Peter D Nellist

Angus I Kirkland

Applications of the Oxford-JEOL Aberration Corrected Electron Microscope

Keywords: aberration correction, HRTEM, STEM user supplied): confocal microscopy aberration correction, HRTEM, STEM, confocal microscopy

published or not. The documents may come

The detailed configuration, specification and operation of the OJ1 instrument (Figure 1 (a)) have been described previously [1,2]. This instrument, which was subsequently commercialised by JEOL in a number of optical variants, was fitted with: a thermally-assisted Schottky field-emission gun; pre-specimen optics containing an aberration corrector, capable of illuminating the sample with either plane-wave illumination, for HRTEM, or providing a focused probe for STEM; a large area EDX detector; a post-sample aberration corrector for HRTEM; an incolumn omega energy-filter [3]; STEM bright-field and dark-field detectors; a TV- The purpose of this paper is to provide a review of the operation and application of both instruments. Their performance in both HRTEM and STEM modes will be illustrated with examples taken from research themes addressed by a number of different researchers in the Oxford EM group. Research at Oxford into novel methods that made use of the unique capabilities of these instruments will also be reviewed.

Aberration-Corrected High Resolution TEM Imaging and Exit Wavefunction Reconstruction

The aberration correctors fitted to both OJ1 and OJ2 instruments were supplied by CEOS GmbH, based on a design originally due to Rose [4,5]. In this arrangement, two sextupole elements are used to generate negative spherical aberration, and are coupled via a round-lens doublet such that one is conjugate to the other, but with a 180º rotation, produced by the transfer doublet such that the first-order effect of threefold astigmatism is cancelled. Crucially, the sextupoles are also coupled to the objective lens by a further round-lens doublet so that off-axial aberrations are also corrected in the aplanatic arrangement that is required for HRTEM imaging. Whilst this configuration results in minimal 5 th -order aberrations, the need for the sextupole elements to be optically conjugate to the objective lens precludes direct, independent control of the 5 th -order aberrations under aplanatic conditions [START_REF] Hartel | Proceedings of EMC 2004[END_REF]. In the OJ2 instrument these residual aberrations have been measured to be approximately 2 mm for both the round fifth order aberration (C 5 ) and six-fold astigmatism (A 5 ).

Phase contrast imaging in the TEM depends critically on the lens aberrations that form the phase-plate used to generate contrast and hence direct interpretation of images can be challenging. A particular focus of the activity at Oxford is the coupling of techniques to reconstruct the complex wavefunction at the exit-surface of the sample from either a focal-series imaging geometry [START_REF] Kirkland | [END_REF], or an illumination tilt-series [8] using electron optically corrected image data.

Direct electron optical correction offers the advantage that it may be achieved on line in a single image, with no requirement for post acquisition processing or acquisition of extended image series. However, as already noted the correctors fitted to both OJ1 and OJ2 instruments are only able to correct aberrations to third order, with partial compensation of selected higher order terms for real recorded image data.

Thus, for aberration corrected instruments computational reconstruction is beneficial as it provides a mechanism for compensation of for higher order aberrations and enables the recovery of the complex specimen exit surface wavefunction.

When used in combination, direct aberration correction and exit wave reconstruction provide additional advantages. For focal series datasets, the elimination of tilt-induced coma relaxes the requirement for using parallel illumination and enables the illumination to be converged onto the specimen area of interest. In this mode, the current density at the sample may be maintained with a reduced source emitter current and consequently lower energy spread in the illumination, providing an improved information limit. For a tilt series dataset, the elimination of tilt induced coma gives rise to less critical focus conditioning for any given tilt magnitude and multiple tilt magnitudes are feasible without a large induced focus change. For either data acquisition geometry, the reduced delocalisation in corrected images is advantageous for quantitative interpretation. Finally, in aberration corrected instruments the voltage centre and axial coma free axis are coincident and hence the temporal coherence envelope is symmetric.

To illustrate the potential benefits of a combined direct / indirect approach to aberration correction and compensation, Figure 2(a) shows the phase of the specimen exit surface wavefunction of a Pt catalyst nanoparticle, viewed close to a <110> direction reconstructed from a focal series of aberration corrected images acquired using the OJ1 instrument [9]. Despite the presence of a partially graphitic carbon support, terraces and steps at the edge of the particle are visible and moreover, the visibility, and spatial resolution of these features are improved significantly compared to a conventional HRTEM image. The monatomic steps that are visible in Figure 2 are conventionally termed A or B type, with either {100} or {111} microfacet atomic risers that have different catalytic properties. The peaks in the recovered phase can be related to the number of Pt atoms in each atomic column and can thus be used to obtain information about the local atomic arrangement on each surface in projection (Figures 2(b) and (c)). Detailed analysis of these provides evidence that the outermost atomic layers consist of irregular islands of atoms, in contrast to the atomic arrangements that have been proposed on the basis of homoepitaxial growth experiments on extended Pt surfaces. Our next example demonstrates experimental verification of the resolution improvement possible using aberration corrected tilted illumination images. This requires a specimen with a real space lattice separation beyond the axial information limit and a gold foil oriented along a <123> direction satisfies this requirement.

Figure 3 compares the phases and moduli of exit wavefunctions restored using both focal series and tilt defocus aberration corrected data sets. The only Fourier space detail present in the phase and modulus of the exit wavefunction (Figure 3 (c)) recovered from the focal series dataset relates to the {111} planes, with a 0.235 nm lattice spacing. In contrast, the wavefunction recovered using tilted illumination data (Figure 3 (f)) contains information at the positions of the {331}, {420} and {242} lattice reflections, corresponding to real space distances of 0.093 nm, 0.091 nm and 0.083 nm .

Although the data presented in this section clearly demonstrates that a combination of direct aberration correction and exit wave reconstruction is advantageous and moreover, that the tilt azimuth acquisition geometry enables superresolution, it is useful to examine the limiting factors in this approach. At large beam tilt magnitudes partial temporal coherence causes a loss of transfer at the centre of the tilted transfer function. For a beam tilt, τ 0 = λk 0 , beams at the same angle τ 0 with respect to the tilt axis are perfectly transferred and the temporal coherence envelope has no effect. However, for larger beam tilts the transfer falls away significantly inside this ring of perfect transfer (the achromatic circle). Reducing the tilt angle prevents this central transfer loss but reduces the resolution potential improvement. This reduction in transfer for tilted illumination can in principal be overcome either by increasing the number of azimuthal angles or by including more than one tilt magnitude in the restoration dataset. However, either of these approaches increases the number of images required with consequent implications for experimental stability required and for the overall radiation dose. The second limiting factor is given by the variation in the experimentally measured axial aberration coefficients with illumination tilt which can be estimated by taking the real part of the expansion of the wave aberration function with respect to the complex aberration coefficients and illumination tilt angle. Thirdly, in contrast to exit wavefunction reconstruction using focal series data the individual images within a tilt series data set are not simply different measurements of the same projected specimen potential. For tilted illumination the specimen potential is projected along a slightly different direction compared to the axial case and this introduces an additional phase shift. This can be estimated using a simple geometric parallax model and choosing a maximum phase variation for a resolvable distance, d as ±π/2, the maximum parallax allowed between the top and bottom specimen surfaces relative to the middle of the specimen is d/4.

For a beam tilt, τ 0 , this leads to a maximum specimen thickness of d/(2τ 0 ). This simple geometric parallax argument is independent of the imaging conditions but has been found to be more stringent than the tilt limit determined from a full dynamical calculation.

These limits for exit wavefunction reconstruction using tilted illumination images are summarised in Table 1. This clearly demonstrates that the aperture synthesis approach to exit wavefunction reconstruction can be carried out using datasets recorded at larger tilt magnitudes under aberration corrected imaging conditions, giving rise to the possibility of resolution improvement substantially beyond the axial information limit. However at extremely high resolution, parallax considerations become increasingly significant leading to a new regime in which sample thickness limits the resolution achievable.

Aberration Corrected STEM Imaging

The advantages of aberration-correction in a STEM geometry are that it provides smaller illuminating probes together with higher beam currents as has been described previously [10,11]. A sextupole corrector, identical to the one described earlier for HRTEM imaging, is used to correct the pre-specimen optics operating in an effectively inverted configuration. However, for STEM imaging it is not necessary to correct for off-axial aberrations since the image is scanned and the scan coils can be placed between the corrector and the objective lens such that the corrector is always axially aligned relative to the illuminating beam. The relaxation of this constraint allows the round 5 th -order aberration (C 5 ) to be nulled by adjusting the transfer lenses such that the negative C 5 combination aberration [START_REF] Hartel | Proceedings of EMC 2004[END_REF], generated by the deliberate misprojection of the corrector with respect to the objective lens, compensates for the inherent C 5 arising from pre-field of the objective lens. The limiting aberration for STEM is thus six-fold astigmatism, as is clearly demonstrated in the Ronchigram shown in Figure 4. In the STEM geometry, alignment of the corrector is achieved using an automated procedure developed by CEOS that relies on annular dark-field (ADF) images of nanoparticles recorded at various, known illumination tilts through the corrector and objective lens system. In practice we have found that 3 rd -order aberrations are stable over weeks of operation (as is also the case for alignment of the lower corrector) and consequently day-to-day tuning of the lower order aberrations is generally performed manually by direct observation of the Ronchigram. In addition to the aberration corrector and associated coupling optics, a three condenser lens system completes the illumination optics of which two are the standard JEOL condenser lenses, with an additional third adaptor lens inserted before the corrector to ensure correct coupling. These three pre-corrector lenses allow independent adjustment of the source demagnification and beam convergence angle enabling beam currents of up to 2 nA and convergence angles of up to 32 mrad to be readily be achieved in the OJ2 instrument.

ADF imaging is the most widely used mode in the STEM geometry as it enables direct incoherent imaging of materials with a recorded intensity that depends strongly on atomic number [12]. This method was successfully applied by Cockayne and co-workers using the 2200FS (OJ1) instrument in an investigation of the distribution of rare earth dopant atoms in intergranular ceramic films [13]. These intergranular films play an important role in controlling the physical properties of many ceramic materials and doping with rare-earth metals has been shown to act as a sintering aid improving crack resistance. In order to explain observed differences in grain anisotropy in Si 3 N 4 ceramics doped with different rare-earth elements, Cockayne and co-workers [13] used the 2200FS (OJ1) instrument to record aberration corrected ADF STEM images along different crystallographic directions for sample containing La and Lu (Figure 5). The aim of the study was to explain why La is more effective than Lu in elevating toughness. The Z-contrast nature of the ADF images allow the dopant atoms to be observed in the intergranular film, and preferential sites are seen for the dopants in the notionally amorphous volume. As can be seen in Fig. 5, data was recorded for the same system along two perpendicular directions, thereby allowing the three-dimensional position of the additive elements to be measured quantitatively. Such quantitative measurements, quoted along with estimated errors, had not achieved before in earlier studies on this system. More recently, progress has been made in improving the quantification of ADF STEM images. In addition to quantitatively measuring atom positions, there has been progress in making quantitative use of image intensities by calibration of the detector in terms of the fraction of the incident beam detected [14] and using quantitative comparisons with simulations [15]. A crucial step in this approach is to quantitatively map the efficiency of the detector as a function of detection position. In both the OJ1 and OJ2 instruments the STEM detectors are located in the same optical plane as the HRTEM imaging detectors and in order to collect a range of scattering angles, as is required for STEM imaging, the post-specimen optics are adjusted to diffraction mode. However, if the post-specimen optics are changed to imaging mode, a magnified image of the STEM probe is formed in the image plane. In this latter configuration, scanning the STEM probe with suitable adjustment of the magnification enables the probe image can be scanned over the ADF detector allowing its efficiency to be mapped. The OJ2 instrument is fitted with two ADF detectors located before and after the omega energy filter. Operation of the pre-filter detector is more straightforward since high-angle scattering does not need to pass through the energy-filter, obviating the need for accurate filter alignment. An efficiency map of this detector is shown in Figure 6 and as is the case for most ADF detectors, it is evident that the response is non-uniform which must be considered when extracting quantitative information from ADF images. For quantitative ADF STEM imaging, calibration of the digitized intensity values in terms of the incident beam is necessary. Typical fractions of the incident beam scattered onto the ADF detector are usually around 10%. On the OJ2 instruments, a routine procedure for quantitative ADF STEM imaging has been developed. In this method the condenser optics are adjusted such that the incident beam current is reduced to ca. 10% of the nominal value used for imaging and this reduced beam current is used to measure the detector map. If the fractional reduction in current is accurately measured using a Faraday cup, the signal measured in any subsequent experiment will be calibrated in terms of fraction of the incident beam current. This approach is valid as long as the detector amplifier contrast (gain) and brightness (offset) are not altered after the initial calibration because experiments have shown that the detector response is linear with respect to beam current [START_REF] Nellist | Proc. EMAG2009[END_REF].

Under these conditions, images can be recorded that are calibrated such that the intensity of the image is expressed as the fraction of the primary beam electrons that are scattered to the ADF detector. Figure 7 shows such an image of a core-shell catalyst nanoparticle comprising a Pd core and Pt shell. The aim of the synthesis is to generate a monolayer of Pt conformally covering the Pd core. There is clear evidence for the Pt shell in the image, but the relatively straight boundaries to the brighter Pt regions suggest that the Pt shell has coverage only on particular facets. The high signal-to-noise ratio available in aberration-corrected STEM allows the terminating atomic columns and several single atoms decorating the surrounding carbon support, to be readily observed. Recording such calibrated data will now allow a direct comparison with simulation allowing more detailed quantitative analysis For quantitative comparison between experiments and simulations, a parameter that can be particularly challenging to quantify is the effective source size measured at the sample [START_REF] Nellist | Science of Microscopy[END_REF]. By the principle of reciprocity, this parameter plays the same role as the modulation transfer function (MTF) of the detector in HRTEM imaging [START_REF] Meyer | [END_REF]. However, as already described, in both the Oxford-JEOL instruments the post-specimen optics allow a aberration free image of the STEM probe to be recorded and this may provide a method for the measurement of STEM imaging parameters more accurately. We also note that accurate imaging of the STEM probe can also be a powerful diagnostic tool for the analysis of instabilities where it provides a sample-independent measurement.

Confocal Imaging

The ability to form an aberration free image of a STEM probe, as outlined in the previous section, enables confocal imaging using trajectories analogous to those used in confocal light microscopy. Scanning confocal electron microscopy (SCEM) was first demonstrated in uncorrected TEM by Frigo and coworkers [19] in an application to the specific problem of imaging thick samples. One of the key advantages of the confocal scanning optical microscope (CSOM), however, is the retrieval of threedimensional imaging through the process of optical sectioning.

In general the depth of focus, ∆z, for an optical system with a wavelength, λ, is given by [START_REF] Born | Principles of Optics[END_REF] α λ 2 sin ≈ ∆z where α is the semi-angle subtended by the focusing lens at the sample. In the case of light optics this value approaches the wavelength of the light. However, in an uncorrected electron microscope α typically takes values close to 0.01 and hence ∆z has values of tens of nanometers, which is larger than the typical thickness of a TEM sample. Aberration correction, however, has a dramatic effect on the depth of focus as an increase of a factor of three in α reduces the depth of focus by a factor of nine, leading to values of just a few nanometers, substantially less than typical sample thicknesses.

Using this reduced depth of focus to retrieve 3D information in STEM has been demonstrated previously [START_REF] Van Benthem | [END_REF][START_REF] Borisevich | Proc. Natl. Acad. Sci[END_REF]. In a non-confocal geometry, however, this approach can only be used for high-resolution imaging as the 3D transfer function for incoherent STEM imaging shows a missing cone which results in significant elongation in 3D images of laterally extended objects [START_REF] Behan | [END_REF]. One of the motivations for the development of SCEM is the removal of this out-of-focus blurring in the image. Various authors [24][25][26] have considered the contrast observed in a confocal geometry and in particular Cosgriff et al. [24] have shown that first-order phase contrast is not transferred in a confocal configuration. Therefore, the conventional "bright-field" imaging mode in the SCEM geometry will have limited application.

However, a more promising imaging mode is available in the 2200MCO instrument that makes use of both aberration correctors together with the in-column energy-filter, whereby it is possible to use the instrument as an energy-filtered scanning confocal electron microscope (EFSCEM). Importantly, it has been shown [25,27] that this mode does not suffer from a missing cone in the transfer function, and therefore extended objects can be imaged with maximum depth resolution.

The instrumental alignment for confocal experiments is challenging. A method for simultaneously tuning the optical elements required to form an aberrationcorrected focused probe illuminating the sample and aberration-corrected postspecimen optics subsequently focusing that probe to the detector plane has been previously described [28]. This involves using the electron Ronchigram to achieve tuned STEM optics, then switching the post-column optics to imaging mode such that a probe image is observed on a CCD camera. Making the assumption that the STEM probe is tuned, the post-specimen optics can be adjusted such that no aberrations are observed in the probe image. Figure 9 shows such a probe where some residual threefold astigmatism can be seen.

A particular difficulty with EFSCEM, however, is that a change in energy of the electrons of interest by inelastic scattering in the sample, combined with the chromatic aberration of the post-sample optics, defocuses the post-specimen optics. A method has been devised [29] in which the incident beam energy is changed by adjusting the high tension supply such that the confocal configuration is regained for electrons that suffer a specific energy loss in the sample (Figure 10), where it is evident that scattering is only brought to focus at the specific energy loss of interest.

Making use of such a configuration, Figure 11 shows the improvement in depth resolution that results from using the EFSCEM mode [29]. A purpose designed piezo-driven stage-scanning system [30] was used to acquire two-dimensional lateral (x-y) scans, or alternatively to slice in depth by scanning along one lateral direction and the depth direction to form an (x-z) scan. A carbon film decorated with Au nanoparticles was used as a test sample and the microscope was adjusted to the confocal mode for the carbon K-edge scattering. Figures 11(a The FWHMs of the intensity profiles L1 and L2 collected in EFSTEM modes (Fig. 11c) are more than 1 µm. In the EFSTEM mode, the out-of-focus scattered electrons are collected so this geometry cannot distinguish the different film thicknesses as expected. In the EFSCEM mode, rejection of scattering away from the confocal plane occurs and the data shows a FWHM in the depth direction of 84±21 nm and 42±21 nm at points 1 and 2, respectively (Fig. 11(f)). Thus direct imaging of the change in thickness of the carbon film can be observed using EFSCEM, and it can be seen that there are significant variations in the height of the carbon film between points 1 and 2. Work is currently underway to explore the range of application of this new technique. [100], respectively. Reproduced from Ref. [14] with permission of the American Institute of Physics.

Conclusions

  The appointment of David Cockayne as Professor in the Physical Examination of Materials in 2000 approximately coincided with the availability of commercial electron optical elements for correction of the inherent spherical aberration present in round electromagnetic objective lenses. Under Cockayne's leadership, the Electron Microscopy and Analysis group at Oxford was at the forefront of this revolution having been successful in obtaining funding for a unique instrument fitted, for the first time, with aberration correctors both before the sample for scanning transmission electron microscopy (STEM) and after the sample for conventional high resolution transmission electron microscopy (HRTEM). The ambitious instrumental development required was undertaken in collaboration with JEOL Ltd and the first instrument, the Oxford-JEOL 2200FS (OJ1), was installed in September 2003 and fully commissioned by the end of that year. This instrument operated successfully until late 2006 when it was replaced with an improved version (the JEOL 2200MCO (OJ2)) which remains in operation.

  CCD camera; and a 2048 x 2048 pixel high-resolution CCD camera. The replacement OJ2 instrument (Figure 1(b)) embodied a number of design modifications including improvements to the stability of the high-tension source (using twin HT tanks) and objective lens power supplies, a new mechanical frame, pre and post specimen high-speed piezo shutters and additional intermediate and projector lenses in addition to a larger 4096 x 4096 pixel CCD camera.

Figure 8

 8 Figure 8 schematically shows that in the confocal configuration, a small collector

  ) and 11(d) compare EFSCEM lateral x-y scans acquired with the sample positioned at the confocal height with and without a collector aperture present. Without a collector aperture, all the scattered intensity collected by the post-specimen lens contributes to the image, and this mode is therefore similar to energy-filtered STEM (EFSTEM) imaging and cannot be regarded as being a confocal. Figures 11(b) and 11(e) compare scans in the x-z plane at positions corresponding to y=0 nm in Figs. 11(a) and (d) respectively.

  In this paper we have reviewed the ways in which the aberration-corrected electron microscopes developed in partnership between Oxford and JEOL have been used since 2003. We firstly describe the advantages and ultimate limitations of combining aberration corrected TEM imaging with exit wavefunction reconstruction with particular reference to a tilt azimuth dataset geometry. Subsequently we review the use of ADF STEM imaging in studies of ceramic grain boundary interfaces carried out by David Cockayne's group and also describe recent progress in quantification of the ADF signal. Differences in the optimum way of operating the correctors for HRTEM and STEM have been discussed. Finally, the development of a new confocal mode of microscopy has been described, making use of the unusual optical configuration of the instrument.

Figure 1 .

 1 Figure 1. The Oxford-JEOL aberration corrected microscopes (a) JEM 2200FS

Figure 2 .

 2 Figure 2. (a) Restored phase of a 6-nm Pt particle obtained by applying spherical

Figure 3 .

 3 Figure 3. Exit wavefunctions for a gold foil oriented along the <123> direction. (a),

Figure 4 .

 4 Figure 4. A Ronchigram demonstrating the accuracy of aberration correction in the

Figure 5 .

 5 Figure 5. Experimental HAADF-STEM images of {100} crystal-intergranular film

Figure 6 .

 6 Figure 6. The ADF detector efficiency as a function of probe position across the

Figure 7 .

 7 Figure 7. An ADF STEM image of a Pd core/Pt shell nanoparticle with an intensity

Figure 8 .

 8 Figure 8. Schematic diagram of the confocal trajectories for SCEM which shows that

Figure 9 .

 9 Figure 9. The image recorded in the detector plane when the microscope is aligned

Figure 10 .

 10 Figure 10. An image of the intensity in the energy dispersion plane with the

Figure 11 .

 11 Figure 11. EFSCEM x-y scans acquired from a carbon film postioned at the confocal

  Figure 1

Figure 4 F

 4 Figure 4

Figure 5 F

 5 Figure 5

Figure 6 F

 6 Figure 6

Figure 7 F

 7 Figure 7

Figure 8 FFigure 11

 811 Figure 8

Table 1 .

 1 Tilt limits determined for an object thickness of 2 nm at 200 kV assuming third order aberration correction and a focal spread = 4 nm.

	Limiting Factor	Maximum Tilt	Resolution Limit
		Magnitude /mrad	/nm
	Incoherent Aberrations	19	0.0629
	Coherent Aberrations	25	0.0547
	Geomatric Parallax	18	0.0645
	.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors would like to acknowledge financial support from the Leverhulme Trust (F/08 749/B) and the EPSRC (EP/F048009/1) and Intel Ireland. Technical assistance from JEOL Ltd and JEOL (UK) Ltd is also acknowledged.


Emilie Roos

Quirine Barth

Hugo A Huijgen

Tania Tempelman

Annemarie Mj Mudrikova

Andy I M Wensing

Hoepelman

Roos E Barth
email: r.e.barth@

Quirine Huijgen

Hugo A Tempelman

Tania Mudrikova

Annemarie Mj

Andy Im Hoepelman

Presence of occult HBV, but near absence of active HBV and HCV infections in people infected with HIV in rural South Africa

Keywords: hepatitis B, hepatis C, occult infection, Sub-Saharan Africa, Human immunodeficiency syndrome

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Human immunodeficiency (HIV), hepatitis B (HBV) and hepatitis C (HCV) viruses are 2 endemic in Sub-Saharan Africa, but data regarding the prevalence of hepatitis co-3 infections in HIV-positive individuals residing there are limited. The aim of the study was 4 to determine the prevalence of HBV, HCV and occult HBV (presence of HBV-DNA in the 5 absence of HBsAg) in a rural, South African cohort. The results were compared to 6 various ethnic groups in a Dutch cohort of people infected with HIV. Antiretroviral-naïve 7 individuals with HIV from both a rural South African clinic (n=258), and a Dutch 8

University hospital (n=782), were included. Both serological (HBV and HCV) and 9 molecular (occult HBV) assays were performed. Logistic regression analysis was used to 10 define independent predictors of a hepatitis co-infection. 11

HBV and HCV prevalence rates in the South African cohort were exceptionally low (0.4%, 12 1/242 and 0.8%, 2/242 respectively), compared to those observed in Caucasians (HBV 13 4.4%, HCV 10.9%) and African immigrants (HBV 8.9%, HCV 4.8%). Conversely, occult HBV 14 was observed in a considerable proportion (10%, 6/60) of South African patients who 15 were anti-HBc-positive but HBsAg-negative. Occult infections were less frequent in [START_REF] Kew | Progress towards the comprehensive control of hepatitis B in Africa: a 17 view from South Africa[END_REF] Caucasians and Africans in the Dutch cohort (3.2% and 1.4% respectively). Independent 17 predictors for occult HBV were not identified, but a trend towards more occult HBV at 18 lower CD4 counts was observed. 19

Local HBV/HCV prevalence data are needed to optimise vaccination and antiretroviral 20 treatment strategies. Occult HBV in patients with HIV may be missed regularly when 21 molecular analyses are not available. Millions of people are infected with human immunodeficiency virus (HIV), hepatitis B 2 virus (HBV) and hepatitis C virus (HCV) globally. Sub-Saharan Africa is affected severely 3 by all three viruses; two-third of the world's 33 million people infected with HIV reside 4 there, as well as an estimated 50 million chronic HBV carriers [START_REF] Modi | Viral hepatitis and HIV in Africa[END_REF]5 WHO, 2002;WHO, 2007]. In addition, three percent of the African population is thought 6 to be infected with HCV [START_REF] Madhava | Epidemiology of chronic hepatitis C virus 30 infection in sub-Saharan Africa[END_REF]. 7

Chronic hepatitis can lead to cirrhosis and hepatocellular carcinoma. Due to shared 8 routes of transmission, the prevalence of HBV and HCV is believed to be high amongst 9

persons infected with HIV [START_REF] Burnett | Hepatitis B virus and human immunodeficiency virus co-infection in sub-17 Saharan Africa: a call for further investigation[END_REF][START_REF] Thomson | Epidemiology of hepatitis C virus infection in HIV-infected 24 individuals[END_REF]. Life 10 expectancy of people living with HIV in Sub-Saharan Africa has improved dramatically 11 since the increased availability of antiretroviral treatment (ART). The long-term effects 12 of HBV and HCV co-infections will therefore become more evident, causing a potential 13 increase in morbidity and mortality [Hoffmann and Thio, 2007;[START_REF] Modi | Viral hepatitis and HIV in Africa[END_REF]. 14 This is stressed by the observation that the progression of HBV-and HCV-related liver 15 disease is more rapid in patients with HIV [START_REF] Chen | Meta-analysis: increased mortality 24 associated with hepatitis C in HIV-infected persons is unrelated to HIV disease 25 progression[END_REF][START_REF] Nikolopoulos | Impact of hepatitis B virus infection on the 41 progression of AIDS and mortality in HIV-infected individuals: a cohort study and 42 meta-analysis[END_REF]. [START_REF] Kew | Progress towards the comprehensive control of hepatitis B in Africa: a 17 view from South Africa[END_REF] Prior to starting ART, it is important to know whether a co-infection with hepatitis is 17

present. An increased hepatotoxicity of antiretrovirals (ARVs) and other drugs is province in the north east of the Republic of South Africa. The centre provides primary 5 health care, including an ART program for people with HIV that has been active since 6

2003. Cohort details have been described elsewhere [START_REF] Barth | Effectiveness of highly active antiretroviral therapy 6 administered by general practitioners in rural South Africa[END_REF]. For this study, 7 all ART-naïve adults (≥15 years) with HIV who enrolled in the Ndlovu ART program from 8

September till November 2008, were included in this study. 9

The University Medical Centre Utrecht is an academic hospital in the city of Utrecht, 10 Utrecht. Serological tests for HBV (anti-HBc, HBsAg and anti-HBs) and HCV (anti-HCV) 5

were performed using the AxSYM ® automated analyzer (Abbott Laboratories, Abbott 6

Park, IL, USA). Anti-HBc was determined using the CORE tm® kit (Abbott Laboratories, 7

Abbott Park, IL, USA). Anti-HBc was considered positive if the ratio was above 1. All anti-8

HBc-positive blood samples were tested for HBsAg and anti-HBs using HBsAg(V2) and 9 AUSAB ® kits (Abbott Laboratories, Abbott Park, IL, USA). Ratios above 4 and 2 were used 10 to define positive anti-HBs and positive HBsAg respectively. In the case of an 11 indeterminate HBsAg result (ratio between 2 and 5) samples were centrifuged using an 12

Eppendorf centrifuge and retested. 13 HBV-DNA load was assessed in samples that tested anti-HBc positive, but did not have 14 detectable HBsAg levels. Quantitative HBV-DNA was determined using the Taqman ® fast 15 real-time nested PCR, as described previously [START_REF] Stuart | Occult 27 hepatitis B in persons infected with HIV is associated with low CD4 counts and 28 resolves during antiretroviral therapy[END_REF]Pas et al., 16 2000]. HBV-DNA was isolated using an automated nucleic acid extraction system 17 (MagNA Pure LC ® ; Roche Diagnostics, Penzberg, Germany). Lower limit of quantification 18 is 200 IU/mL; lower limit of detection is 10 IU/mL. Amplification was performed in 19 duplicate and a three-tiered control was executed (internal, no template and whole 20 procedure control). Outcomes were averaged and in the case of a 3-point difference in 21 DNA levels with the absence of HBV surface antigens were considered to have occult 5

HBV infections. 6

The HCV version 3.0 kit ® (Abbott Laboratories, Abbott Park, IL, USA) was used to 7 determine the presence of anti-HCV. A ratio above 5 was considered positive. Blood 8 samples with indeterminate results (ratio between 1 and 5), were centrifuged and 9 retested. Subsequently, a western blot was performed on all blood samples with a ratio 10 above 1, using the Autoblot 3000 ® (Innogenetics NV, Gent, Belgium). A positive western 11 blot in anti-HCV positive individuals was used to define an HCV infection. 12 13

Statistical analysis: 14

Frequencies were compared using the χ2 and Fischer's exact tests. Continuous data 15 were compared with the independent-sample t-test or Mann Whitney U test for non-16 parametric data. Determinants that were associated with the outcome in univariate 17 analysis (defined as a p-value <0.1) were included in multivariate analysis. Subsequently, 18 logistic regression analysis was done to determine independent predictors of the 19 African immigrants were male. Heterosexual contact was the main route of HIV 8 transmission in African immigrants and was assumed to be the HIV-transmission route in 9

all South African residents. With Caucasian Europeans, HIV was transmitted frequently 10 via homosexual contact. Baseline characteristics are summarised in Table 1. 11 12

Serological evidence of HBV and HCV co-infections: 13

Anti-HBc test results were available for 98% (242/248) of South African residents. Of the 14 242 patients, 68 (28.1 %) tested positive. Most anti-HBc positive patients were anti-HBs 15 positive as well (62.7%). Only one individual tested HBV surface-antigen positive. This 16 patient did not have anti-HBs. Overall, the prevalence of active HBV was only 0.4% 17

(1/242). Serological evidence of an HBV infection was predominantly present in men; 18

HBsAg and anti-HBc prevalence rates were 5.4% and 45.9% in men, compared to 2.3% 19 and 34.1% in women (p=0. Compared to the South African cohort, anti-HBc prevalence rates were significantly 1 higher in both Caucasians and African immigrants (41.4% and 57.6% respectively, 2 p<0.01). Furthermore, a larger proportion of patients was HBsAg-positive, resulting in an 3 active HBV prevalence of 4.4% in Caucasians and 8.9% in African immigrants p<0.01). 4

The prevalence of active HBV remained significantly lower in the South African cohort, 5 as compared to the other groups, when patients infected via heterosexual contact only 6 were included. 7 8 Serological evidence of an HCV infection was also infrequent in the South African cohort. 9

The anti-HCV prevalence rate was 0.8% for African residents, compared to 10.9% and 10

4.8% for Caucasians and African immigrants respectively (p<0.01). 11

All serological outcome data are presented in Table 2. 12 13

Occult HBV infections: 14

The presence of HBV-DNA was assessed in 93% (62/67) of South African patients who 15 were anti-HBc positive but tested HBsAg negative. For the remaining 5 individuals, an 16 insufficient amount of plasma was stored to perform molecular analyses. An occult HBV 17 infection was established in 9.7% (6/62) of tested individuals. HBV-DNA levels were low 18 in patients with an occult HBV infection. In 3 patients, viral loads were below the level of 19 quantification (between 10 and 200 IU/mL), twice just above this threshold (224 and 20 259 IU/mL respectively), and only once clearly higher (5222 IU/mL). 21 were also determined. Occult HBV was present in 2.7% (7/259) of these patients. The 2 prevalence rate was higher in Caucasians (6/187, 3.2%) compared to African immigrants 3

(1/72, 1.4%), but significantly lower compared to the South African cohort (p=0.04). 4

Nearly half of the patients with occult HBV also had anti-HBs (6/13, 46%). 5 6 Predictors for co-infections with HBV or HCV: 7 Age, gender, ethnic origin, HIV transmission route, baseline log HIV-RNA and baseline 8

CD4+ T-cell count were tested as possible determinants for having a HBV or HCV co-9 infection. In univariate analysis, all but the CD4+ T cell count were associated with the 10 presence of HBsAg and with the presence of anti-HCV. In multivariate analysis, only 11 lower age and male gender were independently predictive of an HBV infection. The 12 determinants that remained associated with a positive anti-HCV test were Caucasian 13 ethnicity, HIV transmission route (IVDU or blood products) and a lower baseline log HIV-14

RNA. 15

Among anti-HBc positive individuals, the above mentioned determinants and the 16 presence of anti-HCV and anti-HBs were tested as possible predictors for an occult HBV 17 infection. None remained associated significantly with this outcome in the multivariate 18 analysis, but a trend towards a higher occult HBV prevalence in case of a lower baseline 19 CD4+ T-cell count was observed (Table 3). 20 In this study an exceptionally low prevalence of HBs antigenemia (0.4%) and HCV 2 antibodies (0.8%) was found in a rural South African cohort of people infected with HIV. 3

Occult HBV on the other hand was observed in a considerable proportion (10%) of 4 patients who tested anti-HBc-positive and HBsAg-negative. 5 6

In Western countries, HBV is clearly more prevalent in people with HIV compared to 7 people without HIV [START_REF] Burnett | Hepatitis B virus and human immunodeficiency virus co-infection in sub-17 Saharan Africa: a call for further investigation[END_REF][START_REF] Kellerman | Prevalence of chronic 14 hepatitis B and incidence of acute hepatitis B infection in human 15 immunodeficiency virus-infected subjects[END_REF][START_REF] Thio | Hepatitis B and human immunodeficiency virus coinfection[END_REF]. This 8 relationship is attributed to common routes of transmission. In Sub-Saharan Africa the 9 association between HBV and HIV infections is less apparent, as people generally come 10 into contact with HBV long before the acquisition of HIV. Without making a distinction 11 between people with and without HIV, HBsAg prevalence in Africa is reported to be well 12 over 8% [START_REF] Barth | Hepatitis B/C and HIV in Sub-9 Saharan Africa: an association between highly prevalent infectious diseases. A 10 systematic review and meta-analysis[END_REF][START_REF] Burnett | Hepatitis B virus and human immunodeficiency virus co-infection in sub-17 Saharan Africa: a call for further investigation[END_REF][START_REF] Cdc | Prevalence of chronic infection with hepatitis B virus[END_REF][START_REF] Who | Hepatitis B. Department of communicable diseases surveillance and 27 response[END_REF]. A similar rate was 13 observed among African immigrants in the current study. Published HBsAg prevalence 14 data among South African patients infected with HIV vary (5-23%, [START_REF] Burnett | Increased exposure to hepatitis B virus infection in HIV-positive South 20 African antenatal women[END_REF]15 Firnhaber et al., 2009;[START_REF] Hoffmann | Hepatitis B virus infection and response to 11 antiretroviral therapy (ART) in a South African ART program[END_REF][START_REF] Lukhwareni | Increased 26 detection of HBV DNA in HBsAg-positive and HBsAg-negative South African 27 HIV/AIDS patients enrolling for highly active antiretroviral therapy at a Tertiary 28 Hospital[END_REF]Mphahlele et al., 16 2006]), but all are well above that observed in the cohort presented here. This is 17 remarkable, as HBV prevalence rates are generally assumed to be high in rural African 18 areas [Abdool [START_REF] Karim | High prevalence of hepatitis B virus infection 3 in rural black adults in Mseleni, South Africa[END_REF][START_REF] Kew | Progress towards the comprehensive control of hepatitis B in Africa: a 17 view from South Africa[END_REF]. It is unclear why the observed HBV 19 prevalence in this cohort is so low. Contact with high-endemic areas was possibly limited 20 due to geographical and ethnic factors that may have reduced the amount of migration 21 and number of interracial marriages. The paradigm is that the vast majority of people in Sub-Saharan Africa come into 1 contact with HBV early in life. This would result in the presence of anti-HBc in 70-95% of 2 individuals [START_REF] Burnett | Hepatitis B virus and human immunodeficiency virus co-infection in sub-17 Saharan Africa: a call for further investigation[END_REF][START_REF] Modi | Viral hepatitis and HIV in Africa[END_REF]. However, the majority (73%) of 3 patients infected with HIV in this South African cohort lacked anti-HBc. These results are 4 in line with other studies in South Africa that reported rather low anti-HBc prevalence 5 rates (35-38%, [START_REF] Burnett | Increased exposure to hepatitis B virus infection in HIV-positive South 20 African antenatal women[END_REF][START_REF] Firnhaber | Occult hepatitis B virus infection in patients 42 with isolated core antibody and HIV co-infection in an urban clinic in 1 Johannesburg, South Africa[END_REF][START_REF] Lukhwareni | Increased 26 detection of HBV DNA in HBsAg-positive and HBsAg-negative South African 27 HIV/AIDS patients enrolling for highly active antiretroviral therapy at a Tertiary 28 Hospital[END_REF]). 6

People without anti-HBc are still at risk for acquiring HBV. Therefore, providing them 7 with an HBV vaccine may be worth considering. Such vaccination is common practice in 8 western countries. Future studies will have to establish the risk of acquiring HBV at an 9 older age in African settings and determine whether routine vaccination of patients with 10 HIV who test anti-HBc negative is cost-effective there. 11

Reported anti-HCV prevalence rates for peoples infected with HIV in Africa vary; for 12

South Africa 1 to 13% of people with HIV are believed to be co-infected with HCV 13 [START_REF] Lodenyo | Hepatitis B and C virus infections and 23 liver function in AIDS patients at Chris Hani Baragwanath Hospital, Johannesburg[END_REF][START_REF] Mphahlele | High risk of 34 occult hepatitis B virus infection in HIV-positive patients from South Africa[END_REF][START_REF] Parboosing | Hepatitis C virus seropositivity in a South African 1 Cohort of HIV co-infected, ARV naive patients is associated with renal 2 insufficiency and increased mortality[END_REF]. In LICs, the 14 clinical consequences of being infected with HCV are limited. Treating HCV is generally 15 not feasible due to high costs and frequent adverse events associated with currently 16 available therapies. Moreover, a HCV vaccine is not yet available. 17 an even higher rate of occult HBV (88%) was found [START_REF] Firnhaber | Occult hepatitis B virus infection in patients 42 with isolated core antibody and HIV co-infection in an urban clinic in 1 Johannesburg, South Africa[END_REF]. These 2 observations contribute to the idea that occult HBV represents opportunistic viral 3 reactivation [Cohen [START_REF] Stuart | Occult 27 hepatitis B in persons infected with HIV is associated with low CD4 counts and 28 resolves during antiretroviral therapy[END_REF][START_REF] Lo Re | Prevalence, risk factors, 20 and outcomes for occult hepatitis B virus infection among HIV-infected patients[END_REF][START_REF] Shire | The prevalence and significance of occult hepatitis B virus in a prospective 10 cohort of HIV-infected patients[END_REF]. In the 4 current study, a significant association between the level of immunosuppression and the 5 presence of occult HBV was not found, but a trend towards more occult infections in the 6 case of a lower CD4 count was observed. In accordance with other African cohorts 7 [START_REF] Harania | HIV, hepatitis B and hepatitis C 3 coinfection in Kenya[END_REF][START_REF] Nagu | Hepatitis A, B and C viral co-infections among HIV-37 infected adults presenting for care and treatment at Muhimbili National Hospital 38 in Dar es Salaam, Tanzania[END_REF], no correlation between the CD4+ T-cell count 8 and the presence of an active HBV or HCV infection was found. The idea that HCV-9 infected individuals show the highest occult HBV prevalence [START_REF] Raimondo | Occult hepatitis B virus infection[END_REF]] 10 could not be confirmed by the data presented here. 11

In patients having an occult HBV infection only, occult HBV is linked to the transmission 12 of HBV and the development of liver disease [START_REF] Raimondo | Occult hepatitis B virus infection[END_REF]. However, in the 13 absence of prospective studies, it is unclear whether occult infections are associated 14 with higher liver-related morbidity and mortality rates. Moreover, the clinical relevance 15 of occult HBV is debatable as HBV-DNA loads are generally low and are likely to become 16 identified. Second, due to the low HBV-DNA levels (all but one <1000 copies/mL) and 2 the limited amount of plasma available, in-depth virological characterisation of HBV-3 DNA positive specimens was not performed. The possibility of a false positive result can 4 therefore not be ruled out completely, but is unlikely considering the extensive control 5 procedure that was applied. Third, the cohorts under study vary with regard to gender 6 distribution, nadir CD4 counts and routes of transmission. This may have contributed to 7 the observed difference in co-infection rates. Nonetheless, the results remain of interest 8 as the differences in baseline characteristics represent how the epidemics vary between 9 different geographical areas. 10 11

In conclusion, an exceptionally low frequency of HBV and HCV was observed in a cohort 12 of individuals infected with HIV in rural South Africa. Rates were lower compared to 13 both Caucasians and African immigrants who are receiving care in the Netherlands. 14 Occult HBV on the other hand was observed more often in the South African cohort. 15

This study highlights the need for local HBV/HCV prevalence data in order to guide 16 decisions on future vaccination and treatment strategies. 17

22

 22

16 17

 16 situated in the centre of the Netherlands. All adults (≥18 years) with HIV, who came into 11 care at the University Medical Centre Utrecht from January 1985 till January 2008, were 12 included. Patients were ART-naïve at time of serological HBV and HCV testing. 13For comparisons, analyses were limited to African and Caucasian people. Asians and 14 people originating from the Americas who were in care at the University Medical Centre 15 Utrecht were excluded.Data collection and definitions 18Patient characteristics were extracted from medical charts and clinical databases. Data 19 included in the study were the following: age, gender, ethnicity, country of birth, 20 probable route of HIV transmission and baseline laboratory variables (CD4+ T-cell count, 21 ASAT, ALAT). Venous blood was collected from all patients, and standard methods for 22 were used. Plasma was locally stored at -80 o C and shipped to 1 the Netherlands on dry ice. 2 HBV and HCV analyses were performed at the accredited (according to Dutch guidelines) 3 diagnostic laboratory of the virology department at the University Medical Centre 4

2 A

 2 values (Ct values) or a positive/negative difference, the procedures were 1 repeated. positive HBsAg result was used to define an HBV infection. Patients with detectable 3 HBV-DNA levels were considered to have HBV viremia. Patients with detectable HBV-4

2

 2 outcome. A P-value ≤0,05 was used to define statistical significance. Data were 20 processed and statistical analyses were done using SPSS version 15.Two hundred forty-eight ART naïve adults with HIV from Elandsdoorn, South Africa were 3 included in this study. The majority of included patients were female (73.8%). The Dutch 4 cohort consisted of 846 people infected with HIV. Caucasian Europeans (614, 73%) and 5African immigrants (168, 20%) predominated. In contrast to the South African cohort, 6 more males were included in the Dutch cohort; 535 (87%) of Caucasians and 93 (55%) of 7

21 Page

 21 2 and p<0.01 respectively). No gender difference was seen 20 with regard to the anti-HBs prevalence.

12 For 259

 12259 Dutch patients who were anti-HBc positive and HBsAg negative, HBV-DNA levels 1

22

 22

  Although active HBV was detected rarely in the current study, a considerable proportion 18 of anti-HBc positive patients in South Africa harboured an occult HBV co-infection. 19Previous South African studies reported approximately 23% of people with HIV -who 20 tested anti-HBc positive and HBsAg negative -to have detectable HBV-DNA levels 21[START_REF] Lukhwareni | Increased 26 detection of HBV DNA in HBsAg-positive and HBsAg-negative South African 27 HIV/AIDS patients enrolling for highly active antiretroviral therapy at a Tertiary 28 Hospital[END_REF][START_REF] Mphahlele | High risk of 34 occult hepatitis B virus infection in HIV-positive patients from South Africa[END_REF]. In one study that included severely 22 with HIV (mean CD4+ t-cell count 49 cells/mm 3 ), 1

18 19

 18 undetectable with lamivudine, the only drug with anti-HBV activity included frequently 17 in African ART regimens [Cohen Stuart et al., 2009]. There are potential limitations to this study. First, molecular analyses were limited to 20 anti-HBc positive individuals. Although an occult HBV infection is associated with the 21 presence of anti-HBV antibodies, a minority of occult carriers are negative for all HBV 22

  [START_REF] Torbenson | Occult hepatitis B[END_REF]. Such cases would not have been 1

17 4 Page

 4 This study was performed in compliance with relevant laws and institutional guidelines 1 and in accordance with the ethical standards of the Declaration of Helsinki. 2Due to the retrospective nature of this study, no experiments were performed; only 3 data used for clinical practice were included.

Table 2 :

 2 Serological outcome data according to ethnicity

	Caucasians	African	South African
		immigrants	residents
	(n=607)	(n=168)	(n=242)

Table 3 :

 3 Predictors of an occult HBV infection in anti-HBc positive individuals CD4: number of CD4+ T-cells, %: percentage of patients, IV drug use: intravenous drug use

		Occult HBV	Occult HBV	Univariate	Multivariate
		Yes	No	P value	P value
	CD4 (mean, cells/mm 3 )	214	363	0.06	0.06
	Presence of anti-HCV (%)	7.1	11.8	0.94	-
	Presence of anti-HBs (%)	63.6	67.9	0.77	-
	10log HIV-RNA (mean)	5.0	4.5	0.25	-
	Ethnic origin (%)			0.02	0.18
	• Caucasians	46.2	59.1
	• African immigrants	7.7	24.4
	• South Africans	46.2	16.5
	Male gender (%)	64.3	72.4	0.51	-
	HIV transmission route (%)			0.70	-
	• Heterosexual	57.1	48.3
	• Homosexual • IV drug use • Blood transfusions	35.7 7.1 0	46.1 3.3 2.2

Page 23 of 23 John Wiley & Sons Journal of Medical Virology

John Wiley & SonsJournal of Medical Virology

John Wiley & Sons Journal of Medical Virology John Wiley & Sons Journal of Medical Virology John Wiley & Sons Journal of Medical Virology John Wiley & Sons Journal of Medical Virology John Wiley & Sons Journal of Medical Virology John Wiley & Sons Journal of Medical Virology

Conflicts of Interest: 19

The authors have no conflicts of interest to declare. 20 Ethics Statement: 21

F o r P e e r R e v i e w


Pierre Cardaliaguet
email: cardaliaguet@ceremade.dauphine.fr

Luis
email: luis@math.uchicago.edu

Luis Silvestre

Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side

published or not. The documents may come

Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side

Introduction

We study the regularization effect of Hamilton-Jacobi equations with rough coefficients u t + H(x, t, Du, D 2 u) = 0.

(

Our aim is to show that, if the growth of H with respect to the gradient variable is sufficiently large, then bounded solutions will be Hölder continuous, with a Hölder modulus of continuity independent of the regularity of H with respect to (x, t). The regularization effect is based only on the strong coercivity assumption of H respect to Du, and not on the second order diffusion, which can be degenerate or even nonexistent.

We prove three results. The first one is the simplest case in which H does not depend on D 2 u (there is no diffusion). Theorem 1.1. Let u be a bounded viscosity solution of (1.1) in B 1 × [0, 1] for a function H = H(x, t, Du) that does not depend on D 2 u. Assume that for all x and t,

1 A |Du| p -A ≤ H(x, t, Du) ≤ A|Du| p + A,
for some constants A and p > 1.

Then u is C α in B 1/2 × [1/2, 1]
, where α and the C α norm of u depend on p, A and u L ∞ only.

This result was already proved in [START_REF] Cardaliaguet | A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable[END_REF] and [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF] using representation formulas for Hamilton-Jacobi equations and the reverse Hölder inequality. The proof that we present here is shorter and more elementary. Our new proof does not rely on the path representation of the Hamilton-Jacobi equation. Instead, an improvement of oscillation at all scales is obtained by comparing the solution to explicit super and subsolutions given by the classical formula of Lax and Oleinik.

Using the same idea, we can also address the case of Hamilton-Jacobi equations with diffusion. The proof is slightly longer because we cannot use the formula of Lax and Oleinik and instead we need to construct explicit super and subsolutions.

In order to handle a large class of diffusions, possibly degenerate, we define the extremal operators m ± (X) for any symmetric matrix X. Let m + (X) be the maximum positive eigenvalue of X (zero if none) and m -(X) be the minimum negative eigenvalue of X (zero if none).

Theorem 1.2. Let u be a bounded viscosity solution of (1.1) in B 1 × [0, 1]. Assume that for all x and t,

1 A |Du| p -Am + (D 2 u) -A ≤ H(x, t, Du, D 2 u) ≤ A|Du| p -Am -(D 2 u) + A,
for some constants A and p > 2. Then u is

C α in B 1/2 × [1/2, 1]
, where α and the C α norm of u depend on p, A and u L ∞ only.

Note that the assumption on H allows us to apply Theorem 1.2 to Hamilton-Jacobi equations with degenerate diffusion. In this result, the regularization effect is a consequence of the growth of H respect to Du only. The diffusion plays no role in the proof and in fact since p is assumed to be larger than two, it can be treated as a negligible effect in small scales.

Theorem 1.2 has already been proved in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF] and [START_REF] Cardaliaguet | Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient[END_REF] using a stochastic representation of the equation and the reverse Hölder inequality. Our proof relies on comparison principle only with explicit super and subsolutions. No techniques from probability are used.

Our third result is new and allows us to handle equations with a right hand side that is only bounded below but possibly unbounded above. In order to be able to apply this result we need stronger assumptions on the diffusion. The diffusion can still be degenerate (or even non existent), but we need it to depend on x smoothly. Theorem 1.3. Let u be a bounded viscosity solution of (1.1) in B 1 × [0, 1]. Assume that for all x and t,

1 A |Du| p -Tr(B(x, t)D 2 u) -f (x, t) ≤ H(x, t, Du, D 2 u) ≤ A|Du| p -Am -(D 2 u) + A, where A ≥ 1, p > 2, f ∈ L m for some m > 1 + d/p (d being the space dimension) and B(x, t) is a d × d symmetric nonnegative matrix that is C 1 respect to x. Then u is C α in B 1/2 × [1/2, 1]
, where α and the C α norm of u depend on p, A, u L ∞ , ||B|| C 1 and ||f || L m only.

Note that in this setting we can allow unboundedness of H only from one side. This restriction seems mandatory to ensure the boundedness of the solution.

This extension to unbounded coefficient is new, although related results, for elliptic problems, can be found in [START_REF] Dall'aglio | Local and global regularity of weak solutions of elliptic equations with superquadratic hamiltonian[END_REF]. It is partially motivated by [START_REF] Lasry | Mean field games[END_REF], in which solutions of the so-called mean field game equations are related with some optimal control problems for Hamilton-Jacobi equations of the form

u t -∆u + H(x, t, Du) = f (x, t)
where f ≥ 0 is a distributed control which has to be bounded in some Lebesgue space. Hölder regularity of Hamilton-Jacobi equation with a large growth with respect to the gradient variable has been addressed in several papers: in [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF], [START_REF] Barles | A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF] it is proved that subsolutions of stationary equations are Hölder continuous (even up to the boundary of the domain) as soon as the Hamiltonian is superquadratic. The Hölder regularity of solutions of (1.1) as in Theorem 1.1 or 1.2 has already been obtained in [START_REF] Cardaliaguet | Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient[END_REF]. This paper, based on techniques first introduced in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF], [START_REF] Cardaliaguet | A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable[END_REF], relies on (rather heavy) stochastic control representation for solutions of (1.1) combined with reverse Hölder inequalities. On the contrary, the new proof we present here uses only comparison principle with explicit functions that we can construct. There is no component of probability in the proof.

For Hamilton-Jacobi equations with second order diffusion, if the growth with respect to the gradient variable is at most quadratic, then the second order diffusion dominates in small scales. For uniformly parabolic equations in non divergence form, the solutions are Hölder continuous. In fact, by a simple exponential change of variables (first observed in [START_REF] Trudinger | Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations[END_REF] for elliptic equations), this statement is a simple consequence of the Hölder estimates from Krylov and Safonov for parabolic equations without gradient terms. Hamilton-Jacobi equations with fractional diffusions have also been studied in [START_REF] Silvestre | On the differentiability of the solution to the hamilton-jacobi equation with critical fractional diffusion[END_REF] and [START_REF] Silvestre | Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient[END_REF] in the case where the diffusion is stronger than the effect of the gradient on the Hamiltonian. Our assumption p > 2 in Theorems 1.2 and 1.3 explores the complementary condition in which the second order diffusion is weaker than the gradient terms at small scales. In our results, we treat the diffusion as a perturbation of the equation that plays no role in the regularization of the solutions. The case 1 < p ≤ 2 with degenerate diffusion is unclear, and we do not know if the solutions would become regular in that case.

The proofs we present in this article are based on the iteration at all scales of an improvement of oscillation lemma. This lemma says that the oscillation of a solution u in a cylinder is smaller by a fixed factor than the oscillation in a larger cylinder (also comparable by a fixed factor). The proof of this oscillation Lemma has the interesting feature that it uses the fact that the Hamilton-Jacobi equation sees points (i.e. points have positive capacity, i.e. controlled processes can be forced to hit a given point). The idea is that if u is small at one single point on the bottom of a parabolic cylinder, then the equation forces u to be small everywhere inside the cylinder. In the opposite case, if u is large at every point at the bottom of the cylinder, then naturally u will be large everywhere inside. Both implications are proved comparing u with explicit super and subsolutions.

The organization of the paper is the following. After this introduction, we have a small preliminary section where we analyze the scaling of the equation and we outline a general iterative improvement of oscillation scheme to prove Hölder continuity. Then we have three sections where we prove theorems 1.1, 1.2 and 1.3 respectively. Even though there is a significant overlap between these results, we kept the three sections independent except for one lemma used to prove Theorem 1.2 that is also used in the proof of Theorem 1.3. A reader interested in Theorem 1.1 only, needs to read sections 2 and 3 only (which is simple and very short). A reader interested in Theorem 1.2, needs to read sections 2 and 4 only.

Preliminaries

Scaling

We start this section we exploring the effect of several different scalings of the equation.

Let u be a sub or super solution to

u t + A|Du| p -εm ± (D 2 u) = f
then the function v(x, t) = cu(ax, bt) is a sub(super) solution of the equation

v t + Aa -p bc 1-p |Dv| p -εa -2 bm ± (D 2 u) = bcf (ax, bt)
In order to prove a C α estimate, we need scalings that preserve the constant A in front of the |Du| p and diminishes the effect of diffusion and zero order term.

A first choice to consider is c = 1, b = a p with a < 1. For a < 1, this scaling corresponds to zooming without altering the values of v (L ∞ scaling). If ε = 0 and f = 0 (as needed for Theorem 1.1), the scaling preserves the equation and we do not need to check anything else.

If ε > 0, we would want the scaling not to enhance the second order term. Thus we would need p > 2, so that a -2 b < 1 when a < 1. This will be the case in the proof of Theorem 1.2.

When the right hand side f is unbounded, we need to check the effect of scaling on its L m norm. We have

|bf (a•, b•)| L m = a p(1-1/m)-d/m f m .
Thus, one has a p(1-1/m)-d/m < 1 as soon as p(m -1) > d. This is the case for Theorem 1.3.

Let us now look at the effect of a C α scaling. Let α ∈ (0, 1) and β = p -α(p -1). Then the scaled function u r (x, t) = r -α u(rx, r β t) is a sub(super) solution of

∂ t (u r ) + A|Du r | p -εr -α(p-1)+p-2 m ± (D 2 u) = r p(1-α) f (rx, r β t).
If ε = 0 and f = 0, then the equation is preserved as before. When ε > 0 or f is unbounded, we need to choose α small enough so that the effect of the diffusion and the right hand side decrease as we scale in. For the diffusion part we need r -α(p-1)+p-2 < 1. Regarding the L m norm of f ,

|r p(1-α) f (r•, r β •)| L m = r δ f m with δ = 1 m (p(m -1) -d) + α m ((m -1)p + 1)
and, if p(m -1) > d, one can choose again α small in order to diminish the effect of the right-hand side.

Iterative improvement of oscillation

One of the most common methods to prove an interior Hölder regularity result is by proving an improvement of oscillation Lemma that is iterated at all scales. The improvement of oscillation Lemma says that the oscillation of a solution to a given equation in a parabolic cylinder decreases by a fixed factor (less than one) when the size of cylinder is reduced by another given factor.

The iteration of such result in a decreasing sequence of cylinders produces a Hölder modulus of continuity. We work out this general iterative procedure in the following Proposition.

Proposition 2.1. Let Q r (x, t) denote the cylinder B r (x) × [t -r β , t] for some constant β > 0.
Let u be a function with oscillation 1 in Q 2 (0, 0) such that the following "improvement of oscillation" property holds: there exists λ ∈ (0, 1) and α ∈ (0, 1) such that for all 0 < r ≤ 1 and any cylinder Q r (x, r) contained in Q 2 (0, 0) we have the following property if osc

Qr(x,t) u ≤ r α , then osc

Q λr (x,t)
u ≤ (λr) α .

(2.1)

Then u is C α (Q 1 (0, 0)) and |u(x, s) -u(y, t)| ≤ C |x -y| α + |s -t| α β ∀(x, s), (y, t) ∈ Q 1 ,
where C depends on λ and α only.

Typically, one proves that solutions to an equation are Hölder continuous by showing that (2.1) holds at unit scale r = 1. Then it holds for all r > 0 using the scale invariance of the equation. The proof of Proposition 2.1 is fairly standard. We include it here for completeness.

Proof. Let (x, t) ∈ Q 1 (0, 0). In order to show that u ∈ C α (Q 1 (0, 0)), we must show that osc Qr (x,t) u ≤ Cr α for all r ∈ (0, 1). We will prove by induction with C = 1 and the sequence of radii r k = λ k . Then it follows for all r by using C = λ -α .

For k = 0, we trivially know that osc Q1(x,t) u ≤ 1. Therefore, the desired inequality holds for r 0 . We now proceed with the inductive process. Assume that we know that osc Qr k (x,t) u ≤ r α k up to a certain k ≥ 0. But then the hypothesis (2.1) is exactly what we need to obtain osc

Qr k+1 (x,t) u = osc Q λr k (x,t) u ≤ (λr k ) α = r α k+1 .
And we finish the proof.

First order equations

In this section prove Theorem 1.1 about the Hölder regularity of first order Hamilton-Jacobi equations with superlinear growth in the gradient variable, but without dependence on D 2 u and without an unbounded right hand side. Theorem 1.1 is a consequence of the following more precise result.

Theorem 3.1. Let u be a continuous viscosity solution of the following two equations

u t + A|Du| p + A ≥ 0 (3.1) u t + 1 A |Du| p -A ≤ 0 (3.2) in B 1 × (0, 1), where p > 1. Then |u(x, t) -u(y, s)| ≤ C |x -y| α + |t -s| α p-α(p-1) ∀(x, t), (y, s) ∈ B 1/2 × [1/2, 1]
for some α ∈ (0, p ′ ) (where 1/p + 1/p ′ = 1) and C depending on A, p, u ∞ and dimension. Remark 3.2. Following section 5, the conclusion remains unchanged if u is a continuous viscosity solution of the two equations

u t + A|Du| p + A ≥ 0 (3.3) u t + 1 A |Du| p -f (x, t) ≤ 0 (3.4)
where p > 1 and f ∈ L m (B 1 × (0, 1)) for some m with p(m -1) > d. In this case, the constants α and C depend on A, p, f L m (B1×(0,1)) , u ∞ and dimension.

The proof of the improvement of oscillation lemma uses the Lax-Oleinik formula to construct local approximations of the function u and then apply comparison principle.

The classical Lax-Oleinik formula says that the solution v of the equation

v t + H(Dv) = 0 in B R × [-T, 0]
is given by the formula

v(x, t) = min (y,s)∈∂(BR×[-T,0]) v(y, s) + (t -s)L x -y t -s .
where L is the Legendre transform of H: L(ξ) = max rξ -H(r) and

∂(B R × [-T, 0]) denotes the parabolic boundary B R × {-T } ∪ ∂B R × [-T, 0]
. So, the formula above allows us to compute the values of v everywhere from the values on the parabolic boundary only.

In particular, if

H(ξ) = ε + A|ξ| p , L(r) = -ε + c p A 1-p ′ |r| p ′ , where 1/p + 1/p ′ = 1 and c p is some constant depending on p. For H(ξ) = -ε + 1 A |ξ| p , L(r) = ε + c p A p ′ -1 |r| p ′ Lemma 3.

(improvement of oscillation).

There exist positive constants T (large), θ and ε (small) such that for all continuous functions u which satisfy

u t + A|Du| p + ε ≥ 0 in B 2 × [-T, 0] (3.5) u t + 1 A |Du| p -ε ≤ 0 in B 2 × [-T, 0] (3.6) 0 ≤ u ≤ 1 in B 2 × [-T, 0] (3.7) then osc Q1=B1×[-1,0] u ≤ (1 -θ)
Proof. Recall that the Legendre transform of the function ξ → A|ξ| p is given by the formula c p A 1-p ′ |ξ| p ′ . We start by pointing out how T , θ and ε are chosen so as to make sure there is no circular dependence of the constants. We need the following three inequalities to hold:

c p (T -1) 1-p ′ A p ′ -1 3 p ′ ≤ 1 -4θ (3.8) c p T 1-p ′ A p ′ -1 ≥ 2θ (3.9) εT ≤ θ (3.10)
We can start by choosing T large so that the left hand side of (3.8) is strictly less than 1. Then θ > 0 is chosen small enough to make (3.8) and (3.9) hold. Finally ε > 0 is chosen small so that (3.10) holds. Now we prove that the result of the lemma holds. Let (x, t) be any point in Q 1 . From (3.6) we obtain an upper bound for u(x, t) using Lax Oleinik formula and the comparison principle

u(x, t) ≤ ε(t + T ) + min y∈B2 u(y, -T ) + c p (T + t) 1-p ′ A p ′ -1 |x -y| p ′ ≤ εT + min y∈B2 u(y, -T ) + c p (T -1) 1-p ′ A p ′ -1 3 p ′ ≤ 1 -3θ + min y∈B2 u(y, -T )
The last inequality above follows from (3.8). We see that if there is one point y ∈ B 2 such that u(y, -T ) ≤ 2θ, then u ≤ 1 -θ in Q 1 and we would finish the proof.

We must now analyze the case in which u(y, -T ) ≥ 2θ for all y ∈ B 2 . For this case, we apply the Lax Oleinik formula and comparison principle again, but this time with (3.5).

u(x, t) ≥ min (y,s)∈∂(B2×[-T,0]) -ε(-s + t) + u(y, s) + c p (-s + t) 1-p ′ A 1-p ′ |x -y| p ′ Recall that, by ∂(B 2 × [-T, 0]) we denote the parabolic boundary B 2 × -T ∪ ∂B 2 × [-T, 0]
. We divide the estimate for u(x, t) depending on what part of the boundary (y, s) lies on.

If y ∈ B 2 and s = -T ,

-ε(-s + t) + u(y, s) + c p (-s + t) 1-p ′ A 1-p ′ |x -y| p ′ ≥ -εT + min y∈B2 u(y, -T ) ≤ θ
On the other hand, if y ∈ ∂B 2 and s ∈ [-T, 0] we use (3.9) and u ≤ 1,

-ε(-s + t) + u(y, s) + c p (-s + t) 1-p ′ A 1-p ′ |x -y| p ′ ≥ εT + c p T 1-p ′ A 1-p ′ ≤ θ
In any case we obtain u(x, t) ≥ θ for all (x, t) ∈ Q 1 . This finishes the second case of the proof.

We now restate Lemma 3.3 in a more convenient form.

Corollary 3.4. There exists a λ ∈ (0, 1) such that if u satisfies (3.1) and (3.2) in the unit cylinder

Q 1 = B 1 × [-1, 0] and osc Q1 u ≤ 1, then osc B λ ×[-λ p/2 ,0] u ≤ (1 -θ).
Proof. Consider ũ(x, r) = u(r -1 x, r -p t) -min Q1 u where r = max(2, T 1/p ). The function ũ satisfies

ũt + A|D ũ| p + r -p ε ≥ 0, ũt + 1 A |D ũ| p -r -p ε ≤ 0, in the cylinder B r × [-r p , 0] which includes B 2 × [-T, 0]. According to Lemma 3.3, if ε is small enough and λ = r -2 , osc B λ ×[-λ p/2 ,0] u ≤ osc B1×[-1,0] ũ ≤ 1 -θ.
Applying Corollary 3.4 at all scales with Proposition 2.1, we prove Theorem 3.1.

Proof of Theorem 3.1. Without loss of generality, we prove the estimate for (x, t) = (0, 0) and (y, s) sufficiently close to (0, 0). Also without loss of generality we assume that 0 ≤ u ≤ 1.

Otherwise we consider (u -inf u)/(osc u) instead, which also solves inequalities with (3.1) and (3.2) with a slightly modified A (see section 2.1). We start by scaling the equation so that the zeroth order term is negligible and the oscillation of the function is one. We consider ũ(x, t) = u(ρx, ρ p t) which solves

ũt + A|D ũ| p + ρ p A ≥ 0, ũt + 1 A |D ũ| p -ρ p A ≤ 0,
We choose ρ small enough so that ρ p A < ε, for the constant ε of Corollary 3.4. Note that this rescaling is just zooming in around a neighborhood of (0, 0). If we prove that ũ is Hölder continuous at (0, 0), that is equivalent to u being Hölder continuous at (0, 0) and the norms are related by a constant depending on ρ.

We now show that ũ satisfies the iterative hypothesis of Proposition 2.1. Let α ∈ (0, p 2(p-1) ) be a small positive constant that will be chosen below and Q r = B r (0) × [-r p-α(p-1) , 0]. Suppose that osc Qr ũ ≤ r α . Let ũr (x, t) = r -α ũ(rx, r p-α(p-1) t). The function ũr satisfies osc Q1 ũr ≤ 1 and the inequalities

ũr t + A|D ũr | p + r p(1-α) ρ p A ≥ 0, ũr t + 1 A |D ũr | p -r p(1-α) ρ p A ≤ 0.
Note that r p(1-α) ρ p A ≤ ρ p A < ε. Thus, we can apply Corollary 3.4 and obtain osc B λ ×[-λ p/2 ,0] ũr ≤ 1 -θ. Let us choose α such that λ α ≥ 1 -θ. Therefore we have osc

Q λr ũ = r α osc B λ ×[-λ p-α(p-1) ,0] ũr ≤ r α osc B λ ×[-λ p/2 ,0] ũr ≤ (1 -θ)r α ≤ (λr) α .
And thus, we proceed with Proposition 2.1.

Second order equations with bounded right-hand side

In this section we prove Theorem 1.2 about Hamilton-Jacobi equations with possibly degenerate diffusions. Theorem 1.2 is a consequence of the following more precise statement.

Theorem 4.1. Let u be a continuous function that satisfies the following two inequalities in the viscosity sense:

u t + A|Du| p + Am -(D 2 u) + A ≥ 0 and (4.1) u t + 1 A |Du| p + Am + (D 2 u) -A ≤ 0 (4.2) in B 1 × (-1, 0). Then |u(x, t) -u(y, s)| ≤ C |x -y| α + |t -s| α p-α(p-1) ∀(x, t), (y, s) ∈ B 1/2 × [-1/2, 0]
for some α ∈ (0, p ′ ) (where 1/p + 1/p ′ = 1) and C depending on A, p, u ∞ and dimension.

The key to prove Theorem 4.1 is to prove an improvement of oscillation lemma. More precisely we will see that if the oscillation of a function u in space-time cylinder is one and u satisfies the two inequalities

u t + A|Du| p -εm -(D 2 u) ≥ -ε u t + 1 A |Du| p -εm + (D 2 u) ≤ ε
then, if ε is small enough, the oscillation of u in a smaller cylinder is less than 1 -θ for some θ > 0 depending only on A, p and dimension.

The assumption of ε small is not restrictive, since any function u satisfying (4.1) and (4.2) can be rescaled by ũ(x, t) = u(rx, r p t) in order to make the diffusion term and right hand side as small as we wish.

Along this section, we will find constants R > 0, r > 0, ε 0 > 0 and θ > 0 depending only on A, p and dimension, so that if ε ≤ ε 0 and the oscillation of u in B R+r × 1] is less than one, then the oscillation of u in B R/2 × [1/2, 1] is less than (1 -θ). Iterating this improvement of oscillation lemma at all scales leads to a Hölder continuity result following Proposition 2.1.

We start by constructing the supersolution that sees a point, which we will use to prove the improvement of oscillation in the first case. The supersolution is given explicitly and the proof that it is a supersolution amounts only to replace the values on the equation and check the inequality. It is a somewhat lengthy but elementary computation. Lemma 4.2. For any p > 2 and A > 0, there is a constant C (large) and ε 0 > 0 (small), depending on p, A and dimension only, such that the function

U (x, t) = Ct -1 p-1 (|x| 2 + t) p ′ /2
is a supersolution of

U t + 1 A |DU | p -ε m + (D 2 U ) ≥ 0 in R d × (0, +∞), if ε < ε 0 .
Proof. Just compute.

U t = Ct -p ′ (|x| 2 + t) p ′ /2 - 1 p -1 + p ′ 2 t |x| 2 + t 1 A |DU | p = 1 A C p c p t -p ′ (|x| 2 + t) p ′ /2 |x| p (|x| 2 + t) p/2 ε m + (D 2 U ) = Cεt -1 p-1 (|x| 2 + t) p ′ /2 c ′ p 1 |x| 2 + t + c ′′ p |x| 2 (|x| 2 + t) 2 ≤ Ct -1 p-1 (|x| 2 + t) p ′ /2 εc ′′′ p t |x| 2 + t
Where c p , c ′ p , c ′′ p and c ′′′ p are constants depending only on p and dimension. Therefore

U t + 1 A |DU | p -ε m + (D 2 U ) ≥ Ct -p ′ (|x| 2 + t) p ′ /2 ( p ′ 2 -εc ′′′ p )t |x| 2 + t + C p-1 c p |x| p A(|x| 2 + t) p/2 - 1 p -1
all we are left to do is to show that if C is large and ε is small enough (depending only on p and dimension) then the last factor in the right hand side is nonnegative.

Let

r = t |x| 2 + t s = |x| 2 |x| 2 + t
So we have r + s = 1 and moreover

U t + 1 A |DU | p -ε m + (D 2 U ) ≥ Ct -p ′ (|x| 2 + t) p ′ /2 ( p ′ 2 -εc ′′′ p )r + 1 A C p-1 c p s p/2 - 1 p -1
We first choose ε 0 small so that ( p ′ 2 -εc ′′′ p ) > 1/(p -1). Note that we can make this choice because p > 2. Next, we choose C large so that

1 A C p-1 c p s p/2 -( p ′ 2 -εc ′′′ p )s ≥ -( p ′ 2 -εc ′′′ p ) + 1 p -1
for all values of s. Then the desired inequality follows by writing r = 1 -s.

Corollary 4.3. For any η > 0 the function

U (x, t) = Ct -1 p-1 (|x| 2 + ηt) p ′ /2
is a supersolution of

U t + 1 A |DU | p -ε m + (D 2 U ) ≥ 0 if ε < ηε 0 ,
where C and ε 0 are the same constants as in Lemma 4.2 (depending only on A, p and dimension).

Proof. Let U be the function of Lemma 4.2. For r < 1, consider

U r (x, t) = U (rx, r p t) = Ct -1 p-1 (|x| 2 + r p-2 t) p ′ /2
which by scaling solves

U t + 1 A |DU | p -εr p-2 m + (D 2 U ) ≥ 0
so we choose r small so that r p-2 = η and ηε 0 instead of the ε 0 of Lemma 4.2.

The following Corollary reminds us to the Lax-Oleinik formula, except that it is not an identity but an inequality instead.

Corollary 4.4. Let R > 0 and u be a subsolution of

u t + 1 A |Du| p -εm + (D 2 u) -ε ≤ 0 in B R+r × [0, 1]
where r = (1/C) Proof. The function

V (x, t) = min y∈BR u(y, 0) + U (x -y, t) + εt
is a minimum of translated supersolutions, then it is also a supersolution of

V t + 1 A |DV | p -εm + (D 2 V ) -ε ≥ 0 in R d × [0, 1]. It is easy to check that V ≥ u on the boundary of the cylinder B R+r × [0, 1] since V = u in B R × {0}, V = +∞ in the rest of R d × {0} and V ≥ 1 ≥ u on ∂B R+r × [0, 1].
Proof of Theorem 4.1. Without loss of generality, we prove the estimate for (x, t) = (0, 0) and (y, s) sufficiently close to (0, 0). Also without loss of generality we assume that 0 ≤ u ≤ 1.

Otherwise we consider (u -inf u)/(osc u) instead, which also solves inequalities with (4.1) and (4.2) with a slightly modified A (see section 2.1). We start by scaling the equation so that the zeroth order term and the diffusion term are negligible. We consider ũ(x, t) = u(ρx, ρ p t) which solves

ũt + A|D ũ| p + ρ p A -ρ p-2 Am -(D 2 u) ≥ 0, ũt + 1 A |D ũ| p -ρ p A -ρ p-2 Am + (D 2 u) ≤ 0,
We choose ρ small enough so that ρ p-2 A < ε, for the constant ε of Corollary 4.9. Note that we can make the second order term negligible because p > 2.

We now show that ũ satisfies the iterative hypothesis of Proposition 2.1. Let α ∈ (0, p 2(p-1) ) be a small positive constant that will be chosen below and set Q r = B r (0) × [-r p-α(p-1) , 0]. Suppose that osc Qr ũ ≤ r α . Let ũr (x, t) = r -α ũ(rx, r p-α(p-1) t). The function ũr satisfies osc Q1 ũr ≤ 1 and the inequalities

ũr t + A|D ũr | p + r p(1-α) ρ p A -(rρ) p-2 Am -(D 2 ũr ) ≥ 0, ũr t + 1 A |D ũr | p -r p(1-α) ρ p A -(rρ) p-2 Am + (D 2 ũr ) ≤ 0.
Note that r p(1-α) ρ p A ≤ ρ p A < ε and (rρ) p-2 A ≤ ρ p-2 A < ε. Thus, we can apply Corollary 4.9 and obtain osc

B λ ×[-λ p/2 ,0] ũr ≤ 1 -θ. Let us choose α such that λ α ≥ 1 -θ. Therefore we have osc Q λr ũ = r α osc B λ ×[-λ p-α(p-1) ,0] ũr ≤ r α osc B λ ×[-λ p/2 ,0] ũr ≤ (1 -θ)r α ≤ (λr) α .
And thus, we proceed with Proposition 2.1.

Second order equations with unbounded right hand side

In this section we prove Theorem 1.3 about Hamilton-Jacobi equations with an unbounded right hand side. Note that, with the assumptions of Theorem 1.3, f is actually bounded from below by -A. An example of a situation in which the theorem is applicable is for hamiltonians H of the form H(x, t, q, X) = H 1 (x, t, X) + H 2 (x, t, q) with

H 1 (x, t, X) = sup a∈A - 1 2
Tr (σσ * (x, t, a)X) and 1

A |q| p -f (x, t) ≤ H 2 (x, t, q) ≤ A|q| p + A
where sup t,a σ(•, t, a) C 1 is bounded and f is as above. Such Hamiltonian are often encountered in stochastic control. Theorem 1.3 is a consequence of the following more precise statement.

Theorem 5.1. Let u be a continuous function which satisfies the following two inequalities in the viscosity sense

u t + A|Du| p -Am -(X) ≥ -A and
(5.1)

u t + 1 A |Du| p -Tr(B(x, t)D 2 u) ≤ f (x, t) (5.2) in B 1 × (-1, 0]. Then |u(x, t) -u(y, s)| ≤ C |x -y| α + |t -s| α p-α(p-1) ∀(x, t), (y, s) ∈ B 1/2 × [-1/2, 0]
for some α ∈ (0, p ′ ) (where 1/p + 1/p ′ = 1) and C depending on sup t B(•, t) C 1 , f L m (B1×(0,1)) , A, p, u ∞ and dimension. Remark 5.2. We were not able to find a simple proof of Theorem 5.1 using an explicit construction of sub and supersolutions, like in the previous sections for the case of bounded right-hand side. In the inviscid case (B ≡ 0) it would be possible to build a supersolution of the form

U (x, t) = C min s<t,y s -1 p-1 |y| p ′ + (t -s) -1 p-1 |x -y| p ′ + s 0 f r t y, r dr + t s f y + r -s t -s (x -y), r dr ,
which could be shown to be bounded by ||f || L m , and then we could proceed like in section 3. However, it seems very difficult to find an estimate on the second derivative of such function U (x, t), and therefore we need a different method in order to be able to handle equations with bounded right-hand side and second order diffusion. We are not able to handle diffusions as general as in the bounded coefficient case. In fact we currently do not know if the result of Theorem 5.1 would hold true if the inequality (5.2) was replaced by

u t + 1 A |Du| p -Am + (D 2 u) ≤ f (x, t).
In order to show some estimates on subsolutions, we need an approximation argument. We use here ideas from [START_REF] Ishii | On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions[END_REF], [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II.Viscosity solutions and uniqueness[END_REF] adapted to our context. Lemma 5.3. Let u be a continuous viscosity subsolution of

u t + 1 A |Du| p -Tr(B(x, t)D 2 u) ≤ f (x, t) in B R × (a, b)
Then, for any ε > 0, we can find two sequences of smooth maps (u n ) and (f n ), which converge uniformly in B R-ε × (a + ε, b -ε) to u and f respectively, such that

u n t + 1 2A |Du n | p -div(B(x, t)Du n ) ≤ f n (x, t) + C B p ′ C 1 + B C 1 in B R-ε × (a + ε, b -ε)
where C only depends on A, p and d.

Proof. Let us start as usual by regularizing u by supconvolution: let δ > 0 small and

u δ (x, t) = sup (y,s) u(y, s) - 1 2δ |(y, s) -(x, s)| 2 .
Then it is well-known (see for instance Theorem 3 of [START_REF] Ishii | On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions[END_REF]) that u δ satisfies

u δ t + 1 A |Du δ | p -Tr(B(x, t)D 2 u δ ) ≤ f δ (x, t) in B R-δ × (a + Cδ, b -Cδ)
for some C = C( u ∞ ) and for some f δ which converge uniformly to f as δ → 0. Recall that u δ is semiconvex and therefore Du δ has bounded variations. We denote by

D 2 u δ = (D 2 ij u δ ) the distributional derivative of Du δ . From Alexandrov Theorem, singular part ν δ = (ν δ ij ) of D 2 u δ is nonnegative while its regular part ∂ 2 u δ = (∂ 2 ij u δ ) satisfies (u δ t , Du δ (t, x), ∂ 2 u δ (t, x
)) ∈ P 2,+ u δ (x, t) a.e. (where P 2,+ is the parabolic superjet of u δ , see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]). In particular

u δ t + 1 A |Du δ | p -Tr(B(x, t)∂ 2 u δ ) ≤ f δ (x, t) a.e. in B R-δ × (a + Cδ, b -Cδ) (5.3)
Next we consider a space-time nonnegative convolution kernel φ with support in B 1 and such that |Dφ| p ′ /φ p ′ -1 is smooth (one can take for instance φ(x, t) = ce -1/(1-|(x,t)| 2 ) if |(x, t)| < 1 and φ(x, t) = 0 otherwise, c being such that φ = 1). We set φ γ (t, x) = 1 γ d+1 φ((x, t)/γ) and denote by u δ,γ = u δ ⋆ φ γ the space-time convolution of u δ . Integrating (5.3) one gets

u δ,γ t (x, t) + 1 A φ γ (x -y, t -s)|Du δ (y, s)| p dyds -φ γ (x -y, t -s)Tr(B(y, s)∂ 2 u δ (y, s)) dyds ≤ f δ,γ (x, t) in B R ′ × (a ′ , b ′ ) where B R ′ × (a ′ , b ′ ) = B R-δ-γ × (a + δ + γ, b -δ -γ), f δ,γ = f δ ⋆ φ γ
. Now we note that, since B can be written as B = σσ * , for some continuous square matrix σ, we have by positivity of ν, i,j

φ γ B ij dD 2 ij u δ = i,j (σ i φ γ )(σ j φ γ )dν ij + i,j φ γ B ij ∂ 2 ij u δ ≥ φ γ Tr(B∂ 2 u δ ) .
On the other hand, by integration by parts, we get

i,j φ γ B ij dD 2 ij u δ = - i,j ∂B ij ∂x i φ γ ∂u δ ∂x j - ∂φ γ ∂x i B ij ∂u δ ∂x j Therefore -div(BDu δ,γ ) + φ γ Tr(B∂ 2 u δ ) ≤ - i,j ∂B ij ∂x i φ γ ∂u δ ∂x j + B ij ∂φ γ ∂x i ∂u δ ∂x j + ∂B ij ∂x i φ γ ∂u δ ∂x j - ∂φ γ ∂x i B ij ∂u δ ∂x j
where, from our smoothness assumption on

|Dφ| p ′ /φ p ′ -1 = (|Dφ|φ -1/p ) p ′ , B ij ∂φ γ ∂x i ∂u δ ∂x j - ∂φ γ ∂x i B ij ∂u δ ∂x j ≤ |B ij (y, s) -B ij (x, t)|| ∂φ γ ∂x i |φ -1/p γ φ 1/p γ | ∂u δ ∂x j | ≤C B C 1 φ γ |Du δ | p 1/p while i,j ∂B ij ∂x i φ γ ∂u δ ∂x j + ∂B ij ∂x i φ γ ∂u δ ∂x j ≤ C B C 1 φ γ |Du δ | . So - 1 2A φ γ |Du δ | p -div(BDu δ,γ ) + φ γ Tr(B∂ 2 u δ ) ≤ C B p ′ C 1 + B C 1 .
Putting our estimates together, we get

u δ,γ t + 1 2A φ γ |Du δ | p -div(BDu δ,γ ) ≤ f δ,γ (x, t) + C B p ′ C 1 + B C 1 in B R ′ × (a ′ , b ′ ) where 1 2A φ γ |Du δ | p ≥ 1 2A |Du δ,γ | p .
Choosing δ and γ sufficiently small then gives the result.

We now have the first step of the improvement of the oscillation. Proof. Using Lemma 5.3 it is enough to prove the result for a smooth map u which satisfies

u t + 1 2A |Du| p -div(B(x, t)Du) ≤ f (x, t) + C B p ′ C 1 + B C 1 in B 1+R × (0, 1)
We set g = f + C B p ′ C 1 + B C 1 . Let y ∈ B R be such that u(y, 0) ≤ θ and let us fix (x, t) ∈ B R × [1/2, 1]. Let also φ = φ(z) be a smooth, nonnegative convolution kernel, with support in the unit ball. We assume as before that |Dφ| p ′ /φ p ′ -1 is smooth. Since p > 2 and d < p(m-1), we can fix some β ∈ (1/p, (1/p ′ )∧(m-1)/d). For δ > 0 to be chosen later we set ϕ(z, t) = 1 (ξ δ (s)) d φ We note that ρ is absolutely continuous, ρ(0) = u(y, 0) and ρ(t) = u(x, t). Then ρ t (s) =

t p ′ + |ξ ′ δ | p ′ |z| p ′ + B p ′ ∞ |Dφ| p ′ φ p ′ -1 |ξ δ | -p ′ ) So, if we set ε = B ∞ , u(x, t) ≤ u(y, 0) + C|x -y| p ′ t 1-p ′ + t 0 R d φg + C t 0 |ξ ′ δ | p ′ + ε p ′ |ξ δ | -p ′ .
Let us estimate the right-hand side: ũr ≤ (1 -θ)r α ≤ (λr) α .

And thus, we proceed with Proposition 2.1.

Lemma 5 . 4 .

 54 Let θ ∈ (0, 1/4). Assume that u is a continuous subsolution tou t + |Du| p -Tr(B(x, t)D 2 u) ≤ f (x, t) in B R+1 × (0, 1) with 0 ≤ u ≤ 1 and u ≤ θ at some point in B R × {0}. If ε, B C 1 and f L m (BR+1×[0,1]) are sufficiently small (depending on p and dimension but not on θ), then u ≤ 1 -θ in B R × [1/2, 1].

( 1 -

 1 s/t)y + (s/t)x + z ξ δ (s)whereξ δ (s) = δ min{s β , (t -s) β }.Then we considerρ(s) = R d ϕ(z, s)u(z, s)dz = R d φ(z)u ((1 -s/t)y + (s/t)x + ξ δ (s)z, s) dz .

  t + Du.( x -y t + ξ ′ δ z))dzwhere in the above equality and the (in)equalities below, φ = φ(z) and u (as well as B and g) and its derivative have for argument ((1 -s/t)y + (s/t)x + ξ δ (s)z, s). Henceρ t (s) = R d φ(-|Du| p + g + Du.( x -y t + ξ ′ δ z)) -(B *Dφ).Du ξ δ where φ(-|Du| p + Du.( x -y t + ξ ′ δ z)) -(B * Dφ).Du ξ δ ≤ C(φ( |x -y| p ′

t 0 R d φg = t 0 R≤ C g m 2 t/ 2 0 2 t/ 2 0

 02222 d ϕ(z, s)g(z, s)dzds ≤ g m t 0 B1 ξ d(1-m ′ ) δ φ m ′ 1/m ′ δ d(1-m ′ ) s βd(1-m ′ ) ds 1/m ′ ≤ C g m δ -d/m (where 1/m + 1/m ′ = 1, since βd(1 -m ′ ) > -1 because β < (m -1)/d) and t 0 |ξ ′ δ | p ′ + ε p ′ |ξ δ | -p ′ = δ p ′ β p ′ s (β-1)p ′ + ε p ′ δ -p ′ s -βp ′ ds ≤ C(δ p ′ + ε p ′ δ -p ′ )Therefore ||g r || L m stays bounded for r < 1 as long as α is chosen small enough so that p(1 -α -1/m) -d/m ≥ 0, which is possible since by assumption p(1 -1/m) -d/m > 0. Thus, we can apply Corollary 5.5 and obtain osc B λ ×[-λ p/2 ,0] ũr ≤ 1 -θ. Let us choose α also smaller than 1/2 and such that λ α ≥ 1 -θ. Therefore we have osc λr ũ = r α osc B λ ×[-λ p-α(p-1) ,0] ũr ≤ r α osc B λ ×[-λ p/2 ,0]

The following Lemma proves the first case of improvement of oscillation: when the function u is small at one single point on {t = 0}. In this case we will show that the maximum of u is less than one in a smaller cylinder. We only need to assume that u is a subsolution. Lemma 4.5. Let θ ∈ (0, 1/4). If u is a subsolution of

such that u ≤ θ at some point in B R × {0} and 0 ≤ u ≤ 1 everywhere in that cylinder. If ε and R are small enough (depending on A, p and dimension but not on θ),

Proof. Let u(y 0 , 0) ≤ θ. We apply the formula from Corollary 4.4.

Then we choose ε small so that η is small, and R small in order to obtain that the sum of the last two second terms is less than 1/2, which is in turn less than (1 -2θ). Now we construct the subsolution which we will use to prove the other half of the improvement of oscillation lemma.

If C is large and θ is small enough depending p, A, R and dimension, the function

Proof. We just compute

and the right hand side is non positive if we choose θ p-1 smaller than R p /(4A max |b ′ | p-1 ). Now we prove the other half of the improvement of oscillation. If the function u is not small at any point on {t = 0} then it will not be small anywhere in a smaller cylinder. We only assume u to be a supersolution here. Lemma 4.7. Let u be a supersolution of

such that u ≥ θ in B R × {0} (for θ as in Lemma 4.6) and u ≥ 0 everywhere in that cylinder. If ε is small enough (ε < 1/(2c) for c as in Lemma 4.6) and

Proof. We compare the function u with the supersolution L of Lemma 4.6. We observe that u ≥ L in the parabolic boundary of the cylinder, therefore u ≥ L everywhere. Therefore, for (x, t)

for ε small enough. Now we can combine Lemmas 4.5 and 4.7 to obtain the full improvement of oscillation result.

Lemma 4.8. Assume that in B R+r × [0, 1] the function u satisfies the two inequalities

] is less or equal to 1 -θ, for constants θ, r and R depending only on A, p and dimension.

Proof. We start by defining v(x, t) = u(x, t) -min

u so that v satisfies the same equations as u but 0 ≤ v ≤ 1.

We choose the value of r from Corollary 4.4 and the value of R from Lemma 4.5 (which is independent of θ). Then, for that value of R, we use the value of θ from Lemma 4.6. We choose ε small enough so that all lemmas apply.

If v(y, 0) ≤ θ for some point y ∈ B R , we apply Lemma 4.5 and obtain

We rephrase Lemma 4.8 to a more convenient form. Corollary 4.9. Assume that in B 1 × [-1, 0] the function u satisfies the two inequalities

There is a λ > 0 (depending only on A, p and dimension), such that if ε is small enough, osc B λ ×[-λ p/2 ,0] u ≤ 1 is less or equal to 1 -θ.

Proof. Just apply Lemma 4.8 to u(x, t -1). Recall that the constants R and r were chosen small enough, so we can assume that R + r < 1 so that B R+r ⊂ B 1 and we choose λ = R/2.

We prove Theorem 4.1 following the procedure of Proposition 2.1. This proof is identical to the proof of Theorem 3.1 in section 3 except that we have to check the scaling of the second order term (for which we use the assumption p > 2) and we apply the improvement of oscillation given by Corollary 4.9 instead of Corollary 3.4.

(since (β -1)p ′ > -1 because β > 1/p and β < 1/p ′ ). Therefore, assuming that B ∞ < ε 0 and g m < ε 0 for some ε 0 ∈ (0, 1) to be fixed below, and choosing δ = min{ε

We can now fix R and ε 0 sufficiently small so that the remaining term is less than 1/2: in this case u(x, t) ≤ θ + 1/2 ≤ 1 -θ .

Since the control of supersolutions remains unchanged, one can conclude exactly as in the case of bounded right-hand side that the improvement of oscillations holds: Corollary 5.5. Assume that in B 1 × [-1, 0] the function u satisfies the two inequalities

Assume also osc B1×[-1,0] u ≤ 1. There is a λ > 0 (depending only on A, p and dimension), such that if ε is small enough and if

The proof of Theorem 5.1 is identical to the proof of Theorem 3.1 or Theorem 4.1 but using Corollary 5.5 instead of Corollary 3.4 or 4.9, and verifying that the assumptions of Corollary 5.5 hold at small scales. This is the point where the assumption m > d/p + 1 plays a role.

Proof of Theorem 5.1. Without loss of generality, we prove the estimate for (x, t) = (0, 0) and (y, s) sufficiently close to (0, 0). Also without loss of generality we assume that 0 ≤ u ≤ 1.

We start by scaling the equation so that the zeroth order term and the diffusion term are negligible. We consider ũ(x, t) = u(ρx, ρ p t) which solves

We choose ρ small enough so that ρ p-2 A < ε, for the constant ε of Corollary 5.5 and also

We now show that ũ satisfies the iterative hypothesis of Proposition 2.1. Let Q r = B r (0) × [-r p-α(p-1) , 0] and suppose that osc Qr (x,t) ũ ≤ r α , where α is a small positive constant that will be chosen below. Let ũr (x, t) = r -α ũr (rx, r p-α(p-1) ). The function ũr satisfies osc Q1 ũr ≤ 1 and the inequalities