Marc Josien
email: marc.josien@enpc.fr

Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

Keywords: Green function, periodic homogenization, multiscale problems

published or not. The documents may come

Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

Introduction

In this article, we consider the periodic Green function G n (x, y) associated with the multiscale problem

       -div (A(nx) • ∇u n (x)) = f (x) -ˆQ f for x ∈ Q, ˆQ u n = 0 and u n is Q-periodic, (1)
where n ∈ N is expected to be very large and Q = [-1/2, 1/2] d is the unit cube in dimensions d ≥ 2. Hereafter, we write "periodic" for "Q-periodic". Here A satisfies the classical assumptions of ellipticity, periodicity and Hölder continuity (see [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]). We first derive pointwise estimates for G n and its derivatives ∇ x G n , ∇ y G n and ∇ x ∇ y G n . Although these estimates are seemingly classical, we have not found them elsewhere in the literature in the special case of periodic boundary conditions. In particular, we refer the reader to [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF], which collects similar estimates, but concerning the Green function of elliptic problems with periodic coefficients in R d (and not the periodic Green function). We also express the periodic Green function G n in terms of the Green function of the operator -div (A(n•) • ∇) in R d . This latter result, which yields an alternative proof of estimates on the periodic Green function, is of independent interest. Our proofs crucially make use of homogenization tools of Avellaneda and Lin [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF].

Our study is motivated by the fact that, in numerical applications, numerous homogenization problems are set on cubes with periodic boundary conditions (rather than on an infinite domain). For example, we refer the reader to [START_REF] Anantharaman | Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments[END_REF], where the homogenized matrix of a random medium is approximated, by appealing to a periodic problem (such a strategy has been recently used in [START_REF] Legoll | Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem[END_REF][START_REF] Legoll | On the numerical approximation of fluctuations in stochastic homogenization[END_REF]).

Estimating the behavior of the Green function of elliptic problems has attracted much attention, as the Green functions are a useful tool for getting estimates. Indeed, the solution u n to (1) can be written as an integral of the forcing term (e.g., f in [START_REF] Anantharaman | Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments[END_REF], which can be in the form f = div(H)), multiplied by the Green function. Thus, if the Green function (or its derivatives) is controlled, then one can estimate the solution u n (or its derivatives) directly from the forcing term, using the Young inequality (see, e.g., [START_REF] Bergh | Interpolation spaces. An introduction[END_REF], and see [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF] for such manipulations). However, let us remark that, by duality, such estimates can also be used to get back to the properties of the Green function. We refer the reader to [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF], which goes back and forth from properties of the Green function to estimates on the solution to the oscillating problem.

The behavior of the Green function G of the following Dirichlet problem:

-div (A(x) • ∇u(x)) = f (x) in Ω, u = 0 on ∂Ω,
with elliptic and bounded matrix A has been explored in the seminal article [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] (here Ω is a sufficiently regular bounded domain). Without any regularity assumption on A (and without any hypothesis about the structure of A), the authors derive optimal pointwise estimates on G. But they need to assume that the matrix A is continuous and sufficiently regular in order to obtain pointwise estimates on the gradients ∇ x G, ∇ y G, and on the second derivatives ∇ x ∇ y G.

Loosely speaking, they show under suitable assumptions that these quantities behave as if G was the fundamental solution to the Laplace equation (see [9, Chap. II, p. 13-30]), namely (in dimension d ≥ 3):

|G(x, y)| ≤ C|x -y| -d+2 , (2)
|∇ x G(x, y)| ≤ C|x -y| -d+1 , |∇ y G(x, y)| ≤ C|x -y| -d+1 , (3)
|∇ x ∇ y G(x, y)| ≤ C|x -y| -d . (4)
Since the domain of interest Ω is bounded, the above quantity |x-y| is bounded; hence, the difficulty in the above estimates is obviously when x is close to y.

Their results have been generalized to systems of elliptic equations. In particular, the same type of results is proved in [START_REF] Dolzmann | Estimates for Green's matrices of elliptic systems by L p theory[END_REF], provided that the matrix A is sufficiently regular. On the opposite side, problems like (1) have the specificity that the coefficients A(n•) are more and more oscillating when n increases, since the caracteristic scale 1/n of the microstructure goes smaller and smaller. Therefore, the results that rely on the regularity of the coefficient do not apply uniformly: the constants C of the estimates (3) and (4) blows up when n goes to infinity. With a totally different approach than above, Avellaneda and Lin have proved that the solutions to oscillatory elliptic problems enjoy Hölder and Lipschitz regularity properties, if the matrix A is elliptic, periodic, and Hölder continuous (see [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]). For that purpose, they introduced a so-called compactness method, showing that the oscillatory problems inherit regularity from the homogenized problem. Applying their results to the Green function in R d itself, they derived the same type of estimates as (2), (3), and (4). We refer the reader to [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF] for a review on the pointwise estimates on multiscale Green functions in R d , for matrices A that are elliptic, bounded, periodic and sufficiently regular.

In [START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF], Avellaneda and Lin described the asymptotic behavior, in the limit where the small scale vanishes, of the Green function in R d of periodic elliptic equations by using the Green function of the homogenized problem. Using the same techniques, the authors of [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF][START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF] established the same kind of asymptotics for the Green function of the multiscale problem set in a bounded domain, for Dirichlet and Neumann boundary conditions.

Periodic Green functions can sometimes be expressed thanks to the associated Green functions in the whole space R d . For example, such a decomposition can be found in [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF] for the case of the Laplacian. This consists in a series involving the Green function in R d , translated on the grid Z d . The main difficulty of this decomposition is to ensure that the series actually converges; in the case of the Laplacian, this is shown by resorting to the local symmetries of the Green function of the Laplacian. We address the question of building a similar decomposition for the case [START_REF] Anantharaman | Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments[END_REF].

Most of the theoretical material and ideas used in the present article are borrowed from [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF][START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF][START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF] for the homogenization aspect, and from [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF] for the decomposition of the periodic Green function.

Main results

Before getting to the oscillatory problem, we first establish the existence and the uniqueness of the periodic Green function for general periodic, elliptic and bounded coefficients. Henceforth, we denote by a subscript "per" the functional spaces of periodic functions: for example, L 2 per (Q) is the set of functions defined on R d that are periodic and square integrable on the cube Q. We consider the operator

T : f → u,
where f ∈ L 2 per (Q) and u is the unique periodic solution with zero mean to

-div (A(x) • ∇u(x)) = f (x) -ˆQ f for x ∈ Q, (5)
in which the matrix A is periodic, elliptic and bounded. Namely, there exists a constant µ > 0 such that A satisfies

µ|ξ| 2 ≤ A(x) • ξ • ξ ≤ µ -1 |ξ| 2 ∀x, ξ ∈ R d , (6)
A(x + z) = A(x) ∀x ∈ R d , z ∈ Z d . (7)
The operator T admits the following integral formulation, involving the so-called periodic Green function G associated with the operator -div (A • ∇):

T f (x) = ˆQ G(x, y)f (y)dy. (8)
By classical arguments (see, e.g., [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF]), such a Green function G exists and is unique (see Section 2 for a precise statement). It satisfies the following equation:

-div x (A(x) • ∇ x G(x, y)) = δ y (x) -1 in Q. ( 9
)
Using a method that can be found in [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Th. 13] (see also the proof of [START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF]Th. 3.3]), we show a pointwise estimate on the periodic Green function G associated with the operator -div (A • ∇). In dimension d = 2, this estimate on G(x, y) is logarithmic, which introduces some technicalities. 6) and [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF]. Let G be the periodic Green function associated with the operator -div (A • ∇). Then there exists a constant C > 0 that only depends on d and µ such that the following estimates are satisfied, for all x ∈ R d and y ∈ x + Q, with x = y:

Proposition 1.1. Let the dimension be d ≥ 2. Assume that A ∈ L ∞ per Q, R d 2 satisfies (
if d ≥ 3, |G(x, y)| ≤ C|x -y| -d+2 , ( 10
)
if d = 2, |G(x, y)| ≤ C log(2 + |x -y|). (11)
The proof of Proposition 1.1 is postponed until Section 3.

One can apply the above result to the multiscale problem [START_REF] Anantharaman | Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments[END_REF]. 6) and ( 7), then the periodic Green function G n associated with the operator -div (A(n•) • ∇) satisfies [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] or [START_REF] Josien | Thèse de l'Université Paris Est: Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux[END_REF] (depending on the dimension d), for a constant C that only depends on d and µ (and not on n). Now, we consider a matrix A that is elliptic, periodic, and also Hölder continuous:

Thus, if A ∈ L ∞ per Q, R d 2 satisfies (
A ∈ C 0,α Q, R d 2 , (12)
for α ∈ (0, 1). Using the results of [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF], we derive pointwise estimates on the gradients ∇ x G n and ∇ y G n and on the second derivatives ∇ x ∇ y G n of the periodic Green function G n associated with the operator -div [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF] and [START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF]. Let G n be the periodic Green function associated with the operator -div (A(n•) • ∇). Then, there exists a constant C > 0 such that, for all n ∈ N\{0}, x ∈ R d and y ∈ x + Q, with x = y,

(A(n•) • ∇). Proposition 1.2. Let the dimension be d ≥ 2. Assume that A ∈ L ∞ per Q, R d 2 satisfies (6),
|∇ x G n (x, y)| ≤ C|x -y| -d+1 , (13)
|∇ y G n (x, y)| ≤ C|x -y| -d+1 , (14)
|∇ x ∇ y G n (x, y)| ≤ C|x -y| -d . ( 15
)
The proof of Proposition 1.2 is postponed until Section 4.

Let us underline that the salient point of Proposition 1.2 is that the constant C does not depend on the characteristic scale 1/n of the microstructure. The latter estimates are not unexpected; see, e.g., [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]Prop. 8] for similar estimates on the Green function in the whole space R d .

On the first hand, as is shown in [10, Th. 1.1], in the case of Dirichlet boundary conditions, Estimate [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] does not require any regularity assumption on A. As expected, it is also the case for periodic boundary conditions. On the other hand, Estimates ( 13), ( 14) and ( 15) critically rely on the fact that A is periodic and sufficiently regular.

Using another approach, reminiscent of [7, p. 130-131], we show a decomposition for the periodic Green function G. This formula extensively uses the corresponding Green function G in R d of the operator -div (A • ∇), which satisfies 7) and [START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF]. Let G be the periodic Green function associated with the operator -div (A • ∇). Then, the function G can be decomposed as

-div (A(x) • ∇ x G(x, y)) = δ y (x) in R d . ( 16
) Proposition 1.3. Let the dimension be d ≥ 3. Assume that A ∈ L ∞ per Q, R d 2 satisfies (6), (
G(x, y) = +∞ m=0 k∈Γm H k (x, y) , ( 17
)
where the functions H k are defined by

H k (x, y) := G(x, y -k) -ˆQ G(x, y + y -k)dy -ˆQ G(x + x , y -k)dx + ˆQ ˆQ G(x + x , y + y -k)dy dx , (18)
and the function G by (16), and the sets Γ m by

Γ 0 = k ∈ Z d , 0 ≤ k • (A s ) -1 • k < 2 2 , (19)
Γ m = k ∈ Z d , 2 2m ≤ k • (A s ) -1 • k < 2 2m+2 if m ≥ 1, (20)
where A s is the symmetric part of the homogenized matrix A associated with the matrix A.

The proof of Proposition 1.3 is postponed until Section 5.

The above decomposition (17) naturally appears as a reasonable candidate, being close (but not equivalent) to the decomposition [7, p. 130-131]. But the difficulty is to ensure that the series actually converges, in the sense that

+∞ m=0 k∈Γm H k (x, y) < +∞ for x = y. (21)
In [7, p. 130-131], where the Laplacian with periodic boundary conditions is studied, the convergence is obtained by appealing to the local symmetries of the Green function of the Laplacian. This cannot be applied to our case. Here, the convergence is a consequence of the long-range behavior of the Green function G.

Thanks to the periodicity of A, the function G can be efficiently approximated at large scale by the Green function of the homogenized problem (see [START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF][START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF]). Hence, taking advantage of the long-range symmetries of the Green function of the homogenized problem, one can prove the convergence of the series in (17).

In this regard, we underline that, in general, the series (17) does not converge absolutely with respect to k:

k∈Z d H k (x, y) = +∞ for x = y.
This fact appears as a byproduct of the proof. Last but not least, it should be underlined that the above decomposition provides an alternative way for showing the pointwise estimates [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] on the multiscale periodic Green function G n of the operator -div (A(n•) • ∇). Indeed, the proof of Proposition 1.3 (in Section 5) implies that the series (17) converges uniformly with respect to n.

Extension to systems

Our proof of the existence and the uniqueness of the Green function uses the De Giorgi-Nash-Moser theorem. In dimension d ≥ 3, this ingredient can be replaced by the W 1,p and L ∞ estimates in [8, Lem. 2 & Lem. 3]. Hence, there also exists a unique periodic Green function of the operator Lu := (L α u) α∈[ [1,m]] defined by

L α u := -div   m β=1 A αβ • ∇u β   = - d i,j=1 ∂ i   m β=1 A αβ ij ∂ j u β   , where A = A αβ ij , for i, j ∈ [[1, d]] and α, β ∈ [[1, m]], m ∈ N, is continuous, periodic,
and elliptic in the following sense:

µ|ξ| 2 ≤ d i,j=1 m α,β=1 A αβ ij (x)ξ α i ξ β j ≤ µ -1 |ξ| 2 ∀x ∈ R d , ξ = (ξ α i ) ∈ R dm , (22)
In this case, the periodic Green function G (which is a matrix) satisfies, for all α, γ ∈

[[1, m]], -div x   m β=1 A αβ (x) • ∇ x G βγ (x, y)   = δ αγ (δ y (x) -1) in Q, (23)
where δ αβ is the Kronecker symbol.

As can be seen in Sections 3 and 4, the proofs of Propositions 1.1 and 1.2 involve arguments that are also valid if we study periodic oscillatory systems instead of equations (note that the seminal article [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] dealt with systems). More precisely, if d ≥ 3, the periodic Green function G n associated with the operator -div (A(n•) • ∇) satisfies ( 10), ( 11), ( 13), [START_REF] Legoll | Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem[END_REF], and (15), provided that A = A αβ ij is periodic, satisfies (22), and is Hölder continuous. Notably, the Hölder estimate [2, Lem. 9] can be used instead of the De Giorgi-Nash-Moser theorem in the proof of Proposition 1.1. Besides, the Lipschitz estimate borrowed from [2, Lem. 16] that we use in the proof of Proposition 1.2 also applies.

Finally, the formula (17) can also be generalized to the case of systems, using appropriate sets Γ α,β m while decomposing G αβ (see Section 5.3).

Outline

Our article is articulated as follows. In Section 2, we precisely explain in which sense there exists a unique periodic Green function. We briefly justify this fact by classical arguments. Next, in Section 3, we proceed with the proof of Proposition 1.1. In dimension d ≥ 3, the proof is based on a duality argument involving the De Giorgi-Nash-Moser theorem. In dimension d = 2, using a trick from [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF], it reduces to expressing the 2-dimensional periodic Green function as the integral of a 3-dimensional periodic Green function. In Section 4, combining Estimate (10) of Proposition 1.1 and the Lipschitz estimates of [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF], we show (13) (and similarly ( 14)), from which we deduce [START_REF] Legoll | On the numerical approximation of fluctuations in stochastic homogenization[END_REF]. Finally, in Section 5, we prove Proposition 1.3, which, under suitable hypotheses, yields a decomposition for the periodic Green function. For the sake of simplicity, we first study the case where the homogenized matrix is the identity in Section 5.1, and then the general case in Section 5.2. Additional materials about such a decomposition in the case of systems can be found in the Section 5.3.

Existence, uniqueness and basic properties of the Green function

In this section, we justify that there exists a unique periodic Green function G associated with the operator -div (A • ∇). It lies in the functional space E containing all the functions G(x, y) satisfying, for all p ∈ 1, d d-2

(by convention, 6) and [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF]. Then there exists a unique periodic Green function G(x, y) associated with the operator -div (A • ∇) -namely, G satisfies (8)-that is in the space E, defined by (24) and (25). Moreover, the function G † (x, y) := G(y, x) is the periodic Green function associated with the operator

if d = 2, then d/(d -2) = +∞) and q ∈ 1, d d-1 , sup y∈Q G(•, y) L p (Q) < +∞ sup y∈Q ∇ x G(•, y) L q (Q) < +∞, (24)
sup x∈Q G(x, •) L p (Q) < +∞, sup x∈Q ∇ y G(x, •) L q (Q) < +∞. ( 25
) Proposition 2.1. Let the dimension be d ≥ 2. Assume that A ∈ L ∞ per Q, R d 2 satisfies (
-div A T • ∇ . Last, G is the unique periodic solution in E to (9) satisfying ˆQ G(x, y)dy = 0 ∀x ∈ Q, ( 26
)
and ˆQ G(x, y)dx = 0 ∀y ∈ Q. ( 27
)
Actually, if d ≥ 3, the Green function is expected to satisfy the following estimates:

sup y∈Q G(•, y) L d d-2 ,∞ (Q) < +∞, sup y∈Q ∇ x G(•, y) L d d-1 ,∞ (Q) < +∞, (28)
sup x∈Q G(x, •) L d d-2 ,∞ (Q) < +∞, sup x∈Q ∇ y G(x, •) L d d-1 ,∞ (Q) < +∞, ( 29
)
since it is the case when homogeneous Dirichlet boundary conditions are considered (see [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF]Th. 1.1]). Here L p,∞ denote the Marcinkiewicz spaces (see [4, Chap. I p. [START_REF] Catto | The mathematical theory of thermodynamic limits: Thomas-Fermi type models[END_REF][START_REF] Dolzmann | Estimates for Green's matrices of elliptic systems by L p theory[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF][START_REF] Josien | Thèse de l'Université Paris Est: Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux[END_REF] for a reference on such functional spaces). Thus, the proposed space E is not optimal. However, for the purpose of the present article, it not useful to find the optimal function space, since (28) and ( 29) are a straightforward corollary of Propositions 1.1 and 1.2 below.

Proposition 2.1 can be proven by standard arguments (see, e.g., [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF]) that we briefly describe here (see [START_REF] Josien | Thèse de l'Université Paris Est: Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux[END_REF]Chapter 3]).

First, we establish existence and uniqueness by using the regularizing properties of [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]. Then, we show that (x, y) → G(y, x) is the Green function of the transposed problem by considering the adjoint operator of T . In a third step, we use the variational formulation of ( 5) and establish that the Green function G satisfies ( 9), ( 26) and ( 27). Finally, we show the uniqueness of the solution to ( 9), ( 26) and ( 27), using a variational argument. The detailed proof of each above argument can be found in [START_REF] Josien | Thèse de l'Université Paris Est: Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux[END_REF]Chapter 3].

Pointwise estimates on the periodic Green function

This section is devoted to the proof of Proposition 1.1. For technical reasons, we proceed first with the case of dimension d ≥ 3, and then with the case of dimension d = 2.

The case of d ≥ 3

Pointwise estimates on the Green functions of elliptic problems with Dirichlet boundary conditions have been established in the seminal article [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] of Grüter and Widman. Their proof makes use of the comparison principle. The latter is an appropriate tool for an elliptic equation with homogeneous Dirichlet boundary conditions: in this case, the Green function is positive. But, such an argument fails when considering the periodic Green function, the sign of which varies (it has zero mean). As a consequence, here, we resort to a duality argument and to the De Giorgi-Nash-Moser theorem (see [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Th. 13]). Also, when considering multiscale periodic elliptic systems, the latter theorem does not hold and the Hölder estimates of [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] are necessary for concluding the proof (see Section 1.2).

The proof below is a straightforward adaptation of the proof of [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Th. 13]. The fact that we study periodic boundary conditions do not raise substantial difficulties, since the strategy involves local estimates.

Let us first explain in a few words the ingredients of the proof of Proposition 1.1 in the case d ≥ 3. The first step of the proof consists in combining the De Giorgi-Nash-Moser theorem, the classical Hilbert theory and the Sobolev injections in order to obtain an optimal L ∞ estimate on the periodic solution u to (5) for localized right-hand terms f . By a duality argument used in [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Th. 13], this provides a local L 2 bound on the Green function G(x, y). Using once more the De Giorgi-Nash-Moser theorem, this proves Estimate [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF].

Proof of Proposition 1.1 in dimension d ≥ 3. The proof falls in two steps. Let x 0 ∈ R d , y 0 ∈ x 0 + Q, x 0 = y 0 , and 2r := |x 0y 0 | (by periodicity, it is the only relevant case).

× x 0 × y 0 × y 0 × y 0 × y 0 -div (A(x) • ∇u(x)) = -´Q f Support of f x 0 + Q Figure 1: Illustration of Step 1 of proof of Proposition 1.1.
Step 1: Let us consider f ∈ L 2d d+2 per (Q) that has a support contained in Z d + B(y 0 , r/4) (see Figure 1). Define u as the periodic solution with zero mean to [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]. Since L 2d d+2 per (Q) is in the dual of H 1 per (Q) (see [6, Th. 9.9 p. 278]), then, by the Lax-Milgram theorem, u ∈ H 1 per (Q) and there obviously holds

∇u L 2 (Q) ≤ C f L 2d d+2 (Q) . ( 30
)
As the support of f is contained in Z d + B(y 0 , r/4), then u satisfies in B(x 0 , r) the following equation:

-div (A(x) • ∇u(x)) = -ˆQ f = - ˆB(y0,r/4) f div x -x 0 d .
Therefore, as a consequence of the De Giorgi-Nash-Moser theorem (see [9, Th. 8.24 p. 202]), there exists β ∈ (0, 1) and C > 0 depending only on d and µ such that

|u(x 0 )| ≤Cr -d/2 ˆB(x0,r) |u(x)| 2 dx 1/2 + Cr 2 ˆB(y0,r/4) f . (31)
We now bound the two terms on the right-hand side. For the second term, by the Hölder inequality, we obtain

ˆB(y0,r/4) f ≤ Cr d-2 2 ˆB(y0,r/4) |f (x)| 2d d+2 dx d+2 2d . ( 32
)
For the first term, again using the Hölder inequality, we have

ˆB(x0,r) |u(x)| 2 dx 1/2 ≤Cr ˆQ |u(x)| 2d d-2 dx d-2 2d . ( 33
)
By Sobolev injection of

H 1 (Q) in L 2d d-2 (Q) (and since u has zero mean), ˆQ |u(x)| 2d d-2 dx d-2 2d ≤C ˆQ |∇u(x)| 2 dx 1/2 , ( 34
)
whence, we deduce from (33), ( 34) and (30) that

ˆB(x0,r) |u(x)| 2 dx 1/2 ≤Cr ˆB(y0,r/4) |f (x)| 2d d+2 dx d+2 2d . ( 35
)
Finally, since 2r ≤ √ d, (31), (32), and (35) yield

|u(x 0 )| ≤ Cr -(d-2)/2 ˆB(y0,r/4) |f (x)| 2d d+2 dx d+2 2d . ( 36
)
Step 2: The function u can be expressed thanks to the Green function as

u(x) = ˆQ G(x, y)f (y)dy.
As a consequence, by duality, (36) implies that there exists a constant C > 0 such that

ˆB(y0,r/4) |G(x 0 , y)| 2d d-2 dy d-2 2d ≤ Cr -(d-2)/2 . ( 37
)
As G(x 0 , •) satisfies

-div y A T (y) • ∇ y G(x 0 , y) = -1, (38)
in B(y 0 , r/4), then, using once more [9, Th. 8.24 p. 202], we obtain (in the same manner as (31))

G(x 0 , •) L ∞ (B(y0,r/8)) ≤ r -d/2 ˆB(y0,r/4) |G(x 0 , y)| 2 dy 1/2 + Cr 2 .
By the Hölder inequality, we deduce from the above estimate that

G(x 0 , •) L ∞ (B(y0,r/8)) ≤ r 2-d 2 ˆB(y0,r/4) |G(x 0 , y)| 2d d-2 dy d-2 2d + Cr 2 .
Finally, since 2r = |x 0y 0 | ≤ √ d and thanks to (37), we deduce [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF]. This concludes the proof of Proposition 1.1 in dimension d ≥ 3.

The case d = 2

We now turn to the dimension d = 2. According to an idea that can be found in [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Th. 13] (see also [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]Prop. 4]), the 2-dimensional periodic Green function G associated with the operator -div (A • ∇) can be expressed as

G(x, y) = ˆ1 0 G((x, t), (y, 0))dt (39)
where x, y ∈ R 2 and t ∈ R in the equation above, and G is the 3-dimensional periodic Green function of the following operator:

Lu := -div x (A • ∇ x u) -∂ tt u. ( 40
)
Indeed, it is easily shown (by an integration argument) that the function G defined by (39) satisfies ( 9), ( 26) and ( 27). Hence, G is the (2-dimensional) periodic Green function associated with the operator -div (A • ∇).

Next, applying Proposition 1.1 -which we have already proved in dimension d ≥ 3to the Green function G, we obtain

|G(x, y)| ≤ C ˆ1 0 1 |t| + |x -y| dt ≤ C log (2 + |x -y|) ,
which is [START_REF] Josien | Thèse de l'Université Paris Est: Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux[END_REF]. This concludes the proof of Proposition 1.1 in the case d = 2.

4 Pointwise estimates on the derivatives ∇ x G n , ∇ y G n and ∇ x ∇ y G n of the multiscale periodic Green function

Proposition 1.2 relies on the Lipschitz theory of [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]. Indeed, if u n satisfies

-div (A(nx) • ∇u n (x)) = a in B(x 0 , r), (41)
for some a ∈ R, then [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]Lem. 16] shows that

|∇u n (x 0 )| ≤ Cr -1-d/2 ˆB(x0,r) |u n | 2 1/2 + C|a|r, (42)
where C > 0 is a constant independent of n. As is detailed below, we treat separately the 2-dimensional case.

Proof of Proposition 1.2. Assume first that the dimension satisfies d ≥ 3. Let x 0 ∈ R d , y 0 ∈ x 0 + Q, x 0 = y 0 , and 2r := |x 0y 0 | (once more, by periodicity, this is the only relevant case). Since G n (x, y 0 ) satisfies

-div x (A(nx) • ∇ x G n (x, y 0 )) = -1 in B(x 0 , r),
and thanks to [2, Lem. 16], id est (42), there holds

|∇ x G n (x 0 , y 0 )| ≤ Cr -1 G n (•, y 0 ) L ∞ (B(x0,r)) + Cr. (43)
As a consequence, Estimate [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] and ( 43) yield (13) (for x = x 0 and y = y 0 ). By transposition, ( 14) is also established. By differentiating [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] with respect to y, we obtain

-div x (A(nx) • ∇ x ∇ y G n (x, y 0 )) = 0 in B(x 0 , r/2).
Therefore, thanks to (42),

|∇ x ∇ y G n (x 0 , y 0 )| ≤Cr -1 ∇ y G(•, y 0 ) L ∞ (B(x0,r/2))
.

By using [START_REF] Legoll | Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem[END_REF] and since 2r = |x 0y 0 |, we conclude that

|∇ x ∇ y G n (x 0 , y 0 )| ≤ C|x 0 -y 0 | -d ,
and establish [START_REF] Legoll | On the numerical approximation of fluctuations in stochastic homogenization[END_REF]. Now, let the dimension d = 2. As shown in the proof of Proposition 1.1, the 2-dimensional periodic Green function G n can be expressed as

G n (x, y) = ˆ1 0 G n ((x, t), (y, 0))dt (44)
where x, y ∈ R 2 and t ∈ R in the equation above, and G n is the 3-dimensional periodic Green function of the operator Lu :

= -div x (A(n•) • ∇ x u) -∂ tt u.
Hence, we deduce the 2-dimensional versions of ( 13), ( 14) and ( 15) by integrating their 3-dimensional versions applied to G n (in the same manner as in Section 3.2).

A decomposition of the periodic Green function

In this section, we prove Proposition 1.3. For the sake of simplicity, we first assume that the homogenized matrix A is the identity. We postpone the proof in the general case until Section 5.2.

Case where the homogenized matrix is the identity

For convenience, we have split the proof of Proposition 1.3 in two parts: first, we show that the series in (17) actually converges; second, we check that its limit is the periodic Green function G.

The two main steps of the proof of convergence are the following: first a Taylor expansion allows for expressing the terms H k in (17) as functions of ∇ x ∇ y G. Then, we approximate the Green function of the multiscale problem with the Green function of the homogenized problem (see [START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF]). Second, we take advantage of the long-range symmetries of the Green function of the homogenized problem and establish the convergence of the series in (17). There, the sets Γ m are crucial, since the convergence in (17) is not uniform in k.

Proof of convergence of the series in (17). We denote by G (x, y) the fundamental solution in R d to the homogenized problem. The homogenized matrix is the identity; therefore G is explicitly expressed as

G (x, y) = C d |x -y| 2-d , (45)
where C d is a constant (see [9, (2.12) p. 17]). Since G (x, y) only depends on |x -y|, we henceforth redefine G (x) := G (x, 0).

Step 1: Assume that x ∈ Q, yx ∈ Q and k / ∈ 4Q. We reformulate H k using the Taylor formula:

H k (x, y) = -ˆQ x • ˆ1 0 ∇ x G(x + tx , y -k) -ˆQ ∇ x G(x + tx , y + y -k)dy dtdx = ˆQ x • ˆ1 0 ˆQ y • ˆ1 0 ∇ y ∇ x G(x + tx , y + τ y -k)dτ dy dt dx . (46)
We approximate ∇ x ∇ y G by ∇ x ∇ y G . More precisely, thanks to [3, Corollary p. 905], there exists constants C > 0 and β ∈ (0, 1) such that, for all x = y ∈ R d ,

∇ x ∇ y G(x , y ) - d i,j=1 ∂ xi ∂ yj G (x -y ) (e i + ∇w i (x )) ⊗ e j + ∇w † j (y ) ≤ C|x -y | -d-β . (47)
In (47), w i and w † j denote the correctors associated to the matrix A, respectively A T . That is, w i is the periodic function of zero mean satisfying -div (A(x) • (∇w i (x) + e i )) = 0, for x ∈ Q. Identity (46) and Estimate (47) imply

H k (x, y) =H 1,k (x, y) + H 2,k (x, y), (48)
where

H 1,k (x, y) ≤ C|k| -d-β , (49)
and

H 2,k (x, y) := d i,j=1 ˆQ2 ˆ[0,1] 2 (x • (e i + ∇w i (x + tx ))) y • e j + ∇w † j (y + τ y -k) ∂ xi ∂ yj G (x + tx -(y + τ y -k)) dτ dtdy dx .
All the correctors w i and w † j are bounded; furthermore, Formula (45) implies that the third-order derivatives of G evaluated at xy are bounded by C|x -y| -d-1 , where C is a constant independent of x and y. Therefore, a Taylor expansion yields a constant C such that

H 2,k (x, y) - d i,j=1 ∂ xi ∂ yj G (k)Q ij (x, y) ≤ C|k| -d-1 , (50)
where

Q ij (x, y) := ˆQ2 ˆ[0,1] 2 (x • (e i + ∇w i (x + tx )))
y • e j + ∇w † j (y + τ y ) dτ dtdy dx .

Then, a straightforward integration yields

Q ij (x, y) = ˆQ (x i + w i (x + x ) -w i (x)) dx ˆQ y j + w † j (y + y ) -w † j (y) dy ,
which can be simplified as

Q ij (x, y) =w i (x)w † j (y),
since the correctors w i and w † j are of zero mean. Note that Q ij defined above does not depend on k ∈ Z d because the correctors w i and w † j are periodic. As a consequence, collecting (48), (49), and (50) yields k∈Γm

H k (x, y) ≤C2 -mβ + |Q ij (x, y)| d i,j=1 k∈Γm ∂ xi ∂ yj G (k) . ( 51
)
Step 2: Remark that Q ij (x, y) = 0 in general and that ∂ xi ∂ yj G (k) scales like |k| -d . Therefore, by (50), in general, the series in (17) does not converge absolutely with respect to k.

Invoking once more (45), we obtain

∂ xi ∂ yj G (k) =        C d d(d -2) k i k j |k| d+2 if i = j, C d (d -2) dk 2 i -|k| 2 |k| d+2 if i = j.
Thanks to the symmetry of Γ m with respect to the hyperplane x i = 0, in the case i = j, and thanks to the invariance of Γ m under the relabeling of the components of the vector k, in the case i = j, we deduce that k∈Γm

∂ xi ∂ yj G (k) = 0, ∀i, j ∈ [[1, d]]. (52)
As a consequence, recalling (51),

k∈Γm H k (x, y) ≤ C2 -mβ . (53)
Moreover, by [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]Prop. 4], there exists a constant C such that for any x = y , there holds

|G(x , y )| ≤ C|x -y | -d+2 . (54)
Hence, for any k ∈ Z d ,

H k (x, y) ≤C|x -y| -d+2 . ( 55
)
As a consequence of ( 53) and ( 55), the series (17) converges absolutely in m as follows:

+∞ m=0 k∈Γm H k (x, y) ≤ C|x -y| -d+2 ,
for all x = y, yx ∈ Q. Thus, we have recovered [START_REF] Grüter | The Green function for uniformly elliptic equations[END_REF] by an approach different from Proposition 1.1.

Now that we have justified that the series in (17) converges, we prove that its limit, that we denote by G for the moment, id est,

G(x, y) := +∞ m=0 k∈Γm H k (x, y) , (56)
is actually equal to G. It can easily be checked that G is periodic in x and y and satisfies [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. The technical point is (26), the proof of which relies on the former Taylor expansion and on the classical result [START_REF] Blanc | Asymptotic behavior of Green functions of divergence form operators with periodic coefficients[END_REF]Prop. 8]. According to the latter, there exists a constant C > 0 such that, for all x = y ,

|∇ x ∇ y G(x , y )| ≤ C|x -y | -d . ( 57
)
Proof of Identity (17). Obviously, G is periodic in y. Since A is periodic, there even holds

G(x, y -k) = G(x + k, y),
for any k ∈ Z d . Therefore G is also periodic in x. Moreover, we check that

-div A(x) • ∇H k (x, y) =δ 0 (x -y + k) -2χ Q (x -y + k) + ˆQ χ Q (x -y -y + k)dy , where χ Q is the characteristic function of the set Q. Whence -div A(x) • +∞ m=0 k∈Γm ∇ x H k (x, y) = k∈Z d δ 0 (x -y + k) -1 in R d .
To summarize, G(x, y) defined by ( 56) is x-periodic and y-periodic, and satisfies [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

Next, we justify that G satisfies (26). By integrating (46) along the y variable, there holds

ˆQ H k (x, y)dy = ˆ1 0 ˆ1 0 ˆQ ˆy∈∂Q x • ˆQ ∇ x G(x + tx , y + τ y -k) ⊗ y dy • d S(y)dx dτ dt.
Hence, due to cancellations on the boundaries of the translated cubes k + Q,

ˆQ |k|<2 m H k (x, y)dy = ˆ1 0 ˆ1 0 ˆQ ˆy∈Ξm x • ˆQ ∇ x G(x + tx , y + τ y -k) ⊗ y dy • d S(y)dx dτ dt,
where Ξ m is the boundary of the following set:

∪ |k|<2 m+1 (k + Q) .
Now, by Taylor expansion, and thanks to (57), for all τ ∈ [0, 1], x, x , y ∈ Q, and y ∈ Ξ m ,

|∇ x G(x + tx , y + τ y ) -∇ x G(x + tx , y)| ≤ C2 -md . Therefore ˆQ ∇ x G(x + tx , y + τ y ) ⊗ y dy ≤ ˆQ ∇ x G(x + tx , y) ⊗ y dy + C2 -md .
The integral in the right-hand term of the above estimate vanishes since, by symmetry, ´Q y dy = 0. As a consequence, ˆQ ∇ x G(x + tx , y + τ y ) ⊗ y dy ≤ C2 -md .

Whence, since the surface area of Ξ m is bounded by C2 m(d-1) , we have

ˆQ |k|<2 m+1 H k (x, y)dy ≤C2 -m .
As a consequence, letting m → +∞, we deduce that G satisfies (26). By the same arguments transposed from G(x, y) to G(y, x), it can be shown that G also satisfies (27). Therefore, by Proposition 2.1, we have G(x, y) = G n (x, y), which concludes the proof.

General case

As is easily seen in the proof of Proposition 1.3, the fact that the homogenized matrix A is the identity is only used for establishing (52). One also realizes that, would (52) be replaced by the following estimates:

k∈Γm ∂ xi ∂ yj G (k) ≤ C m for all m ∈ [[1, +∞), and +∞ m=1 C m < +∞, (58)
for well-chosen sets Γ m , then the conclusions of Proposition 1.3 would also apply.

We show that the sets Γ m defined by ( 19) and ( 20) are such that Estimates (58) are satisfied. Hence, the conclusions of Proposition 1.3 are true without any assumption on the homogenized matrix A of A.

The homogenized matrix A is (constant) coercive. Henceforth the Green function in R d associated with the operator -div

(A • ∇) is G (x) = C (A s ) x • (A s ) -1 • x (d-2)/2 ,
where C (A s ) is a constant and A s is the symmetric part of the matrix A . Whence

∇ 2 G (x) = C(d -2) d (A s ) -1 • x ⊗ (A s ) -1 • x -x • (A s ) -1 • x (A s ) -1 x • (A s ) -1 • x (d+2)/2
.

Besides, there exists an orthogonal matrix O and positive scalars λ i , i ∈ [ [1, d]] such that

A s = O -1 • diag λ -2 1 , • • • , λ -2 d • O.
Therefore, denoting by f j the orthonormal base related to O, and decomposing

x = d j=1 λ -1 j x j f j , one obtains ∇ 2 G (x) = C(d -2) d d i,j=1 λ i λ j x i x j f i ⊗ f j - d i=1 x 2 i d i=1 λ 2 i f i ⊗ f i d i=1 x 2 i (d+2)/2
.

For r ∈ R + , we define the following set:

Ω r :=    x ∈ R d , d j=1 | x j | 2 = r 2    . ( 59
)
Remark that there obviously holds

Ω r = x ∈ R d , x • (A s ) -1 • x = r 2 .
On the one hand, if i = j, f i ⊗ f j : ∇ 2 G (x) changes sign with respect to the transformation x j →x j . Therefore, if i = j, there holds

ˆΩr f i ⊗ f j : ∇ 2 G (x)dS(x) = 0. ( 60
)
On the other hand, As a consequence,

f i ⊗ f i : ∇ 2 G (x) = C(d -2)
λ 2 i d x 2 i - d j=1 x 2
ˆΩr f i ⊗ f i : ∇ 2 G (x)dS(x) = 0. (61)
Hence, we define By (60) and (61), we deduce that the last integral in the above estimate vanishes. Furthermore, straightforward estimates on the derivatives of G yield k∈Γm

∂ i ∂ j G (k) ≤C2 -m .
This implies the convergence of the series in (17) in the general case.

Case of systems

In the case of systems, the Green function G of -div (A • ∇) in R d reads:

G αβ (x) = C (A s ) αβ   d i,j=1
x i (A s )

αβ -1 ij x j   -d-2 2 , (62)
where C (A s ) αβ is a constant depending on the matrix (A s )

αβ ij i,j
, and

(A s ) αβ ij = 1
where the function G is the Green function in R d of the operator -div (A • ∇). Last, we define the sets Γ αβ m by:

Γ αβ 0 = k ∈ Z d , 0 ≤ k • (A s ) αβ -1 • k < 2 2 , Γ αβ m = k ∈ Z d , 2 2m ≤ k • (A s ) αβ -1 • k < 2 2m+2 if m ≥ 1,
where (A s ) αβ is considered as a matrix in R d×d .

2 .x 2 k

 22 By invariance of Ω r under the relabeling of the coordinates x j , (d+2)/2 dS(x).

Λ m := ∪ r∈[ 2 m

 2 ,2 m+1 ) Ω r , and Γ m := Λ m ∩ Z d . By convergence of Riemann integrals, we deduce that k∈Γm∂ i ∂ j G (k) ≤C |∂Λ m | sup x∈Λm |∂ i ∂ j G (x)| + C |Λ m | sup x∈Λm |∂ i ∂ j ∇G (x)| + C ˆΛm ∂ i ∂ j G (x)dx .

  the above arguments of Sections 5.1 and 5.2, we have the following decomposition:G αβ (x, y) = of each term will be made precise below. The functions H k are defined byH k αβ (x, y) := G αβ (x, yk) -ˆQ G αβ (x, y + yk)dy -ˆQ G αβ (x + x , yk)dx + ˆQ ˆQ G αβ (x + x , y + yk)dy dx ,

Acknowledgement

The author would like to thank both Frédéric Legoll and Pierre-Loïc Rothé, who have brought his attention to this question. Frédéric Legoll has also helped for the proof in the 2-dimensional case. Moreover, the author acknowledges Claude Le Bris for his suggestions concerning the decomposition of the Green functions. Finally, the author gratefully thanks Xavier Blanc for reading the first version of this article.


D E Walter
email: dwalter@usc.edu.au

Olivier Bonato

G R Stirling

Microarthropods in Australian sugarcane soils: A survey with emphasis on the Mesostigmata as potential regulators of nematode populations

Keywords: mites, Acari, Mesostigmata, nematodes, nematophagy, Oribatida, Collembola

Sugarcane is attacked by a suite of nematode pests, but little is known of the mites and other soil microarthropods that feed on them. To address this deficit, we undertook a survey of the soil microarthropod fauna in 60 sugarcane fields in Queensland, Australia. The results showed that oribatid mites and springtails dominated the upper 10 cm of sugarcane mineral soils. Laboratory observations demonstrated avid nematophagy in Galumnidae (Galumna sp., Pergalumna sp.), Scheloribatidae (Scheloribates sp.), and one unidentified Isotomidae (Collembola). Mesostigmata, a group of mites with many known nematophages, were never very abundant. Most of the taxa identified were previously known from Australia, but Protogamasellus sigillophorus Mineiro, Lindquist and De Moraes is reported for the first time and was reared through several generations on nematodes. In addition, we reared ten other cane field mesostigmatans on nematodes: Antennolaelaps sp., Asca garmani, Asca major, Cheiroseius sp., Cosmolaelaps sp., Gaeolaelaps sp., Gamasellodes bicolor, Gamasiphis sp., Holaspulus tenuipes, Protogamasellus mica. Because Mesostigmata were present at higher population densities in mineral soil where a mulch layer was retained after harvest, we assessed the effect of mulching in two sugarcane fields. Mulching did not result in significant differences in total microarthropods in the upper 5 cm of mineral soil. However, the mulch contained large numbers of free-living nematodes and 10% of the microarthropods were nematophagous mites. Results from a pot experiment also suggested that differences in soil properties affect nematophagous mite communities. More microarthropods and greater numbers of Mesostigmata were recovered from a wellstructured clay loam soil than a sandy loam soil, and populations were highest in surface soils with high carbon contents and high levels of microbial activity.

Introduction

'Microarthropod' is a convenient term for describing the minute arthropods that live in soil-litter systems and help regulate many decomposer subsystem processes through their feeding and other behaviours [START_REF] Moore | Arthropod regulation of micro-and mesobiota in belowground detrital foodwebs[END_REF]. Though definitions vary, arthropods more than a few millimetres in length are generally treated as a separate functional group (macroarthropods), although in reality the size distribution is more or less continuous and the smaller size classes interact with much larger animals, particularly in the litter layer. One aspect that links microarthropods worldwide, however, is that they belong to poorly known taxonomic groups such as the mites, springtails, proturans, dipurans, pauropods, symphylans, pseudoscorpions, the tiniest of myriapods, spiders, and insects, especially beetles, ants, parasitoid wasps, and flies. Thus, few areas in the world have the taxonomic infrastructure to deal with microarthropods at the species level. This taxonomic impediment is especially acute in Australia.

Although sugarcane is the most important agricultural crop in tropical and sub-tropical regions of Australia, and a major crop worldwide, the soil microarthropod fauna of cane fields is little studied. Most research has concentrated on the biology and control of relatively large root-feeding pests such as cane grubs and soldier flies, on a few smaller pests such as ground pearls and symphylans, or on arthropod communities at a low level of taxonomic resolution [START_REF] Allsopp | Pests of Australian sugarcane[END_REF][START_REF] Sajjad | Does trash burning harm arthropod biodiversity in sugarcane?[END_REF][START_REF] Shakir | Seasonal abundance of soil arthropods in relation to meteorological and edaphic factors in the agroecosystems of Faisalabad, Punjab, Pakistan[END_REF]; but see Asif et al., 2016, on Collembola). In this paper we present survey results that show that sugarcane soils, at least those that have not been tilled for several years, contain a diverse community of microarthropods with the potential to improve crop health through their role in cycling nutrients and regulating populations of root pests and pathogens. Initial results from this work have been published previously [START_REF] Manwaring | Microarthropods as predators of nematode pests in sugarcane soils: a review and preliminary studies[END_REF].

The number of microarthropods extracted from our survey samples was highly variable, raising questions as to the causes of this variability. Previous studies in natural and agricultural ecosystems have shown that the abundance and diversity of microarthropods is influenced by factors such as soil texture, soil carbon levels, depth in the soil profile and the presence of plant residues on the soil surface, and all these factors could have affected our survey results. Consequently, we set up field experiments to look at the impact of mulching, as it is a common practice in the Australian sugar industry and our survey data suggested that the presence of mulch (the residues remaining after harvest and referred to as a 'trash blanket' in Australia) may have increased the abundance of microarthropods (see also [START_REF] Manwaring | Effects of a mulch layer on the assemblage and abundance of mesostigmatan mites and other arthropods in the soil of a sugarcane agroecosystem in Australia[END_REF]. We also set up a pot experiment to determine whether properties such as the texture of sugarcane soils affected the presence and diversity of mites. Since Mesostigmata seem to be important predators of nematodes [START_REF] Stirling | The mesostigmatid mite Protogamasellus mica, an effective predator of free-living and plant-parasitic nematodes[END_REF], we were particularly interested in the impact of mulching and soil properties on this group of mites.

Methods

A survey of microarthropods in sugarcane soils

During the period from July to December 2014, 60 mineral soil samples were collected from four major cane-growing regions of Queensland (14 samples from the Herbert; 10 from the Burdekin; 15 from the Mackay region and 21 from farms around Bundaberg, Childers and Hervey Bay). Since tillage is known to be detrimental to microarthropods [START_REF] Wardle | Impacts of disturbance on detritus food webs in agro-ecosystems of contrasting tillage and weed management practices[END_REF], samples were, wherever possible, collected from farms where tillage had been minimised and a best-practice farming system had been adopted [START_REF] Garside | Managing yield decline in sugarcane cropping systems[END_REF]. However, third and fourth ratoon crops were always sampled, as this meant that the soil would not have been tilled for at least 3 years. To avoid the negative effects of compaction caused by harvest and haul-out machinery, samples were always collected in the centre of the row and between sugarcane plants. Each sample (about 3 L of soil) was taken with a shovel to a depth of 10 cm from six randomly selected points in a field. Microarthropods were extracted by placing 600 mL of soil on a Tullgren funnel for 5 days [START_REF] Walter | Collecting, rearing, and preparing specimens[END_REF] and the animals were captured in vials containing 70% ethanol. Microarthropods were sorted and counted under a dissecting microscope by feeding guild (see below) and representatives were slide-mounted for further identification when needed.

Guild / Functional Group Classification

Because soil microarthropods are extremely diverse, tend to have patchy distributions, and have feeding behaviours that often cross taxonomic boundaries, we sorted our collections into functional groups (guilds) that were based on our preliminary samples [START_REF] Manwaring | Microarthropods as predators of nematode pests in sugarcane soils: a review and preliminary studies[END_REF] and reflected their importance to the behaviour of interest: nematophagy. Previous research on nematophagy by microarthropods in non-sugarcane soils (reviewed in [START_REF] Walter | Mites: Ecology, Evolution & Behaviour: Life at a Microscale 2nd Edition[END_REF][START_REF] Manwaring | Microarthropods as predators of nematode pests in sugarcane soils: a review and preliminary studies[END_REF] documented that this behaviour was pervasive in one order of mites (Acari: Parasitiformes: Mesostigmata). We call these mites the Mesostigmata Guild (i.e. all Mesostigmata collected except the specialised fungivores in the Ameroseiidae). For some analyses we also abstracted a 'Small Mesostigmata Guild' (combined totals for all species of Asca, Gamasellodes, Protogamasellus, Multidentorhodacarus, Rhodacarus, and Rhodacarellus). These mites are less than 0.35 mm in length, have divided dorsal shields and narrow bodies (Asca species are at the upper length and width of this group) and are capable of using small pore spaces in soil. The larger Mesostigmata were members of the Ascidae, Ologamasidae, Blattisociidae, Laelapidae, Macrochelidae, Parholaspididae, Pachylaelapidae, and Uropodoidea (see results for identified genera) with inflexible dorsal shields and presumably less access to small pore spaces.

The most diverse and abundant taxa of microarthropods present were oribatid mites (Acari: Acariformes: Oribatida [including Astigmata]) and springtails (Hexapoda: Collembola). As far as is known, most species in these groups feed primarily on microbes and decaying organic matter, but many may consume nematodes when they encounter them [START_REF] Walter | Mites: Ecology, Evolution & Behaviour: Life at a Microscale 2nd Edition[END_REF]; also see Results). We combined the counts for these two taxa into the Fungivore/Detrivore Guild because in general oribatid mites and springtails share similar feeding behaviours and are thought to have some impact on nematode numbers. The other microarthropods present are predators of arthropods, have more specialised feeding habits or have unknown feeding behaviours. These include proturans, diplurans (Campodeidae, Parajapygidae), assorted insects (thrips, planthoppers, ants, beetles, flies, etc.), millipedes (mostly Polyxenidae), centipedes, small spiders, pseudoscorpions, palpigrades (Eukoeninia sp.), and certain mite groups (mostly Prostigmata [e.g. Eupodidae, Tydeidae, Tarsonemidae, Cryptognathidae, Cunaxidae: Cunaxinae, Bdellidae, Rhagidiidae] and Endeostigmata). We group these as Miscellaneous Arthropods and consider that they are unlikely to be important predators of nematodes in sugarcane soils, although they may provide other ecosystem services.

Testing for Nematophagy

We tested our assumptions of nematophagy using live extractions (i.e. into vials with a plasterof-Paris/charcoal base instead of ethanol as per [START_REF] Walter | Collecting, rearing, and preparing specimens[END_REF] of microarthropods collected from cane fields in the Bundaberg region, adding water suspensions of a bacterialfeeding nematode cultured on rolled oats and observing the interactions. All observations of nematophagy were recorded. Mesostigmatans observed feeding on nematodes were transferred to small rearing vials with a small paintbrush and nematodes in a water suspension were regularly added to determine if the mites could reproduce and develop to adults on a nematode diet at room temperature. If so, representatives were slide-mounted and identified as far as possible.

Impact of mulching on microarthropods, particularly the nematophagous mites

To assess the impact of green-cane trash blanketing (i.e. mulching) on the microarthropod community, mulched and non-mulched plots were established at two sites in Bundaberg. The soil at site 1 was a clay loam while site 2 had a lighter-textured sandy clay loam soil. Based on the Australian Soil Classification, these soils were classified as a Red Ferrosol and Yellow Dermosol, respectively. Sugarcane was planted at both sites following soybean and the plant crops were harvested in early September 2015, when the sugarcane was about 13 months old. Two weeks after harvest, the treatments at each site were established by marking out 12 plots 10 m long and 1.85 m wide, raking the trash blanket from half the plots and retaining the crop residues in the mulched plots. Thus, each experiment consisted of two treatments (bare and mulch) x 6 replicates. All plots were 2 m apart and contained one row of sugarcane. Any Table 1 Chemical and biological properties of clay loam and sandy loam soils collected from two depths in the soil profile (0-2 cm and 10-15 cm) and used in a pot study to assess the impact of soil texture and depth on microarthropods weeds that grew in the plots (particularly the plots that were not mulched) were either removed periodically by hand or eliminated by the herbicide program used by the grower. On 10 March 2016, nearly six months after the experiment was set up, samples of mulch and soil were collected to assess populations of nematodes and microarthropods. All samples were collected in the bed and were taken within 15 cm of a sugarcane plant. In the mulched plots, mulch was removed from a 20 x 20 cm sampling zone, and after it was weighed and cut into pieces less than 1 cm long, a 10 g sub-sample from each plot was spread on a nematode extraction tray [START_REF] Whitehead | A comparison of some quantitative methods of extracting small vermiform nematodes from soil[END_REF]. After 3 days, nematodes were recovered by sieving twice on a 38 µm-aperture sieve. Microarthropods were recovered from the mulch by placing 15-20 g sub-samples in a Tullgren funnel. Four soil cores 10 cm in diameter were collected to a depth of 5 cm from each of the plots where the mulch was present and from similar positions in the non-mulched plots. The soil in the cores collected from each plot was then mixed gently and nematodes and mites were extracted by placing 230 and 500 g sub-samples of moist soil on trays and funnels, respectively. Soil and mulch samples were also dried at 80°C to determine the dry weight equivalent of each sample. Numbers of microarthropods and nematodes recovered from the sub-samples and the weights of soil and mulch in the processed samples were used to calculate the number of microarthropods and nematodes /kg dry soil and the number of microarthropods and nematodes /10 g dry mulch.

Effects of soil texture and depth on microarthropods in pots

This experiment aimed to determine whether soil properties (in this case soil texture and depth in the soil profile) influenced the microarthropod community when mineral soil is mulched with sugarcane residue. The soil used for the experiment was obtained from six sugarcane fields in Bundaberg that were growing crops ranging in age from third to sixth ratoon and were on two different soil types. Three of the fields had sandy loam soils and three had clay loam (ferrosol) soil. In November 2014, soil was collected from each field at two depths in the soil profile (0-2 cm and 10-15 cm). Equal volumes of soil from each soil type and depth were then mixed together to produce four soils (sandy loam 0-2 cm; sandy loam 10-15 cm; clay loam 0-2 cm; clay loam 10-15 cm). Basic chemical properties such as pH, electrical conductivity, total C and N (Leco method) and labile C (method of [START_REF] Blair | Soil carbon fractions based on their degree of oxidation, and the development of a carbon management index for agricultural systems[END_REF] were assessed by Environmental Analysis Laboratory, Lismore, NSW, while microbial activity was measured using the Solvita™ CO 2 -burst protocol [START_REF] Franzluebbers | Should soil testing services measure soil biological activity?[END_REF]. These analyses showed that the four soils had markedly different properties. Carbon levels and microbial activity were much higher in the clay loam soil than the sandy loam soil and there were similar differences between soils collected from different depths (Table 1).

Four 1.4 L pots were filled with 1 kg dry weight equivalent of each of the soils and pots were planted with pre-germinated single-eye setts of sugarcane (variety Q208). The 16 pots were then placed in a greenhouse and the soil surface was covered with a layer of sugarcane residue (22 g per pot) to simulate the 15.5 t/ha trash blanket that is laid down when a 100t/ha sugarcane crop is harvested [START_REF] Mitchell | A simple method for estimating the return of nutrients in sugarcane trash[END_REF]. After 8 weeks, soil and roots were removed from the pots and nematodes and microarthropods were extracted by placing 210 g and 400 g of moist soil on trays and funnels, respectively.

Statistical analysis

Data were analysed by analysis of variance (ANOVA) using Genstat 8.2. The two experiments on the effects of mulching were analysed separately while the pot experiment was analysed as a soil texture x depth factorial. Microarthropod and nematode numbers were transformed (log 10 no. animals + 1) prior to analysis. Means ± SE were calculated and regression analyses undertaken using features available in Microsoft Excel.

Results

A survey of microarthropods in sugarcane soils

A total of 60 cane fields were sampled and a trash blanket had been retained as mulch in 50 of these sites while the remaining 10 sites lacked a trash blanket. In total, 7517 microarthropods were extracted and assigned to our three guilds. Numbers per 600 mL sample of mineral soil were highly variable (range of 4-539 in trash-blanketed sites vs 6-185 in sites with no trash blanket), but the animals were almost twice as abundant where the shelter of a trash layer was present (mean = 135.6±16.3 vs 73.9±19.5). The Burdekin sites generally had the lowest numbers of microarthropods, but there were no obvious effects of crop stage, irrigation, soil type, soil moisture or row spacing.

Our results indicate that Queensland sugarcane soils are dominated in numbers by our Fungivore/Detritivore Guild composed of oribatid mites (Acariformes: Oribatida) and springtails (Hexapoda: Collembola) (range 2-415 per sample, mean = 77.1±9.9). The oribatid mites typically were representatives of the Galumnidae, Oripodoidea (especially Scheloribatidae, Oribatulidae), Oppioidea, Nothridae, Epilohmanniidae, Lohmanniidae, Tectocepheidae, and Microzetidae. Other oribatid families were rare or sporadic, but a member of the Acaridae (Rhizoglyphus robini Claparéde) was occasionally abundant. Collembola were represented by members of all of the superfamilies (Poduroidea, Entomobryoidea, Neeloidea, Sminthuroidea). The rapid attraction to and consumption of vermiform nematodes was observed repeatedly for members of the oribatid mite families Galumnidae (Galumna sp., Pergalumna sp.) and Scheloribatidae (Scheloribates sp.) in the live extractions. One observation of nematophagy by an unidentified isotomid springtail was observed. Therefore, it seems likely that these animals function in cane soils as opportunistic nematophages, as they are known to do in other soils (see [START_REF] Walter | Mites: Ecology, Evolution & Behaviour: Life at a Microscale 2nd Edition[END_REF], for a recent review of nematophagy in mites).

Members of the Mesostigmata, the order of mostly predatory mites with numerous known nematophages, were much less abundant (range 1-80 per sample, mean = 14.1±1.9), but diverse (approximately three dozen species). These included members of genera with species known to feed on nematodes, e.g. Rhodacarus, Rhodacarellus (Rhodacaridae); Gamasellodes, Protogamasellus, Asca, Antennoseius (Ascidae); Lasioseius, Cheiroseius (Blattisociidae); Gaeolaelaps, Cosmolaelaps (Laelapidae); Holaspulus (Parholaspididae); Zygoseius (Pachylaelapidae); Antennolaelaps, Athiasella, Gamasiphis (Ologamasidae) and also unidentified Uropodina (some species are nematophagous, but most have unknown feeding habits). Nematophagy was observed in 15 mesostigmatan species from the Bundaberg collections and reproduction on a diet of nematodes was obtained for 11 of those species: Antennolaelaps sp., Asca garmani Hurlbutt, Asca major Womersley, Cheiroseius sp., Cosmolaelaps sp., Gaeolaelaps sp., Gamasellodes bicolor (Berlese), Gamasiphis sp., Holaspulus tenuipes Berlese, Protogamasellus mica (Athias-Henriot) and Protogamasellus sigillophorus Mineiro, Lindquist and De Moraes. The latter (misidentified as Protogamasellopsis sp. in [START_REF] Manwaring | Microarthropods as predators of nematode pests in sugarcane soils: a review and preliminary studies[END_REF] was described from a cornfield in Brazil [START_REF] Mineiro | Edaphic ascid mites (Acari: Mesostigmata: Ascidae) from the state of São Paulo, Brazil, with description of five new species[END_REF]) and is reported from Australia for the first time. Although many could not be assigned to a described species, those that were included both semi-cosmopolitan mites such as A. garmani, Gaeolaelaps aculeifer (Canestrini), Gamasellodes bicolor, H. tenuipes, and P. mica; and apparently Australasian endemics such as A. major, Gamasellodes adrianae Walter, Holostaspella moderata Berlese, Gaeolaelaps queenslandicus (Womersley), and Zygoseius sarcinulus Halliday.

Numbers in the Miscellaneous Arthropods group ranged from 2-349 individuals per sample, but most samples contained less than 50 such microarthropods, with the few larger numbers due to sampling at the margins of an ant colony or the presence of large numbers of scale insect crawlers. Predators of arthropods, both mites (e.g. Bdellidae, Cunaxidae, Rhagidiidae, Stigmaeidae) and other arthropods (e.g. diplurans such as Parajapyx sp. and unidentified Campodeidae, small spiders and centipedes) were common. Symphylans, some of which are root pests, were rare. Isopods, small millipedes, fly larvae, psocopterans and a variety of other small insects also were collected. Additionally, microarthropods thought to be fungivores, but not known to eat nematodes (e.g. members of the Ameroseiidae, Eupodidae, Nanorchestidae, Terpnacaridae, Tarsonemidae, Pygmephoridae [all Acari]; Ptilliidae [Coleoptera], Acerentomidae [Protura], Pauropodidae [Pauropoda]) were present in small numbers. With one exception, an unidentified cunaxid mite in the subfamily Cunaxoidinae (see [START_REF] Walter | Observations on Coleoscirus simplex (Acarina: Prostigmata), a predatory mite that colonizes greenhouse cultures of rootknot nematode (Meloidogyne spp.), and a review of feeding behavior in the Cunaxidae[END_REF], no members in these groups were observed feeding on nematodes in the live extractions.

Dominance by oribatid mites and springtails is generally associated with a fungus-based decomposer system and should be strongly influenced by the amount of organic matter in the soil and retention of sufficient water to maintain fungal growth [START_REF] Coleman | Fundamentals of Soil Ecology 2 nd Edition[END_REF]. Both of these parameters should be enhanced by the retention of a trash blanket. This hypothesis is supported by our survey results, as numbers of Fungivore/Detritivores were on average 55% higher under a trash blanket than in the absence of a trash blanket (81.9±11.5 vs 52.9±14.4 animals per sample). The effect of the trash blanket on the Mesostigmata was even more dramatic, with numbers being 206% higher under a trash blanket (15.5±2.1 vs 7.5±2.4). Farming system also seemed to have a strong influence on Mesostigmata with fields under best-practice management averaging 18.6±3.4 Mesostigmatans compared with 10.5±1.7 in fields using other farming systems. Total microarthropods were also much higher in the former system (150.6±23.6 vs 104.5±16.6).

Impact of mulching on microarthropods, particularly the nematophagous mites

Although the two study sites were less than 10 km apart, there were many more microarthropods at site 1 than site 2 (Table 2). However, analyses of the data from each site indicated that mulching did not significantly affect any component of the microarthropod community, as numbers in mulched and bare plots did not differ significantly.

There were more nematodes in the soil under the mulch than the bare soil (Table 2), with statistical analyses indicating that the mulch effect was significant at site 2 and almost significant at site 1. The reason for this was that there were more free-living nematodes (predominantly bacterivores and fungivores) under the mulch. However, there was no relationship between the nematode population and numbers of their microarthropod predators, as numbers of mites in the Small Mesostigmata guild were similar in mulched and bare soil.

The mulch at both sites contained microarthropods but as with the soil, total numbers were much higher at site 1 than site 2 (1123±223 vs 273±30/10 g mulch, respectively). Large numbers of free-living nematodes were also recovered from the mulch at both sites (an average of 8,899 and 12,947 nematodes/10 g mulch at sites 1 and 2 or the equivalent of 222,000 and 324,000 nematodes/m 2 , respectively). About 10% of the microarthropods in the mulch were nematophagous but the composition of the nematophagous community in the mulch differed from that in soil. The Cunaxoidinae (small, predatory prostigmatid mites) were relatively common in the mulch at both sites (comprising 33 and 32% of the nematophagous community at sites 1 and 2, respectively) whereas they comprised only 12% of the nematophagous community in soil at site 1 and were not recovered from the soil at site 2.

Effects of soil texture and depth on microarthropods in pots

Analyses of total microarthropod numbers showed that there were significant soil texture (2-way ANOVA, F 1,9 = 25.6, P < 0.001) and depth effects (F 1,9 = 10.8, P = 0.009). The clay loam soil supported many more microarthropods than the sandy loam soil and there were significantly more microarthropods in the topsoil than in soil collected from a depth of 10-15 cm (Table 3). Numbers in the Small Mesostigmata did not differ significantly between soils (F 1,9 = 1.17, P = 0.308) but there was a significant depth effect (F 1,9 = 55, P< 0.001), with these mites being far more numerous in the topsoil (Table 3). For neither total microarthropods (F 1,9 = 0, P = 0.948 nor small Mesostigmata (F 1,9 = 4.5, P = 0.062) was the interaction between soil texture and depth significant.

When the nematode data were analyzed the soil texture effect was not significant (F 1,9 = 2.75, P = 0.132) but there was a significant depth effect (F 1,9 = 20.9, P = 0.001) and a significant soil texture x depth interaction (F 1,9 = 9.97, P = 0.012). The most obvious effect was the low number of nematodes in pots filled with the sandy loam soil collected from a depth of 10-15 cm (Table 3). Regression analysis using log-transformed data showed that numbers of mesostigmatans increased as the nematode population increased and that there was a significant but relatively poor relationship between these parameters (P = 0.029; R 2 = 0.295).

Discussion

The microarthropod communities in Queensland sugarcane soils tend to be dominated by the Fungivore/Detritivore Guild (oribatid mites and springtails). These tiny arthropods feed primarily on decomposing plant matter and its associated fungi and other microbes, but many species are known to opportunistically prey on nematodes. Little is known about these animals in sugarcane soils, but only a restricted set of families of oribatid mites are present compared to those found in Australian forest soils (Walter, personal observation). These mites are abundant in well-structured sugarcane soils with mulch layers, and as well as contributing to soil health through their effects on decomposition and nutrient cycling, they may sometimes use nematodes as a food source and possibly act as vectors of microbial nematode antagonists (see [START_REF] Walter | Mites: Ecology, Evolution & Behaviour: Life at a Microscale 2nd Edition[END_REF].

In contrast, mesostigmatans actively prey on nematodes. Many species require nematodes in their diets for reproduction and Australian species have been shown to exhibit a high degree of nematophagy [START_REF] Beaulieu | Predation in suspended and forest floor soils: observations on Australian mesostigmatic mites[END_REF]. In cane soils we found about three-dozen species, many in genera known to have nematophagous species, and observed 15 species feeding on nematodes, including 11 species that were reared successfully with only nematode prey. Many of these are general predators that can subsist on alternative prey (e.g. springtails, mites, small insects), but show improved development and increased reproduction when nematodes are eaten [START_REF] Walter | Species, guilds and functional groups: taxonomy and behavior in nematophagous arthropods[END_REF]. As with the oribatid mites, only a subset of the mesostigmatan fauna was found in cane soils and many of the genera and species that we did find are common in other agricultural systems (see Walter and Kaplan, 1990a, b). Some of these mites are voracious nematophages and if large populations are present, they are likely to contribute to the development of nematode-suppressive soils.

In general, results from the 60 sites surveyed showed that the abundance of all microarthropod groups was highly variable. This is to be expected in a complex system where the soil biology is influenced by many factors. Although no obvious effects of crop stage, irrigation, soil type, soil moisture or row spacing were observed, sites where the trash blanket was retained after harvest and those where the farming system included crop rotation, minimum tillage and controlled traffic showed strong positive effects on the abundance of microarthropods. These effects were especially noticeable with the Mesostigmata, which are thought to be the most important microarthropod predators of nematodes.

Although our field experiments were set up to examine the effect of mulching on microarthropods, the results also showed that numbers of microarthropods were much higher in the clay loam (ferrosol) soil than the sandy clay loam. Ferrosols have much higher soil C levels than lighter-textured soils (see Table 1) and as soil organic matter is the energy source that fuels the soil food web, such a result was expected. However, the textural properties of the clay loam soil probably also contributed, as its polyhedral or blocky structure [START_REF] Mckenzie | Major soils used for agriculture in Australia[END_REF] meant that habitable pore space was available for microarthropods. Results from our pot experiment also showed that more microarthropods were recovered from the clay loam soil than a much lighter-textured sandy loam, again emphasising the fact that a soil's textural properties and carbon levels influence microarthropod abundance [START_REF] Vreeken-Buijs | Relationships of soil microarthropod biomass with organic matter and pore size distribution in soils under different land use[END_REF][START_REF] Larsen | The impact of soil compaction on euedaphic Collembola[END_REF][START_REF] Shakir | Seasonal abundance of soil arthropods in relation to meteorological and edaphic factors in the agroecosystems of Faisalabad, Punjab, Pakistan[END_REF].

Since our survey results showed that cane fields with mulch layers tended to have higher populations of microarthropods in mineral soil than fields where the mulch had been removed, and this was consistent with the findings of [START_REF] Manwaring | Effects of a mulch layer on the assemblage and abundance of mesostigmatan mites and other arthropods in the soil of a sugarcane agroecosystem in Australia[END_REF], we assumed that mulching was providing a soil moisture and temperature environment that favoured these animals. Thus, we were surprised that there was no significant difference in the number of microarthropods recovered from mulched and bare soils in our field experiments. However, total numbers of microarthropods at the two sites were 44 and 34% higher in plots that were mulched, suggesting that a mulch effect may have been obtained if more replicates had been included in the experiment or the treatments had been maintained for longer.

One clear result from the two field experiments was that large numbers of microarthropods are able to live in the mulch layer. Since free-living nematodes are also found in high numbers in the mulch, it is not surprising that nematophagous microarthropods are relatively common. However, one interesting observation was that the Cunaxoidinae were much more important component of nematophagous community in the mulch than the soil. This group of mites are comparable in body length to the Small Mesostigmata but have longer legs and are fast moving. We can only speculate as to why they mainly occur in mulch rather than soil but it is possible that they move more easily in the mulch layer due to its greater porosity.

Given the farm to farm variability in microarthropod numbers that we encountered in our survey, we believe that research should be undertaken to understand the main causes of this variability. Soil type and environmental conditions (particularly rainfall and temperature) will certainly be important causal factors but from a sugar industry perspective, it is also important to know whether the practices used to grow the crop are contributing. Our results suggest that microarthropod numbers are higher in soils farmed using best-practice systems but we surveyed only a limited number of sites, future research should focus on confirming that observation.

With regard to nematophagous mites and their role in regulating populations of plantparasitic nematodes, it was encouraging to find a diverse range of these animals in sugarcane soils. However, if we are to determine whether they are helping to reduce populations of key nematode pests more work will be required. Since many potential food sources are available in soil, we need to understand the food preferences of individual species and determine whether plant-parasitic nematodes are an important component of their diet in the field. The other key question is whether a particular predator is able to access pest nematodes such as Meloidogyne and Pratylenchus, as they are commonly found at depths of 5-30 cm rather than in the surface soil where most microarthropods are located [START_REF] Stirling | Soil carbon, root health and nematode pests in sugarcane soils. 1. Root and soil health and its relationship to soil carbon levels[END_REF]. This means that researchers must focus on species that live in deeper layers of the soil profile; determine whether parameters such air-filled porosity, bulk density, and body size or shape influence their distribution; and identify the main pathways used by nematophagous mites to access their prey (e.g. the channels that remain when roots decompose or the macropores made by earthworms and other ecosystem engineers).

Table 2

 2 Mean abundances of microarthropods and nematodes recovered from soil six months after mulched and bare plots were established in two sugarcane fields at Bundaberg, Queensland.Microarthropod and nematode numbers were transformed [log 10 (no. animals + 1)] prior to analysis and backtransformed means are presented. Within sites, there was no difference between bare and mulched soil except for nematodes at Site 2.

				No. animals/kg soil
			Site 1		Site 2
		Bare	Mulch	ANOVA	Bare Mulch	ANOVA
	Fungivore/detritivores 428	518	F 1,5 =0.19, P=0.683 46	38 F 1,5 =0.37, P=0.568
	Small Mesostigmata	34	31	F 1,5 =0.14, P=0.548	5	6 F 1,5 =0.08, P=0.791
	All microarthropods	496	588	F 1,5 =0.18, P=0.686 59	69 F 1,5 =0.27, P=0.626
	All nematodes	8053	18492	F 1,5 =6.3, P=0.053 10092 15703 F 1,5 =13.7, P=0.014

Table 3

 3 Mean numbers of microarthropods and nematodes, after sugarcane was grown in pots for 8 weeks in clay loam and sandy loam soils collected at two depths in the soil profile: 0-2 cm and 10-15 cm.

	Clay loam	Sandy loam	0-2 cm	10-15 cm
	Small Mesostigmata/kg soil	42 a	32 a	101 a	13 b
	Total microarthropods/kg soil	802 a	166 b	608 a	219 b
				0-2 cm	10-15 cm
	Nematodes/kg soil		Clay loam	30,335 a	23,440 a
			Sandy loam	39,895 a	9,745 b
	Microarthropod and nematode numbers were transformed [log 10 (no. animals + 1)] prior to
	analysis and back-transformed means are presented. Values followed by the same letters are
	not significantly different (P = 0.05).

Walter D.E. and Stirling G.R. (2018), Acarologia 58(3): 673-682; DOI 10.24349/acarologia/20184264

Acknowledgements

Sugar Research Australia provided financial support for this study (project 2014/004). We thank Dr Helen Nahrung for help with the statistical analysis.