourable to our approach.

Attribute definition and selection

We can now describe and determine the features based on the IPM function that we used for our evaluation tests. To define these characterisation features, we first set σ to σ 0 value (e.g. σ 0 = 180). Then, for each pattern,

• We consider the IPM points (at the set σ 0 value of σ) and select two consecutive p a and p b points which are, circularly, the furthest apart in the IPM function as illustrated in Fig. 12, • We determine the distance between both points to produce the first d 1 component of the V 0 attribute vector,

• The next components of V 0 are distances coming after d 1 :

V 0 = (d 1 , d 2 ,…, d N-1 , d N ) (6)
Thus, the V 0 attribute vector is of dimension N where N is the number of IPM points at scale σ 0 . Using the most distant IPM points to define the attribute vector enables us to bypass the problem of correspondence between maxima of the interval tree as specified in [START_REF] Mokhtarian | Efficient and Robust Retrieval by Shape Content through Curvature Scale Space[END_REF].

To benefit from multi-scale information of the IPM function, we can define a set of M values of σ (σ 0 , σ 1 , ...,σ M-1 ) and determine the V i attribute vectors (i=0, 1, 2,…, M-1) corresponding to the σ i scales (see equation 6). The V global attribute vector is then produced by a concatenation of the individual V i scale vectors as follows:

V = (V 0 , V 1 ,…, V M-1 ) (7)
Each V i vector is N(σ i ) in length according to the number of IPM points on the σ i scale. Thus, the length of the global attribute vector equals

∑ - = σ = 1 M 0 i i T ) ( N N
. Since V consists of vectors of multiple scales, it is considered to be a multiple scale attribute vector.

Similarity function

To measure the matching rate between two V A and V B attribute vectors associated to two different patterns, we define a similarity function as follows:

) V , V ( Max ) V , V ( Min )) cos( 1 ( 50 ) V , V ( SimScore B A B A B A γ + = ( 8
)
where γ is the angle between both vectors and where . indicates the module of a vector. This function ranges from 0% for very different vectors to 100% for perfectly matching vectors.

Results of pattern retrieval from a database

In this study, we used a set of five values for σ (180, 20, 30, 40, 50). Based on the total number of IPM points of the query contour that equals N T =41, in the case of a query from the kk707 family, 72 contours are retained for subsequent recognition stages. In this situation, the dimension of the V global attribute vector is equal to N T .

After the pre-selection process based on the total number of IPM points, we apply the SimScore function to all pairs composed with the Q 0 query and to each of the selected contours (see Table II). This pre-selection enables us to reduce computation time (at the SimScore function stage) by reducing the number of pairs to process. Fig. 13 shows an example in which the Q 0 query pattern is a contour of the kk707 family i.e. a transformed copy of the kk707 pattern. This figure shows the 8 most similar contours to the queried contour. The similarity scores for these matching contours are 100%. Table II shows the 72 selected contours and their similarity scores in relation to the queried contour.

As can be seen in Table II and Fig. 13, out of the 72 remaining contours, the first 32 have a similarity score of 100%. It is interesting to note that, similarly to the query contour, they all come from the kk707 family. Another interesting point to note is that the highest similarity scores produced by a pattern from a different family is only 5%. The contour of the kk707 family with the lowest similarity score is the R6H1kk707 contour with a similarity score of 2%. This contour results from a (225°) rotation and a (0.25) scaling applied to the kk707 contour: the low similarity score is due to the scale reduction applied to the original contour. Fig. 14 shows the R6H1kk707 contour which is expanded to the original size and placed on top of the original rotated contour (with a rotation angle of 225°). This figure shows the defects introduced by the transformation applied to the original contour. These defects are due to scale reduction and mainly to the fact that the coordinates after transformation are rounded up to integer values (so that one point corresponds to one pixel in a bitmap image) as explained previously. This rounding up operation also introduces step-wise defects. These defects, which can produce peaks or self-intersections, are not dealt with in the present study. If these defects were taken into account, the results could be improved.

Comparison with an adaptation of CSS method

To evaluate our method, we have adapted the CSS (Curvature Scale Space) method. The maxima of the CSS curves and their position are then used like shape descriptor. Only maxima between the major maximum M and the maximum m=0.2M are taken into account to define the attribute vector. We then apply the similarity function of equation ( 8) after a pre-selection based on the length of the attribute vector as it is explained in section 4.3. To compare the retrieval results from both methods, we carried out four experiments :

1) The query shape "vase" is not in the database. The goal of this experiment if to assess the ability to discriminate shapes. This enables us to evaluate the sensitivity to false alarms situations in pattern recognition.

2) The query pattern "kk707" is in the database and there are also 100 members of the same family in the database. With this experiment, we aim at assessing the separability of the methods : how far are the less similar pattern from the query family and the most similar pattern from a different family (a nonparent pattern of the query). 3) We have applied both methods to a sample contours of the MPEG-Core Experiment CE-Shape-1. We applied them to retrieve a pattern from a database consisting of 20 contours of the same pattern that was subject of different kind of rigid or non-rigid transformations. Fig. 18 shows an example of such contours : in this example, contours are extracted from a flying bat in different kind of motion.

From these experiments, one can note the following interesting points: 1) When the query doesn't exist in the database (see Fig. 15), the IPM based method retrieves 12 patterns with 26% maximal similarity score while the adapted CSS-based method retrieves 58 patterns with 50% maximal similarity score. Our method gives lower maximal similarity score (26%) than the CSS based method (50%). In this experiment, the lower the maximal similarity score is and the better is the recognition system: in deed when the similarity score is low, one can decide easily that the query pattern is unknown (doesn't exist in the database). 2) In the second experiment (see Fig. 16), the lowest similarity score obtained from a parent of the query pattern is 91% and the highest score obtained from a non-parent pattern is 43%: the difference between both scores is ∆s 1 =48%. With the adapted CSS method, this difference is ∆s 2 =1% while the lowest similarity score from a parent is 69%. Of course, a large difference indicates a good separation between classes: in other words, the best method is that method which provides large difference. 3) In the third experiment (see Fig. 17), when applied to contours of bats, the adapted CSS method yields better results than our method. As it can be seen on Fig. 17, some changes from a contour to another are not rigid transformations but they are caused by non-rigid deformations due to the motion of the animal. Since our method is essentially a global-driven method, it may fail for more or less non-rigid deformations. It is adapted better for rigid deformations as define for part A experiment in [START_REF] Latecki | Shape Dscriptors for Non-rigid Shapes with a Single Closed Contour[END_REF][START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF]. In this article, the authors have considered three different stages according to the three parts of the MPEG-7 Core Experiment CE-Shape-1: the part A that deals with robustness to scaling and to rotation is defined as a necessary condition to satisfy. In accordance with these experiments, we can reasonably say that our method is a good candidate to satisfy this necessary condition in shape description. In any case, the query, itself, when it is present in the database, is retrieved with a score of 100%. As shown on Fig. 18, bat-16, which is the query, is retrieved with a score of 100%. Bat-17 is retrieved with a similarity score of 89%: this high similarity score indicates that both patterns are similar as it can be seen visually on Fig. 18. The next contour, bat-01, is found to be similar to the query with 40% similarity score. Visually, both contours are very similar but bat-01 has some high frequency details on the upper side of its wings : that explains the low score. Bat-05 and bat-04 present a very weak score (8% and 5%, respectively), because contrarily, with bat-01, their ears are quite apparent: what creates this strong dissimilarity with the query. Table III summarises results that enable us to carry out a comparison of both the adapted CSS-based method and the MSGPR method. In this study, we used three families of contours: bats, bell and children extracted from the MPEG-7 Core Experiment CE-Shape-1 database. In each family, there are twenty contours representing different views or deformations due to the object movements. For a given family, we use each contour as a query and then record the minimal and maximal similarity score and the number of contours that are retained at the first step. The average value of each of these three parameters is then registered in Table III. For each method, the average value of the maximal similarity score is 100% indicating that the query contour is always recognised with 100% similarity score. This parameter shows the capability of a method to recognise a pattern when this one exists in the database. The average value of the minimal similarity score together with the average value of the number of retained patterns indicates the ability of the method to recognise a pattern that has undergone rigid or non-rigid deformations: the higher these parameters are and the better is the method. As can be seen on Table III, the two parameters seem to behave in opposition one of the other. While the IPM-based method provides the higher global average number of retained patterns and lower average of minimal similarity score, the CSSbased method provides the lower global average number of retained patterns and higher global average value of the minimum similarity score. That thus requires a compromise which will be guided by the contours under study.

Conclusion and discussion

We have presented some preliminary results of a new and simple method for planar pattern description within the context of pattern recognition. This paper analysis also the performance of some attributes derived from the IPM function. According to these results it would seem reasonable to believe that this is a good candidate method within the context of (planar) pattern recognition based on analysis of the closed non-occulted contour of the pattern being studied. This analysis method can be accelerated if just the IPM points for the concerned set of σ values are generated rather than generating the "entire" IPM function. This is an advantage as opposed to the CSS method which has to construct the entire CSS map in order to generate the features. Similarly to the CSS method, our approach has some interesting advantages:

• Curves do not have to be convex,

• The pattern can be identified regardless of its orientation (the method is rotation insensitive),

• The method is translation insensitive,

• The method is scale change insensitive for a large range of scales.

The initial results obtained by applying this method to pattern retrieving from a database show that the IPM function produces consistent attributes and can therefore be positively applied to pattern recognition, especially within the context of OCR (or Optical Character Recognition).

It is also interesting to note that the IPM function is very easy to understand and does not require much theory: this makes it very easy to implement. As stated above, our approach focuses on recognition of planar patterns that do not produce self-intersected contours. This aspect is therefore not taken into account in our studies.

Applying this method to noisy contours show that it is resistant to noise for a range of noise energy. From Fig. 19 we can note the following points: the IPM function is not too affected by noise for high scales (see upper part of the IPM functions); the apparition of many other structures in the lower part of the IPM function indicates that the pattern being studied is noisy. As stated in section 2.5, high frequency information is located in the lower part of the IPM function: as the added noise is represented by fine details (with high frequency), this explains the appearance of those structures for low scales.

As stated earlier, the main weakness of our approach is that it fails to handle occulted contours and those having undergone a non-rigid deformation, nevertheless it is advantageous in that it is simple and easy to to implement and to apply. It can also be considered as a useful step in a pattern recognition system, especially when one is concerned with rigid deformations. Thanks to the results of the comparative study carried out between our method and an adapted-CSS method, we think that it is reasonable to say that our method has its own place in the context of planar shapes description and recognition.

Figure captions

Fig. 1 : 4 : 5 : 6 :Fig. 7 :

 14567 Fig.1: MSGPR method diagram Fig. 2: Rotation compensation Fig. 3: A contour and its corresponding separated functions Fig. 4: Filtered functions and their corresponding smoothed contour Fig. 5: Example of the IPM function Fig. 6: Characterisation of convex/concave parts in the IPM function Fig. 7: Contours and IPM functions of letter "M" and "W" from the bold.chr character set from Borland ® Fig. 8: Scaling effect Fig. 9: Rotation effect Fig. 10: Measuring the rotation angle Fig. 11: An example of contours in the database Fig. 12: Attributes definition for a set of fixed σ values.

Fig. 13 :

 13 Fig. 13: kk707 recognition results for a set of 5 σ values. Fig. 14: Illustration of the defects introduced during transformation: step-wise defects, peak defects and selfintersection defects Fig. 15: Retrieval of a pattern that doesn't exist in the database: a) results from IPM-based method and b) results from the CSS-based method. Fig. 16: Retrieval of a pattern from the database containing some derived patterns (by applying rotation and/or scaling) of the query: a) results from IPM-based method and b) results from the CSS-based method. Fig. 17: Retrieval of a pattern from the database containing some derive patterns from the query. The derived patterns are obtained by changes caused by non-rigid motion: a) results from IPM-based method and b) results from the CSS-based method. Fig. 18: Retrieval of a contour from the database of bats contours by using IPM-based features.

Fig. 19 : 8 Fig. 2 θ

 1982 Fig. 19: The noise effect: a) Noise free, b) Additive noise with standard deviation σ n =1.2, c) Additive noise with standard deviation σ n =1.8

Fig. 4 x

 4 Fig.4

Fig. 6 :

 6 Fig.6:

Fig. 10 Fig. 17 :Fig. 18 :

 101718 Fig.10:

Table captions Table I :

 captionsI Rotation angles and scaling factors used in affine transformations Table II: Similarity scores for different contours in pattern retrieval from a database Table III: Comparative results between IPM-based features and CSS-based ones in pattern retrieval.

			Table I Table III
	Rotation angle (R i ) 10° 45° 60° 90° 135° 180° 225° 270° 300° 330° Scaling factor (H i ) 0.2 0.25 0.5 0.75 1 2 25 10 20 30 Index i 0 1 2 3 4 5 6 7 8 9 Method Pattern Average value of minimum similarity score Average value Average number of maximum of retained similarity score patterns
		Bats	36.66	100.00	3.90
	IPM	Bell Children	10.85 11.55	100.00 100.00	20.00 20.00
		Global averages	19.77	100.00	14.63
		Bats	50.93	100.00	4.60
	CSS	Bell Children	42.52 48.17	100.00 100.00	4.50 10.70
		Global average	47.20	100.00	6.60

Pattern Recognition Letters 27 (2006) 1545-1559 (Preprint copy)

Acknowledgements

We are grateful to Mr Sadegh Abasi, Dr Farzin Mokhtarian and Pr Josef Kittler for making their SQUID system publicly available.

of 33


A Boulkhemair
email: boulkhem@math.univ-nantes.fr

On the Fefferman-Phong inequality

. We also give an abstract result. We even show that in the case of classical symbols, that is S 2 1,0 , the number of needed derivatives is bounded by n 2 + 4 + .

Introduction

The classical Fefferman-Phong inequality, [START_REF] Fefferman | On positivity of pseudodifferential operators[END_REF], states that, if a is a non negative symbol on R n × R n satisfying, for all multi-indices α and β,

|∂ α x ∂ β ξ a(x, ξ)| ≤ C α,β ξ 2-|β| ,
there exists a constant C > 0 such that, for all u ∈ S(R n ), Re(a(x, D)u|u)

L 2 + C||u|| L 2 ≥ 0, (1)
where a(x, D) is the standard quantization of the symbol a, that is the pseudodifferential operator defined by a(x, D)u(x) =

R n e 2πixξ a(x, ξ) u(ξ)dξ, u being the Fourier transform of u. Thus, it is a great improvement of both the classical and the sharp G e 2πi(x-y)ξ a( x+y 2 , ξ)u(y)dydξ, the inequality is also equivalent to saying that there exists a constant C > 0 such that Op w (a) + C ≥ 0.

(

An alternate and recent version of the Fefferman-Phong inequality due to J. M. Bony, [START_REF] Bony | Sur l'inégalité de Fefferman-Phong[END_REF], says that inequalities (1) and ( 2) hold if a is a non negative symbol such that ∂ α

x ∂ β ξ a(x, ξ) are bounded for |α| + |β| ≥ 4, which is a remarkable result since it indicates that only the boundedness of derivatives of order larger than or equal to 4 is relevant.

In this paper, we are interested in estimating the number of derivatives of the symbol a needed to obtain the Fefferman-Phong inequality. In [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF] (see also the short version [START_REF] Lerner | A Wiener algebra for the Fefferman-Phong inequality[END_REF]), N. Lerner and Y. Morimoto proved that this number is bounded by 4 + 2n + with an arbitrary > 0, n being the dimension of the base space. Actually, they established the following more precise result :

Inequalities (1) and ( 2) hold if a is a non negative symbol such that

∂ α x ∂ β ξ a ∈ A 0 (R 2n ) for |α| + |β| = 4,
where A 0 (R 2n ) is the Wiener type algebra of symbols studied by J. Sjöstrand in [START_REF] Sjöstrand | An algebra of pseudodifferential operators[END_REF].

Recall that one way to define the algebra A 0 (R d ) is as the set of functions that can be written as the sum of a series like

k∈Z d u k (x)e ixk ,
where the functions u k are bounded, with spectrum in a fixed compact set and

k∈Z d ||u k || L ∞ < ∞ .
Note that the definition of A 0 does not use derivatives. However, in terms of regularity, for a general function on R d to be in A(R d ), it must have d + 1 bounded derivatives, and this fact is optimal, see [START_REF] Boulkhemair | Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators[END_REF]. Hence, the result of Lerner-Morimoto. In this paper, we take back the argument of Lerner-Morimoto and apply more or less known results on L 2 boundedness of pseudodifferential operators to obtain mainly that For a non negative symbol a, inequalities (1) and (2) hold if ∂ α

x ∂ β ξ a (4) are bounded or locally uniformly square integrable for |α| + |β| ≤ n + 1, or, if a (4) satisfies the classical S 0 1,0 estimates up to [ n 2 ] + 1 derivatives, or, if a satisfies the S 2 1,0 estimates up to [ n 2 ] + 5 derivatives.

Here, a (4) stands for the tensor of fourth order derivatives of a. In fact, we shall also use fractional derivatives in such a way that we finally obtain that the number of derivatives of the symbol a needed to obtain the Fefferman-Phong inequality is bounded by 4 + n + (resp. 4 + n 2 + ), where > 0 is arbitrary. The paper begins with an abstract result. Indeed, we remark that the method of proof of Lerner-Morimoto works if one replaces the algebra A 0 by any subalgebra of L ∞ satisfying few assumptions. See Section 2. Then, we indicate some more or less natural algebras that satisfy the assumptions of Section 2 and prove L 2 boundedness for pseudodifferential operators associated to these algebras in the t-quantization. In the fourth section, we state the result that gives the best bounds for the number of derivatives of the symbol needed to obtain the Fefferman-Phong inequality. We conclude by stating the Fefferman-Phong inequalities in the semi-classical setting.

Thanks are due to P. Bolley and N. Lerner for motivating discussions on the subject.

Some notations

"Cst " will always stand for some positive constant which may change from one inequality to the other.

||.|| E denotes the norm in the space E.

L(E) is the space of bounded operators in E.

(u|v) is the scalar product in L 2 . u = F(u) is the Fourier transform of u.

For a function a(x, η), F 1 (a)(ξ, η) and F 2 (a)(x, y) denote the Fourier transforms of x → a(x, η) and η → a(x, η) respectively.

We often use

∂ α 1 ∂ β 2 a for ∂ α x ∂ β ξ a(x, ξ). τ k is the translation operator : τ k u(x) = u(x -k). If x ∈ R n , x = √ 1 + x 2 . [x] denotes the integral part of the real number x. [α, β] stands for the compact interval {x ∈ R; α ≤ x ≤ β}. λ ∼ µ means that λ µ and µ λ are bounded. H s (R d ), s ∈ R, is the usual Sobolev space. B s (R d ), s ∈ N ∪ {∞},
is the space of bounded functions in R d with bounded derivatives up to the order s. For positive non integral s, B s (R d ) is the usual Hölder space.

An abstract result

Let A be a subalgebra of L ∞ (R 2n ) satisfying the following properties :

(H1) ∃C 0 > 0, ∃m ≥ 0, ∀Y ∈ R 2n , ∀b ∈ A, ||τ Y b|| A ≤ C 0 Y m ||b|| A . (H2) The map b → Op w (b) is bounded from A to L(L 2 (R n )).
We have then the following :

Theorem 1 There exists a constant C > 0 such that, for all non negative functions a on R 2n such that a (4) ∈ A(R 2n ), we have Op w (a) + C||a (4)

|| A ≥ 0, (3)
that is, Op w (a) is semi-bounded.

The proof follows the same lines as that of [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF]. So, we shall be brief and refer to that paper for more details.

Lemma 1 For any function a defined on R 2n and such that a (4) ∈ A(R 2n ), we have

Op w (a) = Op wick (a - 1 8π ∆a) + Op w (r),
where r ∈ A(R 2n ) is such that ||r|| A ≤ C||a (4) || A , C being a constant independent of a, and ∆a = 2n j=1 ∂ 2 j a.

Recall that Op wick (a), the Wick quantization of a, is the pseudodifferential operator whose Weyl symbol is b

(X) = a(X + Y )2 n e -2πY 2 dY. (4)
Proof : If b is given by (4), using Taylor formula, we can write

b(X) = a(X) + 1 8π ∆a + 1 6 1 0 (1 -t) 3 a (4) (X + tY ) Y 4 e -2πY 2 2 n dtdY.
Another application of Taylor formula allows us to write the Weyl symbol of Op wick (∆a) as

θ(X) = ∆a(X) + 1 0 (1 -t) (∆a) (X + tY ) Y 2 e -2πY 2 2 n dtdY.
Thus, the Weyl symbol of Op wick (a

-1 8π ∆a) is equal to b -1 8π θ = a -r where r(X) = - 1 6 1 0 (1 -t) 3 a (4) (X + tY ) Y 4 e -2πY 2 2 n dtdY + 1 8π 1 0 (1 -t) (∆a) (X + tY ) Y 2 e -2πY 2 2 n dtdY, that is, r(X) = 1 0 a (4) (X + tY ) P (t, Y )e -2πY
||r|| A ≤ C 0 ||a (4) || A 1 0 Y m |P (t, Y )|e -2πY 2 dtdY = C||a (4) || A .
It follows from the above lemma and assumption (H2) on A that Theorem 1 is a consequence of the following result of [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF] which is independent of the algebra A.

Proposition 1 There exists a constant C > 0 such that for any nonnegative function a defined on R 2n and such that a (4) ∈ L ∞ (R 2n ), we have

Op wick (a - 1 8π ∆a) + C||a (4) || L ∞ ≥ 0.
The proof of this proposition relies on the Wick pseudodifferential calculus and on a precise result on the decomposition of nonnegative functions as sums of squares. We refer to [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF].

This achieves the proof of Theorem 1.

We turn now to the Fefferman-Phong inequality in the standard quantization case. To be able to deduce it from the Weyl quantization case, we have to strengthen the assumption (H2). We shall use (H2) : The map a → Op t (a) is bounded from A to L(L 2 (R n )) for all t ∈ [0, 1] and its norm is uniformly bounded with respect to t ∈ [0, 1].

Recall that Op t (a) is the pseudodifferential operator defined by Op t (a)u(x) = R 2n e 2πi(x-y)ξ a((1 -t)x + ty, ξ)u(y)dydξ, u ∈ S(R n ), so that, Op 1/2 = Op w and Op 0 is the standard quantization. Recall also that Op t (a) = Op 0 (J t a) where J t = e 2πitDxD ξ , t ∈ R.

Theorem 2 Assume that A satisfies (H1) and (H2) . There exists a constant C > 0 such that, for all non negative functions a on R 2n such that a (4) ∈ A(R 2n ), we have

Re(a(x, D)u|u) L 2 + C ||a (4) || A ||u|| 2 L 2 ≥ 0, u ∈ S(R n ). (5)
Proof : We are concerned with the semi-boundedness of the operator A = [a(x, D) + a(x, D) ]/2. We can write 2A = Op w (J -1/2 a + J 1/2 a). Now, by Taylor formula, we have

J -1/2 a = a -πiD x D ξ a -π 2 1 0 (1 -t)e -πitDxD ξ (D x D ξ ) 2 adt and J 1/2 a = a + πiD x D ξ a -π 2 1 0 (1 -t)e πitDxD ξ (D x D ξ ) 2 adt.
Since a is real, we get

J -1/2 a + J 1/2 a = 2a -π 2 1 0 (1 -t)(J -t/2 + J t/2 )(D x D ξ ) 2 a dt.
Hence,

A = Op w (a) -R where R = π 2 2 1 0 (1 -t)(Op (1-t)/2 (b) + Op (1+t)/2 (b))dt (6)
and b = (D x D ξ ) 2 a. Since b ∈ A, it follows from assumption (H2) that R is bounded in L 2 (R n ) with an operator norm bounded by C ||a (4) || A and therefore the result follows from Theorem 1.

On algebras of symbols and boundedness of operators

We present here some more or less known algebras of symbols which give rise to L 2 -bounded pseudodifferential operators and to which we are going to apply the results of the preceding section.

Uniformly local Sobolev algebras

If E is a Banach space of functions or distributions on R d (for example, containing D(R d ), to avoid trivial cases), we shall denote by E ul the set of functions or distributions u which are locally uniformly in E, that is, the set of u such that u τ y χ is in E uniformly in y ∈ R d for some χ ∈ D(R d ) with non zero integral. The space E ul is then naturally normed by ||u|| E ul = sup y∈R 2n ||u τ y χ|| E . We shall apply this procedure to the usual Sobolev space H s (R 2n ) as well as to its anisotropic analogues

H s,s (R n × R n ) and H σ (R 2n ), s, s ∈ R, σ = (σ 1 , . . . , σ 2n ) ∈ R 2n .
Recall that these are also Hilbert spaces and are defined by : -

u ∈ H s,s (R n × R n ) iff u is a tempered distribution such that the integral R n ×R n ξ s ξ s u(ξ, ξ ) 2 dξdξ is finite. -u ∈ H σ (R 2n ) iff u is a tempered distribution such that the integral R 2n 2n i=1 ξ i σ i u(ξ) 2 dξ is finite.
We shall consider the spaces

H s ul (R 2n ) for s > n, H s,s ul (R n × R n ) for s > n 2 , s > n 2
, and

H σ ul (R 2n ) for σ i > 1 2 , 1 ≤ i ≤ 2n. These are Banach subalgebras of L ∞ (R 2n
) and we have the inclusions

H s ul (R 2n ) ⊂ H s 2 , s 2 ul (R n × R n ) ⊂ H ( s 2n ,..., s 2n ) ul (R 2n )
with continuous injections, of course.

Clearly, these algebras satisfy trivially the assumption (H1). They also satisfy assumption (H2) , although this is much less trivial. In fact, we are going to show that, if a ∈ H σ ul (R 2n ),

σ i > 1 2 , 1 ≤ i ≤ 2n, then, for all t ∈ R, Op t (a) is bounded in L 2 (R n
) and its operator norm can be estimated by C(1

+ t 2 ) N ||a|| H σ ul .
Using dyadic decompositions with respect to each variable, this is equivalent to the following :

Theorem 3 There exist positive constants C and M such that, for all a ∈ L ∞ (R 2n ) with supp( a) ⊂ 2n i=1 [-R i , R i ], R i ≥ 1, 1 ≤ i ≤ 2n
, and all t ∈ R, the following inequality holds :

||Op t (a)|| L(L 2 ) ≤ C(1 + t 2 ) M (R 1 R 2 . . . R 2n ) 1 2 ||a|| L 2 ul .
Proof : In fact, the cases t = 0 and t = 1 2 are proved in [START_REF] Boulkhemair | L 2 estimates for pseudodifferential operators[END_REF] and [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF] respectively. Unfortunately, we have not been able to deduce this theorem from these cases (is it possible ?). However, the proof is very similar to that of Theorem 2.1 in [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF], so, we shall be brief and only indicates the changes, refering at each time to [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF] for details.

We have to study

I = (Op t (a)v|u), u, v ∈ S(R n ),
and we can assume that a ∈ S(R 2n ). The first step in estimating I is given by the analogue of Lemma 2.

of [8] :

Lemma 2 There exists a constant C > 0 such that, for all a ∈ S(R 2n ), all χ ∈ S(R n ) with χ(x)dx = 1, and every t ∈ R, we have

||Op t (a)|| L(L 2 ) ≤ C t s sup k∈R n χ(x) ω(x) ω(y) F 2 (a)(x + k, y) 2 dxdy 1 2
,

where ω(x) = n i=1 x i s i , s i > 1, 1 ≤ i ≤ n, s = s 1 + ... + s n , and y → F 2 (a)(x, y)
is the Fourier transform of a(x, η) with respect to η.

Indeed, one writes

I = F 2 (a)(x, y) v(x + (1 -t)y) u(x -ty) dxdy,
and then follows the proof of Lemma 2.2 of [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF]. The t s appears when one applies Peetre inequality.

The second step is

Lemma 3 There exists a constant C > 0 such that, for all a ∈ S(R 2n ) with

F 2 (a)(x, y) = 0 when y / ∈ n i=1 [-R i , R i ], R i ≥ 1, 1 ≤ i ≤ n, all χ ∈ S(R n ) with χ(x)dx = 1
, and every t ∈ R, the following estimate holds :

||Op t (a)|| L(L 2 ) ≤ C t 2n (R 1 . . . R n ) 1 2 sup k∈R n χ(x) ω(x) a(Rx + k, η) 2 dxdη 1 2
, where Rx stands for (R

1 x 1 , . . . , R n x n ) and ω(x) = n i=1 x i 2 .
In estimating

I = F 2 (b)(Rx, y) V (x + (1 -t)y) U (x -ty) dxdy, where we have set b(x, η) = a(x, R -1 1 η 1 , . . . , R -1 n η n ), U (x) = (R 1 . . . R n ) 1 2 u(Rx) and V (x) = (R 1 . . . R n ) 1 2 v(Rx), we follow the proof of Lemma 2.3 of [8].
For the third and last step in the proof of Theorem 3, we write

I = F 1 (b)(ξ, η) V (η -tξ) U (η + (1 -t)ξ) dξdη, where ξ → F 1 (b)(ξ, η) is the Fourier transform of b(x, η) with respect to x, b(x, η) = a(R -1 x, Rη), U (x) = u(R -1 x)/ √ R 1 . . . R n , V (x) = v(R -1 x) / √ R 1 . . . R n and Rx = (R 1 x 1 , . . . , R n x n ).
Then, the proof finishes as that of Theorem 2.1 of [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF]. However, another power of t appears at this stage when we use the formula

η + (1 -t)ξ s η -tξ s = |α|,|β|≤2s c α,β (t) ξ α η β ,
since now the c α,β (t) are polynomial in t of degree |α| ≤ 2s. Hence, we get the estimate of Theorem 3 with M = n + s and s is an even integer greater than n 2 .

Hölder type algebras

A well known algebra of bounded functions in R 2n which also satisfies (H1) and (H2) is the Hölder algebra B s (R 2n ) for s > n. Here, to be simple, when s ∈ N, this will be the Sobolev space W s,∞ and not the Zygmund class even if many of our statements hold with the latter.

One can obtain somewhat more general algebras by considering Hölder anisotropic regularity in the same spirit as in the case of the uniformly local Sobolev spaces. The more general one is defined by means of a 2n-dyadic partition of unity in R 2n , 1 = j∈N 2n ϕ j , where

ϕ j (ζ) = ϕ j 1 (ζ 1 ) . . . ϕ jn (ζ 2n ), j = (j 1 , . . . , j 2n ), based on a dyadic partition of unity in R, 1 = k∈N ϕ k , (for example, ϕ 0 ∈ D(R), ϕ 1 ∈ D(R \ 0), ϕ k+1 (t) = ϕ k ( t 2 ), t ∈ R, k ≥ 1). If σ ∈ (R * + ) 2n , we have, by definition, u ∈ B σ (R 2n ) if and only if u ∈ L ∞ (R 2n ) and (2 jσ ||ϕ j (D)u|| L ∞ ) j∈N 2n is bounded, where jσ = j 1 σ 1 + • • • + j 2n σ 2n . One can define similarly B s,s (R n × R n ), s > 0, s > 0, if
one wants to take derivatives only in the directions of the subspaces R n × {0} or {0} × R n , by using a dyadic partition of unity in R n × R n which is a tensor product of standard dyadic partitions of unity in R n .

These spaces have natural normed structures and, in fact, are Banach algebras of bounded continuous functions. Note also the following inclusions, for s > 0, which are similar to those with the uniformly local Sobolev spaces,

B s (R 2n ) ⊂ B s 2 , s 2 (R n × R n ) ⊂ B ( s 2n ,..., s 2n ) (R 2n ), (7)
and that the associated injections are continuous. The fact that

B s (R 2n ), for s > n, B s,s (R n × R n ), for s, s > n 2 , and B σ (R 2n ), for σ i > 1 2 , 1 ≤ i ≤ 2n
, satisfy the (H2) assumption is a consequence of Theorem 3 and the above inclusions. The fact that they satisfy (H1) is trivial and holds with arbitrary exponents.

S 0 1,0 type algebras

Another algebra which will be important for us is defined as follows. The idea is that of an S 0 1,0 type algebra with a limited regularity. To any Banach space E of functions in R 2n , one can associate the space denoted by S m 1,0 E, m ∈ R, and defined as the set of functions a :

R n × R n → C such that (i) a(x, ξ)χ(ξ) is in E, for all χ ∈ D(R n ). (ii) {λ -m a(x, λξ)χ(ξ); λ ≥ 1} is a bounded subset of E, for all χ ∈ D(R n \ 0).
The reason for such a definition is that when E is formally the space B ∞ (R 2n ), we obtain in fact the usual Hörmander space S m 1,0 . The space S m 1,0 E is at least a normed space since it can be equipped with the norm

||a(x, ξ)ϕ(ξ)|| E + sup{2 -jm ||a(x, 2 j ξ)ϕ 0 (ξ)|| E ; j ∈ N},
where ϕ ∈ D(R n ) and ϕ 0 ∈ D(R n \ 0) are such that they define a dyadic partition of unity in R n :

ϕ(ξ) + j≥0 ϕ 0 (2 -j ξ) = 1. (8)
Here, we are essentially interested by the cases where E is one of the Banach algebras defined in the preceding subsection. In these cases, we obtain spaces S m 1,0 E which are Banach spaces and, when m = 0, even Banach algebras, as one can check easily. The fact that these Banach algebras satisfy the (H1) assumption is not completely trivial and we state it as Proposition 2 There exist constants C > 0 and M > 0 such that, for all (y, η) ∈ R 2n and all a ∈ S 0 1,0 E, τ (y,η) a is in ∈ S 0 1,0 E and

||τ (y,η) a|| S 0 1,0 E ≤ C η M ||a|| S 0 1,0 E .
Here, E stands for one of the three algebras of the preceding subsection :

B s (R 2n ), B s,s (R n × R n ) or B σ (R 2n ), with s, s > 0, σ ∈ (R * + ) 2n .
Proof : We shall treat the case of E = B σ (R 2n ), the others being similar. Let a ∈ E and χ ∈ D(R n ). We can write, using the dyadic partition of unity [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF],

a(x -y, ξ -η)χ(ξ) = a(x -y, ξ -η)ϕ(ξ -η)χ(ξ) + j≥0 a(x -y, ξ -η)ϕ 0 (2 -j (ξ -η))χ(ξ) = τ (y,η) (aϕ)(x, ξ)χ(ξ) + j≥0 τ (y,η) [a j (x, 2 -j ξ)]χ(ξ)
where we have set a j (x, ξ) = a(x, 2 j ξ)ϕ 0 (ξ). By definition, aϕ ∈ E and (a j ) is a bounded sequence of E. It follows from the translation invariance of E, from the lemma below and from the fact that the sum above has a number of non vanishing terms which is finite and does not depend on (y, η), that χτ (y,η) a ∈ E and ||χτ (y,η) a|| E ≤ Cst ||a|| S 0 1,0 E . Note that we also used the fact that E is an algebra.

Lemma 4 Given σ ∈ (R * + ) d , there exists a constant C > 0 such that, for all h ∈ (R * + ) d and all u ∈ B σ (R d ), the function x → u(hx) = u(h 1 x 1 , ..., h d x d ) is in B σ (R d ) and ||u(hx)|| B σ ≤ C h σ ||u|| B σ , where h σ = h σ 1 1 ... h σ d d , h i = max{1, h i }, i ∈ {1, ..., d}.
The proof of this lemma is easy and is left to the reader. Assume now that χ ∈ D(R n \ 0) and let λ ≥ 1. As before, write

a(x-y, λξ-η)χ(ξ) = a(x-y, λξ-η)ϕ(ξ-λ -1 η)χ(ξ)+ j≥0 a(x-y, λξ-η)ϕ 0 (2 -j (ξ-λ -1 η))χ(ξ) = a(x -y, λξ -η)ϕ(ξ -λ -1 η)χ(ξ) + j≥0 τ (y,λ -1 η) [a λ,j (x, 2 -j ξ)]χ(ξ) (9)
where a λ,j (x, ξ) = a(x, 2 j λξ)ϕ 0 (ξ). The number of non vanishing terms in the last sum is finite and does not depend on (y, η, λ), and, clearly, we can estimate that sum in the E norm as before by Cst ||a|| S 0 1,0 E . It remains to treat the first term in [START_REF] Coifman | Au delà des opérateurs pseudodifférentiels[END_REF]. On the support of this term, we have |ξ -λ -1 η| ≤ 1 and γ 1 ≤ |ξ| ≤ γ 2 with some positive constants γ 1 and γ 2 , so that, λ -1 |η| ≤ 1 + γ 2 . Now, if λ -1 |η| is small enough, for example, λ -1 |η| ≤ γ 1 /2, then, we also have |ξ -λ -1 η| ≥ γ 1 /2, so that we can replace ϕ by some ϕ ∈ D(R n \ 0) and we can then estimate the term a(x -y, λξ -η) ϕ(ξ -λ -1 η)χ(ξ)

as before. Finally, we are left with the term a(x-y, λξ-η)ϕ(ξ-λ -1 η)χ(ξ) under the condition that λ ∼ |η|. We can now write it as (τ (y,η) a λ )(x, λξ) χ(ξ) with a λ (x, ξ) = a(x, ξ)ϕ(ξ/λ), and then, apply Lemma 4 and the translation invariance of E to obtain

||(τ (y,η) )a λ (x, λξ) χ(ξ)|| E ≤ Cst λ |σ | ||a λ || E
where |σ | = σ n+1 + ... + σ 2n . The last thing we do is to restrict ourselves to λ = 2 k , to rewrite a λ and then to estimate it as follows

a λ (x, ξ) = a(x, ξ)ϕ(ξ) + k-1 j=0 a(x, ξ)ϕ 0 (2 -j ξ), ||a λ || E ≤ Cst k||a|| S 0 1,0 E ≤ Cst ln λ ||a|| S 0 1,0 E ≤ Cst ln η ||a|| S 0 1,0 E . Hence, ||(τ (y,η) )a λ (x, λξ) χ(ξ)|| E ≤ Cst η |σ | ln η ||a|| S 0 1,0 E .

This finishes the proof of the proposition.

Concerning the L 2 boundedness of operators associated with symbols in S 0 1,0 E, one can prove the following result :

Theorem 4 Let E stands for B s (R 2n ) with s > n 2 , or B s,s (R n × R n ) with s > 0, s > n 2 , or B σ (R 2n ) with σ i > 0 if 1 ≤ i ≤ n, and σ i > 1 2 if n + 1 ≤ i ≤ 2n.
There exist positive constants C and M (M > 2n works) such that, for all functions a ∈ S 0 1,0 E and all t ∈ R, the operator Op t (a) is bounded in L 2 (R n ) with an operator norm estimated by C t M ||a|| S 0 1,0 E .

Proof : In view of the inclusions [START_REF] Boulkhemair | L 2 estimates for pseudodifferential operators[END_REF], it is sufficient to treat the case of E = B σ (R 2n ). We follow ideas of [START_REF] Coifman | Au delà des opérateurs pseudodifférentiels[END_REF], [START_REF] Boulkhemair | L 2 estimates for pseudodifferential operators[END_REF] and [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF]. Unfortunately, the theorem is not a consequence of the results obtained in these papers.

Let a ∈ S 0 1,0 E. Since the usual regularized functions of a are bounded in S 0 1,0 E by a constant times ||a|| S 0 1,0 E , we can assume that a ∈ S(R 2n ). Write a(x, η) = a(x, η)ϕ(η)

+ k≥0 a(x, η)ϕ 0 (2 -k η),
where ϕ(η) + k≥0 ϕ 0 (2 -k η) = 1 is a standard dyadic partition of unity in R n . In order to treat the terms of this decomposition, we need the following Lemma 5 Let K be a compact set in R n . Then, there exists a constant C K > 0 such that, for all a ∈ S(R 2n ) with supp(η → a(x, η)) ⊂ K and for all t ∈ R,

||Op t (a)|| L(L 2 ) ≤ C K t 2n ||a|| E .
To prove the lemma, write the 2n-dyadic decomposition of a, a = j∈N 2n a j , and apply Lemma 3 to each a j . We get

||Op t (a j )|| L(L 2 ) ≤ C t 2n 2 |j | 2 sup z∈R n χ 0 (x) ω(x) a j (2 j x + z, η) 2 dxdη 1 2
, where j = (j n+1 , ..., j 2n ), 2 j x = (2 j n+1 x 1 , . . . , 2 j 2n x n ), ω(x) = n i=1 x i 2 and χ 0 ∈ S(R n ) with χ 0 (x)dx = 1. Now, it follows from the fact that a(x, η) has a compact support in η that each a j (x, η) is rapidly decreasing in η and that, for all α ∈ N n ,

||η α a j (x, η)|| L ∞ ≤ C α 2 -jσ ||a|| E . Hence, χ 0 (x) ω(x) a j (2 j x + z, η) 2 dxdη ≤ Cst 2 -2jσ ||a|| 2 E , so that, ||Op t (a)|| L(L 2 ) ≤ Cst t 2n j 2 |j | 2 -jσ ||a|| E = C K t 2n ||a|| E ,
which proves the lemma. Of course, Lemma 5 applies to the term a(x, η)ϕ(η). Now, let us consider the terms a(x, η)ϕ 0 (2 -k η) = a k (x, 2 -k η), k ≥ 0, where we have set a k (x, η) = a(x, 2 k η)ϕ 0 (η). By definition of S 0 1,0 E, the sequence (

a k ) is bounded in E. Write a k = b k + r k where b k is given by b k (x, η) = 2 kn χ(2 k y) a k (x -y, η) dy .
with χ ∈ S(R n ) , χ = 1 near 0, and supp( χ) is small enough (for example, χ(ξ) = χ 1 (ξ/ ) with supp( χ 1 ) in the unit ball and small enough). Clearly, (b k ) is also a bounded sequence in E and

||b k || E ≤ ||χ|| L 1 ||a k || E = ||χ 1 || L 1 ||a k || E , since E is translation invariant. Set b(x, η) = k b k (x, 2 -k η) and let us estimate I = (Op t (b)v|u) for u, v ∈ S(R n ). We have I = k F 1 (b k )(ξ, 2 -k η) v(η -tξ) u(η + (1 -t)ξ) dξdη.
Since |η| ∼ 2 k and |ξ| ≤ 2 k on the support of integration, with a small enough , we also have |η -

tξ| ∼ |η + (1 -t)ξ| ∼ |η| ∼ 2 k .
Here, α ∼ β means that the ratio α β has (positive) upper and lower bounds. Note that depends on t (in fact, ∼ 1 t ). However, one can choose bounds on 2 -k |η -tξ| and 2 -k |η + (1 -t)ξ| that do not. Therefore, we can take a ψ ∈ D(R n \ 0) such that we can write

I = k Op t (b k (x, 2 -k η))ψ(2 -k D)v ψ(2 -k D)u ; so that, |I| ≤ k ||Op t (b k (x, 2 -k η))|| L(L 2 ) ||ψ(2 -k D)v|| L 2 ||ψ(2 -k D)u|| L 2 ≤ Cst sup k ||Op t (b k (x, 2 -k η))|| L(L 2 ) ||v|| L 2 ||u|| L 2 = Cst sup k ||Op t (b k (2 -k x, η))|| L(L 2 ) ||v|| L 2 ||u|| L 2
To obtain the last equality, we have applied the following lemma whose proof is easy and left out :

Lemma 6 For all a ∈ S (R n × R n ), t ∈ R and λ > 0, Op t (a) is bounded in L 2 (R n ) if
|I| ≤ Cst t 2n sup k ||b k || E ||v|| L 2 ||u|| L 2 ≤ C t 2n sup k ||a k || E ||v|| L 2 ||u|| L 2 , so that, since u, v ∈ S(R n ) are arbitrary, ||Op t (b)|| L(L 2 ) ≤ C t 2n sup k ||a k || E .
We turn now to the study of r(x, η) = k r k (x, 2 -k η). We need here to use the space F = B σ (R 2n ) with 0 < σ i < σ i for 1 ≤ i ≤ n, and σ i = σ i for n + 1 ≤ i ≤ 2n. Applying as above Lemma 4, Lemma 5 and Lemma 6, we can write

||Op t (r k (x, 2 -k η))|| L(L 2 ) ≤ Cst t 2n ||r k || F .
It suffices now to show that ||r k || F ≤ δ k ||r k || E with (δ k ) ∈ 1 to finish the proof of the theorem. Write the 2n-dyadic decomposition of r k , r k = j∈N 2n r k,j . On the support of F 1 (r k,j )(ξ, η), we have

|ξ i | ∼ 2 j i , 1 ≤ i ≤ n, and |ξ| ≥ 2 k .
This implies that there exists (a rather large) k 0 ∈ N (more precisely, one can check that

2 k 0 ∼ 1 ) such that r k,j = 0 if j i < k -k 0 , ∀i ∈ {1, ..., n}.
Therefore, given j ∈ N 2n such that r k,j = 0, there exists i ∈ {1, ..., n} such that j i ≥ k -k 0 , which allows us to estimate ||r k,j || L ∞ as follows :

2 jσ ||r k,j || L ∞ = 2 -j(σ-σ ) 2 jσ ||r k,j || L ∞ ≤ 2 -j i (σ i -σ i ) ||r k || E , so that, ||r k || F ≤ 2 -(k-k 0 )τ ||r k || E ≤ Cst -τ 2 -kτ ||r k || E ≤ Cst t τ 2 -kτ ||r k || E ,
where τ = min{σ i -σ i ; 1 ≤ i ≤ n}. Hence,

||Op t (r)|| L(L 2 ) ≤ Cst t 2n+τ sup k ||r k || E ≤ Cst t 2n+τ sup k ||a k || E .
This achieves the proof of the theorem.

The conclusion of this subsection is that, if E is one of the algebras

B s (R 2n ) with s > n 2 , B s,s (R n × R n ) with s > 0, s > n 2 , or B σ (R 2n ) with σ i > 0 if 1 ≤ i ≤ n, and σ i > 1 2 if n + 1 ≤ i ≤ 2n,
then, the algebras S 0 1,0 E satisfy (H1) and (H2) .

Estimates on the needed number of derivatives

In view of the results of the preceding sections, we can state the following which gives bounds on the number of derivatives needed for the Fefferman-Phong inequality to hold.

Theorem 5 Let a be a non negative function defined on R n × R n . Then, the Fefferman-Phong inequalities (1) and ( 2) hold if a satisfies one of the following conditions, with a constant C that depends linearly on the norm of a in the imposed space :

(i) ∂ α 1 ∂ β 2 a is in L ∞ (R 2n ) or L 2 ul (R 2n ) for 4 ≤ |α| + |β| ≤ n + 5. (ii) For |α| + |β| = 4, ∂ α 1 ∂ β 2 a is in B n+ (R 2n ) or H n+ ul (R 2n
), > 0, or in one of the other algebras of subsections 3.1 and 3.2.

(iii) For |α| + |β| = 4, ∂ α 1 ∂ β 2 a is in S 0 1,0 E where E is B n 2 + (R 2n ) or B , n 2 + (R n × R n ) or B σ (R 2n
) with σ = ( , ..., ; 1 2 + , ..., 1 2 + ), being an arbitrary positive number (see subsection 3.3).

(iv

) |∂ α x ∂ β ξ a(x, ξ)| ≤ C α,β ξ 2-|β| for |α| + |β| ≤ [ n 2 ] + 5. (v) a ∈ S 2 1,0 E with E = B n 2 +4+ (R 2n ). (vi) For |α| + |β| ≤ 4, ∂ α 1 ∂ β 2 a is in S 2-|β| 1,0 E where E is B , n 2 + (R n × R n ) or B σ (R 2n
) with σ = ( , ..., ; 1 2 + , ..., 1 2 + ), being an arbitrary positive number (see subsection 3.3).

Proof : Parts (i), (ii) and (iii) are consequences of Theorem 1 and Theorem 2 since all the considered algebras satisfy the assumptions (H1) and (H2) .

Here, we shall prove (v) and (vi), (iv) being a consequence of (v). The proof is an adaptation of that of Corollary 1.3.2 (i) of [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF], and we refer to that paper for missing details. Let E stands for either

B n 2 +4+ (R 2n ), B , n 2 + (R n × R n ) or B σ (R 2n
) with σ = ( , ..., ; 1 2 + , ..., 1 2 + ). First, let us treat the case of Weyl quantization. If ϕ(ξ)

+ k≥0 ϕ 0 (2 -k ξ) = 1 is a dyadic partition of unity in R n , we can write Op w (a) = Op w (aϕ) + j≥0 Op w [a j (x, 2 -j ξ)],
where

a j (x, ξ) = a(x, 2 j ξ)ϕ 0 (ξ). Since aϕ is in S 0 1,0 E, it follows from Theorem 4 that Op w (aϕ) is bounded in L 2 (R n ). So, let us consider I j = (Op w [a j (x, 2 -j ξ)]v|v), j ≥ 0, v ∈ S(R n ).
By performing a simple change of variables in the integral defining I j , we can write

I j = (Op w (b j )v j |v j ),
where b j (x, ξ) = a j (2 -j/2 (x, ξ)) and v j (x) = v(2 -j/2 x)2 -jn/4 . It follows from the assumptions that, for |α|

+ |β| = 4, the functions ∂ α 1 ∂ β 2 b j (x, ξ) = 2 -2j ∂ α 1 ∂ β 2 a j (2 -j/2 (x, ξ
)) are bounded in E. This is not sufficient a priori for our goal. However, since they are supported in the annuli c 1 2 j/2 ≤ |ξ| ≤ c 2 2 j/2 , they are in fact bounded in S 0 1,0 E. Indeed, if χ ∈ D(R n \ 0) and λ ≥ 1, we have λ ∼ 2 j/2 on the support of the functions ∂ α 1 ∂ β 2 b j (x, λξ)χ(ξ); so that, applying Lemma 4 yields

||∂ α 1 ∂ β 2 b j (x, λξ)χ(ξ)|| E ≤ Cst sup j≥0 2 -2j ||∂ α 1 ∂ β 2 a j || E ≤ Cst β ≤β sup j≥0 2 (|β |-2)j ||(∂ α 1 ∂ β-β 2 a)(x, 2 j ξ) ∂ β ϕ 0 (ξ)|| E .
It follows from this and from the fact that b j is non negative (we use a non negative ϕ 0 , of course) that the Fefferman-Phong inequality holds for Op w (b j ). However, this is not sufficient to conclude since we have to sum constants and the v j are only bounded with respect to j. So, consider the operator B j = Op w (ψ j )Op w (b j )Op w (ψ j ) where ψ j (ξ) = ψ 0 (2 -j/2 ξ), ψ 0 ∈ D(R n \ 0) and ψ 0 = 1 in a (large enough) neighborhood of supp(ϕ 0 ). One can write, using the Weyl pseudodifferential calculus,

B j = Op w (b j ψ 2 j ) + R j = Op w (b j ) + R j ,
with some remainder operator R j . The reason for introducing the operator B j is that

j (B j v j |v j ) = j Op w (b j )ψ j (D)v j ψ j (D)v j ≥ j -Cst ||ψ j (D)v j || 2 L 2 = j -Cst ||ψ 0 (2 -j D)v|| 2 L 2 ≥ -Cst ||v|| 2 L 2 ,
and it is thus sufficient to prove that R j is bounded in L 2 (R n ) and that (||R j || L(L 2 ) ) j is a summable sequence. It follows from the Weyl calculus that R j = Op w (r j ) where r j is given by the following expression (see [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF]) :

r j (x, ξ) = 2 n 8π 2 |α|=2 1 α! 1 0 (1 -t) e -4πyη ∂ α η [ψ j (ξ + η)ψ j (ξ -η)] ∂ α 1 b j (x + ty, ξ) dydηdt.
Clearly, on the support of r j (x, ξ), we have |ξ| ∼ 2 j/2 . Moreover, since the function ψ j (ξ + η)ψ j (ξ -η) is differentiated, we also have |η| ∼ 2 j/2 on the support of integration. Setting ψ 0 (ξ, η) = ψ 0 (ξ + η)ψ 0 (ξ -η), we see that r j (x, ξ) is a finite combination of the integrals

2 -2j 1 0 (1 -t) e -4πyη ∂ α 2 ψ 0 [2 -j/2 (ξ, η)] ∂ α 1 a j [2 -j/2 (x + ty, ξ)] dydηdt,
or, after some integrations by parts whose gains are some negative powers of 2 j , of the integrals

r j,α (x, ξ) = 2 -4j 1 0 (1-t) e -4πyη |2 -j/2 η| -2 ∂ α 2 ψ 0 [2 -j/2 (ξ, η)] ∆ 1 ∂ α 1 a j [2 -j/2 (x+ty, ξ)] dydηdt.
Now, the fact that ||Op w (r j,α )|| L(L 2 ) = ||Op w [r j,α (2 -j/2 x, 2 j/2 ξ)]|| L(L 2 ) , (see Lemma 6), suggests that we consider

r j,α (2 -j/2 x, 2 j/2 ξ) = 2 -4j 1 0 (1 -t) e -4πyη |η| -2 ∂ α 2 ψ 0 (ξ, η) ∆ 1 ∂ α 1 a j (2 -j x + 2 -j ty, ξ) dydηdt = 2 -4j 1 0 (1 -t) e -4πyη 4πy -2n D η 2n [|η| -2 ∂ α 2 ψ 0 (ξ, η)] ∆ 1 ∂ α 1 a j (2 -j x + 2 -j ty, ξ) dydηdt,
where we have performed the change of variables (y, η) → (2 -j/2 y, 2 j/2 η) and, then, integrations by parts. Hence, using the fact that E is an algebra which is translation invariant and Lemma 4, we obtain that r j,α (2 -j/2 x, 2 j/2 ξ) is in E and that

||r j,α (2 -j/2 x, 2 j/2 ξ)|| E ≤ Cst 2 -4j ||∆ 1 ∂ α 1 a j (2 -j x, ξ)|| E ≤ Cst 2 -4j ||∆ 1 ∂ α 1 a j || E ,
and, consequently,

||r j (2 -j/2 x, 2 j/2 ξ)|| E ≤ Cst 2 -4j ||a (4) j || E ≤ Cst 2 -2j sup k≥0 2 -2k ||a (4)
k || E .

It remains to note that r j (2 -j/2 x, 2 j/2 ξ) has a support in ξ which is contained in fixed compact set and then to apply Lemma 5. The result is that

||R j || L(L 2 ) = ||Op w [r j (2 -j/2 x, 2 j/2 ξ)]|| L(L 2 ) ≤ Cst 2 -2j sup k≥0 2 -2k ||a (4)
k || E , and this achieves the proof of the Fefferman-Phong inequality in the Weyl quantization case. The case of the standard quantization can be seen to be a consequence of Weyl quantization case. In fact, if A = [a(x, D) + a(x, D) ]/2, it follows from (6) that we can write

A = Op w (a)-R where R = π 2 2 1 0 (1-t)(Op (1-t)/2 (b)+Op (1+t)/2 (b))dt and b = (D x D ξ ) 2 a.
Clearly, b ∈ S 0 1,0 E and applying Theorem 4 yields the fact that R is a bounded operator in L 2 (R n ) and that its operator norm is estimated by Cst ||b|| S 0 1,0 E . This establishes the Fefferman-Phong inequality in the standard quantization case and, at the same time, completes the proof of Theorem 5.

Semi-classical estimates

One can also prove semi-classical Fefferman-Phong inequalities using an abstract setting. However, here, the algebra A has to satisfy the following additional assumption :

(H3) ∃C 1 > 0, ∀h ∈ [0, 1], ∀b ∈ A, ||b(hX)|| A ≤ C 1 ||b|| A .
The following results are consequences of those of section 2.

Corollary 1 Assume that A satisfies the assumptions (H1), (H2) (resp. (H2) ) and (H3).

(i) There exists a positive constant C such that, for all non negative functions a on R 2n such that a (4) ∈ A(R 2n ), and for all h ∈ [0, 1], we have

Op w [a(x, hξ)] + C h 2 ||a (4) || A ≥ 0, (resp. Re(a(x, hD)u|u) L 2 + C h 2 ||a (4) || A ||u|| 2 L 2 ≥ 0, u ∈ S(R n )). (ii)
There exists a positive constant C such that, for all non negative functions a h (x, ξ) on

R 2n , h ∈]0, 1], such that the functions (∂ α 1 ∂ β 2 a h )(x, ξ/h)h -|β| are bounded in A for |α|+|β| = 4, we have ∀h ∈]0, 1], Op w (a h ) + C M h 2 ≥ 0, (resp. Re(a h (x, D)u|u) L 2 + C M h 2 ||u|| 2 L 2 ≥ 0, u ∈ S(R n )), where M = sup{||(∂ α 1 ∂ β 2 a h )(x, ξ/h)h -|β| || A ; 0 < h ≤ 1, |α| + |β| = 4}. (iii)
There exists a positive constant C such that, for all non negative functions a

h (x, ξ) on R 2n , h ∈]0, 1], such that the functions (∂ α 1 ∂ β 2 a h )(xh 1/2 , ξh -1/2 )h -|β| are bounded in A for |α| + |β| = 4, we have ∀h ∈]0, 1], Op w (a h ) + C M h 2 ≥ 0, (resp. Re(a h (x, D)u|u) L 2 + C M h 2 ||u|| 2 L 2 ≥ 0, u ∈ S(R n )), where M = sup{||(∂ α 1 ∂ β 2 a h )(xh 1/2 , ξh -1/2 )h -|β| || A ; 0 < h ≤ 1, |α| + |β| = 4}.
Proof : One can check easily, using the (H3) assumption, that (iii) implies (ii) which implies (i). The proof of (iii) is formally the same as that of [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF]. In fact, one has just to replace the Wiener-Sjöstrand algebra A 0 by the algebra A and, of course, to apply Theorem 1 (resp. Theorem 2). So, we refer to [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF].

Taking, for example, A = B n+1 (R 2n ), we obtain the following Corollary 2 There exists a positive constant C such that, for all non negative functions a h on R 2n , h ∈]0, 1], such that

|∂ α 1 ∂ β 2 a h (x, ξ)| ≤ C α,β h |β| for h ∈]0, 1] and 4 ≤ |α| + |β| ≤ n + 5, we have ∀h ∈]0, 1], Op w (a h ) + C M h 2 ≥ 0, (resp. Re(a h (x, D)u|u) L 2 + C M h 2 ||u|| 2 L 2 ≥ 0, u ∈ S(R n )), where M = sup{C α,β ; 4 ≤ |α| + |β| ≤ n + 5}.
Proof : It is easily seen that a h satisfies the conditions of Corollary 1(iii). Moreover, the algebra B n+1 (R 2n ) satisfies the assumptions (H1), (H2) and (H3).

Remarks and further results

1. A natural question that can be raised is whether these upper bounds on the number of derivatives needed for the Fefferman-Phong inequality to hold are optimal. In fact, this is not quite clear for us. However, we can at least say that the bounds n+ and n 2 + concerning the number of derivatives needed for the L 2 boundedness of the involved pseudodifferential operators are optimal. See [START_REF] Coifman | Au delà des opérateurs pseudodifférentiels[END_REF], [START_REF] Boulkhemair | L 2 estimates for pseudodifferential operators[END_REF], [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF]. Furthermore, the "4" number of derivatives is reputed "to be optimal", and it would be a great achievement if one can reduce it. Roughly, one can say that the bounds are optimal with respect to the method of proof.

By using the same ideas, one can estimate the number of derivatives needed for the sharp G

• arding inequality to hold. The usual argument for proving this inequality is simpler of course than that needed to establish the Fefferman-Phong inequality, and even works for systems, that is, for matrices of symbols and operators. For example, if a is some function on the phase space such that a ∈ L ∞ , one can write Op wick (a) = Op w (a) + Op w (r) where r(X) = 2 n (1 -t)a (X + tY )Y 2 e -2πY 2 dtdY.

See [START_REF] Lerner | On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators[END_REF]. Clearly, if a is non negative, a ∈ A and A is a subalgebra of L ∞ satisfying the assumptions (H1) and (H2), it follows from the positivity of the Wick quantization that Op w (a) + C||a (4) || A ≥ 0. Of course, the same is true for the standard quantization if A satisfies (H1) and (H2) . Here, one can even show that (H2) is not necessary and that it is enough to assume instead that a → a(x, D) is bounded from A to L(L 2 (R n )). Taking back the arguments developped in the preceding sections, one can prove, for example, that for a non negative symbol a on R 2n , we have 3. The Fefferman-Phong inequalities (that is, Theorem 1 and Theorem 2) also hold when A = S 0 , E where 0 < < 1 and E is one of the spaces B s (R 2n ) with s > n, B s,s (R n × R n ) with s > n 2 , s > n 2 , or B σ (R 2n ) with σ i > 1 2 , 1 ≤ i ≤ 2n, (or even the spaces obtained when B is replaced by H ul ). Here, the space S m , E, m ∈ R, is defined as the set of functions a : R n × R n → C such that (i) a(x, ξ)χ(ξ) is in E, for all χ ∈ D(R n ).

(ii) {λ -m a(λ -x, λ ξ) χ(λ -1 ξ); λ ≥ 1} is a bounded subset of E, for all χ ∈ D(R n \ 0). In fact, one can apply the same argument as that used above to check that the algebra A = S 0 , E satisfies the assumption (H1). The fact that A satisfies (H2) is already proven in [START_REF] Boulkhemair | L 2 estimates for Weyl quantization[END_REF]. The same property is proven in [START_REF] Boulkhemair | L 2 estimates for pseudodifferential operators[END_REF] in the case of the standard quantization. Now, the case of the t-quantization can be handeled similarly by the same methods. One obtains, for example, that the Fefferman-Phong inequalities (1) and (2) hold for the non negative function a on the phase space if it satisfies the estimates Such inequalities with symbols in the classes S 0 , E are to be compared with similar ones obtained by J.-M. Bony, [START_REF] Bony | Sur l'inégalité de Fefferman-Phong[END_REF], and D. Tataru, [START_REF] Tataru | On the Fefferman-Phong inequality and related problems[END_REF], under more or less different assumptions. However, those authors do not consider the limited regularity of the symbols as we do.

4. In the case of symbols of type (1,0), D. Tataru proved in [START_REF] Tataru | On the Fefferman-Phong inequality and related problems[END_REF] the sharp G • arding and the Fefferman-Phong inequalities with a limited regularity in the x variables by means of the FBI transform. He used 2 derivatives for the first one and 4 for the other one but did not limit the regularity with respect to the frequency variables. These are to be compared with Theorem 5 (vi) above which says that the Fefferman-Phong inequality holds if one uses 4 derivatives in (x, ξ) plus " " derivative in x and n 2 + derivatives in ξ. It is likely that, by doing a paradifferential decomposition of the symbol as did Tataru, one can remove the " " in the case of the x regularity.

•

  arding inequality. Using the Weyl quantization Op w (a)u(x) = R 2n

  Op w (a) + C ≥ 0, and Re(a(x, D)u|u)L 2 + C||u|| L 2 ≥ 0, u ∈ S(R n )), if ∂ α 1 ∂ β 2 aare bounded or locally uniformly in L 2 for |α| + |β| ≤ n + 1 or |α|, |β| ≤ [ x, ξ) are bounded for |α| + |β| ≤ [ n 2 ] + 3.

|∂ α x ∂ β ξ a ( 4 )

 4 (x, ξ)| ≤ C α,β ξ (|α|-|β|) for |α|, |β| ≤ [

  and only if Op t (a(x/λ, λη)) is, and we have ||Op t (a(x/λ, λη))|| L(L 2 ) = ||Op t (a)|| L(L 2 ) .

	Now, it remains to apply Lemma 5 in conjunction with Lemma 4 to obtain


Valery Timkov

Serg Timkov

Vladimir Zhukov

Ti Mkov

S V Zhukov

Ti mkov V. F. The Office o

Keywords: PLANCK UNIVERSAL PROPORTIONS. GRAVITATIONAL -ELECTROMAGNETIC RESONANCE proportion of Planck's constant, the gravitational potential, gravitational -electromagnetic resonance

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

Introduction

The proportions of the Planck constant-mass, length and time sets universal limits of space, energy and temporal parameters of all objects in the Universe that are endowed with mass.

The maximu m possible transmission rate of physical interaction in the Universe -the speed of light depends only on the spatial and energy parameters of the Universe and is independent of its temporal characteristics. Gravitational and electro magnetic waves can enter into resonance.

Planck restrictions

Module of gravitational potential  of spherical body with mass mis equal to [1]:

 = Gm R , (1)
where R -is the distance from the body, G -is the gravitational constant.

It is known that the gravitational potential has the dimension of square of the velocity. Then, if c -is the speed of light, then for the limit ing case and some value of

S R  is true: 2 Gm c S  , ( 2
)
or 2 cm GS  , or p p m m lS  , (3)
where p m and p l -is the mass and length of the Planck.

Fro m (3) follows: where p t -is the Planck t ime, then the total energy of the body E with mass m is equal to:

p p l Sm m  , (4)
p p m E mc Sc F S l    , (6)
where p FS -is the wo rk, which is performed by the gravitational field of the body with mass m on the deformation of space, what is more, S -the radius of curvature of the deformed space by the gravitational field of the body with mass m , or actuallyit is the length of a gravitational wave of the body's gravitational field with mass m . The length of a gravitational wave is half the Schwarzschild radius [START_REF] Weinberg | Grav itation and Cosmology[END_REF].

If the observable Universe has a radius 0 R and mass M , based on its model of a mathematical pendulum, the following is true:

2 0 M cG R  (7) or 0 p p m M lR  . (8)
This means that the observable Universe is limited, and 0 R -is the length its gravitational waves.

Fro m (2, 3, 7, 8) follows, that the maximu m possible transmission speed of physical interactionis the speed of light, which clearly only on its own spatial and energy of the observable Universe, and is independent of its temporal characteristics.

In the conditionally limited space, where there is no mass (energy), the time factor does not exist. At occurrence of mass (energy) in the space, the time factor appears, as its integral characteristic of spatial and energy parameters. In this case, time parameter, wh ich uniquely characterizes both observable Universe and its separate object with mass m , will be assumed as the delay d t in distributing the light signal at a distance equal to the length of a gravitational wave, respectively, of the Universe:

0 0 p dU p t R tR cl  , (9)
and body with mass mwith a length of the gravitational wave S , that is equal: R fro m each other, the law of Un iversal gravitation is:

p dm p t S
2 1 2 1 2 1 2 2 2 2 dd pp m m S S t t F G F F c R R R    . ( 15
)
Conclusion: universal proportions -Planck restrictions clearly define the main spatial and energy parameters of any object of the Universe, wh ich have mass.

Gravitational -electro magnetic resonance

On the basis of data on mass of objects [START_REF]Planetary Fact Sheet -Metric[END_REF] and the formulas (4, 5), (10,11) The presence of gravitational-electro magnetic resonance of the Earth was testedby practical consideration on the experimental measuring system on the base of Ukrainian State Enterprise "Research Institute "Orion". Experimental conditions: air temperature 25 ° C, relat ive humid ity 35%, at mospheric pressure 755 mm Hg. A rt., the height of the measuring system is 15 m above ground level. These conditions do not require the introduction of amendments to the measure.

Experimental measuring system consists of:

-MS signal generator in the frequency range 57 GHz -77 GHz and output power up to 10 mW; -Measurement channel consisting of: a waveguide; calibrated cylindrical cavity with a graduated movable rod stimulated by wave H011; attenuator; -Oscilloscope in the frequency range 57 GHz -77 GHz, for visual display and control of signals from the output of the generator and the output of the measuring channel;

-Power meter of signal K M3-75v with frequency range 57 GHz -77 GHz. A block diagram of the measurement setup is shown in Figure 1.

Figure 1 -Flo w diagram of the measurement setup

It functions as a resonant wave meter [START_REF] Valitov | Rad io measurements at microwave frequencies[END_REF][START_REF] Burdun | Radio measurements at millimeter wavelengths[END_REF] with the setting of the resonance frequency at a minimu m output power of the measuring channel.

The experiment algorithm is the following: measurements were carried out in the frequency range of 65.2 GHz to 68.8 GHz with a step of 0.6 GHz, 0.1 GHz, 0.01 GHz at several fixed values of the power levels of the output signal fro m the signal generator MS, namely : 3.62 mW; 4.73 mW; 4.5 mW, 5.04 mW, 6.1 mW . Signal with strictly defined power and frequency from output of the frequency generator is supplied to a measuring channel (waveguide cavity). At the same time for the visual control of the pres ence of a resonance, signal fro m the output of the generator and the measuring channel is supplied to a mu lti-channel oscilloscope. With the help of the graduated movable rod in the resonator with a waveguide measuring channel resonance is established. Presence of resonance is registered with an oscilloscope and power meter at a minimu m power of the output signal of the measuring channel. Further by the previously set fixed value of the power at a predetermined pitch the frequency of the signal changed at t he output of the generator frequency, and in the resonator with the waveguide via a graduated rod resonance was mounted and the measurements were repeated. At the end of measurements in a given frequency range signal power level at the output of the genera tor frequency changes and re-measurement of minimu ms signal power is carried out at the mo ment of resonance in the resonator with the waveguide at the output of the measuring channel. To prevent accidental emissions of experimental data, measurements were carried out repeatedly, and the data was then averaged.

The graphs were made according to experimental data: Figure 2 -Figure 6. Analysis of the graphs shows the presence of minimu m-min imoru m power of signals at the output of measuring channel at the frequency 67.6 GHz for all fixed power levels of the signal generator MS, which confirm the presence of gravitational-electro magnetic resonance.

On the basis of expressions (4, 5), (10, 11) and (12, 13), it can be argued that the emission spectrum of any astronomical object will contain a co mponent, due to its gravitational-wave modulation. Since the mass of an object and its gravitational wave are connected by unique dependence, then by allocating this component, it can be clearly determined the mass of the object. This means that in astronomy, it can appear an accurate tool for determining the mass of distant objects of the Universe. Spectra of signals fro m objects, that reflect this signal (but not emit), for examp le planets, -will contain two modulating gravitational waves. For the solar system -it is gravitational wave of the Sun and the gravitational wave of own planet.

Gravitational-electro magnetic resonance confirms the possibility of lev itation. Gravitat ional-electro magnetic resonance allows you to create new sources of energy. Conclusion: open the law of nature -Planck universal proportions. On the basis of this law opened a new property of mattergravitational-electro magnetic resonance.

Figure 2 -

 2 Figure 2 -Graph of dependency of signal power on frequency in the frequency range fro m 65.2 GHz to 68.8 GHz with a step of 0.6 GHz at the output of the measuring channel. The signal power at the output of the generator is 3,62mW .

Figure 3 -

 3 Figure 3 -Graph of dependency of signal power on frequency in the frequency range fro m 65.2 GHz to 68.8 GHz with a step of 0.6 GHz at the output of the measuring channel. The signal power at the output of the generator is 4,73mW .

Figure 4 -

 4 Figure 4 -Graph of dependency of signal power on frequency in the frequency range fro m 65.2 GHz to 68.8 GHz with a step of 0.6 GHz at the output of the measuring channel. The signal power at the output of the generator is 5,04mW .

Figure 5 -

 5 Figure 5 -Graph of dependency of signal power on frequency in the frequency range from 65.7 GHz to 68.6 GHz with 0.1GHz at the output of the measuring channel. The signal power at the output of the generator is 4,5mW .

Figure 6 -

 6 Figure 6 -Graph of dependency of signal power on frequency in the frequency range fro m 67.71 GHz to 67,51 GHz with a step of 0.01 GHz at the output of the measuring channel. The signal power at the output of the generator is 6,1mW.

  and (12, 13) it is made calculations of the gravitational wave length and frequency, respectively:

	Name	Mass, kg	Gravitationalwavelength,	Frequency, GHz
			m
	Earth	5.9722 x 10 24	0.00443474	67.6
	Moon	7.3477 x 10 22	0.000054547302	5495.94
	Venus	4.8673 x 10 24	0.0036143131	82.95
	Mars	6.4169 x 10 23	0.00047451718	631.78
	Jupiter	1.8981 x 10 27	1.40948454472	0.2127
	Saturn	5.6832 x 10 26	0.42201429314	0.7104
	Sun	1.989 x 10 30	1477.036	2,0297 x 10 -4