Antoine Ayache
email: antoine.ayache@univ-lille.fr

Narn-Rueih Shieh
email: shiehnr@ntu.edu.tw

Yimin Xiao
email: xiao@stt.msu.edu

Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets

Keywords: Harmonizable Fractional Stable Sheets 2000 AMS Classification numbers: 60G52, 60G17, 60G18, 60G60 Harmonizable fractional stable sheets, Wavelet series representation, LePage representation, Strong local nondeterminism, Local times, Hausdorff dimension, Inverse image. 1

Let Z H = {Z H (t), t ∈ R N } be a real-valued N -parameter harmonizable fractional stable sheet with index H = (H 1 , . . . , H N ) ∈ (0, 1) N . We establish a random wavelet series expansion for Z H which is almost surely convergent in all the Hölder spaces C γ ([-M, M ] N ), where M > 0 and γ ∈ (0, min{H 1 , . . . , H N }) are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure.

Also, let X = {X(t), t ∈ R N } be an R d -valued harmonizable fractional stable sheet whose components are independent copies of Z H . By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image X -1 (F ) holds for all Borel sets F ⊆ R d . This is referred to as a uniform Hausdorff dimension result for the inverse images.

Introduction

For any given 0 < α < 2 and H = (H 1 , . . . , H N ) ∈ (0, 1) N , let Z H = {Z H (t), t ∈ R N } be a real-valued harmonizable fractional α-stable sheet (HFSS, for brevity) with index H, defined by

Z H (t) := Re R N N j=1
e it j λ j -1

|λ j | H j + 1 α M α (dλ), (1.1)
where M α is a complex-valued rotationally invariant α-stable random measure with Lebesgue control measure; we refer to Chapter 6 of [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF], for a detailed presentation of such random measures as well as the corresponding stochastic integrals.

From (1.1) it follows that Z H has the following operator-scaling property: for any N × N diagonal matrix E = (b ij ) with b ii = b i > 0 for all 1 ≤ i ≤ N and b ij = 0 if i = j, we have

Z H (Et), t ∈ R N d = N j=1 b H j j Z H (t), t ∈ R N . (1.
2)

The property (1.2) is also called "multi-self-similarity" by Genton, Perrin and Taqqu (2007). By using (1.1), one can verify that, along each canonical direction of R N , Z H becomes a realvalued harmonizable fractional stable motion; a detailed presentation of the latter process, as well as other self-similar stable processes is given in [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF]. When the indices H 1 , . . . , H N are not the same, Z H has different scaling behavior along different directions of R N . Namely, Z H is anisotropic in the "time" variable. If one replaces in (1.1) the parameter α by 2, then Z H becomes a fractional Brownian sheet (FBS, for brevity) denoted by B H . Sample path properties of FBS and several related classes of anisotropic Gaussian random fields have been studied by several authors; see, for example, [START_REF] Kamont | On the fractional anisotropic Wiener field[END_REF], [START_REF] Ayache | Drap brownien fractionnaire[END_REF], [START_REF] Ayache | Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets[END_REF], [START_REF] Ayache | Joint continuity of the local times of fractional Brownian sheets[END_REF], Wu andXiao (2007, 2011), [START_REF] Xiao | Sample path properties of anisotropic Gaussian random fields[END_REF] and the references therein. Observe that, up to a multiplicative constant, the Gaussian field B H , can also be represented as a moving average Wiener integral, in which the complex-valued kernel N j=1 e it j λ j -1 |λ j | -H j -1/2 is replaced by the real-valued kernel N j=1 (t j -s j )

H j -1/2 + -(-s j ) H j -1/2 +
, with the convention that for all real-numbers x and β, (x) β + = x β when x > 0 and (x) β + = 0 else. However, for 0 < α < 2, the harmonizable α-stable field Z H does not have such a property, and it is very different from the real-valued linear fractional stable field (LFSS, for brevity) Y H = {Y H (t), t ∈ R N } which is defined by

Y H (t) := R N N j=1 (t j -s j ) H j -1 α + -(-s j ) H j -1 α + M α (ds), (1.3)
where M α is a real-valued α-stable random measure.

The main purpose of the present article is to study sample path properties of HFSS and compare them with the properties of LFSS obtained in [START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF]. In Section 2 we establish a random wavelet series representation for Z H ; to this end, we expand the corresponding harmonizable kernel in (1.1) in terms of the Fourier transform of the tensor product of a Lemarié-Meyer wavelet basis for L 2 (R). One of the main difficulties in this matter is that, when α < 2 we do not know whether the Fourier transform of a Lemarié-Meyer wavelet basis is an unconditional basis for L α (R). This is in contrast with the case of LFSS in [START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF], where one does not have to use the Fourier transform of a wavelet basis. In order to overcome this difficulty, we use some specific properties of the harmonizable kernel as well as the fact that the Fourier transform of a Lemarié-Meyer mother wavelet is compactly supported and vanishes in a neighborhood of the origin. Also, it is worth noticing that, the behavior of the random coefficients, in our wavelet expansion of HFSS, can be estimated through their LePage representations. This, in turn, leads to the conclusion that the wavelet expansion of Z H is, for any α ∈ (0, 2), almost surely convergent in all the Hölder spaces C γ ([-M, M ] N ), where M > 0 and γ ∈ (0, min{H 1 , . . . , H N }) are arbitrary constants. In the case of LFSS, a weaker result holds: under the condition that min{H 1 , . . . , H N } > 1/α, the wavelet expansion of Y H is almost surely convergent in all the Hölder spaces C γ ([-M, M ] N ), where M > 0 and γ < min{H 1 , . . . , H N } -1/α (see [START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF]). Notice that, in contrast with HFSS, when min{H 1 , . . . , H N } ≤ 1/α, the LFSS sample path is discontinuous and even becomes unbounded on every open set when the latter inequality is strict. The results in this section may be compared with those of [START_REF] Ayache | Stationary increments harmonizable stable fields: upper estimates on path behaviour[END_REF], where regularity properties of harmonizable stable random fields with stationary increments are studied by using the wavelet method, and those of Xiao (2010), Biermé andLacaux (2009, 2015), [START_REF] Panigrahi | Maximal moments and uniform modulus of continuity of stable random fields[END_REF], where uniform modulus of continuity of stable random fields including harmonizable ones are studied by using different methods.

In Section 3, we consider the (N, d) harmonizable fractional stable sheet

X = {X(t), t ∈ R N } with values in R d defined by X(t) = X 1 (t), . . . , X d (t) , (1.4)
where X 1 , . . . , X d are independent copies of Z H . Our main result of this section is Theorem 3.9 which establishes a uniform Hausdorff dimension result for the inverse images X -1 (F ) for all Borel sets F ⊂ R d , which are either deterministic or random. This theorem is new even for fractional Brownian sheets (i.e., α = 2) and solves the problem raised in Remark 2.8 in Biermé, Lacaux and [START_REF] Xiao | Sample path properties of anisotropic Gaussian random fields[END_REF]. The methodology for proving this uniform Hausdorff dimension result is based on the regularity properties (e.g., Hölder conditions in the set-variable) of the local times of X which is of independent interest for (N, d) random fields. The results of this section motivate several questions that will need further investigation. At the end of Section 3, we provide some remarks and open questions.

Throughout this paper we use | • | to denote the Euclidean norm in R N . The inner product and Lebesgue measure in R N are denoted by •, • and λ N , respectively. A vector t ∈ R N is written as t = (t 1 , . . . , t N ), or as

c if t 1 = • • • = t N = c.
For any s, t ∈ R N we write s ≺ t if s j < t j for all j = 1, . . . , N . In this case [s, t] = N j=1 [s j , t j ] is called a closed interval (or a rectangle). We will let A denote the class of all closed intervals in R N . For two functions f and g, the notation f (t) g(t) for t ∈ T means that the function f (t)/g(t) is bounded from below and above by positive constants that do not depend on t ∈ T .

We will use c and c(n) to denote unspecified positive and finite constants, the latter depends on n. Both of them may not be the same in each occurrence. More specific constants are numbered as c 1 , c 2 , . . .. the financial support from CEMPI are appreciated. The research of Yimin Xiao is partially supported by NSF grants DMS-1612885 and DMS-1607089.

Wavelet series representation of Z H

The goal of this section is to establish a random wavelet series representation for HFSS Z H . First we fix some notation that will be extensively used in the sequel.

(i) The function ψ denotes a usual Lemarié-Meyer mother wavelet, see [START_REF] Lemarié | Ondelettes et bases hilbertiennes[END_REF], [START_REF] Meyer | Wavelets and Operators[END_REF] and [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. The function ψ satisfies the following properties:

(a) ψ is real-valued and belongs to the Schwartz class S(R).

(b) ψ, the Fourier transform of ψ, is compactly supported and its support is contained in the ring {ξ ∈ R : 2π 3 ≤ |ξ| ≤ 8π 3 }. Recall that the Fourier transform of a function f ∈ L 2 (R) is the limit of the Fourier transforms of functions of the Schwartz class S(R) converging to f ; throughout this article the Fourier transform over S(R) is defined as (Fg)(ξ) = g(ξ) = (2π) -1/2 R e -is•ξ g(s) ds and the inverse map as

(F -1 h)(s) = (2π) -1/2
R e is•ξ h(ξ) dξ, thus the Fourier transform is a bijective isometry from L 2 (R) to itself.

(c) The sequence {ψ j,k : (j, k) ∈ Z 2 } forms an orthonormal basis of L 2 (R), where

ψ j,k (x) := 2 j/2 ψ(2 j x -k), ∀ x ∈ R. (2.1)
Note that the isometry property of the Fourier transform implies that the sequence ψ j,k : (j, k) ∈ Z 2 forms an orthonormal basis of L 2 (R) as well. Moreover, simple computation shows that, for all ξ ∈ R, ψ j,k (ξ) = 2 -j/2 e -ik2 -j ξ ψ(2 -j ξ).

(2.2)

For the sake of convenience, for each v ∈ (0, 1) and x ∈ R, we set

w v j,k (x) := R f v (x, ξ) ψ j,k (ξ) dξ, (2.3)
where the function f v (x, •) is defined by f v (x, 0) = 0 and, for all ξ ∈ R \ {0}, by

f v (x, ξ) := e ixξ -1 |ξ| v+ 1 α . (2.4)
Observe that when

f v (x, •) ∈ L 2 (R) (this is equivalent to 1 < 2v + 2 α < 3)
, then, by using the fact that ψ j,k : (j, k) ∈ Z 2 is an orthonormal basis of the latter space, one has

f v (x, •) = j,k∈Z w v j,k (x) ψ j,k (•), (2.5)
where the series converges in the L 2 (R) norm. Roughly speaking, the key idea for obtaining a wavelet representation for HFSS consists in showing that the equality (2.5) holds in L α (R), even when f v (x, •) / ∈ L 2 (R). For this purpose, it is convenient to renormalize the functions ψ j,k in such a way that their L α (R) norms be equal to ψ L α (R) .

(ii) For each (j, k) ∈ Z 2 and ξ ∈ R, we set

ψ α,j,k (ξ) := 2 -j( 1 α -1 2 ) ψ j,k (ξ) = 2 -j/α e -ik2 -j ξ ψ(2 -j ξ).
(2.6)

Moreover for every x ∈ R, we set

w v α,j,k (x) := 2 j( 1 α -1 2 ) w v j,k (x) = 2 j( 1 α -1) R e ixξ -1 |ξ| v+1/α e -ik2 -j ξ ψ(2 -j ξ) dξ. (2.7)
Observe that, for all x, ξ ∈ R, one has

w v α,j,k (x) ψ α,j,k (ξ) = w v j,k (x) ψ j,k (ξ). (2.8)
(iii) In order to conveniently express w v α,j,k (x), let us introduce, for each fixed v ∈ (0, 1), the function ψ v , defined as

ψ v (y) := R e iyη ψ(η) |η| v+1/α dη, ∀ y ∈ R. (2.9)
Observe that in view of the properties (a) and (b) of the Lemarié-Meyer mother wavelet ψ, one can easily show that ψ v is a well-defined real-valued function which belongs to S(R). Furthermore, the change of variable η = 2 -j ξ in (2.7) yields that, for all x ∈ R,

w v α,j,k (x) = 2 -jv ψ v (2 j x -k) -ψ v (-k) . (2.10)
(iv) Let { J,K , (J, K) ∈ Z N × Z N } be the sequence of complex-valued SαS random variables defined as

J,K = R N Ψ α,J,K (λ) M α (dλ), (2.11)
where

Ψ α,J,K (λ) := N l=1 ψ α,j l ,k l (λ l ), ∀λ ∈ R N . (2.
12)

It is easy to verify that, for every (J, K) ∈ Z N × Z N , the scale parameter of J,K is

J,K α = ψ N L α (R) .
Hence the random variables { J,K , (J, K) ∈ Z N ×Z N } are identically distributed.

We are now in position to state our first main result of the section. Proposition 2.1 Let (D n ) n∈N be an increasing sequence of finite subsets of Z N × Z N which "converges" to Z N × Z N (i.e., for every n ∈ N, D n ⊂ D n+1 and ∪ n∈N D n = Z N × Z N ). For each n ∈ N and t ∈ R N , we denote by U n (t) the real-valued random variable defined as

U n (t) = Re (J,K)∈Dn 2 -J,H J,K N l=1 ψ H l (2 j l t l -k l ) -ψ H l (-k l ) .
(2.13)

Then U n (t) converges in probability to Z H (t) when n goes to infinity.

In order to be able to prove Proposition 2.1, we need some preliminary results.

Remark 2.2 Recall that, in contrast to the case where α ≥ 1, when α ∈ (0, 1), the map

f → R N |f (λ)| α dλ 1/α
is no longer a norm on L α (R N ); however one can define a metric ∆ on this space by

∆ L α (R N ) (f, g) := R N |f (λ) -g(λ)| α dλ, ∀ (f, g) ∈ L α (R N ) × L α (R N ).
(2.14)

Moreover the resulting metric space is complete. For the sake of convenience, when α ≥ 1, one sets for every (f, g)

∈ L α (R N ) × L α (R N ), ∆ L α (R N ) (f, g) := f -g L α (R N ) = R N |f (λ) -g(λ)| α dλ 1/α . (2.15)
For simplicity we will abuse the notation slightly and write f L α (R N ) as f α , which should not be confused with the scale parameter of a SαS random variable.

We will make use of the following elementary lemma.

Lemma 2.3 Let (E, d) be a complete metric vector space such that the metric d is translation invariant, namely for all x, y, z ∈ E one has,

d(x + z, y + z) = d(x, y). (2.16)
Let (a i ) i∈I be an arbitrary sequence of elements of E which satisfies

i∈I d(a i , 0) < +∞. (2.17)
Then there is a unique element a ∈ E satisfying the following property: Proof Relation (2.16) implies that, for every p ∈ N and m ∈ N,

d i∈D m+p a i , i∈Dm a i = d i∈D m+p \Dm a i , 0 . (2.18)
Moreover, using the triangle inequality and (2.16), one can prove, by induction on the cardinality of D m+p \ D m , that,

d i∈D m+p \Dm a i , 0 ≤ i∈I\Dm d(a i , 0). (2.19)
Putting together (2.17), (2.18) and (2.19), one can easily show that i∈Dn a i n∈N is a Cauchy sequence of (E, d), which in turn implies that it converges to some limit a ∈ E. Let us now show that the limit a does not depend on the choice of the sequence (D n ) n∈N . Let (D n ) n∈N and (D n ) n∈N be two arbitrary increasing sequences of finite subsets of I which converge to I. We denote by a and a , respectively, the limits of the sequences ( i∈D n a i ) n∈N and ( i∈D n a i ) n∈N . Observe that there exists an increasing map φ : N → N such that, for every n ∈ N, one has D n ⊆ D φ(n) . Therefore, it follows from (2.16) that, for each n ∈ N,

d i∈D φ(n) a i , i∈D n a i = d i∈D φ(n) \D n a i , 0 ≤ i∈I\D n d(a i , 0), (2.20)
where the last inequality is derived in the same way as the inequality (2.19). Finally, letting n go to infinity, one obtains, in view of (2.17) and (2.20), that d(a , a ) = 0.

In the following lemma, recall that w H l α,j l ,k l (t l ) is defined in (2.7) and Ψ α,J,K in (2.12).

Lemma 2.4 For each fixed t ∈ R N , one has:

(J,K)∈Z N ×Z N ∆ L α (R N ) N l=1 w H l α,j l ,k l (t l ) Ψ α,J,K , 0 < +∞. (2.21)
Consequently, in view of Lemma 2.3, there exists a function

F (t) ∈ L α (R N ) such that, for every increasing sequence (D n ) n∈N of finite subsets of Z N × Z N which converges to Z N × Z N , one has lim n→+∞ ∆ L α (R N ) (J,K)∈Dn N l=1 w H l α,j l ,k l (t l ) Ψ α,J,K , F (t) = 0. (2.22)
In order to show (2.21), we first estimate the decreasing rate of the coefficients w v α,j,k (x). Lemma 2.5 For any constant L > 0, the following two results hold:

(i) There exists a constant c 1 > 0, depending only on L, v and α, such that for each x ∈ R, j ∈ Z + and k ∈ Z, one has

|w v α,j,k (x)| ≤ c 1 2 -jv 2 + |2 j x -k| -L + 2 + |k| -L . (2.23)
(ii) For any positive number M , there exists a constant c 2 > 0, depending only on L, v, α and M , such that for each x ∈ [-M, M ], j ∈ Z -and k ∈ Z, one has

|w v α,j,k (x)| ≤ c 2 2 (1-v)j 2 + |k| -L . (2.24)
Proof Part (i) follows easily from (2.10) and the fact that ψ v ∈ S(R). Next we prove Part (ii). Denote by (ψ v ) the derivative of ψ v , then by using again the fact that ψ v ∈ S(R), one has, for all y ∈ R,

(ψ v ) (y) ≤ c 3 2 + M + |y| -L , (2.25)
where c 3 is a constant depending on L, v, α and M only. On the other hand (2.10) and the Mean Value Theorem imply that, for each x ∈ [-M, M ], j ∈ Z -and k ∈ Z, one has

w v α,j,k (x) = 2 j(1-v) x(ψ v ) (2 j d -k), (2.26)
where d ∈ (-M, M ). Thus, combining (2.25) with (2.26) and the triangle inequality, one gets (2.24).

Proof of Lemma 2.4 We will only give the proof for the case where α ∈ (0, 1), the proof in the case α ∈ [1, 2) is rather similar. First, notice that, for every (J, K) ∈ Z N × Z N , one has,

R N Ψ α,J,K (λ) α dλ = R | ψ(ξ)| α dξ N := c 4 .
Therefore, using (2.14), one gets that, for all t ∈ R N ,

(J,K)∈Z N ×Z N ∆ L α (R N ) N l=1 w H l α,j l ,k l (t l ) Ψ α,J,K , 0 = c 4 N l=1 (j l ,k l )∈Z×Z |w H l α,j l ,k l (t l )| α . (2.27)
On the other hand, it follows from (2.23), (2.24), and the inequality (x + y) α ≤ x α + y α for x, y ≥ 0 that, for every l = 1, . . . , N ,

(j l ,k l )∈Z×Z |w H l α,j l ,k l (t l )| α ≤ c 5 +∞ j l =0 2 -j l αγ +∞ k l =-∞ (2 + |2 j l t l -k l |) -Lα + (2 + |k l |) -Lα , (2.28)
where γ = min{H l , 1 -H l } and c 5 is a constant independent of (j l , k l ). Moreover for every constant L which satisfies Lα >

F (t, λ) = N l=1 f H l (t l , λ l ) := N l=1 e it l λ l -1 |λ l | H l + 1 α .
(2.30)

Proof For any constant M > 0 and n ∈ N, we set

D n,M := {(j, k) ∈ Z 2 : |j| ≤ n and |k| ≤ M 2 n+1 } (2.31) and D N n,M := (J, K) ∈ Z N × Z N : (j l , k l ) ∈ D n,M for each l = 1, . . . , N . (2.32)
Similarly to (2.21), one can show that, for every fixed t ∈ R N and l = 1, . . . , N ,

(j l ,k l )∈Z×Z ∆ L α (R) w H l α,j l ,k l (t l ) ψ α,j l ,k l , 0 < +∞.
By applying Lemma 2.3 with D n = D n,M , we see that there exists a function

f l (t l ) ∈ L α (R) such that lim n→+∞ ∆ L α (R) (j l ,k l )∈D n,M w H l α,j l ,k l (t l ) ψ α,j l ,k l , f l (t l ) = 0. ( 2
= D N n,M that, for almost all λ ∈ R N , F (t, λ) = N l=1 f l (t l , λ l ).
Thus, in order to show (2.30), it is sufficient to prove that, for every l = 1, . . . , N and for almost all λ l ∈ R, one has

f l (t l , λ l ) = f H l (t l , λ l ). (2.34)
For every m ∈ N, let h m be the function defined, for all λ l ∈ R, as

h m (λ l ) = 0, if |λ l | ≤ 2 -m+1 π 3 , 1, else. (2.35)
Observe that (2.4) and (2.35)

imply that f H l (t l , •)h m (•) ∈ L 2 (R) and, since ψ j l ,k l : (j l , k l ) ∈ Z 2
is an orthonormal basis of the latter space, one gets

lim n→+∞ R f H l (t l , λ l )h m (λ l ) - (j l ,k l )∈D n,M w j l ,k l (t l ) ψ j l ,k l (λ l ) 2 dλ l = 0, (2.36)
where

w j l ,k l (t l ) := R f H l (t l , λ l )h m (λ l ) ψ j l ,k l (λ l ) dλ l . (2.37)
On the other hand, the property (b) of ψ (given at the beginning of this section) and (2.2) entails that

supp ψ j l ,k l ⊆ λ l ∈ R : 2 j l +1 π 3 ≤ |λ l | ≤ 2 j l +3 π 3 . (2.38)
Putting together (2.37), (2.35), (2.38) and (2.3), we see that, for every j l ≥ -m and k l ∈ Z,

w j l ,k l (t l ) = w H l j l ,k l (t l ). (2.39)
Denote by C m the ring,

C m := λ l ∈ R : 2 -m+3 π 3 ≤ |λ l | ≤ 2 m+3 π 3 . (2.40)
From now on, we assume that λ l ∈ C m . Notice that (2.38) implies that, for all j l ≤ -m and k l ∈ Z, one has that

ψ j l ,k l (λ l ) = 0.
Therefore, in view of (2.39), one obtains that

(j l ,k l )∈D n,M w j l ,k l (t l ) ψ j l ,k l (λ l ) = (j l ,k l )∈D n,M w H l j l ,k l (t l ) ψ j l ,k l (λ l ). (2.41)
On the other hand (2.35) entails that

f H l (t l , λ l )h m (λ l ) = f H l (t l , λ l ) for λ l ∈ C m . ( 2
f H l (t l , λ l ) - (j l ,k l )∈D n,M w H l j l ,k l (t l ) ψ j l ,k l (λ l ) 2 dλ l = 0.
Since C m is a bounded set, Hölder's inequality and (2.8) imply that

lim n→+∞ Cm f H l (t l , λ l ) - (j l ,k l )∈D n,M w H l α,j l ,k l (t l ) ψ α,j l ,k l (λ l ) α dλ l = 0. (2.43)
On the other hand, (2.33) entails that

lim n→+∞ Cm f l (t l , λ l ) - (j l ,k l )∈D n,M w H l α,j l ,k l (t l ) ψ α,j l ,k l (λ l ) α dλ l = 0. ( 2
λ l ∈ C m , one has f H l (t l , λ l ) = f l (t l , λ l ).
Finally, by using the latter equality and the fact that R \ {0} = ∪ m∈N C m , one gets (2.34).

We are now in position to show Proposition 2.1. Proof of Proposition 2.1 This proposition is a straightforward consequence of Lemmas 2.4 and 2.6 as well as of the following standard result on integrals with respect to stable measures: if [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF]).

g n converges to g in L α (R N ) then R N g n (λ) M α (dλ) converges to R N g(λ) M α (dλ) in probability (see
The second main result of this section is the following theorem.

Theorem 2.7 For any constant M > 0 and n ∈ N we denote by U * n the real-valued random function defined, for every t ∈ R N , as

U * n (t) = Re (J,K)∈D N n,M 2 -J,H J,K N l=1 ψ H l (2 j l t l -k l ) -ψ H l (-k l ) ,
where D N n,M is defined in (2.32). Then, the following two results hold:

(i) with probability 1, {U * n } n∈N is a Cauchy sequence in the Hölder space C γ [-M, M ] N of any order γ < min{H 1 , . . . , H N }, the limit is denoted by U * ;

(ii) the random field

U * = {U * (t)} t∈[-M,M ] N is a version of the HFSS {Z H (t)} t∈[-M,M ] N .
Proof Part (i) of the theorem can be proved by using the wavelet method similar to that in the proof of Proposition 6 in Ayache et al (2009). It also follows from Proposition 2.8 below, which provides a sharp "deterministic" upper bound for the random variables | J,K |. Part (ii) of the theorem is an easy consequence of Part (i) and of Proposition 2.1 with

D n = D N n,M .
Proposition 2.8 There is an event Ω * of probability 1 such that for all fixed η > 0 there exists a random variable C > 0 (depending on Ω * , η and α) which satisfies the following property:

(i) If α ∈ (0, 1), then for every ω ∈ Ω * and for all (J, K)

∈ Z N × Z N , | J,K (ω)| ≤ C(ω) N l=1 (1 + |j l |) 1/α+η . (2.45) (ii) If α ∈ [1, 2)
, then for every ω ∈ Ω * and for all (J, K)

∈ Z N × Z N , | J,K (ω)| ≤ C(ω) N l=1 (1 + |j l |) 1/α+η log(2 + |j l |) log(2 + |k l |) . (2.46)
The proof of Proposition 2.8 mainly relies on a LePage series representation of the complexvalued SαS process J,K : (J, K) ∈ Z N × Z N . We skip it since it is similar to that of Lemma 2.7 in Ayache and Boutard (2017).

Uniform Hausdorff dimension result for the inverse images

Let X = {X(t), t ∈ R N } be an (N, d) harmonizable fractional stable sheet defined in (1.4). For any Borel set F ⊂ R d , we define the inverse image X -1 (F ) by

X -1 (F ) = {t ∈ (0, ∞) N : X(t) ∈ F }.
Notice that we have avoided the boundary of R N + on which X(t) ≡ 0 a.s. This causes little loss of generality.

When α = 2 (i.e., X is a fractional Brownian sheet in R d ) and F is fixed, Biermé, Lacaux and Xiao (2009, Theorem 2.3) proved the following result on the Hausdorff dimension of X -1 (F ):

dim H X -1 (F ) L ∞ (P) = min 1≤k≤N k j=1 H k H j + N -k -H k d -dim H F , (3.1)
where for any function Y : Ω → R + , Y L ∞ (P) is defined as

Y L ∞ (P) = sup θ : Y ≥ θ on an event E with P(E) > 0 .
Observe that in (3.1), the probability P{dim H X -1 (F ) > 0} depends on F . In Remark 2.8 in Biermé, Lacaux and Xiao (2009), they asked the following question: If N j=1 1/H j > d, does there exist a single event Ω 1 ⊆ Ω of positive probability such that on Ω

1 , dim H X -1 (F ) = min 1≤k≤N k j=1 H k H j + N -k -H k d -dim H F
holds for all Borel sets F ⊆ R d ? This is referred to as a uniform Hausdorff dimension problem for the inverse images of X. Our objective of this section is to solve this problem for the (N, d) harmonizable fractional α-stable sheet X with α ∈ [START_REF] Adler | The Geometry of Random Fields[END_REF][START_REF] Ayache | Stationary increments harmonizable stable fields: upper estimates on path behaviour[END_REF]. Our main result is Theorem 3.9 below.

Some preliminaries results

Let us collect some known results on harmonizable fractional stable sheets, which will be useful for later sections.

For any n ≥ 1 and t 1 , . . . , t n ∈ R N , the characteristic function of the joint distribution of Z H (t 1 ), . . . , Z H (t n ) is given by

E exp i n j=1 u j Z H (t j ) = exp - n j=1 u j F (t j , •) α α , (3.2)
where

u j ∈ R (1 ≤ j ≤ n), F (t, λ) is the function in (2.30). Recall that for every f ∈ L α (R N ), f α denotes f L α (R N ) .
It follows from (3.2) that the scale parameter of the SαS random variable n j=1 u j Z H (t j ) is

n j=1 u j Z H (t j ) α := n j=1 u j F (t j , •) α . (3.3)
This allows us to describe some probabilistic properties of Z H by the analytic properties of the functions F (t, •) and the geometric structures of the space L α (R N ). Lemma 3.1 is proved in Xiao (2011) for α ∈ [1, 2) and in [START_REF] Wu | Geometric properties of the images of fractional Brownian sheets[END_REF] for α = 2. Part (i) shows that for all 0 < a < b and s, t ∈ [a, b] N the scale parameter of Z H (s) -Z H (t) is comparable with ρ(s, t), which is the metric ρ on R N defined by

ρ(s, t) = N j=1 |s j -t j | H j , ∀ s, t ∈ R N . (3.4)
Part (ii) says that Z H has the property of sectorial local nondeterminism.

Lemma 3.1 Suppose α ∈ [1, 2] and 0 < a < b are constants. Then there exist constants c 6 ≥ 1 and c 7 > 0, depending on a, b, H and N only, such that the following properties hold:

(i) For all s, t ∈ [a, b] N . c -1 6 ρ(s, t) ≤ Z H (s) -Z H (t) α ≤ c 6 ρ(s, t). (3.5)
(ii) For all positive integers n ≥ 2 and all t 1 , . . . , t n ∈ [a, b] N , we have

Z H (t n ) Z H (t 1 ), . . . , Z H (t n-1 ) α ≥ c 7 N j=1 min 0≤k≤n-1 t n j -t k j H j , (3.6)
where Z H (t n ) Z H (t 1 ), . . . , Z H (t n-1 ) α is the L α (R N )-distance from F (t n , •) to the subspace generated by F (t j , •) (j = 0, 1, . . . , t n-1 ).

Remark 3.2 We believe that (3.6) still holds if α ∈ (0, 1), but we have not been able to prove this. The Fourier analytic method in Ayache and Xiao (2016) works well for proving the property of local nondeterminism for harmonizable fractional α-stable fields with stationary increments for all α ∈ (0, 2). Unfortunately, the last part of the proof of Theorem 2.1 in Ayache and Xiao (2016) breaks down for harmonizable fractional stable sheets.

In order to make use of the property of sectorial local nondeterminism, we need the following useful lemma from [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF]. The case of α = 2 is included for completeness and in this case it can be shown that c(n) does not depend on n. Lemma 3.3 Assume α ∈ [1, 2] and 0 < a < b are constants. For all integers n ≥ 2 there exists a positive constant c(n) such that for all t 1 , . . . , t n ∈ [a, b] N and u 1 , . . . , u n ∈ R,

n j=1 u j Z H (t j ) α ≥ c(n) |v 1 | Z H (t 1 ) α + n j=2 |v j | Z H (t j ) Z H (t 1 ), . . . , Z H (t j-1 ) α .
(3.7)

In the above, (v 1 , . . . , v n ) = (u 1 , . . . , u n ) A, (3.8)
where A = (a ij ) is an n × n lower triangle matrix (which depends on t 1 , . . . , t n ) with a ii = 1 for all 1 ≤ i ≤ n.

Remark 3.4 Roughly speaking, in (3.7) we expand n j=1 u j Z H (t j ) α by repeatedly "conditioning" Z H (t j ), given Z H (t 1 ), . . . , Z H (t j-1 ). Moreover, this "conditioning" can be done in an arbitrary order of the random variables Z H (t 1 ), . . . , Z H (t n ). This observation will be useful for studying regularity properties of the local times of Z H below.

The sample function of Z H is continuous and has the following uniform modulus of continuity, which was proved in [START_REF] Xiao | On uniform modulus of continuity of random fields[END_REF]. A similar result for operator-scaling stable random fields with stationary increments was proved by Biermé andLacaux (2009, 2015). Lemma 3.5 Let Z H = {Z H (t), t ∈ R N } be a real-valued harmonizable fractional stable sheet defined by (1.1) Then for any constants 0 ≤ a < b and ε > 0, one has

lim h→0 sup s,t∈[a,b] N |s-t|≤h |Z H (s) -Z H (t)| ρ(s, t) log ρ(s, t)
1/α+ε = 0, a.s.

(3.9)

Local times

Many geometric properties of a random field are related to the existence and analytic properties of its local times. We recall briefly some aspects of the theory of local times. For further information we refer to [START_REF] Geman | Occupation densities[END_REF] and [START_REF] Dozzi | Occupation density and sample path properties of N -parameter processes[END_REF]. Let X : R N → R d be a (deterministic or stochastic) Borel vector field. For any Borel set T ⊆ R N , the occupation measure of X on T is defined as

µ T (•) = λ N t ∈ T : X(t) ∈ • ,
which is a Borel measure on R d . If µ T is almost surely absolutely continuous with respect to the Lebesgue measure λ d , then X(t) is said to have a local time on T . The local time, L(•, T ), is defined as the Radon-Nikodým derivative of µ T with respect to λ d , i.e.,

L(x, T ) = dµ T dλ d (x), ∀x ∈ R d .
In the above, x is called the space variable, and T is the time variable. It is clear that if X has local times on T , then for every Borel set S ⊆ T , L(x, S) also exists. It follows from Theorems 6.3 and 6.4 in Geman and Horowitz (1980) that the local times of X have a version, still denoted by L(x, T ), such that it is a kernel in the following sense:

(i) For each fixed S ⊆ T , the function x → L(x, S) is Borel measurable in x ∈ R d .
(ii) For every x ∈ R d , L(x, •) is a Borel measure on B(T ), the family of Borel subsets of T .

Moreover, L(x, T ) satisfies the following occupation density formula: For every measurable function f : It is known from [START_REF] Adler | The Geometry of Random Fields[END_REF] that, when a local time is jointly continuous, L(x, •) can be extended to be a finite Borel measure supported on the level set

R d → R + , T f (X(t)) dt = R d f (x)L(x, T ) dx. (3.10) Suppose we fix an interval T = N i=1 [a i , a i + h i ] in A. If we can choose a version of the local time, still denoted by L x, N i=1 [a i , a i + t i ] , such that it is a continuous function of (x, t 1 , • • • , t N ) ∈ R d × N i=1 [0, h i ],
X -1 T (x) = {t ∈ T : X(t) = x}. (3.11)
In order to use this fact for proving Theorem 3.9 below, we also need to obtain the Hölder conditions for the local time L(x, •) in the set variable.

For convenience we assume in the rest of this paper that

0 < H 1 ≤ • • • ≤ H N < 1. (3.12)
Of course, there is no loss of generality in the arbitrary ordering of H 1 , . . . , H N . Assume that d < N j=1 1/H j holds and let τ ∈ {1, . . . , N } be the integer such that

τ -1 =1 1 H ≤ d < τ =1 1 H (3.13)
with the convention that 0 =1 1 H := 0. We denote in the sequel

β τ = τ =1 H τ H + N -τ -H τ d. (3.14)
As shown by Theorem 7.1 in Xiao (2009), for any x ∈ R d , we have dim H X -1 (x) = β τ with positive probability. We will make use of the following uniform Hölder condition for the local time L(x, •) in the set variable. We refer to [START_REF] Nolan | Local nondeterminism and local times for stable processes[END_REF], [START_REF] Kôno | Local times and related sample path properties of certain self-similar processes[END_REF], [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF] for earlier results of this type for self-similar stable processes and harmonizable fractional stable motion. The new feature of Proposition 3.7 is that the uniform Hölder condition is given in terms of (H 1 , . . . , H N ) and the side-lengths of I, so it characterizes the anisotropy of X, and the exponents of r are almost optimal. Proposition 3.7 Let X = {X(t), t ∈ R N } be an (N, d) harmonizable fractional stable sheet defined by (1.4) and let T ⊂ (0, ∞) N be an interval. Suppose α ∈ [1, 2] and τ ∈ {1, . . . , N } is the integer so that (3.13) holds. Let L(x, •) be the jointly continuous local time of X on T . Then, for any ε > 0, almost surely there exist positive and finite constants c 8 = c 8 (ω) and r 0 = r 0 (ω) such that

max x∈R d L(x, I) ≤ c 8 τ =1 r 1- H d p N =τ +1 r 1-ε (3.15) for all intervals I = [a, a + r ] ⊆ T with r ≤ r 0 . Here p ≥ 1 (1 ≤ ≤ τ ) are positive numbers such that τ =1 p -1 = 1.
The proof of Proposition 3.7 relies on the moment estimates in Lemma 3.8 below and a chaining argument in [START_REF] Ehm | Sample function properties of multi-parameter stable processes[END_REF] [see also Geman and Horowitz (1980, pp 49-50)]. Since this proof is standard, we omit it.

We remark that, unlike the Gaussian cases considered in [START_REF] Xiao | Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields[END_REF] and Wu and Xiao (2011), we are not able to replace the exponent 1 -ε in (3.15) by 1 together with a logarithmic factor. This is due to the inability in taking full advantage of the sectorial local nondeterminism in Part (ii) of Lemma 3.1. Instead, our proof of Lemma 3.8 below makes use of Lemma 3.3. Lemma 3.8 Assume the conditions of Proposition 3.7 hold. Then for any positive constants p ≥ 1 (1 ≤ ≤ τ ) such that τ =1 p -1 = 1 the following statements hold. (i) For all integers n ≥ 1, there exists a positive and finite constant c 9 = c 9 (n) such that for all intervals I = [a, a + r ] ⊆ T with side-lengths r ∈ (0, 1) and all

x ∈ R d E L(x, I) n ≤ c 9 τ =1 r 1- H d p N =τ +1 r n . (3.16)
(ii) For all even integers n ≥ 2 and all γ ∈ (0, 1) small enough, there exists a positive and finite constant c 10 = c 10 (n) such that for all intervals

I = [a, a + r ] ⊆ T , x, y ∈ R d with |x -y| ≤ 1, E L(x, I) -L(y, I) n ≤ c 10 |x -y| nγ τ =1 r 1- H d p N =τ +1 r n . (3.17)
Proof This is essentially proved in Lemmas 4.3 and 4.7 in Xiao (2011). The difference is that I is assumed to be a cube in [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF] and here it is an arbitrary interval, which is needed for proving Theorem 3.9 below. Again we start with the following identities about the moments of the local time and its increments from Geman and Horowitz (1980): For all x, y ∈ R d and all integers n ≥ 1,

E L(x, I) n = (2π) -nd I n R nd exp -i n j=1 u j , x E exp i n j=1
u j , X(t j ) du dt (3.18) and for all even integers n ≥ 2,

E L(x, I) -L(y, I) n =(2π) -nd I n R nd n j=1 e -i u j ,x -e -i u j ,y × E exp i n j=1 u j , X(t j ) du dt, (3.19)
where u = (u 1 , . . . , u n ), t = (t 1 , . . . , t n ), and each u j ∈ R d , t j ∈ I ⊆ (0, ∞) N . In the coordinate notation we then write u j = (u j 1 , . . . , u j d ). Then upper bounds for (3.18) and (3.19) can be derived by using Lemmas 3.1 and 3.3. More precisely, the proof of (3.16) is the same as the proof of Lemma 4.3, up to (4.28) in [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF], and the proof of (3.17) is the same as the proof of Lemma 4.7, up to (4.47) in [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF]. This finishes the proof of Lemma 3.8.

Inverse image X -1 (F )

In this section we prove the following uniform result for the Hausdorff dimensions of the inverse images X -1 (F ), where F ⊆ R d are Borel sets, of the harmonizable fractional stable sheet defined in (1.4). For definition and basic properties (such as σ-stability and Frostman's lemma) of Hausdorff dimension, we refer to [START_REF] Falconer | Fractal Geometry[END_REF], or [START_REF] Khoshnevisan | Multiparameter Processes: An Introduction to Random Fields[END_REF]. Theorem 3.9 Let X = {X(t), t ∈ R N } be an (N, d) harmonizable fractional stable sheet defined in (1.4). We assume that α ∈ [1, 2] and N j=1 1

H j > d. Denote the random open set O = s,t∈Q N ; 0≺s≺t x ∈ R d : L(x, [s, t]) > 0 .
(3.20)

Then the following statements hold:

(i) With probability 1, dim H X -1 (F ) = min 1≤k≤N k j=1 H k H j + N -k -H k (d -dim H F ) = k j=1 H k H j + N -k -H k (d -dim H F ), if k-1 j=1 1 H j ≤ d -dim H F < k j=1 1 H j (3.21)
for all Borel sets F ⊆ O. In the above we use the convention

0 j=1 H -1 j = 0. (ii) P{O = R d } > 0.
In order to prove Theorem 3.9, we will make use of the following lemma where Y can be quite general. Lemma 3.10 Let Y = {Y (t), t ∈ R N } be a function with values in R d . We assume that for every bounded interval T ⊆ R N and every ε > 0,

Y (s) -Y (t) ≤ ρ(s, t) 1-ε ∀s, t ∈ T (3.22)
and Y has a local time L(x, T ) which is bounded in the space variable x. Then for every Borel set

F ⊆ R d , dim H Y -1 (F ) ≤ min 1≤k≤N k j=1 H k H j + N -k -H k (d -dim H F ) . (3.23)
Proof Our argument is reminiscent to the proof of a corresponding result in [START_REF] Monrad | Local nondeterminism and Hausdorff dimension[END_REF]. For any integer n ≥ 2, we divide T into sub-rectangles R n,i of side-lengths 2 -n/H j (j = 1, . . . , N ). Note that there are at most c 2 nQ such rectangles (recall that Q = N j=1 1 H j ), and each R n,i is equivalent to a ball of radius 2 -n in the metric ρ.

For any (open or closed) ball B(y, Note that the constant c 11 is independent of y and n. Now we prove (3.23) by using (3.25) and a covering argument. Note that it is sufficient to show that dim H Y -1 (F ) ∩ T is bounded by the right hand side of (3.23). The general result follows from the arbitrariness of T and the σ-stability of dim H .

2 -(1-ε)n ), let N ρ (y, n, T ) be the number of R n,i 's such that R n,i ∩ X -1 (B(y, 2 -(1-ε)n )) = ∅.
Given any Borel set F ⊆ R d , we choose and fix an arbitrary constant γ > dim H F . Then there exist a constant δ > 0 and a sequence of balls {B(y j , r j ), j ≥ 1} in R d (in Euclidean metric) such that r j ≤ δ for all j ≥ 1,

F ⊆ ∞ j=1 B(y j , r j ) and ∞ j=1 (2r j ) γ ≤ 1. (3.26)
For every integer j ≥ 1, let n j be the integer such that 2

-(1-ε)(n j +1) ≤ r j < 2 -(1-ε)n j . Since X -1 (F ) ⊆ ∞ j=1 X -1 B(y j , r j ) ,
we obtain a covering of X -1 (F ) ∩ T by a subsequence of intervals {R n j ,i } (i.e., those satisfying R n j ,i ∩ X -1 B(y j , r j ) = ∅). For simplicity, we set N j = N ρ (y j , n j , T ).

For every k ∈ {1, . . . , N }, each rectangle R n j ,i can be covered by at most N =k 2

n j ( 1 H k -1 H )
+1 cubes of side-length 2 -n j /H k . In this way, we obtain a covering of X -1 (F ) ∩ T by cubes of side-length 2 -n j /H k which can be used to bound the Hausdorff measure of X -1 (F ) ∩ T . Let

β k = k j=1 H k H j + N -k -H k (d -γ).
It follows from (3.26) and (3.25) that

∞ j=1 N j N =k 2 n j ( 1 H k -1 H ) + 1 2 - n j H k β k ≤ c ∞ j=1 2 -n j γ ≤ c 12 . (3.27)
This implies that H β k X -1 (F ) ∩ T ) ≤ c 12 . Hence we have dim H X -1 (F ) ∩ T ≤ β k for every k ∈ {1, . . . , N }. Letting γ ↓ dim H F yields the desired upper bound.

We are ready to prove Theorem 3.9.

Proof of Theorem 3.9 First we prove Part (i). Since it is elementary to verify the second equality in (3.21), we will only prove the first equality in (3.21).

For any bounded interval T ⊆ (0, ∞) N , by Lemmas 3.5 and 3.6, X satisfies a uniform Hölder condition in the metric ρ on T of order 1 -ε and has a jointly continuous (hence bounded) local time on T . Hence the upper bound in (3.21) follows from Lemma 3.10 and the σ-stability of Hausdorff dimension.

For proving the lower bound in (3.21), let γ < dim H F be an arbitrary constant. Then by Frostman's lemma [cf. [START_REF] Falconer | Fractal Geometry[END_REF]] there exists a Borel probability measure ν on R d supported by F such that ν(B(y, r)) ≤ c 13 r γ , ∀ y ∈ R d and r > 0.

(3.28)

We define a random Borel measure on (0, ∞) N by

µ(I) = R d L(x, I) ν(dx), for all I ⊂ (0, ∞) N . (3.29)
Since ν is supported by F and L(•, I) vanishes outside the closure X(I), we can see that µ is a finite measure supported on X -1 (F ). Moreover, for every F ⊂ O, we have µ(X -1 (F )) > 0.

Let k ∈ {1, . . . , N } be the integer satisfying

k-1 j=1 1 H j ≤ d -dim H F < k j=1 1 H j . (3.30)
For any r ∈ (0, 1), there is an integer n ≥ 0 such that

2 -(n+1)/H k ≤ r < 2 -n/H k . (3.31)
Hence, for every t ∈ (0, ∞) N , the Euclidean ball U (t, r) can be covered by at most N n intervals

{I i = [t i , t i + 2 -n/H ], 1 ≤ i ≤ N n } (each I i is a subset of the ball centered at t i in the metric ρ of radius √ N 2 -n ),
where

N n ≤ c N j=k+1 2 -n H k + n H j = c 2 n N j=k+1 1 H j -N -k H k . (3.32)
Let us fix an ω ∈ Ω such that the conclusions of both Lemma 3.5 and Theorem 3.7 hold. Hence, by (3.15) and (3.9), we have that a. s. for all n large enough, max

x∈R d L(x, I i ) ≤ c 12 2 -n(1-ε) N =1 1 H -d (3.33) and max s,t∈I i X(s) -X(t) ≤ c 2 -n(1-ε) (3.34)
for every 1 ≤ i ≤ N n . It follows from (3.32), (3.33), (3.28), and (3.34) that for any ε > 0 small

µ(U (t, r)) ≤ Nn i=1 R d L(x, I i ) ν(dx) ≤ c 17 Nn i=1 max x∈R d L(x, I i ) diamX(I i ) γ ≤ c 2 -n k j=1 1 H j -N -k H k -d+γ-ε = c r η , (3.35)
where

η = k j=1 H k H j -N + k -H k (d -γ) -ε ,
where ε = c 18 ε. This, together with the Frostman's theorem [cf. [START_REF] Falconer | Fractal Geometry[END_REF]], implies that almost surely for every

F ⊂ O, dim H X -1 (F ) ≥ k j=1 H k H j -N + k -H k (d -γ) -ε
Letting ε ↓ 0 yields the desired lower bounds for dim H X -1 (F ). This proves (3.21).

In order to prove Part (ii), we first notice that, for any bounded interval I ⊂ (0, ∞) N with non-empty interior, we have E L(0, I) > 0. This follows directly from (3.18), (3.2) and (3.5). Hence P L(0, I) > 0 > 0. Using the a.s. continuity of x → L(x, I), we see that there are constants r > 0 and δ > 0 such that P L(x, I) > 0 for all x ∈ B(0, r) ≥ δ. This implies that where E is the N × N diagonal matrix diag(n H -1 1 , . . . , n H -1 N ) and Q = N j=1 H -1 j . It follows from (3.36) and (3.37) that P B(0, n N r) ⊂ O ≥ δ. By Fatou's lemma, we have P B(0, n N r) ⊂ O for infinitely many n s ≥ δ.

P B(0, r) ⊂ O ≥ δ. ( 3
This proves P O = R d ≥ δ.
We remark that, even though in the present paper we have focused on harmonizable fractional stable sheet X, the methods in this section are applicable to stable random fields that have jointly continuous local times such that their uniform moduli of continuity satisfy (3.9) in Lemma 3.5 and (3.15) in Proposition 3.7, respectively. With some extra effort, one can verify that Part (i) of Theorem 3.9 holds for a large class of (N, d) stable random fields X defined as in (1.4) such that X 1 is a harmonizable stable random field with stationary increments. For example, X 1 can be taken as an operator-scaling harmonizable stable random field constructed by Biermé, Meerschaert and Scheffler (2007) (in this case, Part (ii) of Theorem 3.9 also holds) or a stable random field in Sections 3.1 and 3.2 of [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF].

This paper also raises several interesting open questions. We list some of them below.

• We believe that P O = R d = 1 for an (N, d) harmonizable fractional stable sheet in Theorem 3.9, but we have not been able to prove this conjecture in general. When α = 2, this can be proved by applying Theorem 2 of [START_REF] Monrad | Local nondeterminism and Hausdorff dimension[END_REF] to the stationary Gaussian random field obtained from a fractional Brownian sheet via the Lamperti transform (cf. [START_REF] Genton | Self-similarity and Lamperti transformation for random fields[END_REF] for the latter). For α ∈ (1, 2), the conjecture would be proven if one can establish a Hewitt-Savage zero-one law for harmonizable fractional stable sheets. See Orey and Pruitt (1973, p.141) and Tran (1976, p.32) for results for the Brownian sheet.

• By checking the proofs in Section 3, one can see that the results on local times and uniform Hausdorff dimension of X -1 (F ) would hold for harmonizable fractional α-stable sheets with α ∈ (0, 1), if the property of local nondeterminism of Z H had been proved when α ∈ (0, 1). See Remark 3.2.

• In light of Biermé, Lacaux and Xiao (2009) and the present paper, it is a natural question to determine the packing dimension (cf. [START_REF] Falconer | Fractal Geometry[END_REF]) of the inverse image X -1 (F ) for F ⊂ R d and to establish a corresponding uniform packing dimension result. Solving these problems would require new results on the local times of harmonizable fractional stable sheets. More precisely, instead of Proposition 3.7 which is concerned with uniform modulus of continuity of the local times, it will be essential to study the liminf behavior of the local times. No results of this kind have been proven for stable random fields.

• It would be interesting to establish a similar uniform dimension result for linear fractional stable sheets (LFSS) considered in [START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF]. The method for proving Theorem 3.9 breaks down for a LFSS due to the fact that its uniform modulus of continuity is different from that of Z H in Lemma 3.5, a new method will be needed.

• In (1.4), the real-valued random fields X 1 , . . . , X d are assumed to be independent copies. This technical assumption brings convenience for studying sample path properties of X but it is obviously too restrictive. Recently, Xiao and Li (2011), Kremer and Scheffler (2018), [START_REF] Sönmez | Fractal behavior of multivariate operator-self-similar stable random fields[END_REF][START_REF] Sönmez | The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields[END_REF] have introduced and studied several classes of operator-selfsimilar (N, d) stable random fields whose components are dependent in general. It would be of interest to further investigate the fractal properties and local times of these stable random fields.

  For each increasing sequence (D n ) n∈N of finite subsets of I which converges to I (i.e., D n ⊂ D n+1 for every n ∈ N and ∪ n∈N D n = I), one has lim n→+∞

  then X is said to have a jointly continuous local time on T . The following result follows from Lemmas 3.1 and 3.3 above and Theorem 4.2 in Xiao (2011). Lemma 3.6 Let X = {X(t), t ∈ R N } be an (N, d) harmonizable fractional stable sheet defined by (1.4). If α ∈ [1, 2] and N j=1 1/H j > d, then for any bounded interval T ⊆ (0, ∞) N , X has a jointly continuous local time on T almost surely.

  For each such rectangle R n,i , the uniform Hölder condition(3.22) implies that X(R n,i ) ⊆ B(y, 2 -(1-ε)n+1). It follows from the occupation density formula (3.10) thatN ρ (y, n, T ) 2 -Qn ≤ B(y, 2 -(1-ε)n+1 ) L(x, T ) dx ≤ c 2 -(1-ε)dn . (3.24) This yields N ρ (y, n, T ) ≤ c 11 2 (Q-(1-ε)d)n . (3.25)

. 36 )

 36 On the other hand, by the scaling property (1.2) and the occupation formula (3.10), one can verify that the local times L(x, I) have the following property: For any integer n ≥ 2,L(x, I) d = n N d-Q L(n N x, n E I),(3.37)

  For every t ∈ R N and for Lebesgue almost all λ ∈ R N , one has

	Lemma 2.6

1, we have sup y∈R +∞ k=-∞ (2 + |y -k|) -Lα < +∞. (2.29) Finally, putting together (2.27), (2.28) and(2.29), one gets (2.21). The following lemma shows that the function F (t) in (2.22) is in fact the kernel function corresponding to HFSS (see (1.1)).

  .33) It follows from (2.12), (2.33), (2.32) and (2.22) with D n

  .44) It follows from (2.43) and (2.44) that, for almost all

Acknowledgements. This work was finished during Yimin Xiao's visit to Université de Lille sponsored by CEMPI (ANR-11-LABX-0007-01). The hospitality from this university and


Claudia J Stocks

Jessica B Von Pein

James E B Curson

James Rae

Minh-Duy Phan

Darren Foo

Nilesh J Bokil

Taiho Kambe

Kate M Peters

Robert G Parton

Mark A Schembri

Dr Ronan Kapetanovic
email: r.kapetanovic@imb.uq.edu.au

Matthew J Sweet
email: m.sweet@imb.uq.edu.au

Prof Matthew Sweet

Duy Phan

Frontline Science: LPS-inducible SLC30A1 drives human macrophage-mediated zinc toxicity against intracellular Escherichia coli

Keywords: antimicrobial, E. coli, host-pathogen, metal ions, zinc toxicity, zinc transporters BMM, bone marrow-derived macrophages, CLEM, correlative light-electron microscopy, EV, empty vector, HMDM, human monocyte-derived macrophages, MCS, multiple cloning site, p.i., post-infection, qPCR, quantitative PCR

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

INTRODUCTION

macrophages and other innate immune cells utilize an array of antimicrobial mechanisms to defend against pathogenic bacteria. 1

One such strategy, nutritional immunity, refers to the competition between host cells and invading microorganisms for essential nutrients and metal ions. [START_REF] Hood | Nutritional immunity: transition metals at the pathogen-host interface[END_REF] For example, neutrophils secrete the zinc and manganese-binding protein calprotectin to restrict availability of these critical trace elements to bacteria, thus limiting their growth and survival. [START_REF] Kehl-Fie | Nutrient metal sequestration by calprotectin inhibits bacterial superoxide defense, enhancing neutrophil killing of Staphylococcus aureus[END_REF] Several pathogens counteract these and other innate immune responses by employing sophisticated mechanisms to acquire essential metal ions. [START_REF] Diaz-Ochoa | Transition metal ions at the crossroads of mucosal immunity and microbial pathogenesis[END_REF] For example, Salmonella uses the ZnuABC zinc uptake system for host colonization [START_REF] Ammendola | High-affinity Zn(2+) uptake system ZnuABC is required for bacterial zinc homeostasis in intracellular environments and contributes to the virulence of salmonella enterica[END_REF] and to defend against innate immune-mediated nitrosative stress. [START_REF] Fitzsimmons | Zinc-dependent substrate-level phosphorylation powers Salmonella growth under nitrosative stress of the innate host response[END_REF] In recent years, it has become clear that specific trace elements such as zinc can also be harnessed by innate immune cells as antimicrobial agents to combat bacterial infection. [START_REF] Stafford | Metal ions in macrophage antimicrobial pathways: emerging roles for zinc and copper[END_REF] In Mycobacteria-infected macrophages exposed to inflammatory cytokines, phagosomal zinc concentrations were reported to reach millimolar concentrations. [START_REF] Wagner | Elemental analysis of Mycobacterium avium-, Mycobacterium tuberculosis-, and Mycobacterium smegmatis-containing phagosomes indicates pathogen-induced microenvironments within the host cell's endosomal system[END_REF] Mycobacterium tuberculosis upregulates heavy metal efflux P-type ATPases in the intramacrophage environment, with an accumulation of zinc within bacteria-containing phagosomes also being observed. [START_REF] Botella | Mycobacterial p(1)-type ATPases mediate resistance to zinc poisoning in human macrophages[END_REF] Consistent with this, TLR-mediated macrophage activation triggers the mobilization of zinc into vesicular-like structures that co-localize with engulfed E. coli, with intracellular bacteria producing a transcriptional response consistent with zinc poisoning. [START_REF] Kapetanovic | Salmonella employs multiple mechanisms to subvert the TLR-inducible zinc-mediated antimicrobial response of human macrophages[END_REF] The zinc toxicity response is also deployed by the soil amoeba Dictyostelium discoideum, suggesting this is an ancient host defense pathway. [START_REF] Barisch | Localization of all four ZnT zinc transporters in Dictyostelium and impact of ZntA and B knockout on bacteria killing[END_REF] In keeping with a central role of zinc poisoning in innate immune antimicrobial responses, the bacterial pathogens Salmonella [START_REF] Kapetanovic | Salmonella employs multiple mechanisms to subvert the TLR-inducible zinc-mediated antimicrobial response of human macrophages[END_REF] and uropathogenic E.

coli [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] are able to both resist and evade this response. Mechanisms by which mobilized zinc exerts antimicrobial effects within innate immune cells are unknown. However, studies on bacteria alone have implicated induced copper deficiency, [START_REF] Hassan | Zinc stress induces copper depletion in Acinetobacter baumannii[END_REF] manganese deficiency resulting in increased sensitivity to oxidative stress, [START_REF] Eijkelkamp | Extracellular zinc competitively inhibits manganese uptake and compromises oxidative stress management in Streptococcus pneumoniae[END_REF][START_REF] Mcdevitt | A molecular mechanism for bacterial susceptibility to zinc[END_REF] the replacement of other cations in essential enzymes, [START_REF] Stafford | Metal ions in macrophage antimicrobial pathways: emerging roles for zinc and copper[END_REF] and disruption of iron-sulfur (4Fe-S) biogenesis with consequential arrest of key metabolic pathways [START_REF] Xu | Silver(I), mercury(II), cadmium(II), and zinc(II) target exposed enzymic iron-sulfur clusters when they toxify Escherichia coli[END_REF][START_REF] Li | Zinc toxicity and iron-sulfur cluster biogenesis in Escherichia coli[END_REF] as factors that contribute to the antimicrobial effect of zinc. Interestingly, zinc can synergize with reactive oxygen species to limit E. coli growth. [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] Thus, zinc toxicity could also act in a combinatorial fashion with other innate immune antimicrobial pathways.

Despite existing knowledge of innate immune-mediated zinc toxicity, we still have a very limited understanding of the molecular processes underpinning this response. Mammalian zinc homeostasis is tightly regulated by 2 families of transporters, the ZIP/SLC39A family that delivers zinc to the cytoplasm and the ZnT/SLC30A family that moves zinc from the cytoplasm to the extracellular space or to the lumen of organelles. [START_REF] Kambe | Zinc transporter proteins: a review and a new view from biochemistry[END_REF] SLC39A zinc importers likely contribute to the uptake and thus availability of intracellular zinc for the zinc toxicity pathway. However, macrophage-expressed SLC39A transporters have mainly been studied in the context of regulation of inflammatory responses. For example, SLC39A8 is up-regulated in human monocytes and macrophages in response to either TNF or LPS, [START_REF] Liu | The zinc transporter SLC39A8 is a negative feedback regulator of NF-𝜅B through zinc-mediated inhibition of IKK[END_REF] with this being important for control of inflammatory cytokine production. [START_REF] Pyle | Zinc modulates endotoxin-induced human macrophage inflammation through ZIP8 induction and C/EBP𝛽 inhibition[END_REF] SLC39A8 functions as a negative regulator of NF-𝜅B, as zinc can directly bind and inhibit I𝜅B kinase. [START_REF] Liu | The zinc transporter SLC39A8 is a negative feedback regulator of NF-𝜅B through zinc-mediated inhibition of IKK[END_REF] Some studies have also investigated the roles of SLC30A zinc exporters in innate immune functional responses. LPS up-regulates expression of Slc30a1, Slc30a4, and Slc30a6 in murine dendritic cells, with regulated zinc trafficking linked to antigen presentation. [START_REF] Kitamura | Toll-like receptor-mediated regulation of zinc homeostasis influences dendritic cell function[END_REF] GM-CSF increased mRNA levels of Slc30a4 and Slc30a7 in murine peritoneal macrophages, with the subsequent sequestration of zinc in the Golgi apparatus being associated with impaired survival of the intracellular fungal pathogen Histoplasma capsulatum. [START_REF] Vignesh | Granulocyte macrophage-colony stimulating factor induced Zn sequestration enhances macrophage superoxide and limits intracellular pathogen survival[END_REF] In contrast, while IL-4 stimulation of bone marrow murine macrophages similarly induced Slc30a4 expression, this induction promotes a SLC30A4-dependent increase in intracellular zinc that favors survival of H. capsulatum, [START_REF] Vignesh | IL-4 induces metallothionein 3-and SLC30A4-dependent increase in intracellular Zn(2+) that promotes pathogen persistence in macrophages[END_REF] suggesting complex roles for individual zinc transporters in host defense. In this study, we focused on the SLC30A zinc exporter family and demonstrate that SLC30A1 is sufficient to deliver a zinc toxicity response against intramacrophage bacteria.

MATERIALS AND METHODS

Extended methods describing bacterial strains, correlative lightelectron microscopy (CLEM), infection assays, lentiviral construct generation, immunoblotting, flow cytometry, quantitative real time PCR (qPCR), and selective gene silencing can be found in the Supplementary material.

Ethics statement

Mammalian cell culture

To generate human monocyte-derived macrophages (HMDM), CD14 + monocytes were isolated from human buffy coats (Australian Red Cross Blood Service) and cultured for 7 days with 150 ng/ml recombinant human CSF-1 (University of Queensland Protein Expression Facility), as previously described. [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] To generate mouse bone marrowderived macrophages (BMM), bone marrow cells from the femurs and tibias of C57BL/6 mice were cultured for 7 days as previously described. [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] Human monocytic THP-1 cells were obtained from the American Tissue Culture Collection. Non-adherent THP-1 cells were differentiated into macrophages in presence of 30 ng/ml PMA (Sigma-Aldrich) for 48 h.

Inducible gene expression in THP-1 cells

The lentiviral system utilizing pF_TRE3G_PGK_puro (hereafter pLenti_EV: kindly provided by James Murphy, Walter and Elizabeth

Hall Institute of Medical Research) for doxycycline-inducible gene expression has previously been described. [START_REF] Moujalled | TNF can activate RIPK3 and cause programmed necrosis in the absence of RIPK1[END_REF][START_REF] Murphy | The pseudokinase MLKL mediates necroptosis via a molecular switch mechanism[END_REF] Details on specific lentiviral expression constructs can be found in the Supplementary material and Supplementary Table 1. pLenti_MCS (Empty Vector) or pLenti_SLC30A1_V5 was introduction into THP-1 cells by lentiviral transduction, as previously described. [START_REF] Murphy | The pseudokinase MLKL mediates necroptosis via a molecular switch mechanism[END_REF][START_REF] Vince | IAP antagonists target cIAP1 to induce TNF𝛼-dependent apoptosis[END_REF] Briefly, Lipofectamine 2000 (Invitrogen) in OptiMEM was combined with pCMV-dR8.2 dvpr and pCMV_VSV-G (Addgene), along with the appropriate DNA transfer plasmid (pLenti_EV or pLenti_SLC30A1_V5). Viral particles were harvested at 24 and 48 h post-transfection and applied to non-differentiated THP-1 cells via a 2 h, 1000 × g spinfection at 35 • C.

Cells were subsequently maintained in 1 𝜇g/ml puromycin (Sigma-Aldrich). Inducible overexpression of SLC30A1 was confirmed through treatment with 100 ng/ml doxycycline (Sigma-Aldrich) for 24 h, with subsequent experiments proceeding as described.

Confocal microscopy

Immunofluoresence imaging of fixed cells was performed using a Zeiss

Statistical analyses

Individual experiments were typically performed in experimental duplicate or triplicate, with mean values taken from each experiment for combining data from at least 3 experiments for statistical analyses.

Prism 5 and/or 7 software was used to perform the specific statistical tests that are indicated in the individual figure legends.

RESULTS AND DISCUSSION

Zinc-stressed E. coli reside within membrane-bound compartments in human macrophages

E. coli zntA encodes a zinc efflux system that confers zinc resistance, as originally demonstrated through transposon mutagenesis, targeted gene deletion and gene complementation studies. [START_REF] Beard | Zinc(II) tolerance in Escherichia coli K-12: evidence that the zntA gene (o732) encodes a cation transport ATPase[END_REF][START_REF] Rensing | The zntA gene of Escherichia coli encodes a Zn(II)-translocating P-type ATPase[END_REF] We previously showed that the E. coli K-12 strain MG1655 is exposed to zinc stress during macrophage infection, as evidenced by intra-macrophage expression of zinc-inducible zntA and reduced intramacrophage survival of a zntA mutant strain. [START_REF] Kapetanovic | Salmonella employs multiple mechanisms to subvert the TLR-inducible zinc-mediated antimicrobial response of human macrophages[END_REF][START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] To track this antimicrobial response, we also developed a zinc stress reporter strain (MG1655 pGcCzntAp), which constitutively expresses GFP and inducibly expresses mCherry when bacteria are subjected to zinc stress. [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] To further investigate this phenomenon, we utilized CLEM to visualize MG1655 pGcCzntAp within HMDM (Fig. 1). Live imaging of this zinc stress reporter strain confirmed the deployment of zinc toxicity against MG1655 pGcCzntAp at 8 h post-infection (p.i.) (Fig. 1A). Re-imaging the same cells via serial section transmission electron microscopy enabled visualization of intracellular E. coli, as well as their correlation with initial confocal images. This analysis revealed that both non-zinc-stressed (GFP +ve /mCherry -ve ) and zinc-stressed (GFP +ve /mCherry +ve ) MG1655 pGcCzntAp reside within membranebound compartments (Fig. 1B). Thus, there is likely to be a role for a member (or members) of the zinc exporter family of transporters in the delivery of antimicrobial zinc from the cytoplasm to these vesicular compartments. Furthermore, by comparison to mCherry -ve MG1655 (Fig. 1B, top inset), mCherry +ve bacteria (Fig. 1B, bottom inset) exhibited a distorted morphology, perhaps reflecting stress and/or membrane perturbation.

Zinc transporter SLC30A1 is constitutively expressed and further upregulated by LPS in human macrophages

To gain mechanistic insights into the delivery of antimicrobial zinc, we next profiled the LPS-regulated expression of the family of Slc30a/SLC30A zinc transporters in murine BMM (Fig. 2A) and HMDM (Fig. 2B). LPS significantly up-regulated Slc30a4 and Slc30a7 mRNA levels in BMM, with a trend for increased Slc30a5 and Slc30a9 expression also observed (Fig. 2A). Slc30a2, Slc30a3, Slc30a8, and Slc30a10

were all expressed at very low levels in these cells, but LPS treatment also caused a consistent, though non-significant, elevation in their mRNA expression (data relative to unstimulated control, Supplementary Fig. S1A). In HMDM, LPS significantly increased the expression of SLC30A1 and SLC30A7, with SLC30A5 mRNA levels being transiently down-regulated at 8 h post-stimulation (Fig. 2B). Although absolute mRNA levels were very low, there was also a trend for increased expression of SLC30A10, the most closely related SLC30A family member to SLC30A1 [START_REF] Huang | The SLC30 family of zinc transporters -a review of current understanding of their biological and pathophysiological roles[END_REF] (data relative to unstimulated control, Supplementary Fig. S1B). Given that many Slc30a transporters were LPS-inducible at the mRNA level in BMM (Fig. 2A; Supplementary Fig. S1A), while the regulated expression of SLC30A family members was more restricted in HMDM, we subsequently focused on human macrophages. In HMDM, SLC30A1 was particularly highly expressed and LPS-inducible at both the mRNA (Fig. 2B) and protein level (Fig. 2C). LPS-inducible SLC30A1 was detected as ∼75 and ∼150 kDa in size in HMDM, as confirmed by gene silencing in these cells (Fig. 2D). This raises the possibility that SLC30A1 in macrophages may exist as a tightly bound multimeric complex and/or glycosylated dimer in addition to a glycosylated monomer. [START_REF] Nishito | Zinc transporter 1 (ZNT1) expression on the cell surface is elaborately controlled by cellular zinc levels[END_REF] macrophages. [START_REF] Liu | The zinc transporter SLC39A8 is a negative feedback regulator of NF-𝜅B through zinc-mediated inhibition of IKK[END_REF] We note that the 75 kDa monomer was the most abundantly expressed form of SLC30A1 in HMDM (Fig. 2C, high exposure blot). Interestingly, monomeric SLC30A1 consistently ran at a slightly higher molecular weight in LPS-treated versus control HMDM, suggestive of an LPS-inducible post-translational modification. A previous study demonstrated that SLC30A1 mRNA levels were increased in human macrophages following Mycobacterium tuberculosis infection. [START_REF] Botella | Mycobacterial p(1)-type ATPases mediate resistance to zinc poisoning in human macrophages[END_REF] Thus, it is likely that multiple microbial stimuli up-regulate the expression of this zinc transporter.

Ectopic expression of SLC30A1 in THP-1 cells drives zinc-containing vesicle formation

In contrast to observations in HMDM, LPS did not up-regulate SLC30A1 mRNA expression in PMA-differentiated macrophage-like THP-1 cells (Supplementary Fig. S2A), despite these cells being LPSresponsive as assessed by TNF expression (Supplementary Fig. S2B).

Thus, we used a doxycycline-inducible system to ectopically express SLC30A1 in these cells for investigation of its contributions to the macrophage zinc toxicity response. Doxycycline induced SLC30A1 mRNA expression in THP-1 cells transduced with a lentiviral construct for SLC30A1, but not empty vector (EV), as expected (Supplementary Fig. S2C). Doxycycline-inducible epitope-tagged (V5) SLC30A1 expression in these cells was confirmed by both immunoblotting (Fig. 3A) and flow cytometry (Fig. 3B). In these cells, overexpressed SLC30A1_V5 was detectable at ∼75 kDa (Fig. 3A). However, a weakly expressed higher order complex (∼150 kDa, high exposure blot) was again detected, similar to our observation in HMDM (Fig. 2C).

We next examined the localization of SLC30A1 in macrophages, particularly in relation to LPS-inducible zinc-containing vesicles.

SLC30A1 has been reported to localize exclusively to the plasma membrane of baby hamster kidney cells, [START_REF] Palmiter | Protection against zinc toxicity by metallothionein and zinc transporter 1[END_REF][START_REF] Palmiter | Cloning and functional characterization of a mammalian zinc transporter that confers resistance to zinc[END_REF] rat astroglial cells, [START_REF] Nolte | ZnT-1 expression in astroglial cells protects against zinc toxicity and slows the accumulation of intracellular zinc[END_REF] and rat cortical neurons. [START_REF] Qin | Silencing of ZnT1 reduces Zn2+ efflux in cultured cortical neurons[END_REF] In the absence of ectopic SLC30A1 expression, LPS induced the formation of zinc vesicles in PMA-differentiated THP-1 Upon induction of V5-tagged SLC30A1 in THP-1 cells, we observed that SLC30A1 localized to both the plasma membrane and intracellular vesicular structures (Fig. 3D). Interestingly, treatment with doxycycline to induce SLC30A1 expression increased the percentage of FluoZin-3-positive THP-1 cells, with the effect being similar to that observed with LPS stimulation (Fig. 3E). Importantly, doxycycline alone did not have this effect in pLenti_EV-transduced control cells (Fig. 3F).

F I G U R E 2
This suggests that the overexpression of SLC30A1 alone is sufficient to mobilize zinc in THP-1 cells in a similar fashion to the response trig-gered by LPS. This hypothesis is further supported by the observation that FluoZin-3-stained zinc was often detected within SLC30A1_V5positive vesicles (Fig. 3D). We note, however, that this co-localization was not uniform within a population of macrophages, likely because of the dynamic nature of zinc mobilization.

Although SLC30A1 can prevent cellular zinc toxicity via plasma membrane efflux, [START_REF] Nishito | Zinc transporter 1 (ZNT1) expression on the cell surface is elaborately controlled by cellular zinc levels[END_REF][START_REF] Palmiter | Protection against zinc toxicity by metallothionein and zinc transporter 1[END_REF] examination across a broader range of cell types also supports a contribution to intracellular zinc redistribution. In a human keratinocyte cell line, SLC30A1 localized to the endoplasmic reticulum, nuclear membrane, and Golgi, and was found to regulate the intracellular distribution of zinc, but not the overall cellular concentration. [START_REF] Lazarczyk | Regulation of cellular zinc balance as a potential mechanism of EVERmediated protection against pathogenesis by cutaneous oncogenic human papillomaviruses[END_REF] SLC30A1 was also reported to be expressed at both the plasma membrane and in intracellular compartments in rat [START_REF] Kelleher | Zn transporter levels and localization change throughout lactation in rat mammary gland and are regulated by Zn in mammary cells[END_REF] Indeed, a recent study examining the regulation and modification of SLC30A1 in human pancreatic cancer cells in response to zinc fluctuations concluded that the molecular mechanisms underlying control of zinc homeostasis by SLC30A1 are more complex than previously thought. [START_REF] Nishito | Zinc transporter 1 (ZNT1) expression on the cell surface is elaborately controlled by cellular zinc levels[END_REF] Thus, the localization and functions of SLC30A1 appear to be highly dependent on both cell type and the specific cellular signals received.

F I G U R E

SLC30A1 subjects engulfed E. coli to a zinc stress antimicrobial response in PMA-differentiated THP-1 cells

We next investigated whether the induction of SLC30A1 increased zntA mRNA expression within macrophages. In pLenti_SLC30A1_V5-transduced THP-1 cells, doxycycline treatment prior to infection with MG1655 caused an ∼2-fold increase in zntA mRNA at 8 h p.i., with this effect not observed in pLenti_EV-transduced control cells (Fig. 4A). This is consistent with SLC30A delivering a zinc stress response to intracellular E. coli in macrophages. We then assessed if SLC30A1 overexpression affects the intracellular survival of MG1655 within THP-1 cells. By comparison to wild-type MG1655, intracellular loads of the zinc-sensitive MG1655ΔzntA mutant that is deficient in the ZntA zinc efflux system [START_REF] Kapetanovic | Salmonella employs multiple mechanisms to subvert the TLR-inducible zinc-mediated antimicrobial response of human macrophages[END_REF][START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] were reduced in EV THP-1 cells in the presence or absence of doxycycline at 24 h p.i. (Fig. 4B; Supplementary Table 3). This confirms that zinc export contributes to the survival of MG1655 in THP-1 cells, as was previously observed in E. coliinfected HMDM, where zntA was both deleted and complemented. [START_REF] Botella | Mycobacterial p(1)-type ATPases mediate resistance to zinc poisoning in human macrophages[END_REF] We note, however, that the effect of zntA deletion on survival within THP-1 cells was less pronounced than previously observed for primary (Supplementary Fig. S2D).

We next utilized the MG1655 zinc stress reporter strain [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] (Fig. 1), in conjunction with our inducible THP-1 cell system, to further investigate the capacity of SLC30A1 to elicit an antimicrobial response. As expected, we noted heterogeneity in SLC30A1_V5 expression across the population of pLenti_SLC30A1-transduced THP-1 cells after treatment with doxycycline. We took advantage of this heterogeneity, quantifying numbers of MG1655 bacteria within SLC30A1-positive (V5 +ve ) versus SLC30A1-negative (V5 -ve ) cells within the same population of THP-1 cells after doxycycline-mediated induction of SLC30A1_V5 expression. This revealed that there was no difference in intracellular bacterial loads between V5 +ve and V5 -ve cells at 4 h p.i (Fig. 4C). At this time point, SLC30A1_V5 localized either predominantly at the plasma membrane or at both the plasma membrane and on intracellular vesicular structures (Fig. 4D) of MG1655-infected THP-1 cells. MG1655 found within these SLC30A1_V5-marked structures were consistently zinc-stressed (mCherry +ve ) (Fig. 4D). By 8 h p.i., significantly reduced bacterial loads were observed in V5 +ve cells as compared to V5 -ve cells (Fig. 4E). At this time point, SLC30A1_V5

predominantly localized to the plasma membrane of cells that did not contain bacteria, with the majority of MG1655 within V5 -ve THP-1 cells found to be only weakly mCherry +ve (Fig. 4F). These data represent the first localization studies of SLC30A1 within macrophages, and, combined with the heightened zinc stress gene signature (Fig. 4A) and enhanced clearance (Fig. 4B,E) of MG1655 in SLC30A1_V5-expressing THP-1 cells, strongly support a role for SLC30A1 in the delivery of zinc toward engulfed E. coli to initiate a late-stage zinc toxicity antimicrobial response.

SLC30A1 is not essential for bacterial killing in primary human macrophages

Somewhat surprisingly, silencing of SLC30A1 in HMDM (Fig. 5A, see also Fig. 1D) did not have any impact on the clearance of either wildtype MG1655 or MG1655ΔzntA (Fig. 5B andC; Supplementary Table 4). Furthermore, reduced survival of the MG1655ΔzntA compared to wild-type MG1655 at 24 h was still apparent (Fig. 5C), suggesting that the zinc toxicity response was effectively engaged in SLC30A1-silenced HMDM. The lack of effect of SLC30A1 silencing on bacterial clearance could reflect the complexities of the zinc toxicity pathway (see Section 3.6 below) or be a consequence of the transfection protocol affecting SLC30A1 function in HMDM. It seems more likely, however, that it reflects redundancy in the functions of mammalian zinc exporters in delivering the zinc toxicity response in human macrophages. We note that many SLC30A family members are expressed and regulated in HMDM (Fig. 2B; Supplementary Fig. S1B). Furthermore, LPS still induced zinc-containing vesicles in PMA-differentiated THP-1 cells (Fig. 3C, E, andF), despite the fact that this stimulus did not increase SLC30A1 mRNA in this cell line (Supplementary Fig. S2A). Interestingly, biomolecular fluorescence complementation 37 in MCF-7 breast cancer cells reported the homodimerization of SLC30A1, SLC30A2, SLC30A3, SLC30A4, and SLC30A7. [START_REF] Lasry | In situ dimerization of multiple wild type and mutant zinc transporters in live cells using bimolecular fluorescence complementation[END_REF] As a homodimer, SLC30A1 localized to the plasma membrane, [START_REF] Lasry | In situ dimerization of multiple wild type and mutant zinc transporters in live cells using bimolecular fluorescence complementation[END_REF] however subsequent studies revealed that SLC30A1 also heterodimerized with SLC30A2 and SLC30A4, and that this altered its sub-cellular localization. [START_REF] Golan | Heterodimerization, altered subcellular localization, and function of multiple zinc transporters in viable cells using bimolecular fluorescence complementation[END_REF] Thus, SLC30A1 may also function in conjunction with other zinc transporters to achieve vesicular delivery of zinc to engulfed bacteria.

Additional factors that may contribute to innate immune-mediated zinc toxicity

The role of zinc within macrophage antimicrobial responses is likely complex and multi-layered, extending beyond direct metal ion poisoning. For example, we previously found that reactive oxygen species increased the sensitivity of a uropathogenic E. coli zntA mutant to zinc. [START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] This suggests that the zinc toxicity response may cooperate with other innate immune antimicrobial pathways, such as the phagocyte oxidase system. Another innate immune antimicrobial pathway that could intersect with zinc toxicity is TLR-inducible production of nitric oxide (NO), which is generated by inducible nitric oxide synthase. [START_REF] Stocks | For when bacterial infections persist: Toll-like receptor-inducible direct antimicrobial pathways in macrophages[END_REF] In Salmonella, NO triggers the mobilization of zinc from a range of zinc metalloproteins, including those involved in sensing nitrosative stress, [START_REF] Henard | The 4-cysteine zincfinger motif of the RNA polymerase regulator DksA serves as a thiol switch for sensing oxidative and nitrosative stress[END_REF] DNA binding, [START_REF] Schapiro | Inhibition of bacterial DNA replication by zinc mobilization during nitrosative stress[END_REF] protein synthesis and cell metabolism. [START_REF] Frawley | Nitric oxide disrupts zinc homeostasis in salmonella enterica serovar typhimurium[END_REF] Consequently, effective zinc efflux is required to maintain homeostasis and permit survival of Salmonella under conditions of nitrosative stress. [START_REF] Frawley | Nitric oxide disrupts zinc homeostasis in salmonella enterica serovar typhimurium[END_REF] Thus, delivery of zinc to intramacrophage bacteria via the actions of SLC30A1 and/or other transport mechanisms may enhance the antimicrobial effects of NO. We note, however, that our studies were performed in primary human macrophages that generate substantially less NO than primary mouse macrophages upon activation, because of regulatory differences between the human NOS2 and mouse Nos2 genes. [START_REF] Vitek | Characterization of NO and cytokine production in immune-activated microglia and peritoneal macrophages derived from a mouse model expressing the human NOS2 gene on a mouse NOS2 knockout background[END_REF] Nonetheless, interplay between innate immune zinc delivery systems and NO may be important during NOS2independent nitrosative stress responses in human cells and/or the Nos2-mediated antimicrobial response in mice.

In summary, the LPS-inducible zinc exporter SLC30A1 permits mobilization of zinc toward intracellular E. coli and the generation of a zinc-mediated antimicrobial response in macrophages. On the basis of studies on the E. coli homologue, YiiP, it is likely that SLC30A1 requires a proton gradient to deliver zinc in a pH-driven, sodium-independent, and calcium-sensitive manner (1:1, Zn 2+ /H + ). [START_REF] Shusterman | ZnT-1 extrudes zinc from mammalian cells functioning as a Zn(2+)/H(+) exchanger[END_REF] In the context of macrophage infection, acidification of the phagolysosome by vacuolar

  Similar observations were previously made for the LPS-inducible expression of the zinc importer SLC39A8 in human F I G U R E 1 Zinc-stressed E. coli resides within membrane-bound compartments in infected human macrophages. HMDM were infected with MG1655 pGcCzntAp (MOI 100) for 1 h, followed by gentamicin exclusion. (A) At 8 h p.i., live cells were imaged via phase contrast and for GFP and mCherry fluorescence. (B) Cells were fixed in 2.5% glutaraldehyde, embedded, and sectioned, then imaged by transmission electron microscopy. Green arrows indicate GFP +ve only bacteria, while red arrows indicate GFP +ve , mCherry +ve bacteria. In both cases, bacteria indicated in (A) correlate with putative bacteria indicated in (B) and are shown in a green or red inset, respectively. Black arrows indicate visible membrane surrounding bacteria. Images are of a single cell and are representative of 3 cells from 1 experiment

  LPS-regulated expression of the Slc30a/SLC30A zinc exporter family in mouse and human macrophages. (A) BMM or (B) HMDM were stimulated with LPS for the indicated time points. Expression levels of Slc30a/SLC30A family members (relative to Hprt/HPRT) were determined by qPCR. (C and D) HMDM were (C) stimulated with LPS or (D) transfected with siRNAs against SLC30A1 or HDAC10 (control) for 24 h, after which cells were lysed and analyzed by western blot. (A and B) Data (mean + SEM, n = 3) are combined from 3 independent experiments and were analyzed by 2-way ANOVA with Dunnett's multiple comparisons test. * Denotes P < 0.05, ** P < 0.01, **** P < 0.0001, all other comparisons were not significant. (C) and (D) depict representative blots from 4 and 3 independent experiments, respectively. Arrows indicate 75 kDa (orange) and 150 kDa (red) protein sizes. High and low exposures of the same SLC30A1 blot are shown in (C) and (D) to highlight the 150 kDa band cells, as detected by FluoZin-3 staining (Fig.3C). Notably, the staining pattern of these LPS-inducible zinc vesicles was not as pronounced as we have previously observed in HMDM,[START_REF] Kapetanovic | Salmonella employs multiple mechanisms to subvert the TLR-inducible zinc-mediated antimicrobial response of human macrophages[END_REF][START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] being much more sparse and heterogeneous on a population and individual cell level.

F I G U R E 4

 4 SLC30A1_V5 promotes a zinc stress response and bacterial killing in THP-1 cells, and localizes to both the plasma membrane and compartments containing zinc-stressed E. coli. (A-F) PMA-differentiated THP-1 cells stably transduced with lentivirus expressing either empty vector (EV) or SLC30A1_V5 were left unstimulated (-) or stimulated with (A and B) 100 ng/ml or (C-F) 500 ng/ml doxycycline (Dox) for 24 h. After washing, THP-1 cells were infected with (A) MG1655 or (B) MG1655 and MG1655ΔzntA (MO1 100) for 1 h, with gentamicin exclusion used to remove extracellular bacteria. (A) At 8 h p.i., cells were lysed, total RNA was isolated, and mRNA levels of zntA (relative to the bacterial housekeeping gene gapA) determined by qPCR. In each case (EV or SLC30A1_V5), zntA mRNA levels in E. coli within THP-1 cells not stimulated with doxycycline were used to calculate the fold change response to enable comparisons across different experiments. (B) Cells were lysed and CFU/mL determined at 24 h p.i. For each experiment, data were normalized to the number of wild-type bacteria within unstimulated EV THP-1 cells. (C-F) SLC30A1_V5 THP-1 cells were washed and infected with MG1655 pGcCzntAp (MOI 100) for 1 h, using an initial spinfection (5 min at 500 × g). Cells were fixed with 4% paraformaldehyde, permeabilized, and then stained with mouse anti-V5 followed by anti-mouse Alexa-647 and DAPI. The average numbers of bacteria within infected cells (V5 -ve and V5 +ve ) were determined via blinded quantification of bacterial GFP fluorescence at (C) 4 and (E) 8 h. Experiments were performed simultaneously, and data were normalized to the number of bacteria within V5 -ve cells at each time point. Data are combined from 6 (A) or 5 (B, C, E) independent experiments. Data were analyzed by (A) Wilcoxon matched-pairs signed rank test, (B) 2-way ANOVA with Sidak's multiple comparisons test or (C and E) paired t-test. * P < 0.05, all other comparisons were not significant. (D and F) Depict individual V5 +ve THP-1 cells infected with MG1655 pGcCzntAp at (D) 4 or (F) 8 h p.i., representative of 5 independent experiments. Scale bar represents 10 𝜇m, white arrows indicate plasma membrane, yellow box indicates inset displayed on right F I G U R E 5 Targeted genetic knockdown of SLC30A1 does not prevent the zinc toxicity response. HMDM were transfected with siRNAs targeting SLC30A1 or HDAC10 (control) or with no siRNA (-) and incubated for 24 h, before undertaking the analyses described in (A-C). (A) Cells were lysed and total RNA extracted, and mRNA levels of SLC30A1 (compared to housekeeping gene HPRT) determined. Data (mean + range) represent one matched experiment to cells used in (B) and are representative of 3 independent experiments. (B-C) Cells were infected with either MG1655 or MG1655ΔzntA (MOI 100) for 1 h. Extracellular bacteria were then removed by gentamicin exclusion, with cells lysed and CFU determined at (B) 2 h and (C) 24 h p.i. Fold change was calculated based on the wild-type MG1655 CFU in HMDM transfected with no siRNA in each experiment. Data (mean + SEM, n = 4) are combined from 4 independent experiments. Data were analyzed by 2-way ANOVA with Sidak's multiple comparisons test, ** P < 0.01, all other comparisons were not significant human macrophages.[START_REF] Stocks | Uropathogenic Escherichia coli employs both evasion and resistance to subvert innate immunemediated zinc toxicity for dissemination[END_REF] Interestingly, doxycycline-mediated induction of SLC30A1 expression reduced the intracellular bacterial loads of wild-type MG1655 to levels similar to those observed with the zntA mutant strain. This suggests that the overexpression of SLC30A1 may overcome the resistance of wild-type MG1655 to the macrophage zinc toxicity response. This effect was not due to a reduction in the initial uptake, as there was no difference in intracellular bacterial loads for either wild-type MG1655 or MG1655ΔzntA from either pLenti_EV-or pLenti_SLC30A1-transduced cells at 2 h p.i.

  ATPases[START_REF] Beyenbach | The V-type H+ ATPase: molecular structure and function, physiological roles and regulation[END_REF] might be predicted to subsequently permit SLC30A1mediated delivery of zinc into phagolysosomal compartments from the cytoplasm. This could provide a means of temporal diversification of macrophage antimicrobial responses. At this stage, our understanding of the processes regulating the formation and/or stability of the zinc-containing compartment remain quite rudimentary, but further characterisation of the SLC30A1 +ve zinc-containing compartment should generate new insights into these processes. In the longer term, our findings may lead to new avenues for manipulating SLC30A1 expression or function to enhance innate immune-mediated zinc poisoning as an anti-infective strategy.

3 Ectopically-expressed SLC30A1_V5 in THP-1 cells localizes to the plasma membrane and zinc-containing intracellular compartments.

	PMA-differentiated THP-1 cells stably transduced
	with lentivirus expressing either empty vector (EV)
	or SLC30A1_V5 were left unstimulated (-, Con) or
	were stimulated with (A and B) 100 ng/mL or (C-
	F) 500 ng/mL doxycycline (Dox) for 24 h. (A) Cells
	were simultaneously stimulated ± 100 ng/ml LPS for
	24 h, after which samples were lysed, processed and
	analyzed by western blot. Blots were probed with
	Abs against V5 or GAPDH as a loading control. The
	displayed immunoblots are from a single experiment
	and are representative of 3 independent experiments.
	Arrows indicate 75 kDa (orange) and 150 kDa (red)
	protein sizes. (B) Samples were lifted and fixed, before
	being permeabilized and stained with an anti-V5 Ab,
	followed by anti-mouse Alexa-647, and analyzed
	via flow cytometry. Data, which depict histograms
	from a single experiment, are representative of 2
	independent experiments. MFI, median fluorescence
	intensity. (C-F) Cells were simultaneously stimulated
	± 100 ng/ml LPS for 24 h, after which cells were fixed
	in 4% paraformaldehyde, permeabilized and stained
	sequentially with mouse anti-V5, anti-mouse Alexa-
	647, DAPI, and FluoZin-3. (C and D) Images depict
	individual SLC30A1_V5 THP-1 cells from the same
	experiment and are representative of 5 independent
	experiments. White arrow indicates plasma mem-
	brane, yellow arrow indicates intracellular vesicle.
	Scale bar represents 10 𝜇m. (E and F) The percentage
	of Fluozin-3 positive THP-1 cells was determined
	by blinded quantification of multiple images, with a
	minimum of 75 cells/experiment counted. Data are
	combined from 5 (E) or 3 (F) independent experiments,
	with the horizontal line depicting mean (data for each
	experiment have a distinct symbol). Experiments in
	(F) were performed simultaneously to 3 experiments
	in (E), and data in (E and F) were analyzed by 1-way
	ANOVA with Tukey's multiple comparisons test,

* P < 0.05 mammary glands at different stages of lactation, as well as in mouse mammary epithelial cells.

ACKNOWLEDGMENTS

Confocal microscopy was performed at the Institute for Molecular Bioscience Dynamic Imaging Facility for Cancer Biology at The University of Queensland, which was established with the support of the Australian Cancer Research Foundation. The authors acknowledge the use of the Protein Expression Facility, as well as the Microscopy Australia Research Facility at the Centre for Microscopy and Microanalysis at The University of Queensland. We thank the Australian Red Cross Blood Service for providing buffy coats for the isolation of human monocytes, James Murphy (Walter and Elizabeth Hall Institute of Medical Research, Melbourne, Australia) for providing the pF_TRE3G_PGK_puro construct, and Nicholas Condon (Institute for Molecular Bioscience, The University of Queensland) for assistance with live cell imaging in the CLEM experiments. This work was supported by National Health and Medical Research Council of Australia (NHMRC) project grants (APP1065391 and APP1144967). M.J.S. and M.A.S. are supported by NHMRC Senior Research Fellowships (APP1107914 and APP1106930, respectively), and R.K. was supported by an Australian Research Council Discovery Early Career Researcher Award Fellowship (DE1310470). R.G.P. was supported by grants and a fellowship from the NHMRC (grants APP1140064 and APP1150083 and fellowship APP1156489), and the Australian Research Council (Centre of Excellence in Convergent Bio-Nano Science and Technology CE140100036). The funders had no role in study design, data collection and interpretation, or the decision to submit this work for publication.

DISCLOSURE

The authors declare no conflict of interest.

AUTHORSHIP

C.J.S., J.v.P., J.B.C., and J.R. performed experiments, with assistance from R.K. and M.D.P.; D.F. and N.J.B. performed key initial studies and contributed to project design; K.M.P. and T.K. provided essential reagents (bacterial strains and anti-SLC30A1 Ab, respectively); M.J.S., R.K., M.A.S., and R.G.P. designed and supervised the project; and C.J.S., J.v.P., R.K., and M.J.S. wrote the manuscript. All authors read and approved the final manuscript.

SUPPORTING INFORMATION

Additional information may be found online in the Supporting Information section at the end of the article.


Muriel Valantin-Morison

Safia Médiène

EcosysteMIX: an interactive web app to design crop mixtures combining empirical and scientific knowledge

View project Raphael Paut French National Institute for Agriculture, Food, and Environment (INRAE)

EcosysteMIX: an interactive web app to design crop mixtures combining empirical and scientific knowledge

Raphaël PAUT 1 , Safia MEDIENE 1 and Muriel VALANTIN-MORISON 1 1 INRAE, UMR Agronomie,AgroParisTech,78850 Thiverval-Grignon, France

ABSTRACT

Crop mixture is often presented as a relevant lever for redesigning dominant agricultural systems towards sustainable systems [START_REF] Bedoussac | Ecological principles underlying the increase of productivity achieved by cereal-grain legume intercrops in organic farming. A review[END_REF]. However, their implementation is still facing difficulties due to the lack of knowledge on (i) their complex biological functioning (ii) on species selection and (iii) on their insertion in the crop rotation (Bybee-Finley & Ryan, 2018). In addition, the great diversity of crops in terms of phenology, the number of combinations, the various biological interactions to consider (and the nature of these interactions) make it very difficult to acquire scientific references on these systems. While research is underway to better understand crop mixtures functioning and the services they provide, the nearly unlimited modalities of crop combinations make this work very tedious.

In parallel, there is a substantial "empirical literature" on these systems. This mainly includes reports (annual, technical, project, etc.) and farmers on-site experimentations. It compiles empirical knowledge resulting from the capitalization of site-specific experiments. The objective of the present work is therefore to combine scientific and empirical literature in order to provide a decision support tool for the design of crop mixtures. We carried out a review on crop mixtures in scientific literature and empirical experimentations. With regard to scientific literature, 46 papers were analysed, resulting in 122 experiments (an experiment was defined by a unique crop combination in a site and year). With regard to farmers on-site experimentations, surveys were carried out on ReMix (H2020 project) multi-actor platforms and on a diagnosis of uses conducted on a tool for oilseed rape-based mixture. This resulted in 160 additional experiments.

Data from our literature review were combined in a web-based interactive platform. The user can enter a query by crops, expected service, location or mixture type. Several forms of graphic representations are then available. The platform also allows for a more exploratory approach by generating a crop network representation. The web-based application is available at: https://raphaelpaut.shinyapps.io/EcoSystemix/