Bérénice Froment
email: fromentb@obs.ujf-grenoble.fr

Michel Campillo
email: michel.campillo@obs.ujf-grenoble.fr

Philippe Roux
email: philippe.roux@obs.ujf-grenoble.fr.

Pierre Gouédard
email: gouedard@mit.edu.

Arie Verdel
email: arie.verdel@shell.com.

Richard L Weaver
email: r-weaver@uiuc.edu

Estimation of the effect of non-isotropically distributed energy on the apparent arrival time in correlations

Correlations of random seismic noise are now widely used to retrieve the Green's function between two points. Whereas this technique provides useful results in tomography and monitoring studies, it is mainly limited by an uneven distribution of noise sources. In that case, theoretical requirements are not completely fulfilled and we may wonder how reliable the signals are reconstructed, in particular for the purpose of estimating traveltime from correlations. We present in this study a way to quantify effects of a non-isotropic noise 1 field by estimating the arrival time error due to a particular non-isotropic distribution of recorded wave intensity. Our study is based on a theoretical prediction of this bias and we successfully test the theory by comparing the theoretical expectation to real measurements from seismic-prospecting data. In particular, we distinguish between the effects of source distribution and effects of medium heterogeneity between the sources and the region of receivers. We find relative errors in the order of a percent which may affect monitoring results especially where smaller relative velocity variations (smaller than 10 -3 for some applications) are investigated. Finally we see that correlation of coda waves helps mitigate the effects of a non-isotropic field, hence making the estimation of traveltime quite stable irrespective of the source distribution.

INTRODUCTION

The Green's function of a medium between two points A and B represents the signal recorded at B if we would apply an impulse source at A: therefore the Green's function between any two given points A and B corresponds to the earth response between them. It has been demonstrated that with the crosscorrelation of random seismic wavefields, it is possible to retrieve the Green's function between two points.

Two acoustic experiments have initially demonstrated the possibility to reconstruct the Green's function by correlating diffuse wavefields [START_REF] Weaver | Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies[END_REF][START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF]. Afterwards, different theoretical approaches have formalised the link between correlation of random diffuse fields and Green's function retrieval [START_REF] Weaver | Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies[END_REF][START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF]Derode et al., 2003a,b;[START_REF] Van Tiggelen | Green function retrieval and time reversal in a disordered world[END_REF][START_REF] Wapenaar | Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by crosscorrelation[END_REF][START_REF] Snieder | Extracting the Green's function from the correlation of coda waves: a derivation based on stationary phase[END_REF][START_REF] Roux | Ambient noise cross-correlation in free space: theoretical approach[END_REF]. The crosscorrelation technique has been used successfully in many applications in helioseismology (Duvall et al., 1993;[START_REF] Gilles | A subsurface flow of material from the sun's equator to its poles[END_REF], acoustics [START_REF] Weaver | Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies[END_REF][START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF][START_REF] Larose | Imaging from one-bit correlation of wide-band diffuse wavefield[END_REF], oceanography [START_REF] Roux | Extracting coherent wave fronts from acoustic ambient noise in the ocean[END_REF]Sabra et al., 2005c), active seismic experiments [START_REF] Bakulin | The virtual source method: theory and case study[END_REF][START_REF] Mehta | Improving the virtual source method by wavefield separation[END_REF] and seismology with the correlation of seismic coda [START_REF] Campillo | Long-range correlations in the diffuse seismic coda[END_REF] or seismic noise [START_REF] Shapiro | Emergence of broadband Rayleigh waves from corre-lations of the ambient seismic noise[END_REF].

In seismology, it has been demonstrated that the correlation converges towards the Green's function as long as the different components of the elastic field follow the equipartition condition (Sanchez-Sesma et al., 2006[START_REF] Gouédard | Techniques de corrélation : aspects méthodologiques et applications à la sub-surface[END_REF]. Practically, this corresponds to a selfaveraging process provided by randomization of the noise sources when considering long time series as well as scattering [START_REF] Campillo | Phase and correlation in 'random' seismic fields and the reconstruction of the Green function[END_REF]Larose et al., 2006;Gouédard et al., 2008b).

Even though correlations of seismic noise have provided useful information in tomo-graphic imaging [START_REF] Shapiro | High-resolution surface-wave tomography from ambient seismic noise[END_REF]; Sabra et al. (2005a) and recently [START_REF] Draganov | Reflection images from ambient seismic noise[END_REF] in which body wave reflections are retrieved) and monitoring studies (Brenguier et al., 2008a,b;[START_REF] Sens-Schonfelder | Passive image interferometry and seasonal variations of seismic velocities at Merapi volcano, Indonesia[END_REF][START_REF] Wegler | Fault zone monitoring with passive image interferometry[END_REF], the seismic noise is usually not perfectly diffuse and the conditions of the theory are not completely fulfilled. In that case, we may wonder how reliable the signals are reconstructed by the correlation especially for the purpose of estimating traveltimes: how does a non-isotropic distribution of sources affect the traveltime estimated from these noise correlations ? Tomographic studies based on noise correlations use the traveltime of reconstructed Rayleigh waves to compute surface wave dispersion curves. Therefore, errors in traveltime lead to errors in tomographic results. Concerning monitoring studies, the evolution of traveltimes gives information about changes in the medium. But a change of noise sources may imply a change in traveltimes estimated from correlations which is not representative to a physical variation of the medium. Recent studies based on noise-based monitoring, have detected relative temporal changes of velocity smaller than 10 -3 (Brenguier et al., 2008a,b) and further work is done to improve the accuracy of noise-based measurements.

Considering such tiny variations, the influence of source distribution might be significant and should be quantified to correct it from measurements.

Previous studies have examined bias caused by non-isotropic source distribution in correlations. In an exploration application, van der Neut and [START_REF] Van Der Neut | Estimating and correcting the amplitude radiation pattern of a virtual source[END_REF] studied the amplitude distortions in correlations caused by an imperfect source illumination and proposed a method to correct results from this bias. Besides, [START_REF] Tsai | On establishing the accuracy of noise tomography travel-time measurements in a realistic medium[END_REF] and Yao and van der Hilst (2009) have shown that an uneven distribution of source intensity and medium heterogeneities can produce traveltime bias in noise-based measurements. From a tomographic point of view, Yao and van der Hilst (2009) develop an iterative procedure to correct tomo-graphic results for this bias. We adopt a somewhat different approach as we consider the forward problem and focus on the study of a direct formulation of this bias. In the next section, we illustrate the variability of apparent traveltime in correlations through results of a passive seismic noise study in the Parkfield area, California. Thereafter we present a theoretical expression for the error in arrival time due to a particular non-isotropic distribution of noise sources. In the last section, we compare the theoretical prediction to the real measurements on seismic prospecting data. In particular, we examine the effects of wave propagation in a heterogeneous medium surrounding the receivers and we discuss the advantages to correlate coda waves to mitigate effects of non-isotropy.

ON THE STABILITY OF APPARENT TRAVELTIME IN NOISE CORRELATIONS AT THE SAN ANDREAS FAULT, PARKFIELD

Between July 2001 and October 2002, 30 broadband stations located in an 11-km square, recorded continuous seismic noise in the Parkfield area, California. Plane wave beamforming is performed in [0.1-0.2 Hz] to determine the direction of the seismic noise received by the Parkfield network. Figure 1 presents results of this beamforming analysis done on the entire network. In particular Figure 1(b) shows that the noise field is very directive, with a direction of about 55 • (North is 0 • , positive rotation is clockwise). This direction is consistent with a noise origin in the Pacific Ocean (Sabra et al., 2005b;[START_REF] Stehly | A study of the seismic noise from its longrange correlation properties[END_REF].

Figure 1(a) shows the evolution of the beamformer outputs for 1-h data segments over 31 consecutive Julian Days (JD). This figure indicates daily fluctuations confirming that the seismic noise direction is highly time-dependent.

Consider two particular days (JD18 and JD22) for which the difference in noise direction is about 20 • (see Figure 1(a)): how does this azimuthal difference affect the traveltime in noise correlations ? We compute the 1-day averaged noise correlation function for two receiver pairs. The first pair (stations 8-27) is mostly aligned with the noise direction. By contrast, the second pair orientation (stations 13-30) is nearly orthogonal to the direction of energy flux (see Figure 2(a)). Figures 2(b) and 2(c) show results of the two dailycorrelations of interest (JD18 and JD22) for both station pairs. Whereas the time shift due to the variation in noise direction is negligible for pair 8-27, the effect on the 13-30 receiver pair is significant since there is a relative time shift of about 30% for this case of extreme directive noise.

This particular extreme example shows that variability of apparent traveltime in correlations due to changes of a non-isotropic noise field may be significant. In order to get correct traveltime values, we must be able to quantify the effects of non-isotropy on traveltime estimates. [START_REF] Weaver | On the correlation of non-isotropically distributed ballistic scalar diffuse waves[END_REF] derived a theoretical expression of the error in apparent arrival time in correlations due to a non-isotropic specific distribution of intensity. They consider (i) sources in far-field, (ii) the ballistic case and (iii) a homogeneous medium. The error is then evaluated in an asymptotic limit of large receiver-receiver separation. Results presented in this section correspond to this work. [START_REF] Weaver | On the correlation of non-isotropically distributed ballistic scalar diffuse waves[END_REF] first show that in a fully asymptotic limit in which the two receivers are separated by a distance long compared to a wavelength (x/λ -→ ∞), non-isotropy does not impair estimation of traveltime. However they go further by investigating the effects of smooth but non-isotropic intensity on traveltime estimates, at finite receiver separation (away from the full asymptotic limit).

THEORETICAL ERROR ON APPARENT TRAVELTIME DUE TO A NON ISOTROPIC DISTRIBUTION OF INTENSITY

We select two receivers, at positions -x/2 and x/2 around the origin (see Figure 3).

We consider incoherent impulsive sources over an annular region, at large distance from the receivers. Since attention is confined to the two-dimensional case the field at the receivers may be viewed as a superposition of plane waves.

We consider the case that the recorded wavefield intensity distribution corresponds to the source intensity distribution which is controlled by an angular source weighting B(θ) (θ is defined relative to the strike line connecting the receivers, (see Figure 3)). We consider smooth distributions by writing B as a Fourier series:

B(θ) = B 0 + B 1 cos(θ) + B 2 cos(2θ) + B 3 cos(3θ) + ... (1)
Only cosines are used, by symmetry, as the receiver correlation does not distinguish between positive and negative θ. This means that any distribution may be written as a symmetric distribution.

Considering that the time derivative of field correlation between both receivers

(C ′ x = d dt C x (t)
) converges to the Green's function (Gouédard et al., 2008b), C ′ x gives an estimate of the traveltime t between the two points. The predicted error in this apparent traveltime, due to non-isotropy is then [START_REF] Weaver | On the correlation of non-isotropically distributed ballistic scalar diffuse waves[END_REF]:

δt ∼ B ′′ (0) 2tω 2 0 B(0)
for positive correlation time, (2a)

δt ∼ B ′′ (180) 2tω 2 0 B(180)
for negative correlation time, (2b)

with ω 0 the central angular frequency of the correlation waveform and angles expressed in degree. A positive error means an apparent arrival time earlier than the true arrival time by an amount of δt.

Equation 2 suggests that the error in apparent traveltime due to a non-isotropic distribution of intensity B(θ) decreases for high frequency and large receiver separation which is consistent with results by [START_REF] Tsai | On establishing the accuracy of noise tomography travel-time measurements in a realistic medium[END_REF] and Yao and van der Hilst (2009). Furthermore, we note that this expression gives an error proportional to the second derivative of the intensity B evaluated in the receiver pair direction (θ = 0 • for positive correlation time). The stationary phase theorem helps in interpreting the physics aspect of this observation. [START_REF] Snieder | Extracting the Green's function from the correlation of coda waves: a derivation based on stationary phase[END_REF], [START_REF] Roux | Ambient noise cross-correlation in free space: theoretical approach[END_REF] explain the reconstruction of the Green's function by correlation techniques in the case of a homogeneous body, by using the stationary phase theorem: only sources in the alignment of the receivers contribute to reconstruction. End-fire lobes represent areas where those coherent sources are located. The aperture of this zone depends on the square root of the ratio between wavelength and distance between receivers: the higher the frequency, the narrower the end-fire lobes [START_REF] Roux | Extracting coherent wave fronts from acoustic ambient noise in the ocean[END_REF][START_REF] Larose | Diffusion multiple des ondes sismiques et expériences analogiques en ultrasons[END_REF][START_REF] Spetzler | The Fresnel volume and transmitted waves[END_REF]). In the asymptotic limit considered, coherent reconstruction is then controlled by θ approaching 0 (for positive correlation time). But as we have seen before, we consider the case away from the full asymptotic limit. In that case the intensity near θ = 0 • may be written as B(θ) θ→0 = B(0) + 1 2 B ′′ (0)θ 2 introducing a higher order term in the problem (the symmetry leads to B ′ (0) = 0). Since this term illustrates the difference with the full asymptotic case (and no error), it makes sense that it controls the traveltime error. For simplicity, we have only discussed the positive correlation time, but the argument is the same for negative time.

ESTIMATION OF THE

Estimation of the error with respect to an isotropic distribution of sources

In order to test the theoretical expression derived by [START_REF] Weaver | On the correlation of non-isotropically distributed ballistic scalar diffuse waves[END_REF], we compare the above predictions with measurements on these experimental data. Real measurements consist in estimating the time shift in the non-isotropic case with respect to a reference traveltime. In this section, the reference traveltime stems from the reference waveform obtained for an isotropic distribution of sources around receivers (B(θ) = constant).

Signals from the 240 sources are (i) filtered between 10 and 20 Hz, (ii) windowed into the range 0 to 1.5 seconds to emphasize ballistic waves (see Figure 4 However, these results are based on the hypothesis that an isotropic distribution of sources correspond to a perfect isotropic illumination. But the medium between the sources and the region of receivers is heterogeneous [START_REF] Gouédard | Techniques de corrélation : aspects méthodologiques et applications à la sub-surface[END_REF] and some propagation effects may modify the wavefield making the effective intensity distribution in the vicinity of receivers non-isotropic despite the isotropic source distribution. In that case, the reference taken in these results does not correspond to the actual isotropic case and the previous estimated time shift is not the over-all error.

Effective distribution of intensity for an isotropic distribution of sources

The goal of this section is to determine the effective intensity distribution received at receivers for an isotropic distribution of sources. By doing so, we will be able to highlight and quantify possible propagation effects on traveltime estimates. The analysis is done under assumption of homogeneity between the receivers.

Plane-wave beamforming is performed to get directional features of the wavefield. We These results show that isotropic distribution of sources does not necessarily lead to isotropic distribution of intensity and thus does not correspond to the ideal isotropic illumination. These observations mean that intensity distribution is partly controlled by wave propagation in the heterogeneous medium surrounding the receiver region. The next section is devoted to quantify the propagation effects on traveltime estimates to take them into account in the determination of a total error.

Incorporation of propagation effects in the estimation of the error in traveltime

The previous section has allowed us to determine the contribution of propagation to the intensity distribution. For the sake of simplicity, this contribution (displayed in Figure 7(b))

will be called B p (θ).

In order to quantify the relation between propagation effects and traveltime estimates, we apply the theoretical expression given by equation 2 with B p (θ). Note again that our analysis is done under the assumption of a local homogeneity of the medium in the vicinity of the paths between the receivers. As B p (θ) varies azimuthally, effects on apparent traveltimes will depend on the angular orientation of receiver pairs. We found that relative time shifts may reach 1% for some particular receiver pair orientations (the larger errors are observed for directions of lower intensity in Figure 7(b)). In other words, raypath deflections due to medium heterogeneities may cause an error in the traveltime estimate that is not negligible with respect to errors due to a non-isotropic source distribution as seen in Figure 5. From the estimate of the part of the error due to propagation, how could we estimate the cumulative effect of medium heterogeneity and non-isotropy of sources on apparent traveltime ?

We present here a simple analysis leading to an expression of the total bias. Consider a non-isotropic source distribution (B s (θ) = constant); the actual intensity distribution may be written in first approximation as:

B(θ) = B s (θ) × B p (θ) (3)
leading to an error of the form

δt = δt s + δt p . ( 4
)
Intermediate steps are detailed in Appendix A. A prediction of the actual error in traveltime due to non-isotropy could thus be obtained by summing the predicted time shift due to (i) non-isotropy in source distribution, δt s (estimated in Figure 5) and (ii) propagation effects,

δt p .
In order to control the reliability of our analysis, we would like to compare this bias estimate to reference traveltimes. The presence of active sources near receivers, allows us to directly acquire Green's functions between receivers of interest. These signals may thus be used to obtain reference measurements of traveltimes. The question is to know if the apparent traveltimes from correlations and corrected with δt (given by equation 4) are consistent with this measurement. Since both arrays are shifted by 12.5 m in both directions, we use the closest sources to the receivers considered and we apply a correction on the time axis to correct for this distance difference. However, at the scale of precision considered here, direct measurements are much less stable than correlation measurements, making any comparison difficult. Different causes can explain the instability of direct estimates. First, the sources used in this case, are vibrators producing source signals with a 1-m-side plate.

Considering receivers separated by about 150 m, the spatial precision of the source location corresponds to the order of precision of the traveltime that we investigate (a few percent).

Furthermore, receivers consist of clusters of 12 geophones adding possible imprecisions.

Finally, some spectral aspects may make comparisons difficult between the active signal and the correlation function since the spectra of both signals are quite different and time estimates are strongly sensitive to spectral contents.

Mitigation of the effects of non-isotropy: correlation of coda waves

Coda waves correspond to waves scattered from the heterogeneities in the earth (e.g., [START_REF] Aki | Origin of coda waves: source, attenuation and scattering effects[END_REF]). Scattering tends to make the wavefield more isotropic depending on the distribution of the scatterers in the medium and the lapse time considered in the coda [START_REF] Paul | Empirical synthesis of time-asymmetrical Green functions from the correlation of coda waves[END_REF]. Even if the field is not completely isotropic, the distribution of incoming energy should no longer be controlled exclusively by the source distribution but also by the scatterer distribution in the medium.

In order to study the influence of the source distribution when using coda waves, we investigate how time shifts on correlations of coda waves depend on the source distribution.

We can note that in case of perfect isotropic coda wavefield, the expected bias will be zero (B ′′ (θ) = 0 in equation 2). The idea is the same as with previous time shift measurements (see Figure 5), but instead of correlating direct waves, we select a later part of the signal corresponding to coda waves (see Figure (4(b)). The diffuse regime is indicated by the linear energy decay in decibel scale in this window: to cancel this decay, the decrease of recorded amplitudes in this time window is compensated. As previously, the reference waveform corresponds to the correlation function (here of coda waves) obtained for an isotropic distribution of sources and we compare time shift measurements to the ballistic theory.

However, the signal-to-noise ratio is much lower for correlation of coda waves (Gouédard et al., 2008a). In order to get accurate time measurements, we need to average correlations.

Thus, in this section we measure time shifts on correlation functions averaged over the 18 receiver pairs of different orientations considered before.

Results presented in Figure 8 show that the time shift with respect to the reference is always about zero suggesting that arrival time does not highly depend anymore on source distribution. As in Figure 5, this result does not correspond to a total error, but even if this error is not zero (the scattered field may be still anisotropic) this time shift is almost independent of the distribution of sources. This can be related intuitively to equation 2 by considering that coda waves make the intensity distribution smoother (see Figures 6(a)

and 6(c)). This leads to lower values of B ′′ (θ) and thus smaller traveltime bias. This result confirms the interest of using coda waves in many applications involving uneven source distribution since Figure 8 shows that we mitigate the traveltime bias due to non-isotropic source distributions by a factor of 10.

CONCLUSION

We have shown a way to estimate the error made in traveltime estimated from correlations due to non-isotropic recorded energy provided that the azimuthal distribution is sufficiently smooth.

Non-isotropy in source distribution is one of the main causes of non-isotropic energy.

But in the case of a heterogeneous medium between sources and the region of receivers, propagation may modify the wavefield and make it more complicated, producing an additional error in traveltime. We have seen in this study that this effect is not negligible as compared to the effect of a non-isotropic source distribution. This means that in the case of a very heterogeneous medium, propagation effects should be taken into account in the error estimation. The method proposed in this paper provides a means to estimate the traveltime error taking into account both effects and allows us to predict errors smaller than 1%.

We observe errors in the order of one percent. The influence of this error highly depends on applications: whereas this amount will not introduce misinterpretation in tomographic studies (recently confirmed by results from Yao and van der Hilst ( 2009)), it may affect strongly monitoring results since we investigate much smaller relative velocity variations (smaller than 10 -3 ). Therefore, this work highlights the interest to estimate the effect of non-isotropic distribution of energy to allow to correct noise-based measurements in monitoring studies for this effect.

  TRAVELTIME ERROR ON SEISMIC-PROSPECTING DATA Acquisition geometry and data selection Petroleum Development Oman (PDO) recorded a high-resolution survey in northern Oman. The acquisition grid consists of 1600 25-m-spaced receivers covering a 1×1-km area. The sources (vibrators) are located on a similar square grid shifted with respect to the receiver grid by 12.5 m in both directions. The complete data set then consists of 1600 × 1600 time-domain signals. For this study, two receivers at the center of the array are selected. They are separated by a distance of 155 m, corresponding to a surface wave traveltime of about 0.13 s at 15 Hz (wavelength ∼ 80 m). We consider 240 sources in a 300-m-inner-radius, 70-m-thick ring centered on the receivers (see Figure 4(a)).

  (b)) and (iii) correlated between both receivers. Results are then summed over sources with an angular weighting B(θ). The waveform obtained is thus the correlation between receivers of interest for the specific source distribution B(θ). Time shift δt (that is, the traveltime error) due to non-isotropy of source distribution is then obtained by cross correlating the arrivals as constructed with B(θ) = constant and B(θ) of interest. To give relative time shifts, we estimate the arrival time t by the maximum of the reference wavelet envelope. Finally, to enhance the quality of results, we azimuthally average these measurements. We thus consider 17 other pairs of receivers in the same region, oriented in two main directions (about -130 • and -40 • ) and spaced by distance between 150 and 158 m (θ is redefined for the different pair orientations). Theoretical predictions result from equation 2 using a frequency of 15 Hz and a traveltime of 0.13 s.Results obtained for two different kinds of distribution (see Figures5(a) and 5(c)) are presented in Figure5. We can see that theoretical predictions fit the data measurements well for both examples. The asymptotic theory seems to describe successfully the apparent time shifts in real data, provided an intensity distribution sufficiently smooth (precise quantitative assessment of "sufficient smooth" is an asymptotic issue and has not yet been studied) and despite (i) simplifying assumptions behind equation 2 (plane waves in homogeneous medium) and (ii) receiver separation as small as two wavelengths. It is worth noting that the case presented in Figure2(c) for the Parkfield region, may be explained by this theory considering a steep distribution of the form B(θ) = 1 + B 2 cos(2θ) shown in Figure5(c). In that case, the theoretical time shift obtained with equation 2 using values involved in the Parkfield case (we take t = 3 s and ω 0 = 2π × 0.15 = 0.94 rad/s ) and B 2 = -0.6 corresponds to the very large time shift observed (about 30%).

  use 37 stations of the network centered on the same position as the center of the receiver positions considered before, and the same 240 sources as used previously. The geometry is shown in Figure7(a). Beamforming is computed for each of the 240 different sources on the entire sub-array formed by the 37 receivers. The total distribution may then be seen as the sum of all source contributions. Processing consists in considering [0-1.5 s] whitened signals (i.e., the temporal window corresponding to the ballistic waves) and summing incoherently beamformer outputs over 6 frequencies between 10 and 15 Hz.

Figure 6

 6 Figure6(a) shows an example of beamformer output for a single source whose location

Finally, we have

  seen that scattering may play a useful role in reducing this error since correlating coda waves instead of ballistic waves, allows us to mitigate strongly the influence of source distribution in traveltime estimates. (a) Results of plane wave beamforming on the Parkfield network showing the variation of seismic noise direction between 0.1 Hz and 0.2 Hz, for 31 consecutive days. Direction is represented by angle in degree (North is 0 • , positive rotation is clockwise). The color scale corresponds to the maximum values of the beamformer, 1 corresponding to a perfect match between the data and the plane wave model. (b) Angular distribution of speed of incoming noise on Parkfield network for one day (North is 0 • , positive rotation is clockwise). The color scale corresponds to the agreement between the data and the plane wave model (1 corresponding to a perfect match). The angular spot width matches the expected angular diffraction spot size of the network. 2 (a) Station map of the Parkfield Network. The black arrow shows the average seismic noise direction (∼ 55 • ) indicating that the noise is coming from the Pacific Ocean. Station pairs of interest are indicated by black lines. (b) and (c) 1-day averaged correlation function for two particular days, JD18 (gray line) and JD22 (black dashed line). Two different pairs are considered: the pair 8-27 (b) mostly aligned with the noise direction and, contrariwise, the pair 13-30 (c) perpendicular to the noise direction. 3 Geometry considered. Two receivers at the center of the network are separated by a distance x. Sources are located in an annular region of large radius (compared to the wavelength of interest) centered on receivers. Each source s is located by an angle θ defined relative to the line connecting the receivers (positive rotation is anti-clockwise). 4 (a) Experimental geometry. 240 sources are selected and located in an annulus of inner radius 300 m and thickness 70 m, surrounding two receivers separated by 155 m. (b) Example of intensity (decibel scale) at receivers averaged over the 240 sources selected. The thin black line represents the boundary between both temporal windows considered in this Measured (cross) and predicted (dashed line) relative time shifts for two different kinds of source distribution. Theoretical values result from equation 2. Error bars correspond to standard deviation of the 18 pair measurements. The first case corresponds to a distribution of the form B(θ) = 1 + B 1 cos(θ).

  Figure 7(a). The black line corresponds to the angular position of the source. The an-

  Figure 1:

  Figure 2:

  Figure 4:

  Figure 6:

Figure 8 :

 8 Figure 8:

ACKNOWLEDGMENTS

The authors thank the Ministry of Oil and Gas of the Sultanate of Oman, Petroleum Development of Oman and Shell Research for permission to use the data. One of the authors (B.F.) acknowledges the support from Shell Research. This work is supported by the European Research Council (advanced grant Whisper).

APPENDIX A ACTUAL INTENSITY DISTRIBUTION FORMULATION AND ASSOCIATED TRAVELTIME ERROR

The actual intensity distribution taking into account both source and propagation effects is written in the form

where B * p (θ ′ , θ) describes the contribution of unit sources located in θ ′ to intensity in θ.

In first approximation we write:

and we assume

By making this approximation, we neglect multipathing considering that all energy in the direction θ, i.e. B p (θ), comes from a single source located in θ.

Equation A-1 then simplifies as:

The approximation used here is particularly valid in moderately heterogeneous media or when sources are not too far from receivers (in the case considered, source-receiver separation is approximately 4λ). Such values make this approximation usable in longer period seismology, for example in the primary and secondary microseism band (5-15 s) and station distances in the order of 50-150 km [START_REF] Shapiro | High-resolution surface-wave tomography from ambient seismic noise[END_REF].

From equation A-5, we derive:

Since the correlation function does not distinguish between positive and negative θ, we can write any distribution as a symmetric distribution. That's why we write B s in a Fourier series of cos(θ) and for θ 0 = 0

Then,


Vladimir Kostoglodov

Allen Husker

Nikolai M Shapiro

Juan S Payero

Michel Campillo

Nathalie Cotte

Robert Clayton

2006 Slow Slip Event and Nonvolcanic Tremor in the Mexican Subduction Zone

Last decade features an explosive sequence of discoveries of Slow Slip Events (SSE) and Nonvolcanic Tremor (NVT) in different subduction zones and continental faults. Many observations show that SSE is usually associated with an increased NVT activity but it is not clear yet if those events are the result of the same process or are independent expressions of a common underlying seismotectonic source. A large SSE in Central Mexico occurred in 2006 during the Meso-American Subduction Experiment (MASE) which provided continuous observations of the NVT for the 2005-2007 epoch. GPS and abundant seismic data show that although the NVT energy increased notably during the 2006 SSE, the two phenomena were separated spatially and not consistent in time. Significant NVT episodes occur during the period between SSEs, suggesting again that large slow slip events and NVT observed in the Mexican subduction zone may be of different origins. The

results presented here could contribute in an uncovering the nature of these indistinct in many cases natural phenomena.

Introduction

There is a growing scientific challenge to understand the origin of the Slow Slip Events (SSE) [START_REF] Schwartz | Slow slip events and seismic tremor at circum-Pacific subduction zones[END_REF] and Nonvolcanic Tremor (NVT) [START_REF] Obara | Nonvolcanic Deep Tremor Associated with Subduction in Southwest Japan[END_REF][START_REF] Rubinstein | Non-volcanic tremor: A window into the roots of fault zones[END_REF]. SSE and NVT observed now in different subduction zones may be a very important constituent in the cycle of large subduction thrust earthquakes. Many studies show that the SSE is often associated temporally and spatially with an increased NVT activity. Comprehensive investigations of NVT and short-term SSE in the subduction zones of Japan and Cascadia supported mainly the hypothesis that both these events are occurring simultaneously, and most likely on the same plate interface [START_REF] Rocca | Array analysis and precise source location of deep tremor in Cascadia[END_REF][START_REF] Obara | Non-volcanic deep low-frequency tremors accompanying slow slips in the southwest Japan subduction zone[END_REF][START_REF] Obara | Episodic slow slip events accompanied by non-volcanic tremors in southwest Japan subduction zone[END_REF][START_REF] Peterson | Possible relationship between nonvolcanic tremor and the 1998-2001 slow slip event, south central Alaska[END_REF][START_REF] Rogers | Episodic Tremor and Slip on the Cascadia Subduction Zone: The Chatter of Silent Slip[END_REF]. It results in a view of the problem for which the so called Episodic Tremor and Slip (ETS) events [START_REF] Rogers | Episodic Tremor and Slip on the Cascadia Subduction Zone: The Chatter of Silent Slip[END_REF] are just short-duration SSE and NVT, and are closely associated both in time and space [START_REF] Shelly | Non-volcanic tremor and low-frequency earthquake swarms[END_REF][START_REF] Wech | Seismic and geodetic constraints on Cascadia slow slip[END_REF]. The NVT bursts detected by seismometers in Japan probably consist of a large number of small magnitude low frequency earthquakes [START_REF] Shelly | Low-frequency earthquakes in Shikoku, Japan, and their relationship to episodic tremor and slip[END_REF], which combined slip on the fault may contribute additionally to the crustal deformations produced by concurrent slow slip events. Some observations nevertheless show that SSE and NVT are not systematically associated in the form of ETS. There are examples of NVT episodes happening without any geodetically (GPS) detectable SSE, and the reverse large, long-term SSE not always accompanied by the NVT [START_REF] Delahaye | Microseismicity but no tremor accompanying slow slip in the Hikurangi subduction zone, New Zealand[END_REF][START_REF] Mccaffrey | Slow slip and frictional transition at low temperature at the Hikurangi subduction zone[END_REF]. Unfortunately, and particularly in the regions of steeply dipping subducted slabs, the small magnitude tremor and slow slip events cannot be analyzed in detail. The subduction zone in Central Mexico provides a favorable opportunity for the study of NVT and SSE because of its unusual subhorizontal plate interface and its large periodic slow slip events of M w ~7.5 [START_REF] Cotte | Anticipating the Next Large Silent Earthquake in Mexico[END_REF][START_REF] Kostoglodov | A large silent earthquake in the Guerrero seismic gap, Mexico[END_REF].

Seismic Data and Analysis

Large SSE in Central Mexico coincidentally occurred in 2006 during the Mezo-America Subduction Experiment (MASE) which provided an important set of seismological data making it possible to assess the properties of NVT in the region (Fig. 1).

There were 100 broad-band seismic stations installed approximately every ~6 km along the profile running almost perpendicular to the Pacific coast of Mexico, from Acapulco over 550 km inland [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF]. Unfortunately, in spite of excellent along profile space resolution of the MASE data those do not provide an adequate 2D coverage for accurate hypocentral estimates [START_REF] Payero | Nonvolcanic tremor observed in the Mexican subduction zone[END_REF].

The common approach in the NVT research consists in applying different methods of tremor detection with the records of different stations, then merging them for source location [START_REF] Kao | The Source-Scanning Algorithm: mapping the distribution of seismic sources in time and space[END_REF][START_REF] Rocca | Cascadia Tremor Located Near Plate Interface Constrained by S Minus P Wave Times[END_REF][START_REF] Obara | Nonvolcanic Deep Tremor Associated with Subduction in Southwest Japan[END_REF][START_REF] Shelly | Non-volcanic tremor and low-frequency earthquake swarms[END_REF]. In general, the precise hypocenter estimation of the NVT is still problematic due to the low signal to noise ratio and the lack of identification of coherent seismic phases. In absence of complete set of precise NVT locations a simple way to describe the tremor activity in time is the differential duration of tremor (DDT) episodes (e.g., [START_REF] Payero | Nonvolcanic tremor observed in the Mexican subduction zone[END_REF][START_REF] Shelly | Non-volcanic tremor and low-frequency earthquake swarms[END_REF] ) disregarding their amplitudes. The DDT bears also on the NVT intensity with time since the average tremor amplitudes seems to be invariant to the NVT bursts duration [START_REF] Aguiar | Moment release rate of Cascadia tremor constrained by GPS[END_REF].

To avoid the difficulties related to tremor source location from the MASE data and at the same time to characterize their duration and intensity, we apply an "energy" inference [START_REF] Kostoglodov | Nonvolcanic Tremor Activity is Highly Correlated With Slow Slip Events, Mexico[END_REF]. The tremor wave-train develops mainly between 0.5-15 Hz but has the best signal to noise ratio in the 1-2 Hz frequency range at most of the MASE stations (fig. S1). Therefore, the first step in our energy approach is to filter signals between 1 and 2 Hz and to compute their energy (squared velocity). To obtain comparable NVT energy assessments at every seismic station, it is necessary to correct the records for the site effects after individual background noise reduction. The site effects are estimated using the coda calibration method [START_REF] Husker | Automatic non-volcanic tremor detection in the Mexican subduction zone[END_REF]. We then apply median filter to smooth the entire waveforms and a threshold cutoff procedure to compile a complete register (a list of initial time and duration) of NVT recorded with the MASE seismic stations for 2005-2007.

The NVT register was compiled using the data at 24 MASE stations, which had acceptable signal-to-noise ratio. A space-time distribution of the total NVT radiated energy in 1-2 Hz bandwidth was obtained via integration in time and interpolation in space (Fig 2). It is clearly seen that the NVT seismic energy was radiated to the seismic stations located predominantly above the subhorizontal interplate contact segment, which is at approximately 40 km depth, between 170 and 260 km from the trench. A maximum of the NVT energy attenuation curve (fig. S2) constrains the location of particular tremor bursts train along the MASE profile. Thus, the space distribution of the maximum energy release (Fig. 2) corresponds to the locations of the NVT sources.

GPS data modeling

Surface displacements produced by the 2006 slow slip event were measured by permanent GPS stations, most of them located in Guerrero, Central Mexico [Larson et al., 2007a] along the MASE seismic profile (Fig. 1). The largest displacements, up to 4.4 cm, are observed at the coastal area. The onset and duration of the 2006 SSE were estimated by fitting the sigmoid function to the GPS horizontal components records reduced for the inter-SSE trend at each GPS station (Fig. 2). The total final displacements during the SSE with respect to the North America plate were determined (after appending back the trends) as a difference between maximum and minimum of the fitting function. Dislocation models [START_REF] Savage | A Dislocation Model of Strain Accumulation and Release at a Subduction Zone[END_REF] that best fit the total final displacements during the 2006 SSE (fig. S3) require that most of the slow slip (with a maximum of about 19 cm) occurred on the transition zone of the plate interface, a segment about 80-90 km long located between the seismogenic coupled zone (45-75 km from the trench), and a presumably freely slipping zone (Fig. 2). The model still acceptably fit the data when the inland tip of the SSE zone expands some 10-15 km more into the NVT source area, but the slip on the downdip dislocation patch in this case should be notably reduced.

It should be noted that the forward static modeling of the final GPS displacements along the 1D transect provides better constraints on transient segments location and the amount of fault slip. An attempt to invert GPS time series of the 2006 SSE [Larson et al., 2007b] using the Network Inversion Filter (NIF, [START_REF] Segall | Time dependent inversion of geodetic data[END_REF]) allows excessive inland extension of the slow slip on the plate interface due to very poor approximation of the vertical component time series. Modeling the previous 2002 Guerrero SSE [START_REF] Iglesias | The silent earthquake of 2002 in the Guerrero seismic gap, Mexico (Mw=7.4): inversion of slip on the plate interface and some implications[END_REF][START_REF] Kostoglodov | A large silent earthquake in the Guerrero seismic gap, Mexico[END_REF]Yoshioka et al., 2004] ascertain that almost total slip was released on the transient segment of subduction interface limited by ~170 km inland from the trench.

Results

Fig. 2 summarizes the main results of this study. The energy distribution suggests that the NVT is mainly located within or above the deep free slipping patch of the plate interface (at 170-250 km from the trench). Indeed, the previous study [START_REF] Payero | Nonvolcanic tremor observed in the Mexican subduction zone[END_REF] analyzed more than one hundred strong individual NVT episodes in Guerrero reported locations of those tremors which coincide with the NVT energy release area and the NVT depths are mostly distributed over the lower continental crust and plate contact zone.

The 2006 SSE transient dislocation is determined immediately updip of this zone, which means that the SSE and NVT are spatially separated. This observation agrees with the SSE and NVT study in Oaxaca, Mexico subduction zone [START_REF] Brudzinski | Nonvolcanic tremor along the Oaxaca segment of the Middle America subduction zone[END_REF]. Four large distinct NVT burst episodes occurred during the lone 2006 SSE. Furthermore, there were several comparably long lasting and intensive NVT episodes during the "quiet" epochs when no apparent SSE was detected by GPS (except perhaps a poorly resolved in the GPS time series, short term SSE in March 2005 [START_REF] Vergnolle | Slow slip events in Mexico revised from the processing of 11 year GPS observations[END_REF]). The lack of simultaneity between NVT and long term SSE suggests that they are not an expression of the same underlying process.

We applied a simple method to evaluate an upper limit of the total seismic energy radiated in all NVT episodes during the 2006 SSE from the estimated ratio of the energy power spectrum in 1-2 Hz band and the total energy of the NVT 1 . If all NVT events were occurring on the same fault (plate interface) coherently with similar seismic slip rakes, then the total fault slip produced by all NVT events during the 2006 SSE would be of order 10 -3 -10 -4 cm. This value is negligible compared to the average modeled dislocation (~10 cm) of the SSE. The surface expression of this NVT slip can not be measured by modern GPS technique. As the upper limit of the NVT slip is assessed applying a number of generous assumptions, the real effect can be even smaller. That may explain why GPS data do not exhibit any notable surface displacements produced by large NVT episodes during the inter-SSE epochs. The slip detected during the NVT events in Japan [START_REF] Ito | Slow Earthquakes Coincident with Episodic Tremors and Slow Slip Events[END_REF] by the high sensitivity borehole tiltmeters was apparently produced by concurrent short-term SSE.

Equivalent magnitude of the 2006 SSE (M W ~7) is more than three orders larger than the magnitude of the NVT episodes (M E 3.3) occurring during the period of slip. Remarkably, a difference of the same order was reported for the SSE (M w 6.7) and NVT (M w 3.7) moment magnitudes estimates in the Cascadia subduction zone [START_REF] Kao | Northern Cascadia episodic tremor and slip: A decade of tremor observations from 1997 to 2007[END_REF].

Two types of slow slip events reported in southwest Japan subduction zone [START_REF] Hirose | Repeating short-and long-term slow slip events with deep tremor activity around the Bungo channel region, southwest Japan[END_REF] are the long-term and short-term SSE, both accompanied by NVT. The source area of the short-term SSE is deeper than that of the long-term SSE [START_REF] Ueno | Seismic explosion survey around slow events in the Tokai region, Japan[END_REF]. This observation would be congruent with the results of this study if the short-term and small magnitude SSE recently detected in Guerrero [START_REF] Vergnolle | Slow slip events in Mexico revised from the processing of 11 year GPS observations[END_REF] were occurring in the NVT area of the subduction zone. Reliable modeling of the fault slip locations of these mini-SSE is still problematic because the corresponding displacements recorded by GPS are almost at the noise level of the position time series.

Discussion

To date, there are several reliable observations in different subduction zones which suggest the existence of two distinct phenomena: large aseismic slow slip events (SSE) and the NVT episodes. If the NVT is accompanied by the smaller magnitude, short-term SSE occurring in the same area then this combination is considered as the ETS. Long term SSE and NVT recorded in the Mexican subduction zone are separated spatially and are not strictly correlated in time, although the NVT activity is intensified by large SSE. Triggering of NVT by stress transients has been documented [START_REF] Rubinstein | Non-volcanic tremor driven by large transient shear stresses[END_REF] and the mechanism of this interaction, as well as the origin of SSE and NVT, is still disputable.

There is an obvious progression of seismotectonic segmentation of the subduction plate interface in Mexico (Fig. 2D): the low-temperature, shallow weakly-coupled accretional segment; the seismogenic strongly-coupled section (T~150 C°); then the zone where the transient SSE occur (T 170-400 C°); and finally a supposedly "free slipping" zone where NVT are generated (T = 400-500 C°). Each section has its specific elastic strain accumulation and rebound regime with different recurrence times of the events (order of 10-10 2 years for the subduction thrust earthquakes, 10 -1 -10 years for the SSE and daysmonths for the ETS).

The modeling of the GPS data (Fig. 2 and fig. S3) locates the seismogenic coupled zone at shallow depth, with almost the same width and position as the source area of the 1962 M w ~7.0 Acapulco earthquake doublet [START_REF] Ortiz | Source areas of the Acapulco-San Marcos, Mexico earthquakes of 1962 (M 7.1; 7.0) and 1957 (M 7.7)[END_REF]. The transient SSE segment matches the ultra-slow velocity layer (USL) discovered in Guerrero on the top of the subducting plate [START_REF] Song | Subducting Slab Ultra-Slow Velocity Layer Coincident with Silent Earthquakes in Southern Mexico[END_REF], and the tremor generating zone [START_REF] Payero | Nonvolcanic tremor observed in the Mexican subduction zone[END_REF], corresponds to the higher conductivity area of the continental crust [START_REF] Jödicke | Fluid release from the subducted Cocos plate and partial melting of the crust deduced from magnetotelluric studies in southern Mexico: Implications for the generation of volcanism and subduction dynamics[END_REF] over the "free-slipping" segment of the plate interface. The free slip condition is only an abstraction used for the SSE modeling. In reality the subducting oceanic plate should interact with the overriding continental plate on this weak interplate segment too, probably though a low viscosity layer [START_REF] Hilairet | High-Pressure Creep of Serpentine, Interseismic Deformation, and Initiation of Subduction[END_REF][START_REF] Kostoglodov | Sediment Subduction -a Probable Key for Seismicity and Tectonics at Active Plate Boundaries[END_REF] similar to the USL, and transmit some shear stress to the upper plate. The accumulated strain in the continental plate may be periodically released by the NVT and small magnitude SSE.

Our observations and modeling results suggest the existence of two essentially different stages in the prograding process of the strain rebound on the downward portion of the subduction plate interface: the SSE and NVT. The SSE doesn't produce seismic radiation (at least above the noise level at current instrumentation) and it is a creep-like slip episode.

Slow slip modulates NVT activity by increasing the shear stress and dilatation over the downdip plate interface in the area where appropriate conditions exist for the NVT incidence. Thus, these two phenomena are separated in space and not necessarily consistent in time. Now it is confirmed from a number of observations in different subduction zones including Mexico that some SSE (at least long term events) occur lacking the concurrent detectable tremor at the same area. Nevertheless, based on our results it is not yet clear if the NVT may happen without short term SSEs which are still undetectable by the GPS technique.

Future studies in different subduction zones should verify this hypothesis which is important to understand the origin of the SSE and NVT. Yoshioka, S., T. Mikumo, V. Kostoglodov, K. M. Larson, A. R. Lowry, and S. K. Singh (2004), Interplate coupling and a recent aseismic slow slip event in the Guerrero seismic gap of the Mexican subduction zone, as deduced from GPS data inversion using a Bayesian information criterion, Phys Earth Planet In,, 513-530.

1 Axiliary materials are available in the HTML doi: … [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF]. The background image is a resistivity model [Jödicke et al., 2006]. Inverted pink triangles are annotated with the temperature °C which is modeled on the plate interface [START_REF] Manea | Thermal structure, coupling and metamorphism in the Mexican subduction zone beneath Guerrero[END_REF]. Green antenna-like symbols show projections of the GPS stations locations on the MACE transect. Panels A-C have the same time scale.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. A -Time distribution of all the (1-2 Hz) NVT seismic energy recorded during the

  Figure 1.

Acknowledgments 10 This study was supported by PAPIIT IN103808, CONACYT 84544, SEP-CONACYT-ANUIES-ECOS M06-U02, I832 (G-GAP) ANR, ANR-06-CEXC-005 (COHERSIS), and ERC advanced Number 227507 "WHISPER" grants. The MASE experiment of the Caltech Tectonics Observatory was funded by the Gordon and Betty Moore Foundation. Contribution 82 from the Caltech Tectonics Observatory.

Panels B and D are aligned so that the zone where tremor was detected can be compared to the plate configuration.


Daniel Li
email: daniel.li@euler.univ-artois.fr

Hervé Queffélec
email: herve.queffelec@univ-lille1.fr

Luis Rodríguez-Piazza

Hervé

Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk

Keywords: Mathematics Subject Classification 2010. Primary: 30J99 -Secondary: 30H05 -30C99 -30H20 -46E15 -47B33 Key-words. Calderón-Zygmund decomposition, Carleson measure, weighted Bergman space

We prove that, for every α > -1, the pull-back measure ϕ(A α ) of the measure dA α (z) = (α + 1)(1 -|z| 2 ) α dA(z), where A is the normalized area measure on the unit disk D, by every analytic self-map ϕ : D → D is not only an (α + 2)-Carleson measure, but that the measure of the Carleson windows of size εh is controlled by ε α+2 times the measure of the corresponding window of size h. This means that the property of being an (α + 2)-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces.

Introduction and notation

It is well-known that every analytic self-map ϕ : D → D induces a bounded composition operator f → C ϕ (f ) = f • ϕ from the Bergman space B 2 into itself. By Hastings's version of the Carleson inclusion theorem ( [START_REF] Hastings | A Carleson measure theorem for Bergman spaces[END_REF]), that means that the pull-back measure A ϕ of the normalized area measure A by ϕ is a 2-Carleson measure, that is, for some constant C > 0,

A {z ∈ D ; ϕ(z) ∈ W (ξ, ε)} ≤ C ε 2
for every ε ∈ (0, 1) and every ξ ∈ T, where W (ξ, ε) is the Carleson window centered at ξ and of size ε. It was proved in [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], Theorem 3.1, that one actually has an infinitesimal version of this property, namely, for some constant C > 0:

(1.1) A {z ∈ D ; ϕ(z) ∈ W (ξ, εh)} ≤ C A {z ∈ D ; ϕ(z) ∈ W (ξ, h)} ε 2 , for every ε ∈ (0, 1) and h > 0 small enough. Now, consider, for α > -1, the weighted Bergman space B 2 α . By Littlewood's subordination principle, every analytic self-map ϕ of D induces a bounded composition operator C ϕ from B 2 α into itself (see [START_REF] Mccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF], Proposition 3.4). By Stegenga's version of the Carleson theorem ([9], Theorem 1.2), that means that the pullback measure of A α (see (1.3) below) by ϕ is an (α + 2)-Carleson measure. Our goal in this paper is to show the analog of (1.1) in the following form.

Theorem 1.1 For each α > -1, there exists a constant C α > 0 such that, for every analytic self-map of the unit disk ϕ : D → D, every ε ∈ (0, 1) and every h > 0 small enough, one has, for every ξ ∈ T:

A α {z ∈ D ; ϕ(z) ∈ W (ξ,εh)} ≤ C α ε α+2 A α {z ∈ D ; ϕ(z) ∈ W (ξ, h)} . (1.2)
It should be stressed that the heart of the proof given in [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF] in the case α = 0 cannot be directly used for the other α > -1, and we have to change it, justifying the current paper. Moreover, the present proof is simpler than that of [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF]. We also pointed out that the result holds in the limiting case α = -1, corresponding to the Hardy space H 2 ([5], Theorem 4.19), but the proof is different, due to the fact that one uses the normalized Lebesgue measure on T and the boundary values of ϕ instead of measures on D and the function ϕ itself.

We end the paper by an application to the compactness of composition operators on weighted Bergman-Orlicz spaces.

Another application of Theorem 1.1 is given in [START_REF] Li | On approximation numbers of composition operators[END_REF].

Notation. In this paper, D = {z ∈ C ; |z| < 1} denotes the open unit disk of the complex plane C, and T = ∂D is the unit circle. The normalized area measure dx dy π is denoted by A.

For α > -1, the weighted Bergman space B 2 α is the space of all analytic functions

f (z) = ∞ n=0 a n z n on D such that f 2 α := D |f (z)| 2 dA α (z) < +∞ ,
where A α is the weighted measure

(1.3) dA α (z) = (α + 1)(1 -|z| 2 ) α dA(z) .
The Carleson window centered at ξ ∈ T and of size h, 0 < h < 1, is the set

W (ξ, h) = {z ∈ D ; |z| ≥ 1 -h and | arg(z ξ)| ≤ h} . A measure µ on D is called an α-Carleson measure (α ≥ -1) if sup |ξ|=1 µ[W (ξ, h)] = O h→0 (h α ).
Actually, instead of the Carleson window W (ξ, h), we shall merely use the sets

S(ξ, h) = {z ∈ D ; |z -ξ| ≤ h} ,
which have essentially the same size, so µ is an α-Carleson measure if and only if

sup |ξ|=1 µ[S(ξ, h)] = O h→0 (h α ).
We denote by Π + the right-half plane

(1.4) Π + = {z ∈ C ; Re z > 0} .
To avoid any misunderstanding, we denote by A the area measure on Π + , and not this measure divided by π.

Let T : D → Π + be the conformal map defined by:

(1.5)

T (z) = 1 -z 1 + z ;
we denote by τ α = T (A α ) the pull-back measure defined by:

(1.6) τ α (B) = A α [T -1 (B)]
for every Borel set B of Π + . This is a probability measure on Π + .

We also need another measure µ α on Π + , defined by:

(1.7) dµ α = x α dxdy .

Given two measures µ and ν, we shall write µ ∼ ν when the Radon-Nikodým derivative dµ dν is bounded from above and from below. The pseudo-hyperbolic distance ρ ′ on D is given by

(1.8) ρ ′ (z, w) = z -w 1 -zw , z, w ∈ D .
For every z ∈ D and r ∈ (0, 1),

∆ ′ (z, r) = {w ∈ D ; ρ ′ (w, z) < r}
is called the pseudo-hyperbolic disk with center z and radius r. It is (see [START_REF] Beardon | The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications[END_REF], [START_REF] Duren | Bergman Spaces[END_REF], or [START_REF] Zhu | Operator Theory in Function Spaces[END_REF], for example) the image of the Euclidean disk D(0, r) by the automorphism

ϕ z (ζ) = z -ζ 1 -zζ •
The pseudo-hyperbolic distance ρ on Π + is deduced by transferring the pseudo-hyperbolic distance ρ ′ on D with the conformal map T :

(1.9) ρ(a, b) = ρ ′ (T -1 a, T -1 b) = a -b ā + b ,
and, for every w ∈ Π + and r ∈ (0, 1),

∆(w, r) = {z ∈ Π + ; ρ(z, w) < r}
is the pseudo-hyperbolic disk of Π + with center w and radius r.

Finally, we shall use the following notation:

(1.10) Ω = (0, 2) × (-1, 1) .

Acknowledgement. Part of this work was made during a visit of the second named author at the Departamento de Análisis Matemático of the Universidad de Sevilla in April 2011; it is a pleasure to thank all people of this department for their warm hospitality. The third-named author is partially supported by a Spanish research project MTM 2009-08934.

Transfer to the right half plane

As in [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], we only have to give the proof for ξ = 1 and, by considering g = h/(1 -ϕ), we are boiled down to prove: Theorem 2.1 Let α > -1. There exist constants K 0 > 0, c 0 > 0 and λ 0 > 1 such that every analytic function g : D → Π + with |g(0)| ≤ c 0 satisfies, for every λ ≥ λ 0 :

A α ({|g| > λ}) ≤ K 0 λ α+2 A α ({|g| > 1}) .
As said in the Introduction, this result is an infinitesimal version of the fact that the pull-back measure A α,ϕ of A α by any analytic self-map ϕ of D is an (α + 2)-Carleson measure. In fact, one has the following result. Proposition 2.2 There is some constant

C = C α > 0 such that (2.1) A α ({|g| > λ}) ≤ C λ α+2 |g(0)| α+2
for every analytic function g : D → Π + and every λ > 0.

The goal is hence to replace in the right-hand side the quantity |g(0)| α+2 by A α ({|g| > 1}).

Proof of Proposition 2.2. We may assume that |g(0)| = 1. Hence we may assume that λ > 2, taking C ≥ 2 α+2 , because A α ({|g| > λ}) ≤ 1.

Set

ϕ(z) = [g(z) -g(0)]/[g(z) + g(0)]. Then |g(z)| > λ implies that |ϕ(z) -1| = 2 |Re g(0)|/|g(z) + g(0)| ≤ 2/(λ -1) ≤ 4/λ .
But ϕ maps D into itself, so the measure A α,ϕ is an (α + 2)-Carleson measure and (see the proof of [START_REF] Stegenga | Multipliers of the Dirichlet space[END_REF], Theorem 1.2)

A α ({|g| > λ}) ≤ A α,ϕ [S(1, 4/λ)] ≤ C ′ α C ϕ 2 /(λ/4) α+2 ,
where C ϕ is the norm of the composition operator C ϕ : B 2 α → B 2 α . But ϕ(0) = 0 and hence C ϕ = 1, by using Littlewood's subordination principle and integrating.

For technical reasons, that we shall explain after Lemma 3.4, we need to work with functions defined on Π + . Proposition 2.2 becomes: Proposition 2.3 There exists a constant C = C α > 0 such that, for every analytic function f : Π + → Π + , one has:

(2.2) τ α ({|f | > λ}) ≤ C λ α+2 |f (1)| α+2 . Proof. Set E f (λ) = {|f | > λ} and define similarly E g (λ) = {|g| > λ} where g = f • T : D → Π + . We have g(0) = f (1)
as well as the simple but useful equation:

(2.3) T -1 [E f (λ)] = E g (λ).
So that, by Proposition 2.2:

τ α [E f (λ)] = A α [T -1 (E f (λ)] = A α [E g (λ)] ≤ C λ α+2 |g(0)| α+2 = C λ α+2 |f (1)| α+2 ,
and Proposition 2.3 is proved. Now, to prove Theorem 2.1, it suffices to prove that, when one localizes f on Ω, one may replace the quantity |f (1)| in the right-hand side of (2.2) by

τ α ({|f | > 1} ∩ Ω). This is what is claimed in the next result. Theorem 2.4 There exist constants K = K α > 0, c 1 > 0 and λ 1 > 1 such that every analytic function f : Π + → Π + such that |f (1)| ≤ c 1 satisfies, for every λ ≥ λ 1 : τ α ({|f | > λ} ∩ Ω) ≤ K λ α+2 τ α ({|f | > 1} ∩ Ω).
We shall prove Theorem 2.4 in the next section, but before, let us see why it gives Theorem 2.1 and hence our main result, Theorem 1.1.

Proof of Theorem 2.1. Let E : Π + → D be the exponential map defined by

(2.4) E(z) = e -πz ,
which (up to a radius) maps bijectively Ω onto the annulus

(2.5) U = {z ∈ D ; |z| > e -2π }.
For every g : D → Π + with |g(0)| ≤ (1 -β)/(1 + β) and 0 < β < 1, one has, by Schwarz's lemma (see [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], eq. (3.9)):

|g(z)| > 1 =⇒ |z| > β .
Therefore we only have to work on the annulus U , taking c 0 ≤ tanh π in Theorem 2.1.

Let L = E -1 be the inverse map of the restriction of E to Ω, and

(2.6) σ α = L(A α )
be the pull-back measure of A α by L. This measure is carried by Ω and we have:

Lemma 2.5 On Ω, one has: σ α ∼ µ α ∼ τ α .
Taking this lemma for granted for a while, let us finish the proof of Theorem 2.1 (the measure µ α does not come into play here). Let g : D → Π + be an analytic function and f = g

• E : Π + → Π + (so that g = f • L on E(Ω)). We have |f (1)| ≤ c 1 if |g(0)| ≤ c 0 ,
with c 0 > 0 small enough. In fact, the analytic function h = T • g maps D into itself and hence, by the Schwarz-Pick inequality, h is a contraction for the pseudo-hyperbolic distance on D (see [START_REF] Beardon | The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications[END_REF], eq. (3.3), page 18, for example); hence ρ ′ [h(e -π ), h(0)] ≤ ρ ′ (e -π , 0) = e -π , that is g(e -π )-g(0)

g(e -π )+g( 0 Set:

E g (λ) = {|g| > λ} ∩ U and E f (λ) = {|f | > λ} ∩ Ω.
Observe that, as in (2.3),

L -1 [E f (λ)] = E g (λ) and E -1 [E g (1)] = E f (1).
Hence, in view of Theorem 2.4 and Lemma 2.5:

A α [E g (λ)] = A α L -1 [E f (λ)] = σ α [E f (λ)] ≤ K ′ α λ α+2 σ α [E f (1)] = K ′ α λ α+2 σ α E -1 [E g (1)] = K ′ α λ α+2 (Eσ α )[E g (1)] = K ′ α λ α+2 A α [E g (1)
] , which is exactly what we wanted to prove.

Proof of Lemma 2.5. Let us compute σ α with the change of variable w = E -1 (z). One has z = E(w) and

dA(z) = |E ′ (w)| 2 dA(w) π = 1 π e -2πRe w dA(w).
We get:

Ω h(w) dσ α (w) = U h(Lz) dA α (z) = (α + 1) U h(E -1 z)(1 -|z| 2 ) α dA(z) = α + 1 π Ω h(w) e -2πRe w (1 -e -2πRe w ) α dA(w), so that (2.7) dσ α (w) = α + 1 π e -2πRe w (1 -e -2πRe w ) α 1I Ω (w) dA(w).
Thus, on Ω, we have σ α ∼ µ α . Indeed, the factor e -2Re w is bounded from below and from above, and (1 -e -2Re w ) α ∼ (Re w) α as Re w goes to 0. This proves the first equivalence of Lemma 2.5.

To prove the second equivalence, we use the change of variable formula

z = T w in Ω h(u) dτ α (u) = U h(T z) dA α (z); it gives dτ α (w) = |T ′ (w)| 2 (1 -|T (w)| 2 ) α (α + 1) dA(w)/π, i.e.: (2.8) dτ α (w) = 4 α+1 (α + 1) π (Re w) α |1 + w| 2(α+2) 1I Ω (w) dA(w),
showing that µ α ∼ τ α on Ω.

Proof of Theorem 2.4

Let us split, up to a set of measure 0, the square Ω into dyadic sub-squares

(3.1) Q l = 2j 2 n , 2(j + 1) 2 n × 2k 2 n -1 , 2(k + 1) 2 n -1 of center (3.2) c l = 2j + 1 2 n + i 2k + 1 2 n -1 ,
with n ≥ 0 , 0 ≤ j, k ≤ 2 n -1 and where l = (n, j, k).

Note that Ω = Q (0,0,0) . We are going to use the special form of the measure τ α , taken in (2.8), to get a localized version of Proposition 2.3 as follows.

Proposition 3.1 There is a constant C α > 0 such that, for any analytic function f : Π + → Π + and any dyadic sub-square Q l of Ω, one has, for any λ > 0:

(3.3) τ α ({|f | > λ} ∩ Q l ) ≤ C α λ α+2 τ α (Q l ) |f (c l )| α+2 .
Proof. Using Lemma 2.5, we may replace the measure τ α by dµ α = x α dxdy. This measure is no longer a probability measure, but it has the advantage of being invariant under vertical translations, and, especially, to react to a dilation of positive ratio λ by multiplying the result by the factor λ α+2 .

We first need a simple lemma.

Lemma 3.2 For every 0 ≤ s < 1, there exists a constant M s > 0 such that, for any analytic function f : Π + → Π + and any pseudo-hyperbolic disk ∆(c, s) in Π + , we have, for every z ∈ ∆(c, s):

(3.4) 1/M s ≤ |f (z)|/|f (c)| ≤ M s .
Proof. By the classical Schwarz-Pick inequality, any analytic map f : Π + → Π + contracts the pseudo-hyperbolic distance ρ of Π + (see [START_REF] Beardon | The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications[END_REF], Section 6), so that if z ∈ ∆(c, s), one has:

|u| := f (z) -f (c) f (z) + f (c) ≤ z -c z + c ≤ s.
Inverting that relation, we get Proof. Recall that c l is the center of Q l . We claim that we can find some numerical s < 1 such that Q l ⊂ ∆(c l , s). To show that claim, let l = (n, j, k) and z, w ∈ Q l . We have:

f (z) = uf (c)+f (c) 1-u , whence |f (z)| ≤ |f (c)| 1 + |u| 1 -|u| ≤ |f (c)| 1 + s 1 -
1 -ρ(z, w) 2 = 1 - z -w z + w 2 = 4 Re z Re w |z + w| 2 • But one has 2j/2 n ≤ Re z, Re w ≤ 2(j + 1)/2 n whereas |Im (z + w)| ≤ 2 -n+1 ; hence Re z Re w ≥ 4j 2 4 -n and |z + w| 2 = (Re z + Re w) 2 + [Im (z + w)] 2 ≤ 16(j + 1) 2 4 -n + 4.4 -n ≤ 80j 2 4 -n , because j ≥ 1. Therefore 1 -ρ(z, w) 2 ≥ 4 4j 2 4 -n 80j 2 4 -n = 1 5 ,
so that ρ(z, w) ≤ s = 4/5. In particular, we have Q l ⊆ ∆(c l , s).

τ α ({|f | > λ} ∩ Q l ) ≤ τ α (Q l ) ≤ C α λ α+2 τ α (Q l ).
This is the desired inequality, since we have supposed that |f (c l )| = 1.

Lemma 3.4 Inequality (3.3) holds when the square Q l , of the n-th generation, touches the boundary of Π + , namely when l = (n, j, k) with j = 0.

Proof. This case uses the specific properties of the measure µ α . In view of Lemma 2.5, we have to prove that:

(3.5) µ α ({|f | > λ} ∩ Q l ) ≤ C α λ α+2 µ α (Q l ) |f (c l )| α+2 ,
when the square Q l ⊆ Ω is supported by the imaginary axis. We may again assume that |f (c l )| = 1, and we proceed in three steps. 1) First, (3.5) holds if Q l = Q (0,0,0) = Ω: this is just what we have proved in Proposition 2.3 with (2.2).

2) For h > 0, (3.5) holds when Q l = hΩ = (0, 2h) × (-h, h) is a square meeting the imaginary axis in an interval (-h, h) centered at 0. Indeed, setting E f (λ) = {|f | > λ} as well as f h (z) = f (hz), we easily check that

(3.6) E f (λ) ∩ h Ω = h [E f h (λ) ∩ Ω].
For example, if v ∈ E f h (λ) ∩ Ω, one has |f (hv)| > λ and hence w = hv ∈ E f (λ) ∩ hΩ, giving one inclusion in (3.6); the other is proved similarly. Using the already mentioned (α + 2)-homogeneity of the measure µ α , we obtain, using (2.2) for f h :

µ α [E f (λ) ∩ hΩ] = µ α [h E f h (λ) ∩ Ω ] = h α+2 µ α [E f h (λ) ∩ Ω] ≤ h α+2 C α λ α+2 |f h (1)| α+2 = µ α (Q l ) C ′ α λ α+2 |f (c l )| α+2 , with C ′ α = 4 -(α+2) (α + 1)C α , since the center c l of Q l = hΩ is c l = h. 3) Finally, (3.5) holds if Q l is
any square supported by the imaginary axis. Indeed, this Q l is a vertical translate of the second case, and the measure µ α is invariant under vertical translations, which exchange centers. This ends the proof of the crucial Lemma 3.4 and thereby that of Proposition 3.1.

Remark. We see here why it is better to work with functions f : Π + → Π + instead of functions g : D → Π + ; if the invariance of µ α under vertical translations corresponds to the rotation invariance of A α , the homogeneity of µ α , used in part 2) of the proof, corresponds to an invariance by the automorphisms ϕ a of D, with real a ∈ D, which is not shared by A α , and writing a measure equivalent to A α having these properties is not so simple.

In order to exploit this proposition, we need the following precisions. Lemma 3.5 There exist constants c > 0 and δ 0 > 0, depending only on α, such that for every l, there exists R l ⊆ Q l with τ α (R l ) ≥ c τ α (Q l ) and, for every analytic map f :

Π + → Π + , (3.7) |f (z)| > δ 0 |f (c l )| for every z ∈ R l .
Proof. By Lemma 2.5, it suffices to prove this lemma with µ α instead of τ α . Let us consider two cases. 1) If l = (n, j, k) with j ≥ 1, we can simply take R l = Q l , in view of Lemma 3.2 and Lemma 3.3.

2) If l = (n, j, k) with j = 0, we may assume that Q l = Ω = (0, 2) × (-1, 1), since either vertical translations or dilations of positive ratio are isometries for the pseudo-hyperbolic distance on Π + and, on the other hand, multiply the µ α -measure by 1 or h α+2 respectively. It follows that c l = 1. We are going to check that ∆(1, 1/4) ⊆ Ω = Q l , so that we can take R l = ∆(c l , 1/4). Indeed, set t = 1/4; if |u| := z-1 z+1 ≤ t, we have z = 1+u 1-u and

0 < Re z = 1 -|u| 2 |1 -u| 2 ≤ 1 + t 1 -t < 2 ; |Im z| = 2 |Im u| |1 -u| 2 ≤ 2t (1 -t) 2 = 8 9 < 1.
Moreover, in view of Lemma 3.2, (3.7) holds with δ 0 = M -1 t = 3/5. Finally, the claim on the measures holds with c = µ α [∆(1, 1/4)]/µ α (Ω). Now, we want to control mean values of f on some of the Q l 's. In order to get that, we have to do a Calderón-Zygmund decomposition.

To that end, we need to know that the mean of |f | on Ω is small, namely less than 1, if |f (1)| is small enough. This is the aim of the next proposition. Proposition 3.6 There exists a constant C > 0 such that, for every analytic function f : Π + → Π + , one has:

(3.8) |f (1)| ≤ Ω |f (x + iy)| dxdy π ≤ C |f (1)| .
Moreover, if c is the center of an open square Q contained in Π + , then:

(3.9) π 4 |f (c)| ≤ 1 A(Q) Q |f (z)| dA(z) ≤ C π 4 |f (c)| .
Proof. Let us see first that (3.9) follows from (3.8). Let c = a + ib (a > 0 and b ∈ R) be the center of the square Q = (a -h, a + h) × (b -h, b + h), with 0 < h ≤ a. Consider the function f 1 defined by:

f 1 (z) = f [φ(z)] , where φ(z) = hz -h + a + ib .
Observe that φ : Π + → Π + is an affine transformation sending 1 onto c and that φ(Ω) = Q. Applying (3.8) to f 1 gives:

π 4 |f 1 (1)| ≤ 1 A(Ω) Ω |f 1 (z)| dA(z) ≤ C π 4 |f 1 (1)| .
This yields (3.9) using an obvious change of variable and f 1 (1) = f (c).

The left-hand side inequality in (3.8) is due to subharmonicity: consider the open disk D of center 1 and radius 1; then D ⊆ Ω and, |f | being subharmonic, we have:

|f (1)| ≤ 1 π D |f (x + iy)| dxdy ≤ 1 π Ω |f (x + iy)| dxdy .
We now prove the right-hand side inequality. Using Lemma 2.3 and the fact that µ 0 ∼ τ 0 on Ω (note that µ 0 is just the area measure A on Π + ), we have the existence of a constant κ > 0 such that, for all λ > 0:

(3.10) µ 0 ({|f | > λ} ∩ Ω) ≤ κ λ 2 |f (1)| 2 .
From this estimate (3.10), we can control the integral of |f | over Ω (recall that µ 0 (Ω) = 4):

Ω |f | dµ 0 = +∞ 0 µ 0 ({|f | > λ} ∩ Ω) dλ ≤ 4 |f (1)| + +∞ |f (1)| κ |f (1)| 2 λ 2 dλ = (4 + κ) |f (1)|.
The proposition follows.

Remark. We do not know if the constant π/4 in the left-hand side of (3.9) can be replaced by a better constant; however, it is not possible to replace this factor π/4 by 1. Let us see an example. Let us define f (z) = exp (T z) 4 ) where T z = (1 -z)/(1 + z). Recall that T sends Π + to the unit disk D, and therefore f (z

) ∈ Π + , for every z ∈ Π + because | arg(exp w)| < 1 < π/2, for all w ∈ D. Now let Q be the unit square Q = (-1, 1) × (-1, 1). For 0 < t ≤ 1/2, let Q t be the square, centered in 1, Q t = (1 -t, 1 + t) × (-t, t), which is contained in Π + , and define σ(t) = 1 A(Q t ) Qt |f (z)| dA(z) = 1 4t 2 t -t 1+t 1-t |f (x + iy)| dx dy .
Using a change of variable we have:

σ(t) = 1 4 Q |f (1 + tx + ity)| dx dy .
We are going to prove that there exists t such that σ(t) < 1 = |f (1)| and the average of |f | in the cube Q t is smaller than |f (1)|. Now observe that

f (z) = 1 16 (z -1) 4 + O (z -1) 5 , z → 1.
Consequently, there exists a constant C > 0 such that, for z

∈ Q 1/2 , Re f (z) ≤ 1 16 Re (z -1) 4 + C|z -1| 5
and then, there exists C 1 > 0, such that for every x + iy ∈ Q and t ∈ (0, 1/2),

|f (1 + tx + ity)| ≤ exp 1 16 Re t 4 (x + iy) 4 + C 1 t 5 = exp t 4 16 (x 4 + y 4 -6x 2 y 2 ) + C 1 t 5 .
Integrating over Q, putting:

τ (s) = 1 4 1 -1 1 -1 exp (s/16)(x 4 + y 4 -6x 2 y 2 ) + C 1 s 5/4 dx dy ,
we get that σ(t) ≤ τ (t 4 ), for t ∈ (0, 1/2]. We just need to prove that, for s > 0 close enough to 0, we have τ (s) < 1. But this is easy because τ (0) = 1, and Return now to the proof of Theorem 2.4. Consider, for every n ≥ 0, the conditional expectation of the restriction to Ω of |f | with respect to the algebra Q n generated by the squares

τ ′ (0) = 1 4 1 -1 1 -1 1
Q (n,j,k) , 0 ≤ j, k ≤ 2 n -1 (note that Q n ⊆ Q n+1 ): (3.11) (E n |f |)(z) = 2 n -1 j,k=0 1 A(Q (n,j,k) ) Q (n,j,k) |f | dA 1I Q (n,j,k) (z) ,
and the maximal function M f is defined by:

(3.12) M f (z) = sup n (E n |f |)(z) .
One has

(3.13) M (f )(z) = sup z∈Q (n,j,k) 1 A(Q (n,j,k) ) Q (n,j,k) |f | dA .
Since f is continuous on Ω, one has lim n→∞ E n |f |(z) = |f (z)| for every z ∈ Ω, and it follows that:

(3.14) {|f | > 1} ⊆ {M f > 1} .
Now, the set {M f > 1} ∩ Ω can be split into a disjoint union

{M f > 1} ∩ Ω = n≥1 Z n ,
where

Z n = {z ∈ Ω ; (E n |f |)(z) > 1 and (E j |f |)(z) ≤ 1 if j < n} .
(note that, by Proposition 3.6,

E 0 |f | ≤ 1 if |f (1)| is small enough). Since E n |f | is constant on the sets Q ∈ Q n , each Z n can be in its turn decom- posed, up to a set of measure 0, into a disjoint union E n = (j,k)∈Jn Q (n,j,k) .
By definition, for z ∈ Z n , one has

(E n |f |)(z) ≥ 1 and hence, for (j, k) ∈ J n , 1 A(Q (n,j,k) ) Q (n,j,k) |f | dA ≥ 1 for z ∈ Q (n,j,k) .
But, on the other hand,

(E n-1 |f |)(z) ≤ 1 for z ∈ Z n , and we have, if z ∈ Q (n,j,k) : (E n |f |)(z) = 1 A(Q (n,j,k) ) Q (n,j,k) |f | dA ≤ 1 A(Q (n,j,k) ) Q (n-1,j ′ ,k ′ ) |f | dA ≤ 4 1 A(Q (n-1,j ′ ,k ′ ) ) Q (n-1,j ′ ,k ′ ) |f | dA ≤ 4 ,
where Q (n-1,j ′ ,k ′ ) is the square of rank (n -1) containing Q (n,j,k) .

Finally, we can write {M f > 1} ∩ Ω as a disjoint union, up to a set of measure 0,

(3.15) {M f > 1} ∩ Ω = l∈L Q l ,
where L is a subset of all the indices (n, j, k), for which:

(3.16) 1 ≤ 1 A(Q l ) Q l |f | dA ≤ 4 .
τ α (E λ ∩ Ω) = τ α (E λ ∩ {M f > 1} ∩ Ω) = l∈L τ α (E λ ∩ Q l ) ≤ K α λ α+2 l∈L τ α (Q l ) |f (c l )| α+2 ≤ K α λ α+2 l∈L τ α (Q l ) 16 π α+2 = C α λ α+2 l∈L τ α (Q l ) = C α λ α+2 τ α ({M f > 1} ∩ Ω).
But, on the other hand, the sets R l of Lemma 3.5 are disjoint, since R l ⊆ Q l and we have |f | > δ 0 |f (c l )| > (4/πC) δ 0 := δ 1 on R l , in view of Lemma 3.5 and Proposition 3.6. Therefore:

τ α (|f | > δ 1 ) ≥ τ α l R l = l∈L τ α (R l ) ≥ c l∈L τ α (Q l ) = c τ α l∈L Q l = c τ α ({M f > 1} ∩ Ω). We get hence τ α (E λ ∩ Ω) ≤ C ′ α λ α+2 τ α ({|f | > δ 1 }) for λ ≥ 1, with C ′ α = C α /c. Applying this to f /δ 1 instead of f , we get: τ α ({|f | > λ} ∩ Ω) ≤ C ′′ α λ α+2 τ α ({|f | > 1})
for λ > λ 1 := 1/δ 1 , and that finishes the proof of Theorem 2.4.

An application to composition operators

In this section, we give an application of our main result to composition operators on weighted Bergman-Orlicz spaces.

Recall that an Orlicz function Ψ : [0, ∞) → R + is a non-decreasing convex function such that Ψ(0) = 0 and Ψ(x)/x → ∞ as x goes to ∞. The weighted Bergman-Orlicz space B Ψ α is the space of all analytic functions f : D → C such that 2) Conversely, if

(4.2) lim h→0 Ψ -1 (1/h α+2 ) Ψ -1 (1/h α+2 K µ (h)) = 0 , then B Ψ α is compactly contained in L Ψ (µ).
Here ρ µ is the Carleson function of µ, defined as: We gave in [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], in the non-weighted case, examples showing that conditions (4.1) and (4.2) are not equivalent for general measures µ. However, Theorem 1.1 implies that K ϕ,α+2 (h) ρ ϕ,α+2 (h)/h α+2 and so conditions (4.1) and (4.2) are equivalent in this case. Therefore, we get:

  ) ≤ e -π . It follows that |g(e -π )| -|g(0)| ≤ e -π |g(e -π )| + |g(0)| , i.e. |g(e -π )| ≤ 1 tanh π |g(0)|. Therefore |f (1)| = |g(e -π )| ≤ c 1 if |g(0)| ≤ c 0 with c 0 ≤ c 1 tanh π.

  s and, similarly, |f (z)| ≥ |f (c)| 1-s 1+s . The lemma follows, with M s = 1+s 1-s • Let us now continue the proof of Proposition 3.1. Lemma 3.3 Inequality (3.3) holds when the square Q l , of the n-th generation, does not touch the boundary of Π + , namely when l = (n, j, k) with j ≥ 1. More precisely, we have Q l ⊆ ∆(c l , s) where s < 1 is a numerical constant.

Now, to prove ( 3 . 3 )

 33 , we may assume, by homogeneity (replace f by f /|f (c l )| and λ by λ/|f (c l )|), that |f (c l )| = 1. We then have, by Lemma 3.2, |f (z)| ≤ M s |f (c l )| = M s for every z ∈ Q l . Hence (3.3) trivially holds when λ > M s , since then the set in the left-hand side is empty. So we assume λ ≤ M s . In that case, setting C α = M α+2 s , we have :

16 (x 4 + y 4 -6x 2 y 2 )

 442

Equations ( 3 .

 3 14), (3.15) and (3.16) define the Calderón-Zygmund decomposition of the function f . We are now ready to end the proof of Theorem 2.4. For λ ≥ 1, set E λ = {|f | > λ}; one has, by (3.15), Proposition 3.1 and (3.9):

D

  Ψ(|f |/C) dA α < +∞ for some constant C > 0. The norm of f in B Ψα is the infimum of the constants C for which the above integral is ≤ 1. With this norm, B Ψ α is a Banach space. Now, every analytic self-map ϕ : D → D defines a bounded linear operatorC ϕ : B Ψ α → B Ψ α by C ϕ (f ) = f • ϕ, called the composition operator of symbol ϕ.This is a consequence of the classical Littlewood's subordination principle, using the facts that the measure A α is radial and the function Ψ(|f |/C) is subharmonic for every analytic function f : D → C. Such an operator may be seen as a Carleson embedding J µ : B Ψ α → L Ψ (µ) for the pull-back measure µ = ϕ(A α ). S.Charpentier ([2]), following[START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], has characterized the compactness of such embeddings (actually in the more general setting of the unit ball B N of C N instead of the unit disk D of C):

Theorem 4 . 1 (

 41 S. Charpentier) For every finite positive measure µ on D and for every α > -1, one has:1) If B Ψ α is compactly contained in L Ψ (µ), then (4.1) lim h→0 Ψ -1 (1/h α+2 ) Ψ -1 (1/ρ µ (h)) = 0 .

  When µ = ϕ(A α ) is the pull-back measure of A α by an analytic self-map ϕ : D → D, we denote them by ρ ϕ,α+2 and K ϕ,α+2 respectively.

Theorem 4 . 2

 42 For every α > -1, every Orlicz function Ψ, and every analytic self-map ϕ : D → D, the composition operator C ϕ : B Ψ α → B Ψ α is compact if and only if:

Ψ - 1

 1 (1/h α+2 ) Ψ -1 (1/ρ ϕ,α+2 (h)) = 0 .