Kilian Hett

Hans Johnson

Pierrick Coupé

Jane S Paulsen

Jeffrey D Long

Ipek Oguz

Tensor-based grading: a novel patch-based grading approach for the analysis of deformation fields in Huntington's disease

Keywords: Patch-based grading, tensor-based morphometry, Huntington's disease

INTRODUCTION

Huntington's disease (HD) is a fatal autosomal dominant inherited disorder that causes motor, behavioral and cognitive symptoms. Imaging studies have shown structural changes in the striatum [START_REF] Ross | Huntington disease: natural history, biomarkers and prospects for therapeutics[END_REF]. Moreover, it has been demonstrated that the volume of the putamen is a sensitive imaging-based marker for the tracking of changes that occur in the lifespan of patients with HD [START_REF] J S Paulsen | Clinical and biomarker changes in premanifest huntington disease show trial feasibility: a decade of the PREDICT-HD study[END_REF]. Unlike many other degenerative diseases, gene status can be established via a genetic test well before the onset of first symptoms, which makes HD a good candidate for the evaluation of new imaging-based methods.

Numerous advanced methods have been developed to detect the structural modifications earlier in the course of neurodegenerative diseases using anatomical MRI [START_REF] M R Arbabshirani | Single subject prediction of brain disorders in neuroimaging: promises and pitfalls[END_REF]. On the one hand, studies have proposed to perform analysis based on MRI intensity. Among these methods, the patch-based grading framework has shown attractive characteristics such as low computational cost and competitive performance to recent methods based on deep-learning [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF][START_REF] Gao | A novel grading biomarker for the prediction of conversion from mild cognitive impairment to Alzheimer's disease[END_REF]. In addition, patch-based grading methods also provides the localization of structural differences. Indeed, these approaches aim to measure the local similarity of anatomical patterns by comparing the MRI under study to a template library composed of MRIs representing two distinct populations. Patch-based grading has been successfully applied to various intensity-based features, such as gray matter density maps [START_REF] Komlagan | Anatomically constrained weak classifier fusion for early detection of Alzheimers disease[END_REF], features extracted using texture filters [START_REF] Hett | Adaptive fusion of texture-based grading for Alzheimer's disease classification[END_REF], and diffusion-weighted imaging parameters [START_REF] Hett | Multimodal hippocampal subfield grading for Alzheimers disease classification[END_REF]. On the other hand, many studies have proposed to assess anatomical changes among MRIs with the analysis of the deformation tensor fields resulting from image registration [START_REF] Ashburner | Voxel-based morphometry-the methods[END_REF]. Features derived from deformation tensors are highly interpretable since they provide both localization and geometric characteristic of the brain alterations. Indeed, these methods register the MRIs under study into the same stereotaxic space and use the resulting deformation fields to capture regional changes of brain structures (e.g., local expansion or shrinkage). In particular, the log-Euclidean approach has been proposed as an efficient Riemannian framework for tensor-based morphometry [START_REF] Arsigny | Log-Euclidean metrics for fast and simple calculus on diffusion tensors[END_REF]. The efficiency of tensorbased approaches has been demonstrated in diverse applications such as disease detection, longitudinal studies, clinical trials, and stability control of acquisition protocols [START_REF] Hua | Accurate measurement of brain changes in longitudinal MRI scans using tensor-based morphometry[END_REF][START_REF] Leow | Longitudinal stability of MRI for mapping brain change using tensor-based morphometry[END_REF][START_REF] Koikkalainen | Multi-template tensor-based morphometry: application to analysis of Alzheimer's disease[END_REF]. In this paper, we propose to combine the respective advantages of patch-based grading and tensor-based morphometry in a novel tensor-based grading framework. We evaluate our method with the classification of pre-manifest HD patients and control subjects. The performance of our new method is compared with the volume of the putamen, the original patch-based grading using T1w intensities, and a recent texture-based grading approach. In addition, we also investigate the complementary nature of these different methods. These experiments show that our new method has competitive performance and also suggest complementarity with volumetric features.

MATERIALS AND METHODS

Dataset

All T1-weighted (T1w) MRIs come from the PREDICT-HD study [START_REF] J S Paulsen | Detection of Huntingtons disease decades before diagnosis: the Predict-HD study[END_REF], which is a multi-site longitudinal study of HD. The MRIs have been acquired using 3 Tesla MRI scanners from different vendors (e.g., GE, Phillips, and Siemens). The cohort used in the study includes 683 MPRAGE images from subjects representing three populations: control subjects (CN), pre-manifest HD that is composed of subjects with the expanded cytosine-adenine-guanine (CAG) repeat but who have not yet had a motor diagnosis at the time of the scan, and manifest HD which refers to patients who already have a motor diagnosis by the time of the scan (see Table 1). Only subjects with CAG length from 41 to 43 repeats and at least 2 longitudinal scans at 3T have been embedded in this study.

Preprocessing

The preprocessing has been conducted with the BRAIN-SAutoWorkup pipeline [START_REF] Pierson | Fully automated analysis using BRAINS: AutoWorkup[END_REF]. This pipeline is composed of the following steps: (1) denoising with non-local means filter, (2) anterior/posterior commissure and intra-subject alignments with rigid transformation, (3) bias-field correction, and (4) regional segmentation mask with a multi-atlas method using atlases from Neuromorphometrics1 . After rigid alignment, we take the union of the putamen masks of all subjects. The bounding box of this union is used as a region of interest (ROI). All images are cropped with this common ROI. On the top row: a template library K s , composed of CN (gray color) and manifest HD (blue color), is built for each subject s using an age matching preselection. For each template t of K s the tensor-based grading map is estimated following a leave-k-out cross-validation procedure. Once the grading maps are estimated, a sparse regularization model is computed to estimate the voxels having the most discriminant grading values. On the bottom row: for each subject s the proposed tensor-based grading method is computed using the template library K s . Then the most discriminant voxels based on K s are used to compute a global grading feature which is used to make the final decision.

Deformation-based tensor computation

A non-rigid symmetric normalization registration [START_REF] B B Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF] has been conducted to estimate deformation fields in the ICBM 152 nonlinear stereotaxic space 2 . Next, the Jacobian matrix J of the resulting deformation field, defined at each voxel as:

J =    ∂x-
   , (1)
has been used to compute the deformation-based tensor Φ = √ J T J that describes the geometry of local deformation at each voxel.

Tensor-based grading

Once the deformation-based tensor field is computed, local similarities are estimated at each voxel to estimate the degree of the structural changes (see Fig. 1). We use a patchbased grading approach [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF] instead of analyzing each tensor independently of its neighborhood, in contrast to tensor-based morphometry [START_REF] Ashburner | Voxel-based morphometry-the methods[END_REF][START_REF] Arsigny | Log-Euclidean metrics for fast and simple calculus on diffusion tensors[END_REF]. We hypothesize that such a patch-wise analysis will enable us to better model the deformation patterns.

First, for each subject s, a template library K s composed of the tensor fields t from CN and manifest HD patients is built using an age matching preselection. This preselection technique aims to reduce the bias introduced by age-related differences. Then, at each voxel i (i.e., image coordinate) of the subject s, the patch-based grading is computed as,

g si = t∈Ks exp - d(Ps i ,Pt i ) hi y t t∈Ks exp - d(Ps i ,Pt i ) hi , (2)
where h i = min(d(P si , P ti )), ∀t ∈ K s is used to normalize the patch similarity at each voxel. y t is a binary indicator of the pathological status of the template t, set to -1 for manifest HD and +1 for control. The distance d(P si , P ti ) is based on the log-Euclidean framework [START_REF] Arsigny | Log-Euclidean metrics for fast and simple calculus on diffusion tensors[END_REF]. d describes the similarity of two tensor patches P si and P ti surrounding voxel i and is defined as follows:

d(P si , P ti ) = N n=1 T race((logΦ si (n) -logΦ ti (n)) 2 ) 1 2 , ( 3
)
where N is the number of voxels in each patch, and n is the index (ranging from 1..N ) within the patch of tensors centered around voxel i, such that Φ si (n) and Φ ti (n) are individual tensors in the patches P si and P ti in subject s and template t, respectively.

Feature selection and global grading computation

Since the images have been deformably registered into the same space, we can compare the grading values at a given voxel across the set of images and use these as features for classification. This further allows us to use feature selection techniques to select the most discriminant voxels [START_REF] Gao | A novel grading biomarker for the prediction of conversion from mild cognitive impairment to Alzheimer's disease[END_REF]. In our work, we used an elastic-net regularization model that provides a sparse representation of the most discriminative features, defined as follows:

β = argmin β 1 2 ||Gβ -Y || 2 2 + ρ||β|| 2 2 + λ||β|| 1 , (4)
where β represents the regularization coefficients computed using the tensor-based grading maps G composed of the g ti as defined in Eq. 2, computed on an inner fold of the crossvalidation within the template library K s . Y represents the vector of pathological status composed of the binary indicators y t defined in Sec. 2.4. Finally, a global tensor-based grading feature for the subject s is computed as,

g s = i∈s βi g si i∈s βi . (5)
Thus, g s represents a measure that estimates the global deformation differences of the subject s under study.

Evaluation and implementation details

We evaluate our new method by classifying pre-manifest HD patients and control subjects. For each subject s, the template library K s has been built using 100 deformation-based tensor fields, 50 from CN MRI and 50 from manifest HD MRI pre-selected via age matching to the subject s. Moreover, all longitudinal scans of a given subject s have been removed from its template library K s to avoid double-dipping. The tensor fields have been estimated using a non-rigid diffeomorphic registration computed using ANTs [START_REF] B B Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF]. The registration has been performed using the cross-correlation metric and the smoothing parameter has been set to 2mm. The elasticnet regularization model has been computed using the Matlab Statistic and Machine Learning toolbox, with ρ = 0.2 and λ = 0.09 (see Eq. 4). The classification has been computed using a linear support vector machine with the soft margin parameter set to C = 1. Finally, a stratified cross-validation procedure iterated 100 times has been conducted.

Comparison with state-of-the-art

To evaluate the performance of the proposed tensor-based grading method, we conduct single-feature and two-feature classification experiments. The features compared are the volume of putamen using the multi-atlas label fusion segmentation mask obtained from the BRAINSAutoWorkup pipeline [START_REF] Pierson | Fully automated analysis using BRAINS: AutoWorkup[END_REF], patch-based grading using T1w intensity [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF], and a patch-based grading approach that fuses grading maps estimated from different texture maps [START_REF] Hett | Adaptive fusion of texture-based grading for Alzheimer's disease classification[END_REF]. These latter two features have been aggregated within the putamen segmentation mask to obtain the global grading. Moreover, for these two patch-based grading methods, a piece-wise linear histogram standardization has been used to normalize intensities. The mean accuracy (ACC), sensibility (SEN), and specificity (SPE) over the 100 iterations are reported.

RESULTS AND DISCUSSION

First, we investigated the effect of patch size for the classification of pre-manifest HD patients and control subjects. Table 2 summarizes the obtained results. We compared voxel-wise (i.e., 1×1×1 voxel) and patch-wise (i.e., 3×3×3 and 5×5×5 voxels) tensor-based grading. The results of this experiment demonstrate the importance of embedding neighbor deformations. Indeed, patch-wise tensor-based grading using patches of 3×3×3 voxels obtained in average 87.5% accuracy, which improves the tensor-based grading using single voxel-wise similarity by 5 percentage points in accuracy and specificity, and 7 percentage points in sensitivity. In our experiments, we limited our investigation to patch sizes up to 5×5×5 voxels since the computational cost becomes prohibitive.

Second, the proposed tensor-based grading method was compared to putamen volume, which is provided as baseline, the original patch-based grading using T1w intensities [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF], and a texture-based grading approach [START_REF] Hett | Adaptive fusion of texture-based grading for Alzheimer's disease classification[END_REF]. The results are summarized in Table 3. The classification performance of intensity-based grading is low compared to the volume of putamen. This result is unexpected given the detection performance of this approach obtained for other neurodegenerative diseases (e.g., Alzheimer's disease [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF]). We note that this low performance may be caused by the highly heterogeneous nature of the dataset used to evaluate our method. As described in Sec. 2.1, the PREDICT-HD dataset includes MRIs from diverse scanners and acquisition sequences [START_REF] J S Paulsen | Detection of Huntingtons disease decades before diagnosis: the Predict-HD study[END_REF]. The grading performance is somewhat improved by the use of texture, which obtains comparable accuracy to the volume of putamen. In contrast, our novel tensor-based grading obtains 87.5% accuracy for the classification of pre-manifest HD patients. This improves the accuracy by 5 percentage points compared to putamen volume and texture-based grading and by 13 percentage points compared to intensity-based grading. We hypothesize that the superior performance of tensor-based grading compared to traditional grading approaches may partially come from the use of tensor fields instead of features derived from MRI intensity, since deformation-based tensor fields might be less impacted by the heterogeneous nature of acquisition sequences than intensity-based features. Tensorbased grading also provides a potentially more interpretable description of the structural differences since it indicates not only the localization but also the local geometry of the detected differences. Another advantage of this method is that it does not require an explicit segmentation of the structure.

Next, as described in Table 4, the complementarity of the different methods for the classification of pre-manifest HD patients and control subjects has been investigated. The results of this experiment show that the combination of volume with texture-based grading, and volume with tensor-based grading improve the classification performance compared to the use of single feature. Indeed, the combination of putamen volume and tensor-based grading obtains 89.1% accuracy, which mainly benefits from a higher specificity than the tensor-based grading method alone. In contrast, the combination of texture and tensor-based features does not improve classification performance. Finally, Fig. 2 presents the coefficient map of the left putamen area estimated from Eq. 4. This illustrates the interpretability of the results obtained from the proposed method. Indeed, our method enables us to highlight the areas where the most discriminant deformations occur. In our work, results indicate a discriminant deformation of the superior part of putamen (Fig. 2-a). Our results also show an abnormal deformation on the medial side of the putamen (Fig. 2-b), and around the globus pallidus (Fig. 2-c). These results are in line with previous imaging-based anatomical studies that used volumetric measurements [START_REF] Younes | Regionally selective atrophy of subcortical structures in prodromal HD as revealed by statistical shape analysis[END_REF]. It is noteworthy that while we focused our current experiments on the putamen, nevertheless we also found discriminant features in the surrounding structures such as the globus pallidus. In future work, we will extend this work to whole brain analysis.

CONCLUSION

In this work, we proposed a novel tensor-based grading method for the analysis of tensor fields obtained from nonrigid registration. The proposed method has been evaluated with the classification of pre-manifest HD and control subjects by analyzing the putamen area. Our approach has shown an increase of performance compared to previous patch-based approaches based on MRI intensity and texture. It also outperformed the classification results using the volume of putamen. In addition, our experiments indicate the complementary nature of putamen volume and tensor-based grading.

  Fig.1. On the top row: a template library K s , composed of CN (gray color) and manifest HD (blue color), is built for each subject s using an age matching preselection. For each template t of K s the tensor-based grading map is estimated following a leave-k-out cross-validation procedure. Once the grading maps are estimated, a sparse regularization model is computed to estimate the voxels having the most discriminant grading values. On the bottom row: for each subject s the proposed tensor-based grading method is computed using the template library K s . Then the most discriminant voxels based on K s are used to compute a global grading feature which is used to make the final decision.

Fig. 2 .

 2 Fig. 2. (a-c) Three axial slices showing β (see Eq. 4) as color overlay. Voxels with β = 0 are not shown for clarity. (d) The localization of slices shown in (a-c).

Table 1 .

 1 Demographic description of the PREDICT-HD dataset used in our experiments.

		Control	Huntington's disease
			Pre-manifest Manifest
	Number of MRIs	327	300	56
	Age (years)	49.3±11.9 43.4±10.1 56.8±6.5
	Sex (F/M)	206/121	199/101	28/28
	CAG length	15-35	41-43	41-43

Table 2 .

 2 Patch size influence on the classification performance. All results are expressed in percentage.

	Accuracy Sensitivity Specificity
	1x1x1 (voxel) 82.1±0.5 81.7±0.7	82.5±0.8
	3x3x3 (patch) 87.5±0.5 88.2±0.7	86.9±0.6
	5x5x5 (patch) 86.1±0.5 86.0±0.7	86.2±0.7

Table 3 .

 3 Comparison of different patch-based grading methods. The volume of putamen is given as baseline. All results are expressed in percentage. Accuracy Sensitivity Specificity Putamen volume 82.9±0.5 84.3±0.6 81.3±0.6 Intensity-based grading 73.8±0.7 77.0±1.0 70.8±0.7 Texture-based grading 81.3±0.6 82.1±0.7 80.7±0.9 Proposed method 87.5±0.5 88.2±0.7 86.9±0.6

Table 4 .

 4 Comparison of two-feature classification results. All results are expressed in percentage.

		Accuracy Sensitivity Specificity
	Volume + Texture 84.4±0.6 84.6±0.7	84.3±0.9
	Volume + Tensor	89.1±0.4 88.5±0.6	89.6±0.7
	Tensor + Texture	87.9±0.4 88.4±0.7	87.4±0.6

http://www.neuromorphometrics.com

http://www.bic.mni.mcgill.ca/ServicesAtlases/ ICBM152NLin2009

This work was supported, in part, by the NIH grant R01-NS094456. The PREDICT-HD study was funded by the National Center for Advancing Translational Sciences, and the National Institutes of Health (NIH; NS040068, NS105509, NS103475) and CHDI.org. Vanderbilt University Institutional Review Board has approved this study.


Jessica Guerand
email: jg900@cam.ac.uk

Quantitative regularity for parabolic De Giorgi classes

Keywords: Mathematics Subject Classification: 35B65, 35K10, 35J15 Interior Hölder regularity, De Giorgi method, De Giorgi classes, Intermediate value lemma

We deal with the De Giorgi Hölder regularity theory for parabolic equations with rough coefficients and parabolic De Giorgi classes which extend the notion of solution. We give a quantitative proof of the interior Hölder regularity estimate for both using De Giorgi method. Recently, the De Giorgi method initially introduced for elliptic equation has been extended to parabolic equation in a non quantitative way. Here we extend the method to the parabolic De Giorgi classes in a quantitative way. To this aim, we get a quantitative version of the non quantitative step of the method the parabolic intermediate value lemma, one of the two main tools of the De Giorgi method sometimes called "second lemma of De Giorgi".

Introduction

Let us first introduce the main results and a historical overview of the elliptic regularity theory of De Giorgi [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF].

Main results

The idea of the paper is to give a quantitative proof of the parabolic De Giorgi interior Hölder regularity theorem so that it is possible to compute a lower bound of the Hölder coefficient for both solutions of the parabolic equation and functions in parabolic De Giorgi classes. Roughly speaking, the De Giorgi classes are sets of functions which satisfy energy estimates which contain enough information to get the Hölder continuity. We know that in particular a solution of the parabolic equation is a function of a De Giorgi class (see Proposition 2.4).

The parabolic equation we are interested in is the following

∂ t u = ∇ x • (A∇ x u) + B • ∇ x u + g, t ∈ (T 1 , T 2 ), x ∈ Ω, (1)
where T 1 and T 2 are real numbers, d is a positive integer, Ω is an open set of R d , u is a real-valued function of (t, x), A = A(t, x) a d × d bounded measurable matrix and A satisfies an ellipticity condition for two positive constants λ, Λ,

0 < λI ≤ A ≤ ΛI, (2)
and g = g(t, x), B = B(t, x) are bounded measurable coefficients, and satisfy, |B| ≤ Λ, g ∈ L q ((T 1 , T 2 ) × Ω) where q > max(2, d+2 2 ).

(

) 3
We give the definition of weak solutions and parabolic De Giorgi classes in Definitions 2.1 and 2.3 and explain why a solution of a parabolic equation is in a De Giorgi class (see Proposition 2.4).

We define the parabolic cylinder Q r = (-r 2 , 0) × B r where B r is the ball of radius r centered at 0. Let us state the quantitative Hölder continuity theorem for the parabolic De Giorgi classes (see Definition 2.3).

Theorem 1.1 (Interior Hölder continuity for parabolic De Giorgi classes). Let u :

Q 2 → R be a function in DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p) where 1 ≤ p ≤ d+2 d and γ 1 , γ 2 , γ 3 > 0. Then u ∈ C α (Q 1 ) with u C α (Q 1 ) ≤ C u L 2 (Q 2 ) + 1 ,
where C and α depend only on d, γ 1 , γ 2 , γ 3 and p.

Since the solutions of the parabolic equation [START_REF] Acosta | An optimal Poincaré inequality in L 1 for convex domains[END_REF] are in a De Giorgi class DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p) we deduce the same result for the solutions.

Corollary 1.2 (Interior Hölder continuity for weak solutions). Let u : Q 2 → R be a solution of (1) satisfying [START_REF] Robert | Sobolev spaces[END_REF] and [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF]

such that g L q (Q 2 ) ≤ 1. Then u ∈ C α (Q 1 ) with u C α (Q 1 ) ≤ C u L 2 (Q 2 ) + 1 ,
where C and α depend only on d, λ and Λ.

Remark 1.3. Thanks to the scaling property of the equation and De Giorgi classes, Theorem 1.1 and Corollary 1.2 hold true for all Q = (s, T 2 ) × Ω and Q = (T 1 , T 2 ) × Ω such that Ω ⊂⊂ Ω and T 1 < s < T 2 , instead of Q 1 and Q 2 (see [32, page 16]).

Remark 1.4. Corollary 1.2 is already proven in [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF][START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] in a non quantitative way. The proof is non quantitative because of a non quantitative step, the intermediate value lemma. Concerning Theorem 1.1, the interior Hölder continuity has already been studied in [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF][START_REF] Dibenedetto | Recent results on the Cauchy problem and initial traces for degenerate parabolic equations[END_REF][START_REF] Hou | A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields[END_REF] with a different method than De Giorgi one which doesn't involve a parabolic intermediate value lemma. Our purpose in this paper is to give a simple selfcontained quantitative proof of this theorem so that we could investigate extensions to other equations, for example the kinetic Fokker-Planck equation. We also make the steps explicit so that it is possible to compute a lower bound for the Hölder exponant α.

The main new result of this paper is the quantitative intermediate value lemma which allows to get a quantitative interior Hölder continuity theorem and to compute a lower bound for the Hölder exponant. Let us state this result.

Theorem 1.5 (Parabolic intermediate value lemma).

Let γ 1 , γ 2 , γ 3 > 0 and p. Let u be in DG + (γ 1 , γ 2 , γ 3 , p) such that u ≤ 1 on Q 3 2 . Let Q 1 = (-2, -1) × B 1 . Then for all (k, l) ∈ R 2 such that k < l ≤ 1, we have

(l -k) 2 |{u ≤ k} ∩ Q 1 ||{u ≥ l} ∩ Q 1 | ≤ C|{k < u < l} ∩ Q 2 | 1 4p+2 , ( 4
)
where C depends only on d, k, γ 1 , γ 2 , γ 3 and p.

Remark 1.6. Theorem 1.5 is a step to obtain Hölder regularity with the De Giorgi method (see subsection 3.2). In the subsection 4.3, we will see that the intervals of time must be disjoint in the subsolution or DG + case because there exists counterexamples if they are not.

Historical overview

De Giorgi [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF]9] introduced techniques in 1957 to solve 19 th Hilbert problem about the analytic regularity of local minimizers of an energy functional. In fact, these minimizers are solutions of quasilinear Euler-Lagrange equations. The idea of De Giorgi was to see quasilinear elliptic equations as linear elliptic equation with merely mesurable coefficients. Thus he proved the Hölder regularity of solutions of elliptic equations with rough coefficients which was the last result to obtain to prove the analyticity since we can use Schauder estimates and a bootstrap argument to get the smothness of the solutions. In 1958, Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF] got the result with different techniques for both elliptic and parabolic equations. Then, Moser [START_REF] Moser | A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations[END_REF] proved in 1960 the Hölder regularity with a different approach. These methods are now called the De Giorgi-Nash-Moser techniques.

In his paper [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF], De Giorgi exhibited a class of functions that satisfy energy estimates and he showed that any function in this class is locally bounded and Hölder continuous. These classes of functions are called the De Giorgi classes. Ladyzhenskaya and Uralt'seva [START_REF] Ladyzhenskaya | 'tseva. A boundary-value problem for linear and quasi-linear parabolic equations. I, II, III. Iaz[END_REF] extended this idea to linear parabolic equations with lower order terms and to quasilinear parabolic equations using a different method than De Giorgi [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF]. They introduced the corresponding De Giorgi classes in the parabolic case and proved that Hölder estimate holds when ±u are both in a De Giorgi class. One can find more details in [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF], in [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF] and in Chapter 6 of [START_REF] Gary | Second order parabolic differential equations[END_REF].

There are extensions of the method in degenerate cases, like the p-Laplacian, by Ladyzhenskaya and Ural'tseva [START_REF] Ladyzhenskaya | Linear and quasilinear elliptic equations[END_REF] in the elliptic case. Then DiBenedetto [START_REF] Dibenedetto | On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients[END_REF] covered the degenerate parabolic cases, see also DiBenedetto, Gianazza and Vespri [START_REF] Dibenedetto | Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations[END_REF][START_REF] Dibenedetto | Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations[END_REF][START_REF] Dibenedetto | Harnack's inequality for degenerate and singular parabolic equations[END_REF].

Concerning nonlinear nonlocal time-dependent variational problems, Caffarelli and Vasseur [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation[END_REF] on the first hand and Caffarelli, Chan and Vasseur [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF] on the second hand extended the method of De Giorgi to nonlocal parabolic equations and got a Hölder regularity result for solutions of problems with translation invariant kernels. Also Caffarelli, Soria, Vázquez [START_REF] Caffarelli | Regularity of solutions of the fractional porous medium flow[END_REF] used the De Giorgi method to prove Hölder continuity of solutions of a porous medium equation with nonlocal diffusion effects. This kind of equation has also been studied earlier by Kassmann [START_REF] Moritz Kassmann | A priori estimates for integro-differential operators with measurable kernels[END_REF] using Moser's techniques where he got local regularity results and by Kassmann and Felsinger [START_REF] Felsinger | Local regularity for parabolic nonlocal operators[END_REF] where they obtained a weak Harnack inequality.

Recently, Golse, Imbert, Mouhot, Silvestre and Vasseur proved the Hölder regularity and obtained Harnack inequalities for kinetic equations. More precisely, the Fokker-Planck kinetic equation with rough coefficients was studied by Golse, Imbert, Mouhot, Vasseur [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] and provides the results for the Landau equation. Imbert and Silvestre [START_REF] Imbert | The weak Harnack inequality for the Boltzmann equation without cut-off[END_REF] studied a class of kinetic integro-differential equations and deduced the results for the inhomogeneous Boltzmann equation without cut-off. The quantitative versions of the intermediate value lemmas in those cases are still an open question.

Contribution of this paper and comparison with existing result

The main contribution of this paper is the quantitative proof of the interior Hölder regularity result with De Giorgi method for parabolic De Giorgi classes and parabolic equations. So that we can compute explicitly the Hölder exponant, at least we can give an explicit lower bound. More precisely, there are two main new results. On one hand, we obtain a quantitative version of one key step of the proof, which was the last nonquantitative step in the parabolic De Giorgi method. This step is sometimes called second lemma of De Giorgi or intermediate value lemma. In the other hand, we extend the De Giorgi method for the parabolic De Giorgi classes. The Hölder continuity for these classes was already obtained in [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF][START_REF] Dibenedetto | Degenerate parabolic equations[END_REF] but with a different method than De Giorgi's. Concerning the intermediate value lemma there are many quantitative versions in the elliptic case. De Giorgi [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF]9] obtained a quantitative version using an isoperimetric inequality argument, taken up by DiBenedetto [START_REF] Dibenedetto | Recent results on the Cauchy problem and initial traces for degenerate parabolic equations[END_REF] and Vasseur [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF]. Recently, Hou and Niu [START_REF] Hou | A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields[END_REF] proved a quantitative version of this lemma using a Poincaré inequality. These versions are actually valid for any function in H 1 . About parabolic equations, no quantitative version of this lemma seems to exist. One can find non-quantitative versions, for example in [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF], a version obtained by contradiction with a compactness argument which works only for solutions of the parabolic equation. However, there exists a quantitative version of this lemma for nonlocal time-dependent integral operator [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF]

Aim and applications of the paper

In this paper, we investigate the De Giorgi method in order to provide a detailed selfcontained proof which allows to deal with general assumptions where our aim would be to use this method for other equations. We focus on De Giorgi classes and De Giorgi method to be able to understand the structure and where the relevant information is contained to get the Hölder continuity. The De Giorgi classes make us understand how to get rid of the merely measurable coefficients so that it's not a difficulty anymore. The De Giorgi method consists in two parts. In a first part we see that we can reduce the Hölder continuity theorem with steps which only use the scaling and linear structure of either the equation or the DG classes. So this part is likely to remain similar when we deal with other equations. The second part of the method consists in getting two lemmas called first and second lemma of De Giorgi in order to prove the reduced theorem. We explain how to extract the information from the energy estimate to get those two lemmas.

Moreover the proof is completely quantitative, we can compute explicitly the Hölder exponant, especially the new way of dealing with the intermediate value lemma which is quantitative gives hope to get this lemma for other cases also in a quantitative way. We give a proof which comes from the energy estimate and which is different from the elliptic case so that now we can deal with time dependent equation. We think for example that those techniques would apply to make the second lemma of De Giorgi quantitative for Hamilton-Jacobi equations studied in [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF][START_REF] Stokols | De Giorgi techniques applied to Hamilton-Jacobi equations with unbounded right-hand side[END_REF] since the energy estimates for those equations are very similar to our case. Moreover, being able to compute explicitly the Hölder exponant can be useful for getting explicit rates. For example, it allows to study the behavior of solutions of quadratic reaction diffusion systems: Fellner, Morgan and Tang [17, Theorem 1.1] got a polynomial bound of the solutions in a specific case and the exponant ξ of the polynom depends on the Hölder exponant of a solution of a parabolic equation.

Also dealing with De Giorgi classes, allows to handle equations which are not included in the general case of the equation ( 1) with ( 2)-(3). For example, if the matrix A is not necessarily bounded, then we cannot apply directly the result of the equation to get the Hölder regularity. But in some cases this matrix is explicit and even if it's not bounded, we can get energy estimates which are relevant to define De Giorgi classes for this problem. For example, for this reaction-diffusion equation with self-diffusion

∂ t u -∇ x • (1 + u)∇ x u = u(1 -u),
where u ≥ 0 and in u ∈ L 3 , we can define the corresponding De Giorgi classes and get the Hölder continuity using the same techniques.

Our next purpose would be to apply these techniques to other equations to get Hölder regularity where the difficult part would be to understand what the "good" energy estimates which contains enough information are. As soon as we get the "good" energy estimate, our hope would be that the techniques in this paper would apply to conclude. For example, we would like to be able to treat the case of the following kinetic Fokker-Planck equation

∂ t f + v • ∇ x f = ∇ v • (A∇ v f ) + B • ∇ v f + s.
But exhibiting the relevant De Giorgi classes in this case remains an open question (see subsection 4.3.1). Being able to deal with De Giorgi classes for kinetic Fokker-Planck equation would then allow to handle matrices A which are not necessarily bounded, in a kinetic framework (in a case where we have self-diffusion for example as mentionned previously in the parabolic framework).

Organisation of the paper

In Section 2, we give the notations and the definition that we use in this paper. In Section 3, we extend the steps of the De Giorgi method to get the Hölder regularity of parabolic De Giorgi classes, we prove Theorem 1.1 and deduce Corollary 1.2. In Section 4, we recall and simplify a proof of the intermediate value lemma in the elliptic case obtained in [START_REF] Hou | A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields[END_REF] and prove Theorem 1.5, the parabolic case.

Notations and definitions

We give the notations that are used in this paper. Here in R d , for r > 0 and x 0 ∈ R d , B r (x 0 ) is the ball of radius r center at x 0 , B r the ball of radius r of center 0. We define for r > 0 and (t

0 , x 0 ) ∈ R × R d the parabolic cylinder Q r (t 0 , x 0 ) = (t 0 -r 2 , t 0 ) × B r (x 0 ).
We define as well the cylinder centered at (0, 0) by Q r = (-r 2 , 0) × B r and the cylinder

Q 1 = (-2, -1) × B 1 .
For U an open bounded domain of R d , we denote by C α (U ) the space of Hölder continuous functions u, with the norm

u C α (U ) = u L ∞ (U ) + sup x,y∈U |u(x) -u(y)| |x -y| α .
We define the oscillation of a function u on a set E of R d by osc

E u = sup E u -inf E u.
We define the positive (resp. negative) part of a function u by

u + = max(u, 0) (resp. u -= max(-u, 0)). For X = (t, x) ∈ R × R d with x = (x 1 , . . . , x d ), we define the norm X = max(|t|, x 2 )
where

x 2 = d i=1 x 2 i 1 2 .
Let us introduce the notation for the measure of sets. Let E be a subset of R d or R d+1 , the measure of the set E is denoted by |E|. For u : E → R, and (a, b) ∈ R 2 , the sets {u ≥ a} ∩ E, {u ≤ b} ∩ E and {A < u < b} ∩ E will denote respectively {y ∈ E, u(y) ≥ a}, {y ∈ E, u(y) ≤ b} and {y ∈ E, a < u(y) < b}. This notation is used for the statements. In the proofs, we will use the following shorthand notations. The quantities |u ≥ a, E|, |u ≤ b, E| and |a < u < b, E| will denote respectively |{y ∈ E, u(y) ≥ a}|, |{y ∈ E, u(y) ≤ b}| and |{y ∈ E, a < u(y) < b}|.

Let us give the definition of weak solution, sub-solution and super-solution of the parabolic equation (1). Let T 1 < T 2 be real numbers and Ω be an open set in

R d . Let Q = (T 1 , T 2 ) × Ω.
Definition 2.1 (Weak-solutions). We say that u is a weak subsolution (resp. weak supersolution) of (1) satisfying [START_REF] Robert | Sobolev spaces[END_REF]

and (3), if u ∈ L ∞ ((T 1 , T 2 ); L 2 (Ω)) such that ∇ x u ∈ L 2 (Q) and ∂ t u ∈ L 2 ((T 1 , T 2 ); H -1 (Ω)), and for all ϕ ∈ C ∞ c (Q) nonnegative we have - Q u∂ t ϕ + Q A∇ x u • ∇ x ϕ - Q B • ∇ x uϕ - Q gϕ ≤ 0 (resp. ≥ 0).
We say that u is a weak solution of (1

) if u ∈ L ∞ ((T 1 , T 2 ); L 2 (Ω)) such that ∇ x u ∈ L 2 (Q) and ∂ t u ∈ L 2 ((T 1 , T 2 ); H -1 (Ω)), and for all ϕ ∈ C ∞ c (Q) we have - Q u∂ t ϕ + Q A∇ x u • ∇ x ϕ - Q B • ∇ x uϕ - Q gϕ = 0. Remark 2.2.
In what follows, we will drop the word weak for solutions, subsolutions and supersolutions but it will be implicitly assumed.

Let us give the definition of the parabolic De Giorgi sub-classes and super-classes.

+ (γ 1 , γ 2 , γ 3 , p) (resp. DG -(γ 1 , γ 2 , γ 3 , p)) the set of function u such that u ∈ L ∞ ((T 1 , T 2 ); L 2 (Ω)) such that ∇ x u ∈ L 2 (Q), which satisfies ∀k ∈ R, ∀(s, t) ∈ R 2 such that T 1 ≤ s < t ≤ T 2 , ∀0 < r < R and ∀x 0 ∈ Ω such that B R (x 0 ) ⊂ Ω, we have the following inequality Br(x 0 ) (u -k) 2 ± (t, x)dx + γ 1 t s Br(x 0 ) |∇ x (u -k) ± (τ, x)| 2 dxdτ ≤ B R (x 0 ) (u -k) 2 ± (s, x)dx + γ 2 (R -r) 2 t s B R (x 0 ) (u -k) 2 ± (τ, x)dxdτ +γ 3 t s B R (x 0 ) (u -k) p ± (τ, x)dxdτ 1/p .
In fact we can prove that any weak subsolution (resp. supersolution) is in a De Giorgi sub-class DG + (γ 1 , γ 2 , γ 3 , p) (super-class DG -(γ 1 , γ 2 , γ 3 , p)) for some parameters γ 1 , γ 2 , γ 3 and p. And any solution is in the intersection of a sub and super class DG

+ (γ 1 , γ 2 , γ 3 , p)∩ DG -(γ 1 , γ 2 , γ 3 , p).
Proposition 2.4. Let u be a subsolution (resp. supersolution) of (1) satisfying (2) and (3). Then there exist γ 1 , γ 2 and γ 3 positive such that u

∈ DG + (γ 1 , γ 2 , γ 3 , p) (resp. u ∈ DG -(γ 1 , γ 2 , γ 3 , p)). Moreover if u is a solution then there exist γ 1 , γ 2 , γ 3 and p such that u ∈ DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p).
Proof. Here we deal with u a subsolution of (1). The case of supersolution is very similar and the case of the solution is a combinaison a both cases. It's exactly deriving the energy estimates for the subsolution (u -k) + of (1) with the source term g1 (u-k)

+ . Let us define ϕ ∈ C ∞ c (B R ) such that 0 ≤ ϕ ≤ 1, |∇ x ϕ| ≤ 2 R-r and ϕ = 1 in B r 0 ouside B R ,
and the sequence of functions

ψ ε (τ ) =      1 ε (τ -s) if τ ∈ (s, s + ε) 1 if τ ∈ (s + ε, t -ε) -1 ε (τ -t) if τ ∈ (t -ε, t) The idea is to use the test function (τ, x) → (u-k) + (τ, x)ϕ 2 (x) which is not with compact support in time by first using the function (τ, x) → (u -k) + (τ, x)ψ ε (τ )ϕ 2 (x)
as a test function which is allowed by density arguments and then take ε → 0.

- Q (u -k) + ∂ t (u -k) + ψ ε ϕ 2 = Q ∂ t (u -k) + (u -k) + ψ ε ϕ 2 = 1 2 Q ∂ t (u -k) 2 + ψ ε ϕ 2 = - 1 2 Q (u -k) 2 + ∂ t ψ ε ϕ 2 = - 1 2 s+ε s Ω (u -k) 2 + ϕ + 1 2 t t-ε Ω (u -k) 2 + ϕ 2
By the Lesbesgue differentiation theorem when ε → 0, we have

- Q (u -k) + ∂ t (u -k) + ψ ε ϕ 2 → - 1 2 Ω (u -k) 2 + (s, .)ϕ 2 + 1 2 Ω (u -k) 2 + (t, .)ϕ 2 . (5)
For the other terms, since there is no derivative in time, ψ ε will be a common factor for each term. By using the dominated convergence theorem, when ε → 0, ψ ε → 1 almost everywhere so at the limit, the other terms will be

I := Q A∇ x (u -k) + • ∇ x (u -k) + ϕ 2 1 (s,t) - Q B • ∇ x (u -k) + (u -k) + ϕ 2 1 (s,t) - Q g(u -k) + ϕ 2 1 (s,t) .
We then have by using a Young inequality and the fact that g L q ≤ 1 and defining p = q q-1 ,

I ≥ λ t s B R |∇ x (u -k) + | 2 ϕ 2 -2Λ t s B R |∇ x (u -k) + |ϕ(u -k) + |∇ x ϕ| -Λ t s B R |∇ x (u -k) + |(u -k) + ϕ 2 - t s B R |g|(u -k) + ϕ 2 ≥ λ t s B R |∇ x (u -k) + | 2 ϕ 2 - λ 4 t s B R |∇ x (u -k) + | 2 ϕ 2 - 4Λ 2 λ t s B R (u -k) 2 + |∇ x ϕ| 2 - λ 4 t s B R |∇ x (u -k) + | 2 ϕ 2 - Λ 2 λ t s B R (u -k) 2 + |∇ x ϕ| 2 -g L q t s B R (u -k) q q-1 + ϕ 2 q-1 q ≥ λ 2 t s Br |∇ x (u -k) + | 2 - 5Λ 2 λ(R -r) 2 t s B R (u -k) 2 + -g L q t s B R (u -k) q q-1 + q-1 q (6)
Combining ( 5) and ( 6) we deduce that the subsolution u is in the De Giorgi sub-class DG + ( λ 2 , 5Λ 2 λ , g L q , q q-1 ). In Section 3 and 4, a universal constant will be a constant which only depends on d, γ 1 , γ 2 , γ 3 , and p.

De Giorgi method for parabolic De Giorgi classes

In this section, we are going to prove the interior Hölder continuity of functions in De Giorgi classes (Theorem 1.1) so in particular we deduce the result for weak-solutions of the parabolic equation (1) (see Corollary 1.2), thanks to Proposition 2.4. The idea is to reduce the interior Hölder continuity theorem using the oscillation of the solution.

Step by step in Subsection 3.1 using the oscillation, we prove that it is enough to get a lowering of the maximum property to get the theorem. This lowering of the maximum property states that any function in a De Giorgi class smaller than 1 with enough mass below 0 is in fact far from 1 in a smaller cylinder. After reducing the theorem to the lowering of maximum property, we prove two tools called the first and the second lemma of De Giorgi which are the key ideas of the proof of this lowering of maximum property. In the end, Theorem 1.1 and Collorary 1.2 follow from the lowering of maximum property.

Reduction of the Hölder continuity theorem

In this subsection, we explain how to reduce the interior Hölder continuity theorem to the lowering of the maximum property. There are three steps to do it that we introduce in three lemmas. As the proof for weak-solutions [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF][START_REF] Guerand | Quantitative parabolic regularity à la De Giorgi[END_REF], this reduction only relies on "scaling and linearity properties" of the definition of the De Giorgi classes. More precisely, in the case where we do not have the last term is the definition of the De Giorgi class (for the equation it corresponds to the source term g = 0), for any parameter

(t 0 , x 0 , h) ∈ R × R d × R such that (h 2 (t -t 0 ), h(x -x 0 )) still stays in Q 2 , and for any (a, b) ∈ R 2 , for any function u ∈ DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p), the function u(t, x) = au(h 2 (t -t 0 ), h(x -x 0 )) + b is still in DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p).
Because of the last term of the definition of the De Giorgi class (corresponding to a source term which is not zero) which breaks the linearity of the equation, we need to define a universal constant

β > 0 such that if u ∈ DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p), then the function u(t, x) = au(h 2 (t-t 0 ), h(x-x 0 ))+b is in DG + (γ 1 , γ 2 , γ 3 , p)∩DG -(γ 1 , γ 2 , γ 3 , p)
for the particular constants a we are going to use in the proof.

We define the universal constant β as follow

β = 1 2 2C δ|Q 1 | 6 |Q 2 |+1 (7)
smaller than 1, where δ is the universal constant given in Lemma 3.8 and C is the constant of [START_REF] Felsinger | Local regularity for parabolic nonlocal operators[END_REF].

So the general idea is to get the first and second lemma of De Giorgi (Lemmas 3.8 and 3.13) for DG + (γ 1 , γ 2 , γ 3 , p), to use those lemmas to deduce the lowering of maximum property for DG + (γ 1 , γ 2 , βγ 3 , p), and then to deduce the interior Hölder continuity for

DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p) so for DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p). In fact, if u ∈ DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p) then βu ∈ DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 ,
βγ 3 , p) so βu would be Hölder continuous and then u itself.

We first begin by proving that we can reduce the interior Hölder continuity of

DG + (γ 1 , γ 2 , γ 3 , p)∩DG -(γ 1 , γ 2 , γ 3 , p) to the lowering of maximum of DG + (γ 1 , γ 2 , βγ 3 , p)∩ DG -(γ 1 , γ 2 , βγ 3 , p).

Preliminary step: Reduction of the problem.

We prove step by step that one can reduce Theorem 1.1 to Lemma 3.6. Indeed, the Hölder continuity is a consequence of the following lemma.

Lemma 3.1 (Traduction of the definition). Let

u : Q 2 → R be a function in DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p)
where β satisfies [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF]. Then u satisfies

∀(t 0 , x 0 ) ∈ Q 1 , ∀r ∈ 0, 1 2 , osc Qr(t 0 ,x 0 ) u ≤ Cr α u L 2 (Q 2 ) + 1 ,
where C and α only depend on d, γ 1 , γ 2 , γ 3 , and p.

Remark 3.2. We can define the oscillation thanks to the L 2 -L ∞ estimate (Lemma 3.8).

We assume that Lemma 3.1 is true and prove Theorem 1.1.

Proof of Theorem 1.1. The function βu is in DG

+ (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p). Let X = (t, x) ∈ Q 1 and Y = (s, y) ∈ Q 1 . We define Z = X+Y 2 , X 1 = X+Z 2 and Y 1 = Y +Z 2 , r = X-Y 4 . Using Lemma 3.1, we get |u(X) -u(Z)| ≤ osc Qr(X 1 ) u ≤ C X -Y 4 α u L 2 (Q 2 ) + 1 , |u(Z) -u(Y )| ≤ osc Qr(Y 1 ) u ≤ C X -Y 4 α u L 2 (Q 2 ) + 1 .
So by a triangular inequality, adding the last two inequalities, we deduce Theorem 1.1 for βu and then for u. The previous lemma is a consequence of the following oscillation decrease. This version of the lemma is slightly different from the case without source term [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF].

Lemma 3.4 (Local decrease of the oscillation). Let

u : Q 2 → R be a function in DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p)
where β satisfies [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF]. Then there exists a constant θ ∈ 1 2 , 1 only depending on d, γ 1 , γ 2 , γ 3 , and p, such that

• if osc Q 1 u ≥ 2, then osc Q 1/2 u ≤ θ osc Q 1 u, • if osc Q 1 u ≤ 2, then osc Q 1/2 u ≤ 2θ.
We assume that Lemma 3.4 is true and prove Lemma 3.1.

Proof of Lemma 3.1. Let us define for n ∈ N \ {0} a sequence of function in DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p) where β satisfies (7) (since 1 4θ < 1),

u n (τ, y) = 2θ 1-n max(2, osc Q 3/2 u) u t 0 + τ 4 n , x 0 + y 2 n .
By induction let us prove that for all n ∈ N \ {0},

osc Q 1/2 u n ≤ 2θ. ( 8
)
Indeed for n = 1, we have (8) thanks to Lemma 3.4 since osc

Q 1 u 1 ≤ 2. Assuming that osc Q 1/2
u n-1 ≤ 2θ and using Lemma 3.4, we distinguish two cases. If osc

Q 1 u n ≤ 2, we have (8). If osc Q 1 u n ≥ 2, we have osc Q 1/2 u n ≤ θosc Q 1 u n = osc Q 1/2 u n-1 ≤ 2θ,
and we deduce [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF]. So using [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF] for n -1 we have, osc

Q 1 u n = 1 θ osc Q 1/2 u n-1 ≤ 2.
Thus we deduce by induction and using Lemmas 3.4 and 3.8 that for all n ≥ 1, osc

Q 1 2 n (t 0 ,x 0 ) u = max(2, osc Q 3/2 u)θ n-1 2 osc Q 1 u n ≤ θ n-1 max(2, osc Q 3/2 u) ≤ θ n-1 C u L 2 (Q 2 ) + 1 .
We choose α ∈ (0, 1) such that θ = 1 2 α . Let r ∈ 0, 1 2 . In particular there exists n ∈ N \ {0} such that 1 2 n+1 ≤ r < 1 2 n . So we deduce that osc

Qr(t 0 ,x 0 ) u ≤ osc Q 1 2 n (t 0 ,x 0 ) u ≤ 1 2 n α C2 α u L 2 (Q 2 ) + 1 ≤ r α C4 α u L 2 (Q 2 ) + 1 .
Remark 3.5. To prove Lemma 3.1 we only used the scaling properly of the definition of the De Giorgi classes.

The local decrease of the oscillation is a consequence of the following result.

Lemma 3.6 (Lowering the maximum).

There exists a constant µ ∈ (0, 1) which only depends on d, γ 1 , γ 2 , γ 3 , and p, such that for any function v :

Q 2 → R in DG + (γ 1 , γ 2 , βγ 3 , p) where β satisfies (7), if v verifies    v ≤ 1 in Q 3 2 |{v ≤ 0} ∩ Q 1 | ≥ |Q 1 | 2 , ( 9
)
then v ≤ 1 -µ in Q 1 2 .
These cylinders are represented in Figure 1. We assume that Lemma 3.6 is true and prove Lemma 3.4.

Proof of Lemma 3.4. We distinguish two cases: either osc

Q 1 u ≥ 2 or osc Q 1 u ≤ 2. In the first case, we set v = 2 osc Q 1 u u -sup u+inf u 2
, where the supremum and the infimum are

taken in Q 1 . So v is still in DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p)
where β satisfies [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF]. Moreover -1 ≤ v ≤ 1 in B 1 and either v or -v satisfy (9). We deduce that osc

B 1/2 v ≤ 2 -µ,
and osc

B 1/2 u ≤ 1 - µ 2 osc B 1 u.
We deduce the result taking θ = 1 -µ 2 .

Q 1/2 Q 1 Q 1 Q 3/2 Q 2 -1 -9/4 t x 0 Figure 1: Parabolic cylinders.
In the second case, we set v = u -sup u+inf u

2

. The functions v and -v are still in DG + (γ 1 , γ 2 , βγ 3 , p) ∩ DG -(γ 1 , γ 2 , βγ 3 , p) where β satisfies [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF]. And either v or -v satisfies ( 9). So we have osc

Q 1/2 u = osc Q 1/2 v ≤ 2 -µ ≤ 2 1 - µ 2 .
We deduce the result taking θ = 1 -µ 2 . Remark 3.7. To prove Lemma 3.4, concerning the definition of the De Giorgi classes, we only use the fact that if v is in DG + (γ 1 , γ 2 , βγ 3 , p) then -v is in DG -(γ 1 , γ 2 , βγ 3 , p) and reciprocally.

Lemmas of De Giorgi

In this subsection, we introduce the two lemmas of De Giorgi which strongly rely on the definition of the De Giorgi classes.

First lemma of De Giorgi

Let us state the first lemma of De Giorgi which is a L 2 -L ∞ estimate. Lemma 3.8 (First Lemma of De Giorgi: L 2 -L ∞ estimate). There exists a positive constant δ which depends only on d, γ 1 , γ 2 , γ 3 and p such that for any u :

Q 2 → R in DG + (γ 1 , γ 2 , γ 3 , p) the following implication holds true. If Q 1 u 2 + ≤ δ,
then we have

u + ≤ 1 2 in Q 1/2 .
Remark 3.9. By applying Lemma 3.8 to

√ δu ( u L 2 (Q 1 ) +1
) we get the following inequality

u + L ∞ (Q 1/2 ) ≤ C( u + L 2 (Q 1 ) + 1),
where C > 0 depends only on d, γ 1 , γ 2 , γ 3 and p . Remark 3.10. We can replace Q 1/2 and Q 1 by respectively Q 3/2 and Q 2 so that u is bounded in Q 3/2 and the oscillation of u was well-defined in the previous lemmas. Remark 3.11. By symmetry, we can get the same result for u -and DG -(γ 1 , γ 2 , γ 3 , p) and deduce the result for u and

DG + (γ 1 , γ 2 , γ 3 , p) ∩ DG -(γ 1 , γ 2 , γ 3 , p).
Before doing the proof let us introduce a lemma which will be useful for the proof.

Lemma 3.12. Let (V k ) k≤0 be a sequence of real numbers such that for all k ≥ 1,

V k ≤ C k V α k-1 ( 10
)
where α > 1. Then for

V 0 < C -α 2 (α-1) 2 , the sequence (V k ) converges to 0 when k → ∞.
Proof. By induction we have

V k ≤ C k+(k-1)α+•••+2α k-2 +α k-1 V α k 0 = C S k V α k 0 , where S k = k i=0 iα k-i . Let us prove that for all k ≥ 1, S k ≤ α 2 (α -1) 2 α k-1 . ( 11
)
In fact,

S k = α k-1 k i=0 i 1 α i-1
.

And we know that

k i=0 X i = 1 -X k+1 1 -X ,
so by differentiating we get

k i=0 iX i-1 = X k (kX -(k + 1)) + 1 (1 -X) 2 .
And since α > 1, we deduce

S k ≤ α k-1 1 1 -1 α 2 ,
which gives [START_REF] Dibenedetto | Recent results on the Cauchy problem and initial traces for degenerate parabolic equations[END_REF]. So we have

V k ≤ C α 2 (α-1) 2 α k-1 V α k 0 ≤ C α 2 (α-1) 2 V 0 α k . And for V 0 < C -α 2 (α-1) 2 , we deduce that V k → 0, when k → ∞.
The following proof already exists in [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF][START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF]. Our proof here is a bit different from [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF] since it doesn't use an interpolation inequality and from [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF] since we use a Sobolev inequality instead of the L p gain of integrability relying on averaging lemmas [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF] and we use the energy estimate in a different way.

Proof of Theorem 3.8. In this proof C > 0 will denote a constant which will only depend on d, γ 1 , γ 2 , γ 3 , and p. We define

U k = Qr k (u -c k ) 2 + dxdt,
where

r k = 1 2 (1 + 2 -k ) and c k = 1 2 (1 -2 -k ). We notice that Q r k goes from Q 1 to Q1
2 and c k from 0 to 1 2 . We would like to prove that U k satisfies the following induction formula

U k ≤ C k U α k-2 ,
where C > 0 is a universal constant and α > 1 also. Defining

V k = U 2k , the sequence (V k ) satisfy V k ≤ C k V α k-1
, and as in [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF]Theorem 12], we deduce that V n = U 2n tends to 0 when V 0 = U 0 is small enough. Moreover we have

U 0 = Q 1 u 2 + and U ∞ = Q 1 2 u - 1 2 2 +
= 0 and we deduce the result.

Let us prove the induction formula. Let us define the Sobolev exponant

ρ =      2d d-2 if d > 2 q if d = 2, with q ∈ (4, +∞) +∞ if d = 1
in the following Sobolev inequality, for almost every t ∈ (-r 2 k , 0),

(u -c k ) + (t, •) L ρ (Br k ) ≤ C(d) (u -c k ) + (t, •) H 1 (Br k ) , (12)
where C(d) is a constant which only depends on the dimension d and which can be explicitly computed using [START_REF] Robert | Sobolev spaces[END_REF]. Using an Hölder inequality, we have

U k = Qr k (u -c k ) 2 + ≤ 0 -r 2 k Br k (u -c k ) ρ + (t, •)dx 2 ρ |{u(t, •) ≥ c k } ∩ B r k | 1-2 ρ dt. ( 13
) Since {u(t, •) ≥ c k } = {u(t, •) ≥ c k-1 + 2 -k-1 }, we deduce that |{u(t, •) ≥ c k } ∩ B r k | 1-2 ρ ≤ |{u(t, •) ≥ c k-1 + 2 -k-1 } ∩ B r k | 1-2 ρ ≤ 2 2k+2 Br k (u -c k-1 ) 2 + (t, •) 1-2 ρ ≤ C k   sup t∈(-r 2 k ,0) Br k (u -c k-1 ) 2 + (t, •)   1-2 ρ . ( 14
)
We can use the first part of the inequality defining the De Giorgi class (Definition 2.3) with s integrated in (-r 2 k-1 , -r 2 k ) to bound the supremum and obtain in [START_REF] Dibenedetto | Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations[END_REF],

|{u(t, •) ≥ c k } ∩ B r k | 1-2 ρ ≤ C k   0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 (u -c k-1 ) p + 1 p   1-2 ρ ≤ C k 0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 1 {u≥c k-1 } 1-2 ρ + C k 0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 1 {u≥c k-1 } 1 p (1-2 ρ ) . ( 15
)
where we used that (u

-c k-1 ) p + ≤ (u -c k-1 ) 2 + + 1 {u≥c k-1 } to get the last bound. Since we have 0 -r 2 k-1 Br k-1 1 {u≥c k-1 } = |{u ≥ c k-1 } ∩ Q r k-1 | ≤ |{u(t, •) ≥ c k-2 + 2 -k } ∩ Q r k -1 | ≤ 2 2k Qr k-1 (u -c k-2 ) 2 + ≤ 2 2k U k-2 ,
we deduce using [START_REF] Dibenedetto | Harnack's inequality for degenerate and singular parabolic equations[END_REF], [START_REF] Droniou | Solutions de viscosité et solutions variationnelles pour EDP non-linéaires[END_REF] We notice that the last bound is independent of the variable t so it remains to bound

|{u(t, •) ≥ c k } ∩ B r k | 1-2 ρ ≤ C k (U k-1 + U k-2 ) 1-2 ρ + (U k-1 + U k-2 ) 1 p (1-2 ρ ) ≤ C k U 1-2 ρ k-2 + U 1 p (1-2 ρ ) k-2
0 -r 2 k Br k (u -c k ) ρ + (t, •)dx 2 ρ
dt in [START_REF] Dibenedetto | Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations[END_REF].

Using the Sobolev inequality [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF] and the second part of the inequality defining the De Giorgi class (Definition 2.3) with s integrated in (-

r 2 k-1 , -r 2 k ), we deduce 0 -r 2 k Br k (u -c k ) p + (t, •) 2 p ≤ C 0 -r 2 k Br k (u -c k ) 2 + (t, •) + 0 -r 2 k Br k |∇ x (u -c k ) + | 2 (t, •) ≤ C   0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 (u -c k-1 ) p + 1 p   ≤ C 0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 1 {u≥c k-1 } + C 0 -r 2 k-1 Br k-1 (u -c k-1 ) 2 + + 0 -r 2 k-1 Br k-1 1 {u≥c k-1 } 1 p ≤ C k U k-2 + U 1 p k-2 . ( 17
)
By definition U k is non-increasing so assuming that U 0 < 1, we have U k < 1 for every k ≥ 0. Combining ( 16) and ( 15) and assuming U 0 < 1, we deduce that U k satisfies the formula

U k ≤ C k U k-2 + U 1 p k-2 U 1-2 ρ k-2 + U 1 p (1-2 ρ ) k-2 ≤ C k U α k-2 ,
with α = 1 p 2 -2 ρ > 1 which ends the proof using Lemma 3.12 choosing δ < C

-α 2 (α-1) 2 .

Second lemma of De Giorgi

To prove the result of lowering of maximum (Lemma 3.6) we need also the so-called second lemma of De Giorgi, the intermediate value lemma.

Lemma 3.13 (Second lemma of De Giorgi: Intermediate value lemma). Let u be in

DG + (γ 1 , γ 2 , γ 3 , p) such that u ≤ 1 on Q 3 2 . Let Q 1 = (-2, -1) × B 1 . Then we have |{f ≤ 0} ∩ Q 1 ||{f ≥ 1 2 } ∩ Q 1 | ≤ C|{0 < f < 1 2 } ∩ Q 2 | 1 4p+2 , ( 18
)
where C only depends on d, γ 1 , γ 2 , γ 3 , and p.

Proof. We apply Theorem 1.5 with k = 0 and l = 1 2 .

Proof of the lowering of the maximum lemma

Now we can prove Lemma 3.6 using the first and the second lemma of De Giorgi.

Proof of Lemma 3.6. We introduce a sequence of function v k in DG + (γ 1 , γ 2 , γ 3 , p),

v 0 = v v k = 2 v k-1 -1 2 .
Here v is a function in DG + (γ 1 , γ 2 , βγ 3 , p) where β satisfies [START_REF] Hin | De Giorgi techniques applied to the Hölder regularity of solutions to Hamilton-Jacobi equations[END_REF] and the functions v k are in DG + (γ 1 , γ 2 , γ 3 , p) (it will be explained at the end of the proof why the sequence v k remains in DG + (γ 1 , γ 2 , γ 3 , p)). More precisely, we have

v k = 2 k v -(1 -2 -k ) . So that the sets {0 < v k < 1 2 } = {1 -1 2 k < v < 1 -1 2 k+1
} are disjoints and the sequence v k still satisfies (9). If not, we consider k 0 ≥ 1 an index such that

If Q 1 (v)
Q 1 (v k ) 2 + > δ, for any 0 ≤ k ≤ k 0 .
We have the following inequalities for any 0 ≤ k ≤ k 0 -1,

|{v k ≥ 1 2 } ∩ Q 1 | = |{v k+1 ≥ 0} ∩ Q 1 | ≥ Q 1 (v k+1 ) 2 + > δ,
and

|{v k ≤ 0} ∩ Q 1 | ≥ |{v ≤ 0} ∩ Q 1 | ≥ |Q 1 | 2 .
So by the intermediate value lemma (Lemma 3.13),

|{0 < v k < 1 2 } ∩ Q 2 | ≥ δ C |Q 1 | 2 6 .
By summing all the intermediate measure and using the fact that the sets are disjoints we have,

|Q 2 | ≥ k 0 k=1 |{0 < v k < 1 2 } ∩ Q 2 | ≥ k 0 δ C |Q 1 | 2 6 .
So k 0 is bounded such that

k 0 ≤ 2C δ|Q 1 | 6 |Q 2 |,
and necessarily, there exists k ≤ 2C

δ|Q 1 | 6 |Q 2 | + 1 such that Q 1 (v k ) 2 + ≤ δ so by Lemma 3.8, we have (v k ) + ≤ 1 2 in Q 1/2 so that v ≤ 1 - 1 2 k+1 ≤ 1 - 1 2 2C δ|Q 1 | 6 |Q 2 |+2 in Q 1/2 ,
and we choose µ =

1 2 2C δ|Q 1 | 6 |Q 2 |+2
. So in the end we deal only with the sequence until a universal index, so choosing β =

1 2 2C δ|Q 1 | 6 |Q 2 |+1
, for all k ≤ 2C

δ|Q 1 | 6 |Q 2 | + 1, v k is in DG + (γ 1 , γ 2 , γ 3 , p).

Intermediate value lemma

In this section, we deal with intermediate value lemmas for functions in H 1 and for functions in DG + (γ 1 , γ 2 , γ 3 , p). We first recall the lemma in the H 1 case since we use it in the proof of the DG + (γ 1 , γ 2 , γ 3 , p) case. Then we give the proof of Theorem 1.5, the intermediate value lemma for functions in the De Giorgi class DG + (γ 1 , γ 2 , γ 3 , p).

Functions in H 1

We give a simpler proof of [

(l-k) {u ≤ k}∩B R × {u ≥ l}∩B R ≤ R|B R | {k < u < l}∩B R 1 2 B R |∇(u -k) + (x)| 2 dx. ( 19
)
Proof. We will use the shorthand notations |u ≤ k|, |u ≥ l| and |k < u < l| for the mesures of the sets {x ∈ B R , u(x) ≤ k}, {x ∈ B R , u(x) ≥ l} and {x ∈ B R , k < u(x) < l}.

We define the following truncated function

v(x) =      0 if u(x) ≤ k, u(x) -k if k < u(x) < l, l -k if u(x) ≥ l. ( 20
)
By Stampacchia theorem in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 7.8] or [START_REF] Droniou | Solutions de viscosité et solutions variationnelles pour EDP non-linéaires[END_REF], we have v ∈ H 1 (B R ). By Poincaré inequality since v ∈ W 1,1 (B R ), see for example [1, Theorem 3.2], we have

B R |v(x) -v| dx ≤ R B R |∇v(x)| dx, (21)
where

v = 1 |B R | B R v(x) dx. The sets {x ∈ B R , v(x) = 0}, and {x ∈ B R , v(x) = l -k}
are respectively denoted by {v = 0} and {v = l -k} and their mesures by |v = 0| and |v = l -k|. We have the following inequalities

(l -k) |B R | |v = 0||v = l -k| ≤ {v=0} v dx ≤ {v=0} |v(x) -v| dx ≤ B R |v(x) -v| dx, ( 22
)
and by Cauchy-Schwarz inequality

B R |∇v(x)| dx = {k<u<l} |∇v(x)| dx ≤ B R |∇(u -k) + (x)| 2 dx|{k < u < l} ∩ B R | 1 2 . ( 23

Functions in DG +

In this section, we prove Theorem 1.5. The proof of this theorem deeply uses the definition of the De Giorgi class. We can see the inequality of Definition 2.3 as two inequalities. The second one . Then there exists a constant C > 0 such that for all -2 < s < t < 0, we have

γ 1 t s Br(x 0 ) |∇ x (u -k) ± (τ, x)| 2 dxdτ ≤ B R (x 0 ) (u -k) 2 ± (s, x)dx + γ 2 (R -r) 2 t s B R (x 0 ) (u -k) 2 ± (τ, x)dxdτ + γ 3 t s B R (x 0 ) (u -k) p ± (τ, x)dxdτ
t s B 5 4 |∇ x (u -k) + | 2 (τ, x) dxdτ ≤ C,
where C only depends on d, k, γ 1 , γ 2 , γ 3 and p.

Proof of Lemma 4.2. We use Definition 2.3 for r = 5 4 , R = 3 2 , x 0 = 0 and we deduce

t s B 5 4 |∇ x (u -k) + | 2 (τ, x) dxdτ ≤ (1 -k) 2 γ 1 |B3 2 | + 32 γ 2 γ 1 |B 3 2 |(1 -k) 2 + 2 1 p γ 3 γ 1 |B 3 2 | 1 p (1 -k).
The second lemma is a first step for the proof of Theorem 1.5. It gives "almost" an intermediate value lemma with an error which is small for close times. We will see that the first inequality of Definition 2.3,

Br(x 0 ) (u -k) 2 + (t, x)dx ≤ B R (x 0 ) (u -k) 2 + (s, x)dx + γ 2 (R -r) 2 t s B R (x 0 ) (u -k) 2 + (τ, x)dxdτ + γ 3 t s B R (x 0 ) (u -k) p + (τ, x)dxdτ 1/p
contains the information which quantify the fact that u cannot do an increasing jump in time in term of measures. In fact, (u -k) + is bounded so the previous inequality becomes

Br(x 0 ) (u -k) 2 + (t, x)dx ≤ B R (x 0 ) (u -k) 2 + (s, x)dx + C(t -s) 1/p .

Lemma 4.3 (A key inequality for close times). Let

u : Q 2 → R be a function in DG + (γ 1 , γ 2 , γ 3 , p) such that u ≤ 1 on Q 3 2
.Then for all (k, l) ∈ R 2 such that k < l ≤ 1 and for all (t 1 , t 2 , τ ) ∈ (-2, 0) 3 such that -2 < t 1 < τ < t 2 < 0, we have

(l -k) 2 |u ≥ l, (τ, t 2 )×B 1 ||u ≤ k, (t 1 , τ )×B 1 | ≤ C|k < u < l, (t 1 , τ )×B 2 | 1 2 +C(t 2 -t 1 ) 2+ 1 p ,
where C only depends on d, k, γ 1 , γ 2 , γ 3 and p.

Proof of Lemma 4.3. In this proof, let C > 0 be a constant which only depends on d, k, γ 1 , γ 2 , γ 3 and p which will change from line to line. Thanks to the definition of De Giorgi classes, we have

B 1 (u -k) 2 + (t, x)dx ≤ B 5 4 (u -k) 2 + (s, x)dx + 16γ 2 t s B 5 4 (u -k) 2 + (τ, x)dxdτ + γ 3   t s B 5 4 (u -k) p + (τ, x)dxdτ   1/p . ( 24
)
Simplifying ( 29), we have

(l -k) 2 |u ≥ l, (τ, t 2 ) × B 1 ||u ≤ k, (t 1 , τ ) × B 1 | ≤ C|k < u < l, (t 1 , τ ) × B 2 | 1 2 +C(t 2 -t 1 ) 2+ 1 p , (30)
which ends the proof. Now let us prove Theorem 1.5. The idea of the proof is to understand that the "error" term (t 2 -t 1 ) 2+ 1 p in Lemma 4.3 is negligible compared to the other terms when t 2 -t 1 is small and when the intervals are well-chosen.

Proof of Theorem

1.5. Using that |u ≤ k, (t 1 , τ ) × B 1 | = |u ≤ l, (t 1 , τ ) × B 1 | -|k < u < l, (t 1 , τ ) × B 1 |, we deduce from Lemma 4.3 (l -k) 2 |u ≥ l, (τ, t 2 ) × B 1 ||u ≤ l, (t 1 , τ ) × B 1 | ≤ C|k < u < l, Q 2 | 1 2 + C(t 2 -t 1 ) 2+ 1 p .
We discretize the time interval. Let n ∈ N \ {0}, α n = 1 n , T = -1 and t k = kα n . Necessarily by the pigeonhole principle, there exists

i ∈ [1, n] such that |u ≤ k, (t i-1 , t i ) × B 1 | ≥ |u ≤ k, Q 1 | n , ( 31
)
and there exists j ∈ [n, 2n -1] such that

|u ≥ l, (t j , t j+1 ) × B 1 | ≥ |u ≥ l, Q 1 | n . ( 32
)
But since we would like adjacent intervals of time, we relax the inequalities ( 31) and ( 32) and we still have

|u < l, (t i-1 , t i ) × B 1 | ≥ |u ≤ k, Q 1 | 2n , ( 33
)
and

|u ≥ l, (t j , t j+1 ) × B 1 | ≥ |u ≥ l, Q 1 | 2n . ( 34
)
We distinguish two cases, either there exists m ∈ [i, 2n -1] such that m + 1 does not satisfy (33) (i.e., (33) is false for i = m + 1), or for all m ∈ [i, 2n -1], m + 1 does satisfy (33). In the first case, letting p be the first integer m satisfying "m + 1 does not satisfy (33)", we have

|u < l, (t p , t p+1 ) × B 1 | < |u ≤ k, Q 1 | 2n , so |u ≥ l, (t p , t p+1 ) × B 1 | ≥ |B 1 |α n - |u ≤ k, Q 1 | 2n ≥ |u ≥ l, Q 1 | 2n and |u < l, (t p-1 , t p ) × B 1 | ≥ |u ≤ k, Q 1 | 2n .
In the second case, let p = j. Then in all cases, using Lemma 4.3 we have, (l -k)

(l -k) 2 |u ≤ k, Q 1 ||u ≥ l, Q 1 | ≤ C|k < u < l, Q 2 | 1 4p+2 .
This achieves the proof of the theorem.

Remarks and counterexamples

We remark that Theorem 1.5 is false for subsolutions if we replace Q 1 by Q 1 . For example, the function f (t, x) = 1 for t ∈ (-2, -1] 0 for t ∈ (-1, 0) , is a subsolution of (1) in Q 2 but does not satisfy Theorem 1.5 for k = 0 and l = 1 with Q 1 instead of Q 1 . In fact, the intermediate value lemma does not allow increasing jump in time. In the solution case of for function in DG + ∩ DG -, we can obtain the same inequality with Q 1 instead of Q 1 in (4).

Extension to kinetic equations?

Let us consider the following kinetic Fokker-Planck equation of [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF],

∂ t f + v • ∇ x f = ∇ v • (A∇ v f ) + B • ∇ v f + s, (t, v, x) ∈ Q 2 , ( 35
)
where Q R = (-R 2 , 0)×B R ×B R 3 is a kinetic cylinder. We define Q 1 = (-2, -1)×B 1 ×B 1 .

In dimension d = 1, considering the following subsolution f (t, x, v) = 1 for x + 2t < -2 0 for x + 2t ≥ -2, we notice that it does not satisfy an inequality of the form

|{f ≤ 0} ∩ Q 1 | α |{f ≥ 1 2 } ∩ Q 1 | β ≤ C|{k < f < l} ∩ Q 2 | γ , ( 36
)
for some constants α, β, γ and C which do not depend on the f . In fact, for some parameters c > 1 (to have a subsolution) and a ∈ R, f a,c (t, x, v) = 1 for x + ct < a 0 for x + ct ≥ a, is also a subsolution of (35). Drawing many lines of discontinuity x+ct = a, we notice that to find a valid intermediate value inequality, we must consider two cylinders which cannot be both crossed by the same line of discontinuity x + ct = a. More precisely, we must have a "gap" in time between the two cylinders of the same size (or at least not smaller) than the two cylinders. Let us change the definition of Q 1 by Q 1 = (-3, -2) × B 1 × B 1 .

The two domains Q 1 and Q 1 are never both crossed by the same line of discontinuity x + ct = a. That is why this intermediate value inequality seems to be more accurate,

|{f ≤ 0} ∩ Q 1 | α |{f ≥ 1 2 } ∩ Q 1 | β ≤ C|{k < f < l} ∩ Q R | γ .
In fact, the local energy estimate usually used for this equation (see for example [START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF]Lemma 11]) is too weak to be able to define kinetic De Giorgi classes in this way. We are losing information especially in the variable x and subsolution are not bounded in H 1 so we cannot use directly the proof of the parabolic case. An idea then could be to keep the term ∂ t f + v • ∇ x f as one block and to understand what would be the "good way" to integrate the equation and this term should makes the gap appear.

  ) Using[START_REF] Guerand | Quantitative parabolic regularity à la De Giorgi[END_REF],[START_REF] Hou | A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields[END_REF] and[START_REF] Imbert | The weak Harnack inequality for the Boltzmann equation without cut-off[END_REF] and the equalities |v = 0| = |{u ≤ k} ∩ B R | and |v = l -k| = |{u ≥ l} ∩ B R |, we deduce (19).

  Definition 2.3 (De Giorgi classes DG ± (γ 1 , γ 2 , γ 3 , p)). Let Ω be a bounded open subset of R d and T 1 < T 2 two real numbers. For the positive parameters γ 1 , γ 2 , γ 3 and 1 ≤ p ≤ d+2 d , we define the De Giorgi sub-class (resp. super-class) and denote by DG

  Remark 3.3. Lemma 3.1 is just rewritting the interior Hölder regularity in terms of the oscillation. It doesn't use the definition of De Giorgi classes.

	We can deduce Corollary 1.2.

Proof of Corollary 1.2. It is a consequence of Theorem 1.1 and Proposition 2.4.

  22, Theorem 2.9], about an intermediate value lemma for functions which are bounded in the Sobolev space H 1 . This lemma in an alternative version of the De Giorgi isoperimetric inequality [32,Lemma 10]. As we previously saw, it is a crucial tool in the De Giorgi proof of the Hölder regularity for solutions of elliptic equations. (Intermediate value lemma in H 1 ). Let u ∈ H 1 (B R ). Then for all (k, l) ∈ R 2 such that k ≤ l, we have

	Lemma 4.1

  In fact, it helps us to bound the norm of the gradient of a function by a universal constant since (u -k) + is bounded and to get an intermediate value lemma in H 1 which only depends on the measures and universal constant. So we first get the following lemma. (Universal bound of the L 2 of the gradient). Let u : Q 2 → R be a function in DG + (γ 1 , γ 2 , γ 3 , p) such that u ≤ 1 on Q 3 2

	1/p contains the information which quantify the fact that there is no jump in the space , variable x. Lemma 4.2

  -k) 2 |u ≤ k, Q 1 ||u ≥ l, Q 1 | ≤ Cn 2 |k < u < l, Q 2 | So necessarily |k < u < l, Q 2 | > 0. And taking n such that Cn -1 p ≤ C n 2 |k<u<l, Q 2 |

	Thus, we have
	(l 1 2 + Cn -1 p .
					1
					2	2	, for
	example n =	2 |k<u<l, Q 2 |	p 4p+2	+ 1, we get
				1 2 + C 2	2 n	2+ 1 p .

2 |u≤k, Q 1 | 2n |u≥l, Q 1 | 2n ≤ (l -k) 2 |u < l, (t p-1 , t p ) × B 1 ||u ≥ l, (t p , t p+1 ) × B 1 | ≤ C 1 |k < u < l, Q 2 |

Acknowledgements. The author acknowledges Cyril Imbert for his expert advice and encouragement throughout working on this paper. The author acknowledges the support of partial funding by the ERC grant MAFRAN 2017-2022.

First, we bound the left hand side from below

Second, we bound from above each term of the right hand side. The first term gives

The second term gives

And the third term gives

So combining ( 24), ( 25), ( 26), ( 27) and ( 28) we deduce

Multiplying the last inequality by |{u(s, .) ≤ k} ∩ B5 ) by Fubini's theorem), we get

We integrate the latter over s ∈ [t