Sylvain Prigent
email: sylvain.prigent@inria.fr

Cesar Augusto Valades-Cruz
email: augusto.valades-cruz@curie.fr

Ludovic Leconte

Jean Salamero

Charles Kervrann

STracking: a free and open-source Python library for particle tracking and analysis

Analysis of intra-and extracellular dynamic like vesicles transport involves particle tracking algorithms. The design of a particle tracking pipeline is a routine but tedious task. Therefore, particle dynamics analysis is often performed by combining several pieces of software (filtering, detection, tracking, etc.) requiring many manual operations, and thus leading to poorly reproducible results. Given the new segmentation tools based on deep learning, modularity and interoperability between software have become essential in particle tracking algorithms. A good synergy between a particle detector and a tracker is of paramount importance. In addition, a user-friendly interface to control the quality of estimated trajectories is necessary. To address these issues, we developed STracking, a P ython library that allows combining algorithms into standardized particle tracking pipelines.

Introduction

The study of cell biology dynamics, such as intracellular membrane transport inward (i.e. endocytosis) and outward (i.e. exocytosis/ recycling), has been difficult or at least incomplete until recently, due to the heterogeneity of the motion behavior of the analyzed structures. Different tracking software have been published [e.g. utrack [START_REF] Jaqaman | Robust single-particle tracking in live-cell time-lapse sequences[END_REF], TrackMate [START_REF] Tinevez | TrackMate: an open and extensible platform for single-particle tracking[END_REF], MTT (Serge et al., 2008), MSSEF-TSAKF [START_REF] Jaiswal | Tracking virus particles in fluorescence microscopy images using multi-scale detection and multi-frame association[END_REF], maptrack [START_REF] Feng | Multiple dense particle tracking in fluorescence microscopy images based on multidimensional assignment[END_REF], Cell Tracking Profiler [START_REF] Mitchell | Cell tracking profiler-a user-driven analysis framework for evaluating 4D live-cell imaging data[END_REF], btrack [START_REF] Ulicna | Automated deep lineage tree analysis using a Bayesian single cell tracking approach[END_REF]] to track individual biomolecules or extended objects with a shape, such as cells [e.g. TrackMate [START_REF] Ershov | Bringing TrackMate into the era of machine-learning and deep-learning[END_REF] and CellProfiler [START_REF] Stirling | CellProfiler 4: improvements in speed, utility and usability[END_REF]], to obtain spatial information and to quantify their kinetics. Most of them focus on the accuracy and reproducibility of the analysis but the user interfaces remain complex or even limited to a two-dimensional representation. The development of a user-friendly graphical user interface (GUI) therefore appears necessary to facilitate the selection of parameters, the analysis and the visualization of 3D þ time trajectories estimated from complex 3D videos. The use of Python, a versatile and free programming language is growing rapidly within the bioimaging user community [START_REF] Fernandez-Gonzalez | PyJAMAS: open-source, multimodal segmentation and analysis of microscopy images[END_REF]. Python tools for visualization [e.g. napari (Sofroniew et al., 2021), ipyvolume (Breddels et al., 2018), SeeVis [START_REF] Hattab | SeeVis-3D space-time cube rendering for visualization of microfluidics image data[END_REF]] and analysis [ZeroCostDL4Mic (von Chamier et al., 2021), BioImageIT [START_REF] Prigent | BioImageIT: Open-source framework for integration of image data-management with analysis[END_REF], Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF]] are widely applied to microscopy images.

On the other hand, a lot of particle tracking approaches have been developed over the last decades. Interestingly, although a number of studies aimed at comparing particle tracking performance have been published [START_REF] Carter | Tracking single particles: a user-friendly quantitative evaluation[END_REF][START_REF] Cheezum | Quantitative comparison of algorithms for tracking single fluorescent particles[END_REF][START_REF] Chenouard | Objective comparison of particle tracking methods[END_REF][START_REF] Ruusuvuori | Evaluation of methods for detection of fluorescence labeled subcellular objects in microscope images[END_REF][START_REF] Smal | Quantitative comparison of multiframe data association techniques for particle tracking in time-lapse fluorescence microscopy[END_REF][START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF], none of the tested methods seems to perform in a generic way, regardless of the type of image data. As a consequence, it is critical for users to have the possibility to test different detectors and/or trackers in order to identify the best solution for their application. In addition, efforts in the Python community includes particle tracking packages such as TrackPy (Allan et al., 2021). TrackPy is a complete particle tracking toolkit, but the code can be a barrier for non-expert user.

Here, we present STracking, an open-source Python library for combining algorithms into standardized particles tracking pipelines for biological microscopy images. STracking is distributed under a BSD 3-Clause 'New' or 'Revised' License. STracking combines particle detection, tracking and analysis methods and can be used via a napari plugin. STracking contributes to the recent ecosystem of Python-based plugins for bioimage analysis.

Implementation and application

STracking breaks down a particle tracking pipeline into five components: (i) frame-by-frame particle detection; (ii) particle linking; (iii) analysis of particle properties; (iv) design of track features; and (v) filtering of tracks. Each component is represented as a Python object.

Each component can be implemented separately. This modular design makes it easy to update and facilitates interoperability with other plugins/algorithms. Whenever a new detection or tracker algorithm is added, compatibility is guaranteed with the particle tracking pipeline, and it is versioned within the STracking library. Several particle detectors are available in STracking: Difference of Gaussian, Determinant of Hessian and Laplacian of Gaussian from the Python library scikit-image (van der [START_REF] Van Der Walt | Scikit-image: image processing in Python[END_REF]. In addition, STracking includes a mask detector, called SSegDetector. This detector takes a label or binary mask as input image and returns a list of object positions (centroids of connected components).

Moreover, STracking includes a tracker [START_REF] Matov | Optimal-flow minimum-cost correspondence assignment in particle flow tracking[END_REF] that estimates the optimal tracks as follows: first, a connection graph is created with all the possible connections. Second, tracks are iteratively extracted from the graph using shortest path and graph pruning.

STracking library uses two data structures: 'SParticles' to manage the set of detected particles and 'STracks' to manage the collection of trajectories. These data structures contain SciPy [START_REF] Virtanen | SciPy 1.0: fundamental algorithms for scientific computing in Python[END_REF] objects to store the particles and the tracks. The particles are represented with a 2D numpy array where each row is dedicated to a specific particle and columns are [T, Z, Y, X] for 3D data and [T, Y, X] for 2D data. The properties of particles are stored in a dictionary. Similarly, tracks are stored in a 2D numpy array where each row is dedicated to specific particle and columns are [trackID, T, Z, Y, X] for 3D data and [trackID, T, Y, X] for 2D data. Tracks features and split/merge events are stored using dictionaries. This data representation is the same as napari (Sofroniew et al., 2021) points and tracks layers, making STracking natively compatible with the napari viewer. We thus implemented a STracking napari plugin suite (napari-tracks-reader, naparistracking). It provides a graphical interface to create a STracking pipeline without writing Python code. STracking could be used as script in Python or napari plugin. The STracking library can be combined with other Python packages to extend STracking functionalities. The napari plugin allows one to perform a full STracking pipeline, or to load detections or tracks from another software such as StarDist [START_REF] Schmidt | Cell detection with star-convex polygons[END_REF], TrackMate [START_REF] Tinevez | TrackMate: an open and extensible platform for single-particle tracking[END_REF] or u-track [START_REF] Jaqaman | Robust single-particle tracking in live-cell time-lapse sequences[END_REF] and continue the analysis with STracking and napari. Documentation on STracking library with examples is available at https://sylvainprigent.github.io/stracking/. STracking documentation was created using sphinx and the autodoc extension.

STracking pipeline using the napari plugin is illustrated with data obtained in Lattice Light-Sheet Structured Illumination Microscopy [START_REF] Chen | Lattice light-sheet microscopy: imaging molecules to embryos at high spatiotemporal resolution[END_REF](Fig. 1 and Supplementary Video S1). The STracking workflow could also be implemented using Jupyter notebook (Supplementary Note S1). Additionally, STracking library can be used for cell migration experiments (Supplementary Note S2) using label mask images produced by other software such as CellPose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF] or StarDist [START_REF] Schmidt | Cell detection with star-convex polygons[END_REF]

Conclusions

The STracking library simplifies the design of single particle tracking workflows through a graphical interface using napari and a comprehensive Python library of functions. Unlike previous single particle tracking tools in Python ecosystem, it provides a very flexible solution for processing and visualizing the tracks taking advantage of Napari (Sofroniew et al., 2021) viewer for 3D þ time representation. A similar approach was introduced in TrackMate software [START_REF] Tinevez | TrackMate: an open and extensible platform for single-particle tracking[END_REF] for the visualization and validation of 2D tracks in Fiji [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] java-based environment. Thus, reproducible analysis can be performed without being an expert programmer. For this purpose, STracking library includes a pipeline class to allow executing a tracking pipeline recorded as a json file. We would point out that this plugin-implemented recording technique is not an optimal software architecture since it should be done by the host platform. To overcome this difficulty, we recommend using a powerful data management software such as the recent BioImageIT platform [START_REF] Prigent | BioImageIT: Open-source framework for integration of image data-management with analysis[END_REF].

In summary, the STracking library greatly simplifies the inspection and optimization of single particle tracking algorithms and thus allows the evaluation of new detection and tracker algorithms in this context, which are constantly being developed.

Fig. 1 .

 1 Fig. 1. Overview of STracking library implemented in its napari plugin (a-c) and Jupyter Notebook (Kluyver et al., 2016) (d-f). Fifty-five planes 3D volumes of live RPE1 cells double stained with PKMR for Mitochondria (magenta) and with plasma membrane deep red (PMDR) for endosomal pathway (green) were acquired within 4.3 s per stack using Lattice light-sheet Structure Illumination Microscopy (LLS-SIM). STracking workflow is illustrated here with single particle tracking of endosomal pathway (PMDR). napari-stracking includes spots detection (a) and linking (b) through a GUI. 3D data and tracks are rendered using napari viewer (c). Jupyter notebook (d) allows also spots (e) and tracks (f) analysis. Additionally, they permit to get spots properties and tracks features, as well as tracks filtering. LLS-SIM data were reconstructed using MAP-SIM (K r ı zek et al., 2016) (A color version of this figure appears in the online version of this article.)

  through napari-stracking plugin, Jupyter Notebook or Python scripting. These examples demonstrate the ability of STracking to analyze complex datasets acquired with most advanced microscopy technologies.

Funding

This work was supported by the French National Research Agency (France-BioImaging Infrastructure [ANR-10-INBS-04-07] and LabEx Cell(n)Scale [ANR-11-LABX-0038] as part of the IDEX PSL [ANR-10-IDEX-0001-02]).

STracking is available as a Python library using 'pip install' and the source code is publicly available on GitHub (https://github.com/sylvainprigent/stracking). A graphical interface is available using two napari plugins: napari-stracking and napari-tracks-reader. These napari plugins can be installed via the napari plugins menu or The napari plugin source codes are available on GitHub (https://github.com/syl vainprigent/napari-tracks-reader, https://github.com/sylvainprigent/napari-stracking

Data availability

The data underlying this article are available in FigShare, at https:// doi.org/10.6084/m9.figshare.19322171.


Marc Josien

Stochastic homogenization : a short proof of the annealed Calderón-Zygmund estimate

Keywords: Stochastic homogenization, Calderón-Zygmund estimates AMS classification: 35B27, 35R60, 35J15, 35B45, 42B15

come

Stochastic homogenization : a short proof of the annealed Calderón-Zygmund estimate

Various problems in continuum physics involve elliptic operators in divergence form -∇ • a∇. In many contexts (e.g. electrostatics, elasticity...), it is relevant to assume that the coefficient field a, which may be seen as a metric tensor field on R d , is heterogeneous and random. The large-scale mapping properties of these linear operators are a crucial ingredient for analysis. However, in this task, the main difficulty comes from the fact that a is heterogeneous: this prevents to benefit from most classical elliptic regularity properties, which rely on regularity assumptions. Indeed, for heterogeneous coefficient fields, these are not available on large scales. (The surviving large-scale estimates for heterogeneous coefficient fields are perturbations of the energy estimate, namely variants of the so-called Meyers' estimates [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF].)

Here, we focus on a specific mapping property of the operator ∇(-∇•a∇) -1 ∇•, namely the so-called annealed Calderón-Zygmund (CZ) estimates, which are richer and tighter than the classical energy estimate. The wording annealed CZ estimate reflects a way of norming the space of random fields f on R d by intertwining spatial Lebesgue L p -norms and local L r -moments of f for p, r ∈ (1, ∞). On the one hand, the term CZ refers to the well-known L p -theory for elliptic PDEs due to Calderón and Zygmund. On the other hand, the term annealed comes from statistical physics, and ultimately originates from metallurgy. It opposes to the term quenched, which precisely describes a pathwise way of norming a fixed realization of a random field.

Our result grows out of recent progress in quantitative stochastic homogenization, which is inspired by the classical large-scale regularity theory of Avellaneda and Lin [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]. Hence, our criterion on the validity of the annealed CZ estimates unsurprisingly involves the so-called correctors that describe the deviation of the scalar and vector potentials of the a-harmonic coordinates from affine functions: We assume that the stochastic moments of these correctors are finite. This is an intrinsic property of the coefficient field a, the statistics of which we do not fix (unlike most articles in homogenization).

This article is a companion to the more expanded and detailed paper [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], which has the double objective of providing a new proof of the annealed CZ estimates first shown in [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] and exemplifying their use in the context of quantitative stochastic homogenization. The new approach of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] relies on functional analytics and has the advantage of being less intrusive on the level of the PDE w. r. t. [START_REF] Armstrong | Calderón-Zygmund estimates for stochastic homogenization[END_REF][START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF][START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF] briefly discussed below, which all use large-scale Lipschitz regularity as a crucial input. Here, the aim is to provide a streamlined and simplified version of the proof in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. The heuristics are borrowed from [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] but we employ a particular framework where we may use slightly different arguments, which are hopefully more efficient and intuitive while avoiding some technicalities.

CZ estimates in homogenization have been discussed for the last three decades. Indeed, (deterministic) CZ (or L p ) estimates have been first obtained in the context of periodic homogenization in [START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF] by analyzing the singular integral involving the Green function. Also, they have been deduced from Lipschitz regularity in [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF] by using a CZ decomposition originating from [START_REF] Caffarelli | On W 1,p estimates for elliptic equations in divergence form[END_REF]. Our result applies in these contexts, where no randomness is involved. Very recently, CZ estimates have been established in the context of stochastic homogenization, either in a quenched form or in an annealed form, cf. [START_REF] Armstrong | Calderón-Zygmund estimates for stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF] and [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], respectively.

Annealed CZ estimates play a double role in stochastic homogenization. On the one hand, they are interesting per se because they provide natural estimates on the solution of the heterogeneous problem in norms intertwining moments in space and probability. On the other hand, they are very convenient tools in quantitative stochastic homogenization, which aims at obtaining a precise description of the oscillations and the fluctuations of the solution of the heterogeneous problem: This corresponds to approximating accurately the solution of the heterogeneous problem in strong norms and in weak norms (or on average), respectively. Indeed, as illustrated in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], we may build a self-contained quantitative homogenization theory allowing for a description of the oscillations and the fluctuations just by appealing to functional calculus (i.e. spectral gap estimates, cf. [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF]) and to these estimates. Hence, we may avoid any quenched result related to the large-scale Lipschitz estimates [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]. Since the article [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] has also the nature of a review article, we refer to it for a detailed overview of recent developments in quantitative homogenization as well as for a thorough justification of the usefulness of the annealed CZ estimates.

Our framework

We introduce a framework in which our result holds. We purposely choose not to specify the construction of the coefficient field a (which may be related to a Gaussian field with integrable correlation as exemplified in Section 3.2) but rather assume that it enjoys desirable properties on the level of extended correctors (φ i , σ i ) -which play a crucial role in homogenization. These appear through the following decomposition1 of the coefficient field a:

ae i = āe i -a∇φ i + ∇ • σ i . (1)
Here (e i ) i∈{1,••• ,d} is the canonical basis of R d and:

• ā is a constant coefficient, the so-called homogenized coefficient. The homogenized coefficient is of paramount use in applications, e.g. in mechanics, for it encodes the large-scale homogeneous behavior ∇(-∇ • ā∇) -1 ∇• of the heterogeneous operator ∇(-∇ • a∇) -1 ∇•;

• the functions φ i are the correctors, which correct the ā-harmonic coordinates x i to a-harmonic coordinates in the sense of

-∇ • a∇(φ i + x i ) = 0; (2)
• the skew-symmetric fields σ i are the flux-correctors (they indeed correct the flux a(e i + ∇φ i )). Notice that the flux correctors σ i satisfying (1) are not unique. In general, the following gauge is employed:

-∆σ ijk = ∂ j (e k • a(e i + ∇φ i )) -∂ k (e j • a(e i + ∇φ i )).
Decomposition (1) may be seen as an a-Helmholtz decomposition of (aā)e i since a∇φ i is the product of a and a curl-free vector field (that is, a gradient in R d ) and since ∇ • σ i is divergence-free.

Let d ≥ 1. Assume that we are given λ > 0, α ∈ (0, 1) and a nondecreasing function

κ : [1, ∞) → [1, ∞).
We consider an ensemble • of coefficient fields a : R d → R d×d that satisfy the following assumptions:

A.1 a is a symmetric uniformly λ-elliptic and bounded coefficient field, namely there holds

ξ • a(x)ξ ≥ λ|ξ| 2 and ξ • a -1 (x)ξ ≥ |ξ| 2 for all x, ξ ∈ R d ; (3)
A.2 a admits the decomposition (1) where ā is constant, and where the extended correctors (φ i , σ i ) are uniformly bounded with overwhelming probability in the sense of

2 sup x∈R d |(φ i , σ i )(x)| r 1 r ≤ κ(r) for all r ∈ [1, ∞) and i ∈ {1, • • • , d}; (4)
A.3 a is uniformly Hölder continuous in the sense of

[a] C 0,α (R d ) := sup x,x ∈R d |a(x) -a(x )| |x -x | α ≤ 1. (5)
In the sequel, we do not give the precise dependence of our estimates with respect to the given data d, λ, α and κ. On the contrary, most of the estimates will involve constants depending on γ := (d, λ, α, κ). To emphasize this dependence, we make use of the notation " β1,••• ,βn " which reads "≤ C for a (finite) constant C depending on the tuple

(β 1 , • • • , β n )
of previously defined parameters"3 . Likewise, the notations β1,••• ,βn 1 (and β1,••• ,βn 1) read "sufficiently small (sufficiently large) with respect to a constant depending on the parameters β 1 , • • • , β n ". For simplicity, the subscripts might be omitted in the course of the proofs.

Remark 1. Assumption A.2 is the only assumption related to the homogenization theory -in practice, verifying this assumption requires to appeal to homogenization results, e.g. [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]. As is well-known, controlling the growth of the correctors is the royal road to quantitative homogenization (either stochastic or not, cf. e.g. [START_REF] Fischer | Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space[END_REF][START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF][START_REF] Josien | Quantitative homogenization for the case of an interface between two heterogeneous media[END_REF] and [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Bella | Quantitative stochastic homogenization: local control of homogenization error through corrector[END_REF][START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF], respectively). In particular, notice that stationarity of the ensemble • is not a necessary assumption. Hence, media perturbed by a local defect enter this framework (see [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF]).

Result

The purpose of this article is to provide a simple proof of the annealed CZ estimates. We define the annealed CZ norms by

h p,r := ˆ |h| r p r 1 p (6)
and establish Theorem 1.1. Let the random fields ∇u and f be square-integrable and related by

-∇ • a∇u = ∇ • f. (7)
Then, for any ensemble • of coefficient fields a satisfying Assumptions A.1, A.2, and A.3, there holds ∇u p,r γ,p,r,r f p,r for all exponents 1 ≤ r < r ≤ ∞ and 1 < p < ∞.

We emphasize the special order of the norms in (6): we strive for L p R d estimates for vector-valued functions in the space L r

• (in the parlance of [START_REF] Mcconnell | On Fourier multiplier transformations of Banach-valued functions[END_REF]) and not to control the moments of a quenched CZ estimate, which would write in the functional spaces L r

• (L p R d ). The advantage of Theorem 1.1 is that there is no restriction on the exponents (p, r) ∈ (1, ∞) 2 . In this sense, this results is the natural extension of the classical CZ estimates for homogeneous coefficient, which hold for any p ∈ (1, ∞), in opposition to the perturbative Meyers estimate4 , which requires p being close to 2. This is highly desirable when studying the fluctuations5 of the solution to [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF], cf. [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF].

As already noticed in [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] and explained in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Sec. 7.1], the loss in stochastic integrability is unavoidable when the exponents (p, r) are far from [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]. This is due to the randomness of the coefficient field a, which appears here through the fact that the extended correctors (φ i , σ i ) are not • -almost surely bounded but only satisfy (4). Indeed, since the medium is random, if p is far from 2, then there generically exists a r. h. s. f of [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] such that the constant C in the CZ estimate ´|∇u| p ≤ C ´|f | p is arbitrarily large. In particular, this justifies that (8) cannot hold for p = r = r .

Remark 2. We study the CZ estimates on the level of the a-Helmholtz decomposition, and not on the (somewhat more common) level of the operator ∇ 2 (-∇ • a∇•) -1 . These first ones are indeed useful because, on the one hand, many physical problems take the form [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] and, on the other hand, the quantity ∇u is oscillating on scale 1, so that there is no hope to get a strong estimate on ∇ 2 u as if the heterogeneous operator (-∇ • a∇•) -1 were replaced by the homogeneous operator (-∇ • ā∇•) -1 .

Strategy

As in classical in homogenization (see e.g. [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]), the large-scale regularity theory for the heterogeneous operator ∇ • a∇ is inherited from the constant-coefficient regularity theory for its homogenized counterpart ∇ • ā∇. The two-scale expansion bridges the gap between the two operators (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] and [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF] below for the two-scale expansion). In our proof, we employ extensively functional analysis -in particular real and complex interpolation. Remarkably enough, in opposition to many articles in stochastic homogenization, e.g. [START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF][START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF], this does not require large-scale Lipschitz regularity theory, cf. [16, Th. 1]. Indeed, here, we make use of a single large-scale regularity result for the heterogeneous operator: the energy estimate.

Our goal is to establish [START_REF] Caffarelli | On W 1,p estimates for elliptic equations in divergence form[END_REF]. If a were constant, then we would directly obtain Theorem 1.1 in form of Lemma 1.2 (See Lemma 7.4 of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]). Assume that the constant coefficient ā is elliptic and bounded in the sense of (3). Then, for any square-integrable random functions v and g and square-integrable random field f related through

1 T -∇ • ā∇ v = 1 T g + f, (9)
there holds v √ T , ∇v p,r d,λ,p,r g √ T , f p,r for all p, r ∈ (1, ∞). (10)
(Lemma 1.2 corresponds to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Lem. 7.4], in which we provide a proof based on a straightforward application of the Mikhlin theorem [21, Th. 1.1] for vector-valued functions in UMD spaces.) Notice that the constant in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] does not depend on the massive parameter T , the appearance of which is explained below, so that the statement is preserved for T = ∞ and g = 0. The above result conveniently applies to the homogenized problem, cf. (13) below. The philosophy of our proof is to transfer this result to the level of the a-Helmholtz projection.

A first natural but intractable idea is to directly compare [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] with its homogenized counterpart: -∇ • ā∇v = ∇ • f , for which estimate [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF] holds. However, the function f enjoys no additional regularity which would help to control the error between u and its two-scale expansion.

A more elaborate idea found out in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] is to split ∇u into two contributions: a high-pass part and a low-pass part.

The high-pass part u > is defined by means of the massive a-Helmholtz projection

∇( 1 T -∇ • a∇) -1 ∇• as 1 T -∇ • a∇ u > = ∇ • f (11)
for arbitrary T ∈ [1, ∞), and the remaining low-pass part u < := uu > is thus characterized by

-∇ • a∇u < = 1 T u > . (12)
The term massive comes from quantum field theory and is motivated by the interpretation of 1 T -∇ • a∇ as the generator of a diffusion coupled to desorption at exponential rate 1 T . This massive term is well-known to provide an infra-red cut-off: more precisely, up to exponentially small tails, the operator

∇ 1 T -∇ • a∇ -1 ∇• has locality
properties on scale √ T (as can be guessed from the expression of the fundamental solution of 1 T -∆). As suggested by the terminology, the high-pass and the low-pass parts enjoy locality and regularity properties, respectively. Indeed, on the one hand, the high-pass part ∇u > benefits from the locality properties on scale √ T of the massive a-Helmholtz projection. On the other hand, since the r. h. s. of ( 12) is regular, we may advantageously approximate u < by its homogenized counterpart ū which solves

-∇ • ā∇ū = 1 T u > . (13)
Indeed, approximating u < by its two-scale expansion (1 + φ i ∂ i )ū, we classically have that the associated error

w := u < -(1 + φ i ∂ i )ū (14)
satisfies the following intertwining relation (cf. ( 1), [START_REF] Fischer | Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space[END_REF], and ( 13))

-∇ • a∇w = ∇ • ((φ i a -σ i )∇∂ i ū). (15)
As a consequence, formally, we may control ∇w by means of ∇ 2 ū, which is itself well-behaved. This leads us to the following √ T -dependent splitting of ∇u into high-pass and low-pass parts:

∇u = ∇u > + ∂ i ū(e i + ∇φ i ) + ∇u < -∂ i ū(e i + ∇φ i ) . (16)
For any exponents p ∈ (1, ∞) and 1 ≤ r < r , our goal is to control quantitatively the smallest constant C p,r,r ≥ 1 in the annealed CZ estimate

∇u p,r ≤ C p,r,r f p,r , (17)
where the random square integrable fields ∇u and f are related by the a-Helmholtz projection [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF]. (As discussed in Remark 3 below, we shall make the apriori qualitative assumption that C p,r,r < ∞.) Through the splitting ( 16), we relate [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF] to the corresponding estimate on the level of the massive a-Helmholtz projection. The latter is expressed by defining the smallest constant C p,r,r (T ) ≥ 1 such that

∇u > p,r ≤ C p,r,r (T ) f p,r , (18)
for any square-integrable random fields u > and f satisfying [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]. Therefore, we aim for establishing simultaneously that C p,r,r 1 and C p,r,r (T ) 1 for all (p, r, r ) ∈ (1, ∞) 3 such that r < r .

Our starting point is the classical weighted energy estimate (see [18, Proof of Lem. 7.5, Step 2] for a proof):

Lemma 1.3 (See [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], proof of Lemma 7.5, Step 2). Let T ≥ 1. There holds

C 2,2,2 (T ) d,λ 1. (19)
Moreover, if the square-integrable random fields u > and f are related through [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], there holds

ω T u > √ T , ∇u > 2,2 d,λ ω T f 2,2 , (20)
where the weight is of exponential form

ω T (x) := exp - |x| C √ T for C d,λ 1. (21)
Lemma 1.3 relies only on the ellipticity property (3) of the coefficient field a, and is therefore unrelated to homogenization. (In this regard, we underline that the energy estimate is the only large-scale regularity ingredient we use for the a-Helmholtz projection.) It plays a double role. First, it establishes the result (8) for the case p = r = r = 2 (letting T ↑ ∞), which is used later as an anchor in our iterative procedure. Second, it encloses the locality property on scale √ T of solutions to the massive equation by means of the pivotal estimate [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF]. Estimate ( 20) is a key ingredient to get suboptimal estimates on the massive operator, which are independent of homogenization but, on the other hand, involve the cut-off scale √ T :

Lemma 1.4. Let T ≥ 1. There holds

C p,r,r (T ) γ,p,r √ T d provided p ∈ (1, ∞) and 1 ≤ r < ∞. (22)
Lemma 1.4 and its proof are close to [18, Lem. 7.5] and are displayed in Section 2.4 for self-consistency 6 . It relies on [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF], where we replace the • 2,2 norms in by generic • p,r norms by using the locality property of the exponential weights. In this procedure, we lose the volume factor √ T d associated with the cut-off scale. However, this main argument only applies to the large scales; we tackle the small scales by an independent ingredient: we may gain in local integrability thanks to the (deterministic) elliptic small-scale CZ estimates in which no loss arises, namely Lemma 2.5.

Up to now, all the arguments only rest on ellipticity, boundedness, and local regularity of a, namely Assumptions A.1 and A.3. Then comes the core of the proof, which is devoted to a buckling argument and where the homogenization theory -and thus Assumption A.2 -plays a crucial role through our splitting [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]. First, under a mild quantification of the control on the constants C p,r,r (T ) for the massive a-Helmholtz projection in a given range of exponents, an inner buckling argument provides an optimal estimate on the level of the massless a-Helmholtz projection in a slightly smaller range of exponents, cf. Lemma 2.1. Second, this estimate is upgraded to an optimal estimate for the massive a-Helmholtz projection, cf. Lemma 2.2.

Last, we conclude by an iterative argument which we briefly sketch: By complex interpolation between the optimal energy estimate and the suboptimal estimates of Lemma 1.4, we may ascertain a sufficiently well controlled growth of the constants C p,r,r (T ) for the massive operator in a range of exponents (p, r, r ) not far from (2, 2, 2) (the latter exponents correspond to the energy estimate). Next, we apply the two-step buckling argument described above to get optimal estimates on C p,r,r and C p,r,r (T ) in this zone. Iterating the argument, we gradually enlarge the zone in which optimal estimates are proved and finally reach any 3-tuple of exponents (p, r, r ) ∈ (1, ∞) 3 with r < r in a finite number of steps.

The main subtlety of the proof is that the annealed CZ estimates for both the massive and the massless the a-Helmholtz projection generically suffer from a loss in the stochastic exponent. Such a phenomenon is due to the fact that the correctors (φ i , σ i ) are not almost-surely bounded but only controlled by (4) (see the discussion in Section 1.2).

In practice, it prevents us from utilizing a direct buckling argument. On the contrary, we have to employ a real interpolation argument. Indeed, the loss in stochastic integrability of the low-pass part is counterbalanced by the gain in stochastic integrability of the high-pass part. There, the massive parameter √ T plays the role of a potentiometer: When it grows, it damps the low-pass part and favors the high-pass part. (We refer to Step 3 of the proof of Lemma 2.1 for more details.) Notice that this additional difficulty is inherent to stochastic homogenization and would not appear if we restricted our proof to deterministic cases (e.g. periodic, quasi-periodic, periodic with defects...).

Remark 3 (Finiteness of C p,r,q ). The cautious reader may fear that there is a circular argument since we assume that the massless constants C p,r,r is finite in order to derive a bound on it. However, we underline that the final bound we obtain on C p,r,r only requires its apriori finiteness (but does not depend on the precise value of the apriori bound). Even though we will not justify here that C p,r,r < ∞, there are two ways to circumvent this difficulty:

• Assume that the random medium is periodic of period L, so that periodic homogenization provides C p,r,r < ∞.

Then, our argument shows that the final estimate on C p,r,r is independent of L; hence, we may retrieve the desired result without periodization in the limit L ↑ ∞. In this task, the bounds (4) on the periodic correctors shall obviously be independent of L (this is shown in a particular context in [START_REF] Clozeau | Bias in the representative volume element method: periodize the ensemble instead of its realizations[END_REF]).

• First prove Theorem 1.1 on the level of the massive operators (since, by Lemma 1.4, the constants C p,r,r (T ) are always finite) and then take the limit T ↑ ∞. This is the philosophy of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF].

Organization of the paper

Section 2 contains the proof of Theorem 1.1. First, in Section 2.1, we show how all the ingredients fit together in order to get Theorem 1.1. Then we show the intermediate results: In Section 2.2, we obtain an optimal estimate for the massless a-Helmholtz projection from slightly suboptimal estimates on the massive a-Helmholtz projection, that is Lemma 2.1. In Section 2.3, this estimate is upgraded in Lemma 2.2 to an optimal estimate for the massive a-Helmholtz projection. Last, in Section 2.4, we establish the suboptimal massive estimates, namely Lemma 1.4. We conclude this section by emphasizing in Section 2.5 on the main similarities and differences between this proof and [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. Finally, in Section 3, we briefly discuss our Assumptions A. Lemma 2.1. Let r < r < r < r < r < r be such that for some θ ∈ (0, 1] we have

1 r < 1 s := θ 1 r + (1 -θ) 1 r . ( 23
)
Suppose that for some constant Λ ∈ [1, ∞) there holds

C p,r ,r (T ) + C p, r, r (T ) ≤ Λ √ T 1-θ for all T ∈ [1, ∞). ( 24
)
Then we have

C p,r,r γ,p,r, r, r,r ,r ,r Λ 1 θ . ( 25
)
The proof of Lemma 2.1 crucially relies on the introduced splitting ( 16) as well as on the annealed CZ estimates for the homogeneous problem, namely Lemma 1.2.

The second result is somehow the "converse statement" corresponding to Lemma 2.1 and allows for jumping from massless estimates (and slightly suboptimal massive estimates) to optimal massive estimates. Indeed, it yields an optimal estimate on the massive constant C p,r,r (T ) under the mild assumption that C p,r ,r (T ) √ T . It has no equivalent in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. Lemma 2.2. For r < r < r < r < r we have

C p,r,r (T ) γ,p,r,r ,r ,r ,r C p,r ,r 1 + C p,r ,r (T ) √ T . ( 26
)
The proof of Lemma 2.2 relies on the same ingredients as for Lemma 2.1; however, it requires an additional splitting into high-pass and low-pass parts on the level of the homogenized solution ū (see ( 45) and (46) below).

Then, establishing Theorem 1.1 is equivalent to showing iteratively in n the following:

Statement P n . For all exponents (p, r, r ) belonging to the set E n (see Figure 1) defined by

1 - 1 2d n ≤ 2 p , 2 r ≤ 2 -1 - 1 2d
n and r < r , there holds C p,r,r γ,p,r,r 1 and C p,r,r (T ) γ,p,r,r 1.

(27)

We may take the limit T ↑ ∞ in estimate ( 19) of Lemma 1.3 and obtain C 2,2,2 d,λ 1. Hence, by Jensen's inequality (see (57) below) the Statement P 0 holds. Induction step. We now assume that the Statement P n is satisfied and establish P n+1 . The strategy is the following: First we employ complex interpolation between P n and Lemma 1.4 in order to get a larger range of exponents (q, s, s ) in which we control the massive constants C q,s,s (T ) in a mildly suboptimal way. Then, we turn this suboptimal estimate into an estimate on the massless constants C q,s,s by means of Lemma 2.1. Thanks to Lemma 2.2, the latter is upgraded to an optimal estimate on the massive constants C q,s,s (T ) in the same range of exponents, so that we finally get P n+1 .

1 1 2 0 1 1 2 0 1 p 1 r • E 0 E n E n+1 E n+2
Assume that the exponents (p, r, r ) ∈ E n , and q, s, s , r, p ∈ (1, ∞) satisfy

1 s = θ r + 1 -θ r < 1 s = θ r + 1 -θ r and 1 q = θ p + 1 -θ p for θ ∈ (0, 1). ( 28
)
By complex interpolation between the second inequality of ( 27) and ( 22) for (p, r) ( p, r), we obtain that

C q,s,s (T ) √ T d(1-θ)
.

Imposing that d(1θ) ≤ 3 4 and recalling the constraints (28) on p, r , r, p and r, we get that

C q,s,s (T ) √ T 3 4 (29)
holds for all (q, s, s ) such that

           s < s , 2 q , 2 s > 1 - 3 4d 1 - 1 2d n , 2 q , 2 s < 1 - 3 4d 2 -1 - 1 2d n + 2 3 4d = 2 -1 - 3 4d 1 - 1 2d n . (30)
By (29), we may apply Lemma 2.1 and obtain that for all exponents (q, s, s ) satisfying (30) there holds C q,s,s 1. Thanks to Lemma 2.2, this in turn yields C q,s,s (T )

(29) 1 + √ T 3 4 √ T 1
in the same range of exponents (30). We may restrict this range to E n+1 , so that we have established P n+1 .

By induction, this shows that P n holds for all n and completes the proof of Theorem 1.1.

2.2

From suboptimal massive estimates to massless estimates, i.e. proof of Lemma 2.1

Before proceeding with the proof of Lemma 2.1, we state two elementary results (the proofs of which are given at the end of the section): The first one is a uniform estimate on the gradients of the extended correctors Lemma 2.3. There holds:

sup x∈R d |(∇φ i (x), ∇σ i (x))| r 1 r γ,r 1 for all r ∈ [1, ∞) and i ∈ {1, • • • , d}. (31)
The second one extends the use of the constant C p,r,r (T ) for the massive a-Helmholtz projection (defined by ( 18)) to nondivergence-form r. h. s. and allows for controlling u> √ T as well, thus recovering a statement analogous to [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] in the constant-coefficient case: Lemma 2.4. Assume that the ensemble • satisfies Assumption A.1. Fix T > 0 and (p, r, r ) ∈ (1, ∞) 3 such that r < r and assume that C p,r,r (T ) < ∞. If the square integrable functions u > and g and field f are related through

1 T -∇ • a∇ u > = 1 T g + ∇ • f, (32)
then there holds

u > √ T , ∇u > p,r d C p,r,r (T ) g √ T , f p,r . (33)
Also, we have C p ,(r ) ,r (T ) d C p,r,r (T ) for the dual exponents

1 p = 1 -1 p , 1 r = 1 -1 r , and 1 (r ) = 1 -1 r .
Proof of Lemma 2.1. Recall the splitting [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]. The first (high-pass) part is estimated in Step 1, the second (low-pass) part in Step 2.

Step 1: Estimate on the high-pass part. We claim that the first part in ( 16) is estimated as follows

∇u > + ∂ i ū(e i + ∇φ i ) p,r C p, r, r (T ) f p, r . (34)
Indeed, by the triangle inequality, Jensen's inequality in probability (namely (57)) and (31) combined with the Hölder inequality, there holds

∇u > + ∂ i ū(e i + ∇φ i ) p,r ∇u > p,r + ∂ i ū(e i + ∇φ i ) p,r (57)
By [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], we may rewrite the constant-coefficient equation ( 13) satisfied by ū as -∇ • ā∇ū = ∇ • (a∇u > + f ), on which we may use Lemma 1.2, then recall [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] and make use of (57), to the effect of

∇ū p, r (10)
∇u > p, r + f p, r

C p, r, r (T )

f p, r + f p, r (57)
C p, r, r (T ) f p, r .

Inserting (36) into (35) yields (34).

Step 2: Estimate on the low-pass part. We claim that the second part in ( 16) is estimated as follows

∇u < -∂ i ū(e i + ∇φ i ) p,r C p,r,r 1 √ T C p,r ,r (T ) f p,r . (37)
We indeed reformulate the l. h. s. of (37) as

∇u < -∂ i ū(e i + ∇φ i ) = ∇w + φ i ∇∂ i ū, ( 38
)
where we introduce the error in the two-scale expansion [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], which satisfies [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF]. Appealing to the definition ( 17) of the massless CZ constant, and using the Hölder inequality combined with the corrector estimates (4), we obtain

∇w p,r (17)
C p,r,r |(φ i , σ i )|∇ 2 ū p,r (4)
C p,r,r ∇ 2 ū p,r .

In view of (38) and once more (4), this may be upgraded by the triangle inequality to

∇u < -∂ i ū(e i + ∇φ i ) p,r C p,r,r ∇ 2 ū p,r . (40)
We now appeal to the estimate [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for the constant-coefficient equation ( 13) rewritten as ∇ • ā∂ i ∇ū = 1 T ∇ • (u > e i ), and to the property (33) of C p,r ,r (T ) to obtain

∇ 2 ū p,r (10) 1
T u < p,r (33) 1 √ T C p,r ,r (T ) f p,r . (41)
Inserting this into (40) entails (37).

Step 3: Conclusion via Interpolation. By the triangle inequality applied to ( 16), ( 34) and (37) combine to

∇u p,r C p, r, r (T ) f p, r + C p,r,r 1 √ T C p,r ,r (T ) f p,r ,
into which we insert our assumption (24), to the effect of

∇u p,r Λ √ T 1-θ f p, r + C p,r,r 1 √ T θ f p,r for all T ∈ [1, ∞). ( 42
)
Notice that, if all the stochastic exponents in (42) were equal to r, then it would directly produce

C p,r,r Λ √ T 1-θ + C p,r,r 1 √ T θ ,
in which the second r. h. s. term could be absorbed into the l. h. s. for √ T Λ 1 θ , so that (25) would be directly obtained. However, the different stochastic exponents in (42) prevent us from using a direct buckling argument. To overcome this, we use below the J-method of real interpolation (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Sec. 3.2

p. 42]).

In view of the relation [START_REF] Müller | Nonlinear partial differential equations I[END_REF], we have indeed the inclusion of the Lorentz spaces L r • ⊂ L s,1

• , and L s,1

• is obtained by real interpolation of parameters (θ, 1) between the spaces L r

• and L r • (see Theorem [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Th. 5.3.1. p. 113]). Hence, using the J-functional version of real interpolation there exists a family {f L } L∈(0,∞) of vector fields such that

f = ˆ∞ 0 f L dL and f p,r ˆ∞ 0 L 1-θ f L p, r + 1 L θ f L p,r dL. ( 43
)
By linearity of the a-Helmholtz projection, this induces a family {∇u L } L∈(0,∞) such that ∇u = ˆ∞ 0 ∇u L dL and thus ∇u p,r ≤ ˆ∞ 0 ∇u L p,r dL.

Applying (42) with (∇u, f ) replaced by (∇u L , f L ) we obtain

∇u L p,r ΛC 1-θ p,r,r ( √ T C p,r,r ) 1-θ f L p, r + ( C p,r,r √ T ) θ f L p,r .
Choosing √ T := 1 + C p,r,r L, integrating in L (recall that C p,r,r ≥ 1), using Jensen's inequality in probability (57) in conjunction with r < r , and finally appealing to (43), we get

∇u p,r ΛC 1-θ p,r,r ˆ∞ 0 L 1-θ f L p, r + L -θ f L p,r dL + Λ ˆ1 0 f L p, r dL (57) ΛC 1-θ p,r,r ˆ∞ 0 L 1-θ f L p, r + L -θ f L p,r dL + Λ ˆ1 0 f L p,r dL ΛC 1-θ p,r,r ˆ∞ 0 L 1-θ f L p, r + L -θ f L p,r dL (43) ΛC 1-θ p,r,r f p,r .
Last, by definition [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF], this entails C p,r,r ΛC 1-θ p,r,r . Since θ > 0 and by the qualitative assumption C p,r,r < ∞ (see Remark 3), this yields estimate (25).

Proof of Lemma 2.4. We fix g and f and define u > by (32) and v by

1 T -∆ v = 1 T g + ∇ • f. (44)
We first prove (33) and then the duality result.

Step 1: Argument for (33). Subtracting (32) from (44), we get

1 T -∆ (v -u > ) = ∇ • (∇u > -a∇u > ) and 1 T -∇ • a∇ (v -u > ) = ∇ • (∇v -a∇v) .
Hence, applying [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] as well as [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] on the first and second above equation, respectively, we get

v -u > √ T p,r
∇u > p,r , and ∇v -∇u > p,r ≤ C p,r,r (T ) ∇v p,r .

Whence, applying twice the triangle inequality and the above estimates, there holds

u > √ T , ∇u > p,r v √ T , ∇v p,r + ∇u > p,r + C p,r,r (T ) ∇v p,r v √ T , ∇v p,r + C p,r,r (T ) ∇v p,r .
Anticipating on (57), we also have

v √ T , ∇v p,r v √
T , ∇v p,r . Hence, applying once more (10) for r r yields (33).

Step 2: Duality argument. Assume now that the square-integrable random fields u > and f as well as u > and f are related through [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], respectively, with f p ,r < ∞ and f p,r < ∞. The variational formulation of (11) yields

1 T ˆ u > u > + ˆ∇ u > • a∇u > = - ˆ∇ u > • f = -ˆ∇u > • f .
(Here, we make use of the symmetry of a.) Hence, applying the Hölder inequality and (18), we get

ˆ∇u > • f ∇ u > p,r f p ,r ≤ C p,r,r (T ) f p,r f p ,r .
As a consequence, we obtain by duality that ∇u > p ,(r ) C p,r,r (T ) f p ,r . This directly entails C p ,(r ) ,r (T ) C p,r,r (T ).

Proof of Lemma 2.3. Since the argument is the same for σ i and φ i , we only show it for φ i . Using Assumption A.3, by the classical Schauder theory applied to (2), we have

|∇φ i (x)| a C 0,α (B(x)) + φ i L 2 (B(x)) (3),(5)
≤ 2 + φ i L 2 (B(x)) .
Taking the L r • -norm and using (4) implies the desired (31) for φ i .

2.3

From massless estimates to optimal massive estimates, i.e. proof of Lemma 2.2

Proof of Lemma 2.2. Recall that C p,r,r (T ) is defined by [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. Hence, we fix f such that f p,r < ∞ and define u > by [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]. In order to establish (26), we go beyond the splitting u = u > + u < introduced in ( 12) and also split ū = ū> + ū< defined by ( 13) according to

1 T -∇ • ā∇ ū> = 1 T (ū -u < ), (45) 1
T -∇ • ā∇ ū< = 1 T u. (46)
This induces the following decomposition of ∇u > :

∇u > = ∇u -∂ i ū< (e i + ∇φ i ) -∇u < -∂ i ū< (e i + ∇φ i ) . (47)
We tackle separately the first and second contributions in Steps 1 and 2 by showing

∇u -∂ i ū< (e i + ∇φ i ) p,r C p,r ,r f p,r , (48)
∇u < -∂ i ū< (e i + ∇φ i ) p,r C p,r ,r C p,r ,r (T ) √ T f p,r , (49)
which, by the triangle inequality, yields (26) in form of

∇u > p,r C p,r ,r 1 + C p,r ,r (T ) √ T f p,r . (50)
Step 1: Argument for (48). As a consequence of the triangle inequality, Jensen's inequality in probability (57), and the corrector estimate (31) combined with the Hölder inequality, we get

∇u -∂ i ū< (e i + ∇φ i ) p,r (57),(31)
∇u p,r + ∇ū < p,r .

Then, thanks to the estimate [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for the constant coefficient equation (46

) in form of 1 T -∇ • ā∇ ∂ i ū< = 1 T ∂ i u, we have ∇ū < p,r
∇u p,r . Inserting this into (51) and appealing to (17) (with (r, r ) (r , r )) and (57) yields (48).

Step 2: Argument for (49). We rewrite the second r. h. s. of (47) in terms of w defined by ( 14) as

∇u < -∂ i ū< (e i + ∇φ i ) = ∇w + φ i ∇∂ i ū + ∂ i ū> (e i + ∇φ i ).
Using the triangle inequality and the corrector estimates ( 4) and (31) in conjunction with the Hölder inequality yields

∇u < -∂ i ū< (e i + ∇φ i ) p,r (4),(31)
∇w p,r + ∇ 2 ū p,r + ∇ū > p,r . (52)
Note that (45) can be reformulated as

1 T -∇ • ā∇ ū> = - 1 T (w + φ i ∂ i ū).
Using the constant-coefficient CZ estimate (10) on a differentiated form of this equation and using once more (4) yields

∇ū > p,r (10)
∇w p,r + 1 √ T φ i ∂ i ū p,r (4)
∇w p,r + 1 √ T ∇ū p,r .
Inserting this into (52) and appealing to (39) with (r, r , r ) (r , r , r ) entails

∇u < -∂ i ū< (e i + ∇φ i ) p,r (39)
C p,r ,r ∇ 2 ū p,r + 1 √ T ∇ū p,r .

Last, the constant-coefficient estimates (41) for (r , r ) (r , r ) and (36) for ( r, r) (r , r ) entail (49).

2.4 Suboptimal massive estimates, i.e. proof of Lemma 1.4

As previously announced, we control u> √ T , ∇u > on the scales smaller than 1 thanks to the local regularity of a. In particular, the local CZ estimates can be used in order to gain in local integrability. This is conveniently expressed by means of a nested version of the norms (6), namely

h p,r,q := ˆ - ˆB1(x) |h| q r q p r dx 1 p , (53)
where7 p, r, q ∈ [1, ∞]. These norms advantageously separate the small-scale behavior L q B1(x) from the large-scale behavior L p R d . We make use of the following small-scale regularity result: Lemma 2.5. Let • be an ensemble of coefficient fields a satisfying estimates (3) and (5) for fixed λ > 0 and α ∈ (0, 1). If the square integrable random vector field u > and f are related through [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] for T ≥ 1, then there holds

u > √ T , ∇u > p,r d,λ,α,p,r u > √ T , ∇u > p,r,q + f p,r for all p, r, q ∈ (1, ∞), (54)
u √ T , ∇u p,r,q d,λ,α,p,r,q u √ T , ∇u p,r,2 + f p,r,q for all p, r, q ∈ (1, ∞).

The proof of Lemma 2.5 appeals to the classical CZ and Schauder theories in a slightly intricate way. Since it is essentially a simplification of the proof of [18, Lem. A.2], we postpone it until Appendix A.

Proof of Lemma 1.4. Throughout the proof, the square-integrable random fields u > and f are related by [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF].

The proof relies on two ingredients: the locality on scale √ T of the massive a-Helmholtz projection and the regularity on scale 1 of the coefficient field due to [START_REF] Bergh | Interpolation spaces. An introduction[END_REF], which are captured through a weighted energy estimate in Lemma 1.3 and local annealed CZ estimates in Lemma 2.5, respectively. These two properties allow for estimates between different spatial L p R d -norms, where locality provides the large-scale cut-off, and regularity the small-scale cut-off. In particular, we will jump between the L 2 R d -norm, on which scale we have the weighted energy estimate [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF], and the L ∞ R d -norm, where we may handle the L r

• -norm. The proof is divided into 3 steps. In Step 1, we derive some useful properties of the norms (53), and in particular that they decrease with increasing spatial exponent p, see (56). In Step 2, we make use of the locality property on scale √ T in order to derive a weak form of [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF] on the scale of the norms • p,r,2 defined by (53), see (63) below. In Step 3, finally, we use local regularity (54) and (55) to return to the original norms [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF].

Step 1: Nestedness properties of the norms (53). We claim the following properties of the norms (53):

h p ,r,q d h p,r,q provided p ≤ p , (56)
h p,r ≤ h p,r and h p,r,q ≤ h p,r ,q provided r ≤ r ,

h p,r,q ≤ h p,r,q provided q ≤ q , (58)

h p,r,q ≤ h p,r provided q ≤ min{p, r}, (59)
h 2,2,2 = h 2,2 . (60)
We start with the argument for (56). This is due to the norm equivalence

h p k L r • (L q (B 1+ √ d (k))) d h p,r,q d h p k L r • (L q (B 1+ √ d (k))) for h p k L r • (L q (B 1+ √ d (k))) := k∈Z d - ˆB1+ √ d (k) |h| q r q p r 1 p , (61)
in conjunction with the embedding of the discrete spaces p (Z d ) ⊂ p (Z d ). In turn, the leftmost estimate in (61) follows from the elementary geometric fact that B 1+ √ d (k) can be covered by a d-dependent finite number of balls of radius 1 2 , whereas the rightmost estimate in (61) is obvious by inclusion 57) and (58) are obtained by Jensen's inequalities in probability and in space, respectively. We turn to (59), which is a consequence of Jensen's inequality: Rewriting definition (53) in form of

B 1 (x) ⊂ B 1+ √ d (k) if x ∈ k + [0, 1) d . Inequalities (
h p,r,q = ˆ - ˆB1 |h(x + z)| q dz r q p r dx 1 p = - ˆB1 |h(• + z)| q dz 1 q L p q R d L r q •
, we learn from the convexity and translation invariance of the involved norms (here we use q ≤ min{p, r}) that

h p,r,q ≤ - ˆB1 |h(• + z)| q L p q R d L r q • dz 1 q (6)
= h p,r .

Finally, (60) directly follows from the identity ˆh(x

)dx = ˆ- ˆB1(x) h(y)dydx, (62)
(Notice that the value 2 of the exponent plays no special role and can be replaced by any other value p ∈ [1, ∞].)

Step 2: Suboptimal estimates on the scale of norms (53). Now, we are given p, r ∈ (1, ∞) and establish

u > √ T , ∇u > p,r,2 γ,p,r √ T d f p,r,2 . (63)
By duality (in the same spirit as Lemma 2.4), it is sufficient to show (63) for r ≥ 2. We choose the weight ω T as in Lemma 1.3. The two ingredients for (63) are the following norm relations, to be established below:

sup x - ˆB1(x) |h| 2 1 2 =: h ∞,2,2 h 2,2,2 , (64)
ω T h 2,r,2 √ T max{0,d( 1 2 -1 p )} ω 4T h p,r,2 , (65) ω
T 4 h p,r,2 √ T d p ω T h ∞,r,2 . (66)
The merit of passing to the spatial exponent p = ∞ is that by duality, if r > 2, we have

h ∞,r,2 = sup |F | 2r r-2 ≤1 F h ∞,2,2 . (67)
Estimate ( 64) is immediate from (56). In the case of p ≥ 2, (65) follows from appealing to the relation ω T = ω 2 4T , cf. [START_REF] Mcconnell | On Fourier multiplier transformations of Banach-valued functions[END_REF], which allows to use the Hölder inequality, so that it reduces to the obvious ω

4T 2p p-2 ,∞,∞ √ T d( 1 2 -1 p ) (recall T ≥ 1).
In the case of p ≤ 2, we appeal to (56) and the obvious ω T ≤ ω 4T . For (66), we start from ω T /4 = ω 2 T , use the Hölder inequality, and ω T p,∞,∞ √ T d p .

Here comes the argument for (63). Since the case r = 2 is simpler (in particular, we do not need to introduce the additional random variable F below), we henceforth assume that r > 2. By (66) and (67), we have

ω T u > √ T , ∇u > p,r,2 (66) √ T d p ω T u > √ T , ∇u > ∞,r,2 (67)
= √ T d p sup |F | 2r r-2 ≤1 ω T F u > √ T , ∇u > ∞,2,2 . (68)
Then, (64), (20) (and (60)), and the Hölder inequality yield

ω T F u > √ T , ∇u > ∞,2,2 (64)
ω T F u > √ T , ∇u > 2,2,2 (20)
,(60)

ω T F f 2,2,2 ≤ ω T f 2,r,2 |F | 2r r-2 r-2 2r . (69)
Last, appealing to (65), we obtain

ω T f 2,r,2 (65) √ T max{0,d( 1 2 -1 p )} ω 4T f p,r,2 . ( 70
) u > √ T , ∇u > p,r,2 √ T d ω 4T f p,r,2 .
Since by the definition (53) and the properties of ω T we have

ω T (• -z)h p,r,2 ∼ ˆ ω T (x -z) - ˆB1(x) |h| 2 r 2 1 r p dx 1 p ,
the desired (63) follows from taking the L p -norm in the shift z.

Step 3: Conclusion, namely proof of [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF]. Let p, r ∈ (1, ∞). We first upgrade (63) to

u > √ T , ∇u > p,r,q γ,p,r,q √ T d f p,r,q for all q ∈ (1, ∞). (71)
Then, selecting q := min{p, r} and applying (54), (71), and (59) (since T ≥ 1) entails the desired estimate [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF] via

u > √ T , ∇u > p,r (54)
u > √ T , ∇u > p,r,q + f p,r (71) √ T d f p,r,q + f p,r (59) √ T d f p,r .
Here comes the argument for (71). By duality (in the same spirit as Lemma 2.4), it suffices to establish it for q ≥ 2. By (55), (63), and (58), and since T ≥ 1, we obtain (71) in form of

u > √ T , ∇u > p,r,q (55)
u > √ T , ∇u > p,r,2 + f p,r,q (63) √ T d f p,r,2 + f p,r,q (58) √ T d f p,r,q .
2.5 Differences with respect to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Prop. 7.1] We compare here our strategy with [18, Prop. 7.1], which contains a more general (but more complicated) version of the present article. We first underline that [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] and the present article share the same fundamental ideas; in particular, Section 2.4, and 2.1 are simplified versions of arguments from [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], whereas Sections 2.2 and 2.3 substantially differ from this reference. The very interest of our new simplified proof is that it allows for emphasizing on the fundamental ingredients of the proof while suppressing many technicalities.

The main difference with respect to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] is that we employ homogenization results on the level of the massless operator. Conveniently, this only requires Assumption A.2 on the correctors, which has been verified in the literature for various frameworks (see, e.g. [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]), but it also comes with the disadvantage of being unable to treat the case of sublinear but unbounded correctors (as in the relaxed assumption (72) of Section 3.1). On the contrary, in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], reasoning only on the level of the massive operators comes with the benefit of having always a locality property on the scale √ T , so that (a mild modification of) the case (72) of Section 3.1 may be treated as well. However, this requires to introduce more elaborated objects, namely the massive correctors the growth of which has to be controlled by an independent argument, and to appeal to two different scales related to massive parameters √ τ √ T . In this respect, replacing

( √ τ , √ T ) ( √ T , ∞)
provides a formal correspondence between the two proofs.

As a side-effect, in the proof of Lemma 2.1, we carry out the real interpolation argument on the level of the stochastic exponent related to the r. h. s. of [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]. Thus, we need to appeal to the J-method of interpolation, whereas the K-method was used in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], for the reasoning was conducted on the level of the stochastic exponent related to the l. h. s. of [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF].

Last, we emphasize our use of a simple local regularity assumption, namely [START_REF] Bergh | Interpolation spaces. An introduction[END_REF], instead of the more complicated assumption (73) used in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. Besides numerous simplifications in the proofs, the only visible effect is that the use of local regularity results causes no loss in stochastic exponent (see e.g. (54) and ( 55)), so that the loss in ( 8) is only due to stochastic homogenization, cf. discussion in Section 1.2.

Proof. Since a self-contained proof may be found in a slightly different context in [15, Prop. 1], we only give here the main ingredients and some additional references.

First, by (76) we easily verify that Assumption A.1 is satisfied. Second, by definition (75), we have by the properties of convolution

[a] C 0,α (B1(x)) ≤ η C 0,α (B1) ˆB2(x) |A(g(y))|dy 1.

This shows (5) for a a(µ•) where µ is given by (77), so that Assumption A.3 is satisfied for a a(µ•).

It remains to establish that a satisfies Assumption A.2. (W. l. o. g. we may assume µ = 1.) As remarked in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF]Prop. 1] in the discrete setting and in [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF]Cor. 1] for the continuous setting, it suffices to establish a spectral gap inequality to obtain bounded correctors in dimension d > 2. Here, we make use of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Prop. 4.1]8 , which requires to establish the following spectral gap inequality: Inserting the above estimate into (79) yields (78).

(F -F ) 2 ˆ - ˆB1 (

A Local CZ estimates

Some parts of the proof below are essentially quoted from [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF].

Proof of Lemma 2.5. The proof relies on deterministic local CZ estimates enclosed in Lemma A.1 below. We split it into two independent parts.

Part 1: Proof of (54). Let ρ ≥ 0. We introduce a x,ρ (y) := a(y) if y ∈ B ρ (x), a(x) otherwise.

Accordingly, we freeze the variable x and define v x as the solution to

1 T -∇ • a x ∇ v x = ∇ • 1 Bρ(x) f , (80)
where the operator ∇ acts on the variable y. This combines with ( 11) to

1 T -∇ • a∇ (v x -u > ) = 0 in B ρ (x). (81)
Then, the proof falls in three steps: In Step 1, we establish local and annealed estimates on v x by means of the Meyers strategy, then in Step 2 we establish local estimates on the locally a-harmonic function v xu > , and last, we combine these estimates in order to establish (54).

Part 1, Step 1: Estimate on v x . By (5) we have a xa(x) L ∞ ≤ ρ α ; therefore, by a perturbative argument à la Meyers (see e.g. [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF] and a more recent presentation in [23, Chap. 2, Th. 2.6.2, p. 122]), we may select ρ d,λ,α,p,q,r 1 such that Since the proof is the same for (82) and (83), we only establish (83). To this purpose, we rewrite (80) as

ˆ v x √ T , ∇v x q ˆB1(x) |f | q , (82)
1 T -∇ • a(x)∇ v x = ∇ • ( a x -a(x))∇v x + ∇ • 1 B1(x) f .
Thus, by Lemma 1.2, it suffices to establish that the operator ∇ 1 T -∇ • a(x)∇ -1 ∇ • ( a xa(x)) is a contraction on

L p R d L r
• to obtain (83). Denoting by Cp,r < ∞ the operator norm

Cp,r := ∇ 1 T -∇ • a(x)∇ -1 ∇ • L L p R d L r • ,
there obviously holds

∇ 1 T -∇ • a(x)∇ -1 ∇ • ( a x -a(x)) L L p R d L r • ≤ Cp,r ρ α .
Thus, for ρ 1, the above r. h. s. is strictly smaller than 1, so that we have established the desired contraction property.

Part 1, Step 2: Estimate on v xu > . By (5), we may apply Lemma A.1 to (81), and more precisely a rescaled version of (88) on v xu > in the ball B ρ (x), and make use of the triangle inequality, to the effect of

v x -u > √ T , ∇v x -∇u > L ∞ ( 1 2 Bρ(x)) (88)

1

 1 Our framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Details of proof 2.1 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 From suboptimal massive estimates to massless estimates, i.e. proof of Lemma 2.1 . . . . . . . . . . . 2.3 From massless estimates to optimal massive estimates, i.e. proof of Lemma 2.2 . . . . . . . . . . . . . 2.4 Suboptimal massive estimates, i.e. proof of Lemma 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Differences with respect to [18, Prop. 7.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Short discussion on the assumptions and an example of application 3.1 Relaxing the assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A simple example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Representation of indexes p and r for (p, r, r ) ∈ E n .

  ,(31) ∇u > p, r + ∇ū p, r .

2 dxF

 2 for any random variable F depending on a. (78)Here appears the Malliavin derivative defined bylim h→0 (a + h a) -F (a) h = ˆ∂F (a) ∂a(x) : a(x)dx.Estimate (78) follows from (F -F ) 2 a consequence of (74) (we refer to the argument in[START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] Lem. 3.1]). Indeed, by the chain rule we have y)A (g(y))dz, so that, since η is of compact support in B 1 , there holds∂F ∂g(x) A L ∞ (R) η L ∞ (R d )ˆB1(y) ∂F ∂a(z) dz.

  v xu > √ T , ∇v x -∇u > L min{p,r,q} (Bρ(x)) ≤ v x √ T , ∇v x L min{p,r,q} (B1(x)) + u > √ T , ∇u > L min{p,r,q} (B1(x)) .

  1, A.2, and A.3, and we provide a concrete example of application of our theorem, namely a coefficient field built by convolving a bounded function applied to a Gaussian field. The Appendix is devoted to the proof of a technical result based on classical local CZ estimates.

2 Details of proof 2.1 Proof of Theorem 1.1 Theorem 1.1 requires two upcoming intermediate results. The most important intermediate result allows for jumping from slightly suboptimal massive estimates to massless estimates. It replaces and simplifies [18, Proof of Prop. 7.3(ii), Parts 1 & 2].

DES/IRESNE/DEC/SESC/LM2C, Commissariat à l'Énergie Atomique et aux Énergies Alternatives, Centre de Cadarache, 13108 Saint Paul-lès-Durance, France

By default, the equations hold in R d and we write ´for ´Rd .

For conciseness, in the context of estimates, we may write (f, g) for the vector field obtained by concatenating the coordinates of the vectors fields f and g.

The cautious reader might be surprised that our estimates may depend on the function κ, although we did not introduce any topology for this function. However, in our applications, we only use the notation " γ,β 1 ,••• ,βn " in cases where β 1 , • • • , βn ∈ R and the induced constant C in the estimate depends only on κ through the evaluation of κ(C ), for a sufficiently large constant C depending on (d, λ, α, β 1 , • • • , βn) (recall that κ is a nondecreasing function).

In this regard, the philosophy and the proof of this result are very different from the annealed CZ estimates that can be obtained by the perturbative Meyers strategy and which only holds for (p, r = r ) sufficiently close to (2, 2) (see[START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] Prop. 7.1(i)]).

There, the particular case p = 4 plays a crucial role.

The main difference w. r. t.[START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] Lem.

7.5] is that we make use of the easier estimate (5) instead of (73), and we slightly change the order of the arguments in a more efficient way, using only local CZ estimates in Step 3.

The cases p, r or q = ∞ requires to interpret (53) by using the normsL ∞ R d , L ∞ • , or L ∞ B 1 (x), respectively.

An easy inspection of the proof shows that this result holds under Assumptions A.1 and A.3 provided the spectral gap inequality (78) is satisfied.

Acknowledgements

We acknowledge Pr. Felix Otto for his decisive contribution to this work. This work was mostly done while the author was a postdoctoral researcher at the Max-Planck-Institut für Mathematik in den Naturwissenschaften, which we further acknowledge for funding visits of the author in 2021 and 2022.

3 Short discussion on the assumptions and an example of application

Relaxing the assumptions

We make Assumptions A.1, A.2 and A.3 in order to simplify the proof of Theorem 1.1 below. However, these may be relaxed as explained below (we also refer the interested reader to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Prop. 7

.1(ii)]).

In Assumption A.1, the symmetry of a is convenient but unnecessary. However, it comes with the advantage that the extended correctors (φ i , σ i ) associated with the adjoint a obviously coincide with (φ i , σ i ). If a is not symmetric, we have to assume that these satisfy (4) as well. Notice that we treat with scalar equations for simplicity, but the result of this paper applies as well to elliptic systems without any modification in the proof.

In Assumption A.2, instead of (4), we may only assume that the extended correctors (φ i , σ i ) are quantitatively strictly sublinear with overwhelming probability, in the sense that there exists β ∈ [0, 1) such that

Such a quantified growth is known to be sufficient to get quantitative homogenization properties (see, e.g. [START_REF] Gloria | A Regularity Theory for Random Elliptic Operators[END_REF]) and is indeed sufficient in the proofs of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF] for obtaining annealed CZ estimates -with the slight subtlety it has to be transposed on the level of the massive correctors.

Assumption A.3 on local regularity of a is of lesser importance. For example, we may morally only assume that a is Hölder continuous with overwhelming probability in the sense that there holds

Assumption (73) may ultimately be removed, c.f. [11, Th. 6.1], at the price of downgrading the resulting estimate ( 8) below to ˆ -

where we denote the ball of center x and radius R by B R (x) (if R = 1 or x = 0, these parameters are omitted) and the averaged integral by - ´D = 1

|D|

´.

A simple example of application

We propose here a possible example of application of Theorem 1.1 close to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF].

Let α, β ∈ (0, 1) and λ > 0. Assume that we are given an ensemble • of stationary and centered Gaussian fields g characterized by the covariance function c, the (non-negative) Fourier transform Fc of which is assumed to satisfy an improved integrability condition:

The coefficient a is defined through the convolution

where

) is a Lipschitz application in the set of symmetric, elliptic, and bounded matrices, namely

and where η ∈ C 0,α c (B 1 ) is a Hölder-continuous cut-off function with values in R + satisfying ´η = 1. We claim that we may apply Theorem 1.1 to such an ensemble • of coefficient fields. More precisely, we show the following:

We may estimate the first r. h. s. term by (82), so that, taking the L p R d L r • -norm and using ( 58) and ( 59)

(84)

Part 1, Step 3: Conclusion, namely proof of (54). Covering the balls B 1 (x) by smaller balls 1 2 B ρ (x ), and using the triangle inequality, we may estimate the left-hand side of (54) as follows:

We estimate the first r. h. s. term by means of (83),

f p,r , and the second r. h. s. term by Jensen's inequality and (84), namely

As a consequence, we obtain (54).

Part 2: Proof of (55). We first apply Lemma A.1 on u > in B 1 (x), and more precisely (87) for (p, q) (q, 2). Then, taking the L p R d L r • -norm yields (55).

Lemma A.1. Assume that coefficient field a satisfies estimates (3) and

for λ > 0 and α ∈ (0, 1). For T ≥ 1, we assume that the function u > and the vector field f satisfy the relation

Then, there holds

and, provided f = 0, u > √ T , ∇u > L ∞ (B1) d,λ,α,[a] C 0,α (B 2 ) ,q u > √ T , ∇u > L q (B2) for all q ∈ (1, ∞).

(88)

Lemma A.1 relies on standard CZ and Schauder theory for [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]


Mickael Montassier
email: montassi@labri.fr

Arnaud Pêcher
email: pecher@labri.fr

André Raspaud
email: raspaud@labri.fr

Xuding Zhu
email: zhu@math.nsysu.edu.tw

Edge-partitions of sparse graphs and their applications to game coloring

In this note, we prove that every graph with maximum average degree less than 32 13 (resp. 30 11 , 32 11 , 70 23 ) admits an edge-partition into a forest and a subgraph of maximum degree 1 (resp. 2, 3, 4). This implies that these graphs have game coloring number at most 5, 6, 7, 8, respectively.

Introduction

Let G be a simple graph. The game coloring number of G is defined through a two-person graph ordering game. Alice and Bob take turns choosing vertices from the set of unchosen vertices of G. This defines a linear order L of the vertices of G with x < y, if and only if, x is chosen before y. The back degree of a vertex x with respect to L is the number of its neighbors y in G such that y < x. The back degree of L is the maximum back degree of a vertex of G with respect to L. Alice's goal is to minimize the back degree of L and Bob's goal is to maximize it. The game coloring number col g (G) of G is defined to be k + 1, where k is the minimum integer such that Alice has a strategy for the graph ordering game to ensure that the back degree of L is at most k. Equivalently, k is the maximum integer such that Bob has a strategy for the graph ordering game to ensure that the back degree of L is at least k. This notion was first formally defined in [5] as a tool to find bounds to the game chromatic number [START_REF] Bodlaender | On the complexity of some coloring games[END_REF].

Recently, Zhu [6] proved that the game coloring number of every planar graph is at most 17. This result was improved in the case of planar graphs with large girth, by Borodin et al. [START_REF] Borodin | Decomposing a planar graph with girth 9 into a forest and a matching[END_REF] and He et al. [4]. These results are based on some structural properties of planar graphs with large girth:

Theorem 1 (Borodin et. al. [2] + He et. al. [4])

Let G be a planar graph with girth at least g. 1 1. If g ≥ 9, then G admits an edge-partition into a forest and a matching [START_REF] Borodin | Decomposing a planar graph with girth 9 into a forest and a matching[END_REF].

2. If g ≥ 7, then G admits an edge-partition into a forest and a graph with maximum degree 2 [4].

3. If g ≥ 5, then G admits an edge-partition into a forest and a graph with maximum degree 4 [4]. In this note, we study edge-partitions of sparse graphs, in the meaning of small maximum average degree, and derive bounds on the game coloring number.

The maximum average degree of G, denoted by Mad(G) is:

Mad(G) = max{2|E(H)|/|V (H)|, H G}

Our main result is:

Theorem 2 Let G be a simple graph.

If Mad(G) < 32

13 , then G admits an edge-partition into a forest and a matching. 2. If Mad(G) < 30 11 , then G admits an edge-partition into a forest and graph with maximum degree at most 2.

If Mad(G) < 32

11 , then G admits an edge-partition into a forest and graph with maximum degree at most 3.

If Mad(G) < 70

23 , then G admits an edge-partition into a forest and graph with maximum degree at most 4.

In contrary to Theorem 1, Theorem 2 is not restricted to planar graphs. We note however that we can not infer Theorem 1 from Theorem 2, by using the usual inequality Mad(G) ≤ 2g/(g -2) for every planar graph G of girth at least g.

Combining with Lemmas 1 and 2, we get:

Corollary 2 Let G be a simple graph. 1. If Mad(G) < 32 13 , then col g (G) ≤ 5.

If Mad(G) < 30

11 , then col g (G) ≤ 6.

If Mad

(G) < 32 11 , then col g (G) ≤ 7. 4. If Mad(G) < 70 23 , then col g (G) ≤ 8.
Section 2 is dedicated to the proof of Theorem 2. Section 3 contains some final remarks.

Proof of Theorem 2

Let G be a simple graph. Let d(x) denote the degree of

x in G. A vertex of degree k (resp. at least k, at most k) is called a k-vertex (resp. ≥ k-vertex, ≤ k-vertex). An (a, b)-alternating cycle is an even cycle x 1 x 2 x 3 . . . x 2k x 1 such that d(x i ) = a if i is even and d(x i ) = b
otherwise. An k l -vertex is a vertex of degree k adjacent to exactly l 2-vertices.

Let G be a counterexample of Theorem 2, i.e. a graph that does not admit an edgepartition into a forest and a subgraph with maximum degree k

(k = 1, 2, 3, 4), minimizing σ(G) = |V (G)| + |E(G)|.

Structural properties of G

Discharging procedures

In what follows, we will define an additional structure, called bank, which is a subgraph of G composed of maximal connected components, called agencies. In fact, we will show that each bank is a forest and each agency a tree. These structures will be used, during the discharging procedure, to transfer charges. Usually, the discharging rules operate locally ; agencies will allow us to transfer charges non locally. In our discharging procedures, the vertices adjacent to an agency C will give their excess charge to C which will redistribute this excess charge to the vertices of C which does not have enough charges.

First we assign to each vertex v a charge ω(v) equal to its degree, i.e. ∀v ∈ V (G), ω(v) = d(v). Moreover we assign to each agency C (that will be defined later) a charge ω(C) = 0. We define then discharging rules and redistribute charges accordingly. Once the discharging is finished, a new charge function ω * is produced. However, the total sum of charges is kept fixed when the discharging is in process. Nevertheless, we can show that ω * (v) ≥ 32 13 (resp.

32 13 |V (G)| ≤ X v∈V (G) ω * (v) + X C agency of G ω * (C) = X v∈V (G) ω(v) + X C agency of G ω(C) = X v∈V (G)d(v) = 2|E(G)|
This leads to the following obvious contradiction:

32 13 = 32 13 |V (G)| |V (G)| ≤ 2|E(G)| |V (G)| ≤ Mad(G) < 32 13
and hence demonstrates that no such counterexample can exist (as well for Mad(G) < 30 11 , 32 11 , 70 23 ).

Graphs with Mad < 32 13

Here, the bank of G is the subgraph of G defined as follows: its set of vertices contains all the 3 2 -vertices, 3 3 -vertices and the 2-vertices adjacent to 3 2 -vertices, or 3 3 -vertices ; its set of edges is the set of edges between the 2-vertices and the 3 2 -vertices, 3 3 -vertices. By Claim 1.3, an agency is a tree whose each leaf is a 2-vertex.

We say that a vertex, which does not belong to an agency, is adjacent to an agency if it is adjacent to a 2-vertex belonging to an agency.

The discharging rules are defined as follows:

R1. Every ≥ 3-vertex gives 3 13 to each adjacent 2-vertex.

R2. Every ≥ 3-vertex not belonging to an agency gives 2 13 to each adjacent agency.

R3.

Each agency gives 2 13 to each of its own 3 3 -vertices.

Let us check first that for each vertex v, ω * (v) ≥ 32 13 . Let v be a k-vertex (k ≥ 2 by Claim 1.1).

Case k = 2 Initially, ω(v) = 2. The vertex v receives 3 13 from of each of its neighbors (which are ≥ 3-vertices by Claim 1.2). Hence, ω * (v) = 2 + 2 • 3 13 = 32 13 .

Case k = 3 Initially, ω(v) = 3. If v is adjacent to at most one 2-vertex, then ω * (v) ≥ 3-3 13 -2 13 ≥ 34 13 . If v is an 3 2 -vertex, then v belongs to an agency and gives two times 3 13 by R1 and nothing by R2. Hence ω * (v) = 3 -2 • 3 13 = 33 13 . Finally assume that v is a 3 3 -vertex. The vertex v gives three times 3 13 by R1 and receives 2 13 by R3. Hence ω * (v) = 3 -3 • 3 13 + 2 13 = 32 13 . Case k ≥ 4 Initially, ω(v) = k. The vertex v is adjacent to at most k 2-vertices and to at most k agencies. Hence by R1 and R2, ω

* (v) ≥ k -k • 3 13 -k • 2 13 = 8k 13 ≥ 32 13 if k ≥ 4.
It remains to prove that the charge remaining on each agency is non-negative. Let C be an agency. Let n 33 (C), and n l (C) be the number of 3 3 -vertices, and leaves of C respectively.

Observe that:

n l (C) ≥ n 33 (C) (1)
By R2, the agency C receives 2 13 from its adjacent vertices (i.e. it receives n l (C) • 2 13 ), and gives 2 13 to each of its own 3 3 -vertices by R3 (i.e. it gives n 33 (C)

• 2 13 ). Hence, ω * (C) = n l (C) • 2 13 -n 33 (C) • 2 13 ≥ n 33 (C) • 2 13 -n 33 (C) • 2
13 ≥ 0 by Equation ( 1). This completes the proof of Theorem 2.1.

Here, the bank of G is the subgraph of G defined as follows: its set of vertices contains all the 5 4 -vertices, 5 5 -vertices and the 2-vertices adjacent to 5 4 -vertices, or 5 5 -vertices ; its set of edges is the set of edges between the 2-vertices and the 5 4 -vertices, 5 5 -vertices. By Claim 1.3, an agency is a tree whose each leaf is a 2-vertex.

The discharging rules are defined as follows:

R1. Every ≥ 5-vertex gives 5 11 to each adjacent 2-vertex.

R2. Every ≥ 5-vertex not belonging to an agency gives 2 33 to each adjacent agency. R3. Each agency gives 2 11 to each of its own 5 5 -vertices.

Conclusion

In this note, we established that for every simple graph G, if Mad(G) ≤ 32 13 (resp. 30 11 , 32 11 , 70 23 ) then G admits an edge-partition into a forest and a graph with maximum degree at most 1 (resp. 2, 3, 4), hence the game chromatic of G is at most 5 (resp. 6, 7, 8). In order to study the tightness of Theorem 2, we introduce a function f : N → R defined by f (k) = inf{Mad(H) | H does not admit an edge-partition into a forest and a subgraph with maximum degree k }. It is easy to observe that the complete bipartite graph K 2,2k+2 has Mad(K 2,2k+2 ) = 4(k+1) k+2 and does not admit an edge-partition into a forest and a subgraph with maximum degree at most k. Hence,

Claim 1 3 .

 13 The counterexample G does not contain: 1. 1-vertices, 2. two adjacent ≤ k + 1-vertices, 3. (k + 2, 2)-alternating cycles. Proof 1. By contradiction, assume that G contains an 1-vertex v adjacent to u. By minimality, of G, the graph H = G \ u admits an edge-partition into a forest F and a subgraph D with maximum degree k. We can extend this edge-partition to G by adding the edge uv into F , a contradiction. 2. Assume that G contains two adjacent ≤ k + 1-vertices, say u an v. By minimality, of G, the graph H = G \ uv admits an edge-partition into a forest F and a subgraph D with maximum degree k. If at least one of u and v is incident to k edges in D, then add uv in F ; otherwise, add uv into D. This extends the edge-partition to G, a contradiction. Assume that G contains a (k + 2, 2)-alternating cycle C = x 1 x 2 x 3 . . . x 2l x 1 with d(x i ) = k + 2 if i is even and d(x i ) = 2 otherwise. By minimality of G, the graphH = G \ {x 1 x 2 ,x 2 x 3 , . . . , x 2l-1 x 2l , x 2l x 1 } admits an edge-partition into a forest F and a subgraph D with maximum degree k. We may assume that x 2j is incident to at least one edge of F , for otherwise we can add an arbitrary edge incident to x 2j into F . Now by adding the edges x 2j x 2j+1 into D, adding x 2j x 2j-1 into F , we obtain a required edge-partition of G, a contradiction.

2

 2

  all v ∈ V (G) and ω * (C) ≥ 0 for all agency C of G. Hence the following equation follows:

Lemma 2 (Faigle et al. [3]) Let

  Zhu established this upper bound of the game coloring number: Suppose that the graph G has an edge-partition into two subgraphs G 1 and G 2 , then col g (G) ≤ col g (G 1 ) + ∆(G 2 ). T be a forest. Then col g (T ) ≤ 4.

	Lemma 1 (Zhu [5]) Faigle et al. studied the game coloring number of a forest:
	Hence combining these two lemmas with Theorem 1, we have
	Corollary 1 ([2] + [4]) Every planar graph with girth at least 9 (resp. 7, 5) has game color-
	ing number at most 5 (resp. 6, 8).

  For every k, what are (if any) the graphs which do not admit an edge-partition into a forest and a subgraph with maximum degree at most k, but such that every graph with smaller maximal average degree does?

	2, 461... =	32 13	≤ f (1) ≤	8 3	= 2, 666...
	2, 727... =	30 11	≤ f (2) ≤ 3
	2, 909... =	32 11	≤ f (3) ≤	16 5	= 3, 2
	3, 043... =	70 23	≤ f (4) ≤	10 3	= 3, 333...
	We conclude with the following problem:
	Problem 1