Frédéric Bernicot
email: frederic.bernicot@univ-nantes.fr

BOUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS FR ÉD ÉRIC BERNICOT AND POLONA DURCIK

Keywords: Mathematics Subject Classification, Primary 42B15; Secondary 42B20, 1

.

Introduction

The classical Coifman-Meyer theorem [START_REF] Coifman | Au delá des opérateurs pseudo-différentieles[END_REF][START_REF] Coifman | Wavelets, Calderón-Zygmund Operators and Multilinear Operators[END_REF] is concerned with bilinear operators of the form

T m (F 1 , F 2 )(x) = R 2n
F 1 (ξ) F 2 (η)m(ξ, η)e 2πix•(ξ+η) dξdη, defined for test functions F 1 , F 2 : R n → C and m a bounded function on R 2n . The Coifman-Meyer theorem states that if m, in addition, satisfies

|∂ α ξ ∂ β η m(ξ, η)| ≤ C|(ξ, η)| -|α|-|β| (1.1)
for all multi-indices α, β ∈ N n 0 up to a sufficently large finite order and all (ξ, η) = 0, with 0 ≤ C < ∞, then the operator T m maps L p 1 ×L p 2 to L p 3 whenever 1 < p 1 , p 2 ≤ ∞, 1/2 < p 3 < ∞, and 1/p 1 + 1/p 2 = 1/p 3 . A notable application of this result is to the fractional Leibniz rule by Kato and Ponce [START_REF] Kato | Commutator estimates and the Euler and Navier-Stokes equations[END_REF], which has further applications to nonlinear PDE; see for instance the work by Christ and Weinstein [START_REF] Christ | Dispersion of small amplitude solutions of the generalized KdV equation[END_REF].

Multi-parameter variants of the Coifman-Meyer theory arise by considering multilinear operators with symbols behaving as tensor products of symbols (1.1). A simple bi-parameter example can be obtained by considering the operator T m with m(ξ, η) = m 1 (ξ)m 2 (η), (1.2) where m 1 and m 2 are smooth away from the origin and satisfy the analogous estimates as in (1.1). This case immediately splits into a pointwise product of two linear Calderón-Zygmund operators. It can be observed that the symbol m in (1.2) satisfies

|∂ α ξ ∂ β η m(ξ, η)| ≤ C|ξ| -|α| |η| -|β| . (1.3)
In contrast to this example, a major early contribution to the theory of bi-parameter operators by Grafakos and Kalton [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF] states that the condition (1.3) is in general not sufficient for the L p boundedness of T m . Further developments of the multi-parameter theory were driven by interest in obtaining various fractional Leibniz-type rules such as in the works by Muscalu, Pipher, Tao, Thiele [START_REF] Muscalu | Bi-parameter paraproducts[END_REF][START_REF] Muscalu | Multi-parameter paraproducts[END_REF]. In particular, these papers show bounds for the operators T m with symbols satisfying

|∂ α (ξ 1 ,η 1 ) ∂ β (ξ 2 ,η 2 ) m(ξ, η)| ≤ C|(ξ 1 , η 1 )| -|α| |(ξ 2 , η 2 )| -|β| , (1.4)
where ξ = (ξ 1 , ξ 2 ), η = (η 1 , η 2 ) ∈ R n ×R n . More general operators with so-called flag singularities were studied by Muscalu [START_REF] Muscalu | Flag paraproducts[END_REF][START_REF] Muscalu | Paraproducts with flag singularities I: A case study[END_REF]. Some recent works in the area include the one by Muscalu

and Zhai [START_REF] Muscalu | Five-Linear Singular Integral Estimates of Brascamp-Lieb Type[END_REF], who investigate a certain trilinear operator which falls under the class of singular Brascamp-Lieb integrals with non-Hölder scaling, and some related flag paraproducts studied by Lu, Pipher, and Zhang [START_REF] Lu | Bi-parameter trilinear Fourier multipliers and pseudo-differential operators with flag symbols[END_REF].

In the aforementioned papers, the symbols in question generalize products of Coifman-Meyer symbols (1.1), each symbol acting on one or several fibers of the input functions. For instance, the fiber-wise action in (1.4) means that the first factor on the right-hand side of (1.4) concerns only F 1 (•, ξ 2 ) and F 2 (•, η 2 ), while the second term concerns only the complementary fibers F 1 (ξ 1 , •) and F 2 (η 1 , •). Several recent developments in the theory of singular integrals include the study of multilinear operators with symbols acting fiber-wise on the input functions but with an additional "twist" as compared to (1.4), such as a symbol acting on F 1 (ξ 1 , •) and F 2 (•, η 2 ), or F 1 (•, ξ 2 ) and F 2 (η 1 , •). A fiber-wise action of this kind was first studied by Kovač [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] and the first author [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF] in the one-parameter setting (1.1) and dimension n = 2. In this paper we address such a situation in the multi-parameter setting.

We follow customary practice to model symbols by the convolution-type P -Q operators, see for instance [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] or [START_REF] Muscalu | Uniform estimates on paraproducts. Dedicated to the memory of Thomas H. Wolff[END_REF]. For k ∈ Z and 1 ≤ i ≤ 2, let ϕ i,k and ψ i,k be smooth functions adapted in the interval [-2 k+1 , 2 k+1 ], and let ψ i,k vanish on [-2 k-100 , 2 k-100 ]. A function ρ adapted to an interval I ⊆ R means a function supported in I and satisfying

∂ α ρ ∞ ≤ |I| -α
for all multi-indices α up to order N for some large N ; see [START_REF] Stein | Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF]. Let P i,k and Q i,k denote the one-dimensional Fourier multipliers with symbols ϕ i,k and ψ i,k respectively, i.e.

P i,k f = f * q ϕ i,k , Q i,k f = f * q ψ i,k
for f ∈ L 1 loc (R). When we apply such operators to one-dimensional fibers of a two-dimensional function we use a superscript to denote the fiber on which the action takes place. For instance, P

i,k F (x 1 , x 2 ) = F (•, x 2 ) * q ϕ i,k (x 1 ), P

i,k F (x 1 , x 2 ) = F (x 1 , •) * q ϕ i,k (x 2 ), and similarly for Q

i,k F and Q

i,k F . The central objects of this paper are the two-dimensional bi-parameter bilinear operators

T 1 (F 1 , F 2 )(x) = (k,l)∈Z 2 :k≤l (Q (1) 1,k P (2) 2,l F 1 )(x) (Q (1) 2,l P (2) 1,k F 2 )(x)
and

T 2 (F 1 , F 2 )(x) = (k,l)∈Z 2 :k≤l (P (1) 1,k P (2)
2,l F 1 )(x) (Q

2,l Q

1,k F 2 )(x), defined for test functions F 1 , F 2 : R 2 → C. We prove the following bounds in Section 2.

Theorem 1. The operators T 1 and T 2 are bounded from L p 1 (R 2 ) × L p 2 (R 2 ) to L p 3 (R 2 ) whenever 1 < p 1 , p 2 < ∞, 1 < p 3 < 2, and 1

p 1 + 1 p 2 = 1 p 3 .
Passing to the Fourier side, the operators T 1 and T 2 can be viewed as multiplier operators which map the tuple (F 1 , F 2 ) to the two-dimensional function defined by

x → R 4 F 1 (ξ) F 2 (η)m(ξ, η)e 2πix•(ξ+η) dξdη (1.5)
for a suitable bounded function m on R 4 . In the case of T 1 and T 2 , the function m satisfies

|∂ α (ξ 1 ,η 2 ) ∂ β (ξ 2 ,η 1 ) m(ξ, η)| ≤ C|(ξ 1 , η 2 )| -|α| |(ξ 2 , η 1 )| -|β| (1.6)
for all α, β ∈ N 2 0 up to order N and all (ξ, η) = 0. In general, the estimates (1.6) alone are not sufficient for boundedness of the multiplier operator (1.5). We elaborate on this in Section 2.4 by reducing a special case of (1.5) to the aforementioned counterexample from [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF].

Here and in the sequel, the notion of bi-and multi-parameter is related to the number of parameters of frequency scales. For instance, each factor on the right-hand side of (1.6) gives rise to one parameter. Given the constraint k ≤ l, the symbols of T 1 , T 2 do not split into the tensor products of two symbols m 1 (ξ 1 , η 2 ) and m 2 (ξ 2 , η 1 ). However, bounds for T 1 , T 2 imply bounds on the multiplier operator (1.5) in the case when m is indeed of tensor type, i.e.

m(ξ, η) = m 1 (ξ 1 , η 2 )m 2 (ξ 2 , η 1 ), (1.7)
where m 1 , m 2 satisfy the estimates

|∂ α (ζ 1 ,ζ 2 ) m i (ζ 1 , ζ 2 )| ≤ C|(ζ 1 , ζ 2 )| -α (1.8)
for all α ∈ N 2 0 up to order N and all 0

= (ζ 1 , ζ 2 ) ∈ R 2 .
This can be seen by the classical cone decomposition.

Corollary 2. Let m be given as in (1.7). Then the associated operator (1.5) is bounded from

L p 1 × L p 2 to L p 3 whenever 1 < p 1 , p 2 < ∞, 1 < p 3 < 2, and 1 p 1 + 1 p 2 = 1 p 3 .
The multiplier operator (1.5) with the symbol (1.7) was suggested by Camil Muscalu. The case when m 1 and m 2 are localized to cones in the frequency plane is sometimes called twisted paraproduct. A special case when one of the symbols m i is constantly equal to one, i.e.

x → R 4 F 1 (ξ) F 2 (η)m(ξ 1 , η 2 )e 2πix•(ξ+η) dξdη, (1.9)
has been previously studied by Kovač [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] and the first author [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF]. It is a degenerate case of the two-dimensional bilinear Hilbert transform investigated by Demeter and Thiele [START_REF] Demeter | On the two-dimensional bilinear Hilbert transform[END_REF]. The one-parameter operator (1.9) is known to map L p 1 × L p 2 → L p 3 in a larger range than stated in Corollary 2, namely whenever the exponents satisfy

1 < p 1 , p 2 < ∞, 1 2 < p 3 < 2, and 1 p 1 + 1 p 2 = 1 p 3 .
Indeed, the techniques developed in [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] yield bounds whenever 2 < p 1 , p 2 < ∞, 1 < p 3 < 2. Fiber-wise Calderón-Zygmund decomposition from [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF] can be then used to extend the range of exponents.

Let us also mention that there are further connections of (1.9) with several objects in ergodic theory. Indeed, the techniques developed in [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] have been subsequently refined and used to study larger classes of multilinear forms generalizing (1.9), which are motivated by problems on quantifying norm convergence of ergodic averages: the papers by Kovač [START_REF] Kovač | Quantitative norm convergence of double ergodic averages associated with two commuting group actions[END_REF] and Kovač, Škreb, Thiele and the second author [START_REF] Durcik | Norm-variation of ergodic averages with respect to two commuting transformations[END_REF] study ergodic averages with respect to two commuting transformations, while Škreb [START_REF] Škreb | Norm variation of cubic ergodic averages[END_REF] studies certain cubic averages. Other applications of operators related to the twisted paraproduct are in stochastic integration obtained by Kovač and Škreb [START_REF] Kovač | One modification of the martingale transform and its applications to paraproducts and stochastic integrals[END_REF] and also in Euclidean Ramsey theory when investigating patterns in large subsets of the Euclidean space; see for instance the work by Kovač and the second author [START_REF] Durcik | A Szemerédi-type theorem for subsets of the unit cube[END_REF]. We also refer to the paper by Stipčić [START_REF] Stipčić | T (1) theorem for dyadic singular integral forms associated with hypergraphs[END_REF] and the references therein.

More generally, one can consider the class of all bilinear operators (1.5) where m is of the tensor type

m(ζ 1 , ζ 2 , ζ 3 , ζ 4 ) = m 1 ((ζ a ) a∈S 1 )m 2 ((ζ a ) a∈S 2 )m 3 ((ζ a ) a∈S 3 )m 4 ((ζ a ) a∈S 4 ), (1.10)
where S i ⊆ {1, 2, 3, 4} and m i are symbols on R |S i | , each of them satisfying the estimates analogous to (1.8). Then (1.7) is a special case with S 3 = S 4 = ∅. For the purpose of this paper, let us restrict ourselves to the case when the sets S i are pairwise disjoint and consider (1.5) with such a symbol. If S 1 = {1}, then due to boundedness of the one-dimensional Calderón-Zygmund operators we may replace the function F 1 in (1.5) by

m 1 (ξ 1 ) F 1 (ξ). (1.11)
Performing the analogous step for any singleton S i , we may reduce (1.5) to the case where each non-empty set S i satisfies |S i | ≥ 2. Up to symmetries, mapping properties of the cases that arise are discussed in Table 1 below.

1 m 1 (ξ 1 , η 1 ) 1 < p 1 , p 2 ≤ ∞, 1 2 < p 3 < ∞ 2 m 1 (ξ 1 , ξ 2 ) 1 < p 1 , p 2 < ∞, 1 2 < p 3 < ∞ 3 m 1 (ξ 1 , η 2 ) 1 < p 1 , p 2 < ∞, 1 2 < p 3 < 2 4 m 1 (ξ 1 , ξ 2 )m 2 (η 1 , η 2 ) 1 < p 1 , p 2 < ∞, 1 2 < p 3 < ∞ 5 m 1 (ξ 1 , η 1 )m 2 (ξ 2 , η 2 ) 1 < p 1 , p 2 ≤ ∞, 1 2 < p 3 < ∞ 6 m 1 (ξ 1 , η 2 )m 2 (ξ 1 , η 1 ) 1 < p 1 , p 2 < ∞, 1 < p 3 < 2 7 m 1 (ξ 1 , ξ 2 , η 1 ) 1 < p 1 , p 2 < ∞, 1 2 < p 3 < ∞ 8 m 1 (ξ 1 , ξ 2 , η 1 , η 2 ) 1 < p 1 , p 2 ≤ ∞, 1 2 < p 3 < ∞
The operator corresponding to Case 1 is a classical Coifman-Meyer multiplier [START_REF] Coifman | Au delá des opérateurs pseudo-différentieles[END_REF][START_REF] Coifman | Wavelets, Calderón-Zygmund Operators and Multilinear Operators[END_REF] acting on the first fibers of the functions F 1 , F 2 . Case 2 is a pointwise product of a linear Fourier multiplier with the identity operator and can be treated analogously as (1.11). The estimates for Case 2 stated in Table 1 are obtained by Hölder's inequality. Case 3 is the operator (1.9), which has been studied in [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF]. Case 4 is a pointwise product of two linear operators and the analogous reduction as in (1.11) applies. Case 5 is a bi-parameter version of a Coifman-Meyer multiplier and its boundedness follows by [START_REF] Muscalu | Bi-parameter paraproducts[END_REF]. As remarked earlier, in [START_REF] Muscalu | Bi-parameter paraproducts[END_REF] the authors are able to handle multipliers m which are not necessarily of tensor type. This is in contrast with Case 6, which is the content of Corollary 2. It remains to consider Case 8, which is the classical Coifman-Meyer multiplier, and Case 7. We prove the following bounds for Case 7 in Section 3.

Theorem 3. If m(ξ, η) = m 1 (ξ 1 , ξ 2 , η 1 ), then the associated operator (1.5) is bounded from L p 1 × L p 2 to L p 3 whenever 1 < p 1 , p 2 < ∞, 1 2 < p 3 < ∞, and 1 p 1 + 1 p 2 = 1 p 3 .
We emphasize that extending the range of exponents for p 3 in Cases 3 and 6 remains open. Dualizing the operators in (1.5) with symbols of the form (1.10), the corresponding trilinear forms are particular examples of singular Brascamp-Lieb integrals with several singular kernels, see also the survey [START_REF] Durcik | Singular Brascamp-Lieb: a survey[END_REF]. Indeed, they can be represented by the trilinear forms

R 6 F 1 (x + A 1 s + B 1 t)F 2 (x + A 2 s + B 2 t)F 3 (x)K(s, t)dxdsdt (1.12)
for suitable matrices A 1 , B 1 , A 2 , B 2 ∈ M 2 (R) and a kernel K, whose Fourier transform K is of the form (1.10). In particular, the operator with (1.7) is associated with a tensor product of two Calderón-Zygmund kernels

K(s, t) = K 1 (s)K 2 (t),
where K 1 = q m 1 , K 2 = q m 2 . In this case, if one of the kernels K 1 , K 2 in (1.12) specializes to the Dirac delta, then the object reduces to a one-parameter family of the two-dimensional bilinear Hilbert transforms [START_REF] Demeter | On the two-dimensional bilinear Hilbert transform[END_REF]. Studying (1.12) for an arbitrary choice of matrices A i , B i and any Calderón-Zygmund kernels K 1 , K 2 remains an open problem.

Furthermore, the operators T 1 and T 2 can be viewed as particular fiber-wise versions of the paraproducts studied by Muscalu, Tao, and Thiele in [START_REF] Muscalu | Uniform estimates on paraproducts. Dedicated to the memory of Thomas H. Wolff[END_REF]. Following the setup from [START_REF] Muscalu | Uniform estimates on paraproducts. Dedicated to the memory of Thomas H. Wolff[END_REF], let n ≥ 1 and let Ω n ∈ Z n be a convex polytope of the form

Ω n = {(k 1 , . . . , k n ) ∈ Z n : k iα ≥ k i α for all 1 ≤ α ≤ K},
where i α , i α ∈ {1, . . . , n} and K ≥ 0 an integer. Consider the operators which maps an n-tuple

(F i ) n i=1 of test functions on R 2 to the two-dimensional function x → (k 1 ,...,k 2n )∈Ω 2n n i=1 (Q (1) i,k i Q (2) i+n,k i+n F i )(x), (1.13)
where each Q i,k is a Fourier multiplier with symbol ψ i,k , which is a bump function adapted in [-2 k+1 , 2 k+1 ] and vanishes on [-2 k-100 , 2 k-100 ]. When Ω 2n is of the form Ω n × Z n , bounds for (1.13) follow from [START_REF] Muscalu | Uniform estimates on paraproducts. Dedicated to the memory of Thomas H. Wolff[END_REF]. When n = 1, (1.13) reduces to a classical linear Calderón-Zygmund operator. When n = 2, one can classify the cases similarly as in Table 1. Indeed, this can be seen by summing over Ω 4 in at least two parameters k i in (1.13). Using the fact that k∈Z Q i,k are linear Calderón-Zygmund operators and hence satisfy the desired bounds, the problem is reduced to objects with the summation over Ω 2 , such as T 1 , T 2 , and other bi-parameter analogues in [START_REF] Muscalu | Bi-parameter paraproducts[END_REF]. This yields L p 1 × L p 2 to L p 3 bounds whenever 1 < p 1 , p 2 < ∞ and 1 < p 3 < 2, but the known range may, in particular cases, be larger. However, as it is the case for T 1 and T 2 , the symbols of (1.13) are in general not of tensor type because of the constraint on Ω n . The main idea used in the proof of Theorem 1 is to reduce the problem to the vector-valued estimates for the operator (1.9) with one constant symbol. The key step is in observing that the operator is localized in frequency due to the frequency supports of the fibers of the input functions. This can be seen in sharp contrast with (1.9) itself, where such localization does not occur. Localization of the operator allows replacements of some low-frequency projections, acting on the input functions, with the identity operators. This, in turn, allows for applications of Hölder's inequality. We prove Theorem 1 and Corollary 2 in Section 2. In the case of (1.9), quasi-Banach estimates can be proven using the fiber-wise Calderón-Zygmund decomposition from [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF]. This decomposition does not seem applicable in the context of Theorem 1, as the symbols act on both fibers of the input functions. Extending the range of exponents in Theorem 1 remains an open problem.

Theorem 3 is proven in Section 4 and in the Banach case it relies on the bounds for the operator (1.9). In Theorem 4, the operator acts on only one fiber of the function F 2 ; in this case we are able to use the fiber-wise Calderón-Zygmund decomposition to prove quasi-Banach estimates as stated in Theorem 3.

Multilinear and multi-parameter generalizations. At present there is only partial understanding of the multilinear generalizations of Theorem 1. In the case of (1.7), multilinear operators with only one non-constant symbol can be described in the language of bipartite graphs and were studied by Kovač in [START_REF] Kovač | Bellman function technique for multilinear estimates and an application to generalized paraproducts[END_REF] in a dyadic model.

In this paper, we discuss a particular tri-parameter trilinear example, which can be seen as a natural generalization of (1.7). Let m 1 , m 2 , m 3 be symbols satisfying (1.8) for 1 ≤ i ≤ 3. Define the trilinear operator which maps a triple (F 1 , F 2 , F 3 ) of functions on R 2 to the two-dimensional function given by

x → R 6 F 1 (ξ) F 2 (η) F 2 (τ )m 1 (ξ 1 , η 2 )m 2 (η 1 , τ 2 )m 3 (τ 1 , ξ 2 )e 2πix•(ξ+η+τ ) dξdηdτ, (1.14)
where ξ = (ξ 1 , ξ 2 ), η = (η 1 , η 2 ), τ = (τ 1 , τ 2 ). We prove the following bounds.

Theorem 4. The operator (1.14) is bounded from

L p 1 ×L p 2 ×L p 3 to L p 4 when 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞, 4 i=1
1 p 1 = 1, and 1 p 1 + 1 p 2 > 1 2 , 1 p 2 + 1 p 3 > 1 2 , 1 p 1 + 1 p 3 > 1 2 .
Note that the range in Theorem 4 is non-empty. For example, it contains exponents in vicinity of 3 < p 1 = p 2 = p 3 < 4. The proof of Theorem 4 is detailed in Section 4 and can be seen as an iteration of the steps used in the proof of Theorem 1 and its corollary, by gradually reducing to the vector-valued estimates for the operators with one or more constant symbols. Iterating this procedure and applying estimates for one-and two-parameter operators, which hold in restricted ranges of exponents, is the reason for further restriction of the range in Theorem 4.

The proof of Theorem 4 does not immediately generalize to all higher degrees of multilinearities. Beside facing the issues with the exponent range, objects with constant symbols which arise in the proof may not be localized in frequency, which prevents further iterations of the approach. Similar issues also arise when trying to generalize Theorems 1 and 4 to higher dimensions. Obtaining bounds for a larger class of multi-parameter objects, such as multilinear operators in (1.13), is closely related to studying a large class of maximally truncated singular integrals, one-and multi-parameter. One such instance is the maximally truncated one-parameter twisted paraproduct which maps a tuple (F 1 , F 2 ) of functions on R 2 to the two-dimensional function

x → sup N >0 |k|<N (P (1) 1,k F 1 )(x) (Q (2) 1,k F 2 )(x) .
(1.15)

Its boundedness seems to be out of reach of the currently available techniques. Further motivation for proving such maximal estimates and also stronger variational estimates is provided by questions on pointwise convergence of certain ergodic averages. Establishing L p estimates for the operator which maps a tuple (F 1 , F 2 ) of functions on R 2 to the one-dimensional function

x 2 → sup N >0 |k|<N (P (1) 1,k F 1 )(x 1 , x 2 ) (Q (2) 1,k F 2 )(x 1 , x 2 ) L p x 1 (R)
could be considered as an intermediate case between (1.9) and (1.15). Such bounds also remain open.

Notation. For two non-negative quantities A, B we write A B if there exists an absolute constant C such that A ≤ CB. If C depends on the parameters P 1 , . . . , P n , we write A P 1 ,...,Pn B.

Acknowledgments. The authors thank Cristina Benea, Vjekoslav Kovač, Camil Muscalu, and Christoph Thiele for several motivating discussions. The first author is supported by ERC project FAnFArE no.637510. The second author acknowledges the hospitality of Université de Nantes while this research was performed.

The bi-parameter bilinear operators T 1 and T 2

This section is devoted to the proof of Theorem 1. Throughout this section we will use the shorthand notation k l to denote k < l -200. Similarly, k l will denote k > l + 200 and k ∼ l will mean l -200 ≤ k ≤ l + 200.

It will be evident from the proof that the argument will not rely on the particular choice of the bump functions ϕ i,k , ψ i,k as long as they satisfy the assumptions in the definition of T 1 and T 2 . For simplicity of notation, we shall therefore only discuss the case ϕ

1,k = ϕ 2,k = ϕ k and ψ 1,k = ψ 2,k = ψ k for each k ∈ Z. We shall also write P 1,k = P 2,k = P k and Q 1,k = Q 2,k = Q k .
2.1. Boundedness of T 1 . First we split the summation over k ≤ l into the regimes where k ∼ l and k l respectively, i.e.

T 1 (F, G) = (k,l)∈Z 2 l-200≤k≤l (Q (1) k P (2) l F 1 ) (Q (1) l P (2) k F 2 ) + (k,l)∈Z 2 k l (Q (1)
k P

(2)

l F 1 ) (Q (1) l P (2) k F 2 ). (2.1)
Bounds for the sum over l -200 ≤ k ≤ l follow by Cauchy-Schwarz. Indeed, using Cauchy-Schwarz in l ∈ Z we pointwise bound this term as

-200≤s≤0 l∈Z (Q (1) l+s P (2) l F 1 ) (Q (1) l P (2) l+s F 2 ) ≤ -200≤s≤0 Q (1) l+s P (2) l F 1 2 l (Z) Q (1)
l P

(2)

l+s F 2 2 l (Z) .
We consider the product of terms on the right-hand side for each fixed -200 ≤ s ≤ 0. To estimate the L p 3 norm of the product we apply Hölder's inequality and use bounds for the classical square function. We obtain Q

(2)

l F 1 2 l (Z) Q (1) l P
(2)

l+s F 2 2 l (Z) L p 3 (R 2 ) ≤ Q (1) l+s P
(2)

l F 1 L p 1 ( 2 l ) Q (1) l P
(2)

l+s F 2 L p 2 ( 2 l ) p 1 ,p 2 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 )
whenever 1/p 3 = 1/p 1 + 1/p 2 and 1 < p 1 , p 2 < ∞. In the end it remains to sum the individual contributions of these finitely many terms. It remains to consider the case k l in (2.1). By duality it suffices to study the corresponding trilinear form and show

(k,l)∈Z 2 :k l R 2 (Q (1)
k P

(2)

l F 1 ) (Q (1)
l P

(2)

k F 2 ) F 3 p 1 ,p 2 ,p 3 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 ) F 3 L p 3 (R 2 )
for any choice of exponents 1

< p 1 , p 2 < ∞, 2 < p 3 < ∞ with 1/p 1 + 1/p 2 + 1/p 3 = 1.
By the frequency supports of F 1 , F 2 , the form on the left-hand side can be written up to a constant as

(k,l)∈Z 2 :k l R 2 (Q (1)
k P

(2)

l F 1 ) (Q (1) l P
(2)

k F 2 ) (Q (1) l P
(2)

l F 3 ), (2.2)
where P l and Q l are Fourier multipliers with symbols adapted to [-2 l+3 , 2 l+3 ], the symbol of P l is constant on [-2 l+2 , 2 l+2 ], and the symbol of

Q l vanishes on [-2 l-90 , 2 l-90 ].
Then we write P l = ϕ l (0)I + (P l -ϕ l (0)I) where I is the identity operator, and plug this decomposition into the form (2.2). This yields

(2.2) = (k,l)∈Z 2 :k l M k,l + (k,l)∈Z 2 :k l E k,l , (2.3)
where we have defined

M k,l = R 2 (Q (1) k F 1 ) (Q (1) l P
(2)

k F 2 ) (ϕ l (0)Q (1) l P
(2)

l F 3 ), E k,l = R 2 (Q (1)
k (P (2) l -ϕ l (0)I (2) )F 1 ) (Q (1) l P
(2)

k F 2 ) (Q (1) l P
(2) l F 3 ).

First we consider the term involving M k,l . Since |ϕ l (0)| ≤ 1, up to redefining P l we may assume that ϕ l (0) = 1 for each l. We split the summation as

(k,l)∈Z 2 :k l M k,l = (k,l)∈Z 2 M k,l - (k,l)∈Z 2 :k∼l M k,l - (k,l)∈Z 2 :k l M k,l .
(2.4)

By the frequency support of the first fibers of the functions F i it follows that the term over k l vanishes. To estimate the term with summation over k ∼ l we use Hölder's inequality in l and in the integration, yielding

l∈Z s∼0 M l+s,s ≤ s∼0 Q (1) l+s F 1 L p 1 ( ∞ l ) Q (1)
l P

(2)

l+s F 2 L p 2 ( 2 l ) Q (1)
l P

(2)

l F 3 L p 3 ( 2 l ) .
It remains to use bound on the one-dimensional maximal function and the two square functions, which hold uniformly in s ∼ 0, and finally sum in s. Thus, it suffices to estimate the case when the sum is unconstrained, i.e. over (k, l) ∈ Z 2 . By Cauchy-Schwarz in l and Hölder's inequality in the integration we estimate

(k,l)∈Z 2 M k,l ≤ k∈Z (Q (1) k F 1 ) (Q (1) l P (2) k F 2 ) L p 3 ( 2 l ) Q (1)
l P

(2)

l F 3 L p 3 ( 2 l ) , (2.5)
where 1/p 3 + 1/p 3 = 1. The second term on the right-hand side is a classical square function. To bound the first term we use L

p 1 (R 2 ) × L p 2 ( 2 ) → L p 3 ( 2 )
vector-valued estimates for the operator (1.9), which hold whenever 1 < p 3 < 2 and 1 < p 1 , p 2 < ∞. These vector-valued estimates are obtained by freezing the function F 1 ∈ L p 1 and using the linear inequalities of Marcinkiewicz and Zygmund [START_REF] Marcinkiewicz | Mean values of trigonometrical polynomials[END_REF], together with scalar-valued boundedness of (1.9). We obtain

k∈Z (Q (1)
k F 1 ) (Q (1) l P
(2)

k F 2 ) L p 3 ( 2 l ) p 1 F 1 L p 1 (R 2 ) Q (1) l F 2 L p 2 ( 2 l ) p 1 ,p 2 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 ) ,
where we have used the Littlewood-Paley inequality for the last bound. This yields the desired estimate for the first term in (2.3). It remains to estimate the second term in (2.3). By frequency consideration in the second fiber, one has

(k,l)∈Z 2 :k l E k,l = c (k,l)∈Z 2 :k l R 2 (Q (1) k Q (2) l F 1 ) (Q (1) l P
(2)

k F 2 ) (Q (1) l P
(2)

l F 3 ), (2.6)
where c is an absolute constant and Q l is a Fourier multiplier with symbol adapted to [-2 l+3 , 2 l+3 ] which vanishes at the origin. Indeed, note that this is the case both when ϕ l (0) = 0 and when ϕ l (0) = 0. We split the summation in the regions where (k, l) ∈ Z 2 , k l and k l. By the analogous considerations as in the paragraphs from (2.4) to (2.5) we note that it suffices to instead consider the case when the sum is unconstrained, i.e.

(k,l)∈Z 2 R 2 (Q (1) k Q (2) l F 1 ) (Q (1) l P (2) k F 2 ) (Q (1) l P (2) l F 3 ).
By the Cauchy-Schwarz inequality in l and Hölder's inequality in the integration we bound the last display by

k∈Z (Q (1) k Q 2 l F 1 ) (Q (1) l P (2) k F 2 ) L p 3 ( 2 l ) Q (1) l P (2) l F 3 L p 3 ( 2 l ) .
The second term is a square function. Bounds for the first term follow from L p ( 2 ) × L q ( 2 ) → L r ( 2 ) vector-valued estimates for the twisted paraproduct (1.9) and two applications of the Littlewood-Paley inequality. The vector valued estimates which we need in this case are a consequence of scalar boundedness of the operator (1.9) and a result by Grafakos and Martell [START_REF] Grafakos | Extrapolation of weighted norm inequalities for multivariable operators and applications[END_REF]Theorem 9.1]. This yields the desired bound for the second term in (2.3) and in turn establishes the claim for T 1 .

2.2. Boundedness of T 2 . The proof for T 2 proceeds in the analogous way as the proof for T 1 and we only sketch the necessary ingredients. By duality it suffices to bound

(k,l)∈Z 2 :k≤l R 2 (P (1)
k P

(2)

l F 1 ) (Q (1) l Q (2) k F 2 ) F 3 p 1 ,p 2 ,p 3 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 ) F 3 L p 3 (R 2 )
for any choice of exponents 1

< p 1 , p 2 < ∞, 2 < p 3 < ∞ with 1/p 1 + 1/p 2 + 1/p 3 = 1.
The case when l -200 ≤ k ≤ l is bounded by Hölder's inequality, so it suffices to consider k l. By frequency considerations, the form on the left-hand side is a constant multiple of

(k,l)∈Z 2 :k l R 2 (P (1)
k P

(2)

l F 1 ) (Q (1) l Q (2) k F 2 ) (Q (1)
l P

(2)

l F 3 )
for frequency projections Q l and P l as in (2.2). As above we split P l = ϕ(0)I + (P l -ϕ(0)I). By considerations as in the discussion after (2.3) it remains to estimate the analogue of the term associated with M k,l , i.e.

(k,l)∈Z 2 R 2 (P (1)
k F 1 ) (Q (1) l Q (2) k F 2 ) (ϕ(0)Q (1) l P
(2)

l F 3 ),
and the analogue of the term associated with E k,l , i.e.

(k,l)∈Z 2 R 2 (P (1) k Q
(2)

l F 1 ) (Q (1) l Q (2) k F 2 ) (Q (1) l P
(2)

l F 3 ),
where Q l is as in (2.6). The proofs for each of these terms proceed analogously as for (2.5) and (2.6) respectively, reducing to vector-valued estimates for the operator (1.9).

Remark 5. An alternative way to prove bounds for T 2 is to deduce them from the bounds for T 1 via a telescoping identity, which "swaps" the P -and Q-type operators. Indeed, this can be achieved in a special case when T 1 and T 2 are related by a condition on the bump functions as in Proposition 8 in Section 4 below. Then, one can deduce a general case of T 2 from the special case by an averaging argument. We perform such arguments in a trilinear tri-parameter setting in Section 4 below.

2.3.

Proof of Corollary 2. This corollary can be deduced from Theorem 1 by a classical cone decomposition, as performed for instance in [START_REF] Muscalu | Classical and Multilinear Harmonic Analysis[END_REF]. For completeness we outline the relevant steps.

Let m be a function on R 2 \ {0} satisfying

|∂ α m(ζ)| |ζ| -|α|
for all α up to a large finite order and all ζ = 0 in R 2 . Let ϕ be a smooth function supported in [-2, 2] and constantly equal to

1 on [-1, 1]. Let ψ = ϕ -ϕ(2•). Then k∈Z ψ(2 -k τ ) = 0 for each τ = 0. We can write m(ζ) = (k,l)∈Z 2 m(ζ)ψ(2 -k ζ 1 )ψ(2 -l ζ 2 ).
Splitting the sum into regions when k ≤ l and k > l, respectively, and summing in the smaller parameter we obtain

m(ζ) = k∈Z m(ζ)ϕ(2 -k ζ 1 )ψ(2 -k ζ 2 ) + k∈Z m(ζ)ψ(2 -k ζ 1 )ϕ(2 -(k-1) ζ 2 ).
We consider the first sum; the second is treated analogously. Note that for each k ∈ Z, the summand is supported in

{ζ ∈ R 2 : |ζ 1 | ≤ 2 k+1 , 2 -k-1 ≤ |ζ 2 | ≤ 2 k+1 }.
The smooth restriction of m to that region can be decomposed into a double Fourier series, which yields

k∈Z m(ζ)ϕ(2 -k ζ 1 )ψ(2 -k ζ 2 ) = (n 1 ,n 2 )∈Z 2 k∈Z C k n 1 ,n 2 ϕ(2 -k ζ 1 )e cπi2 -k ζ 1 n 1 ψ(2 -k ζ 2 )e cπi2 -k ζ 2 n 2
where c is a fixed constant and the Fourier coefficients

C k n 1 ,n 2 satisfy |C k n 1 ,n 2 | (1 + |n 1 |) -N (1 + |n 2 |) -N
for any N > 0 up to a large order, uniformly in k ∈ Z. For details we refer to [START_REF] Muscalu | Classical and Multilinear Harmonic Analysis[END_REF], Chapter 2.13.

Normalizing the bump functions and the coefficients, the last display can be written as an absolute constant times

(n 1 ,n 2 )∈Z 2 (1 + |n 1 |) -2 (1 + |n 2 |) -2 k∈Z C k n 1 ,n 2 ϕ k,n 1 (ζ 1 )ψ k,n 2 (ζ 2 ), where ϕ k,n (ζ 1 ) = c 1 ϕ(ζ 1 )e cπiζ 1 n 1 , ψ k,n 2 (ζ 2 ) = c 2 ψ(ζ 2 )e cπiζ 2 n 2 ,
and the constants c 1 , c 2 are such that both functions are adapted to [-2 k+1 , 2 k+1 ]. Moreover,

ϕ k,n 1 (0) is the same constant for each k ∈ Z, ψ k,n 2 vanishes in [-2 k-1 , 2 k-1 ], and | C k n 1 ,n 2 | ≤ 1.
This reduces the matters to considering symbols for each fixed n 1 , n 2 with bounds uniform in n 1 , n 2 . Note that the coefficients C k n 1 ,n 2 can assumed to be equal to 1 by subsuming them into the definitions of ψ k,n 2 .

We perform this decomposition for the symbols m 1 and m 2 in (1.7). Then the bounds for the associated operator follow from bounds on T 1 and T 2 .

A counterexample.

In this section we show that the estimates (1.6) alone are not sufficient for boundedness of the operator (1.5). We restrict ourselves to the Banach regime. More precisely, we show that there exists a bounded function m on R 4 satisfying

|∂ α (ξ 1 ,η 2 ) ∂ β (ξ 2 ,η 1 ) m(ξ, η)| α,β |(ξ 1 , η 2 )| -|α| |(ξ 2 , η 1 )| -|β|
for all multi-indices α, β ∈ N 0 such that the associated operator (1.5) does not satisfy L p 1 × L p 2 to L p 3 estimates for any exponents 1 < p 1 , p 2 , p 3 < ∞.

To see this we first recall a result from [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF]; see the remark at the end of Section 3 in [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF]. Existence is shown of a symbol m 0 on R 2 \ {0} satisfying

|∂ α ξ 1 ∂ β η 1 m 0 (ξ 1 , η 1 )| α,β |ξ 1 | -α |η 1 | -β for all multi-indices α, β ∈ N 0 , such that the one-dimensional operator mapping a tuple (f 1 , f 2 ) of functions on R to a one-dimensional function x → R 2 f 1 (ξ 1 ) f 2 (η 1 )m 0 (ξ 1 , η 1 )e 2πix(ξ 1 +η 1 ) dξ 1 dη 1 , (2.7)
does not satisfy any L p 1 × L p 2 to L p 3 bounds.

To show that the estimates (1.6) are in general not sufficient we reduce a special case of (1.5) to this counterexample. Let us define

m(ξ 1 , ξ 2 , η 1 , η 2 ) = m 0 (ξ 1 , η 1 ) m(ξ 1 , η 2 ) m(η 1 , ξ 2 ), where m is a smooth symbol on R 2 \ {0} supported in the cone {(ζ 1 , ζ 2 ) : |ζ 2 | |ζ 1 |}, satisfying m(ζ 1 , 0) = 1 whenever |ζ 1 | = 0, and
|∂ α m(ζ 1 , ζ 2 )| α |(ζ 1 , ζ 2 )| -|α|
for all α ∈ N 2 0 and all (ζ 1 , ζ 2 ) = 0. Then m satisfies (1.6) and we have m(ξ 1 , 0, η 1 , 0) = m 0 (ξ 1 , η 1 ). We dualize the operator (1.5) associated with this multiplier and consider the corresponding trilinear form, which reads

R 4 F 1 (ξ) F 2 (η) F 3 (ξ + η)m(ξ, η)dξdη.
(2.8)

Let λ > 0 and let 1 < p 1 , p 2 , p 3 < ∞ be such that 1/p 1 + 1/p 2 + 1/p 3 = 1. For 1 ≤ i ≤ 3 we set

F i (x 1 , x 2 ) = f i (x 1 )λ -1/p i ϕ(λ -1 x 2 ),
where f i and ϕ are one-dimensional smooth compactly supported functions and ϕ ≥ 0. Plugging these particular functions F i into the trilinear form (2.8) we obtain

λ -1 R 4 f 1 (ξ 1 )λ ϕ(λξ 2 ) f 2 (η 1 )λ ϕ(λη 2 ) f 3 (ξ 1 + η 1 )λ ϕ(λ(ξ 2 + η 2 )) m(ξ 1 , ξ 2 , η 1 , η 2 )dξ 1 dξ 2 dη 1 dη 2 .
(2.9)

By rescaling in ξ 2 and η 2 it suffices to consider

R 4 f 1 (ξ 1 ) ϕ(ξ 2 ) f 2 (η 1 ) ϕ(η 2 ) f 3 (ξ 1 + η 1 ) ϕ(ξ 2 + η 2 )m(ξ 1 , λ -1 ξ 2 , η 1 , λ -1 η 2 )dξ 1 dξ 2 dη 1 dη 2 .
Letting λ → ∞ and then integrating in ξ 2 , η 2 we obtain a non-zero constant multiple of

R 2 f 1 (ξ 1 ) f 2 (η 1 ) f 3 (ξ 1 + η 1 )m 0 (ξ 1 , η 1 )dξ 1 dη 1 ,
which can be recognized as (2.7) paired with a function f 3 . Therefore, the form in the last display does not satisfy any L p estimates. Since

F i L p i (R 2 ) equals f i L p i (R) ϕ L p i (R)
, it follows that (2.9) does not satisfy any L p estimates as well.

A one-parameter bilinear operator

This section is devoted to the proof of Theorem 3. By a three-dimensional analogue of the cone decomposition outlined in Section 2.3 it suffices to estimate the operators mapping (F 1 , F 2 ) to two-dimensional functions given by

x → k∈Z (Q (1) k P (2) 1,k F 1 )(x) (P (1) 2,k F 2 )(x), (3.1)
x → k∈Z (P

(1) 1,k P (2) 2,k F 1 )(x) (Q (1) k F 2 )(x), (3.2)
and

x → k∈Z (P (1) 1,k Q (2) k F 1 )(x) (P (1) 2,k F 2 )(x), (3.3)
where P i,k , Q k are Fourier multipliers with symbols ϕ i,k and ψ k , respectively, which are adapted to [-2 k+1 , 2 k+1 ], and in addition

ψ k vanishes on [-2 k-1 , 2 k-1 ].
We will first prove bounds for each of them in the open Banach range. Then we will extend the range with a fiber-wise Calderón-Zygmund decomposition from [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF].

3.1.

Boundedness in the open Banach range. In this section we show that the operators (3.1), (3.2) and (3.3) are bounded from L p 1 × L p 2 to L p 3 whenever 1 < p 1 , p 2 , p 3 < ∞ and 1/p 1 + 1/p 2 + 1/p 3 = 1.

To bound the first operator (3.1), we first reduce to the case when the symbol of P 2,k is supported in [-2 k-99 , 2 k-99 ]. Indeed, if this is not the case we split

P 2,k = P 2,k + Q 2,k ,
where the symbols of P 2,k and Q 2,k are supported in [-2 k-99 , 2 k-99 ] and [-2 k+1 , -2 k-100 ] ∪ [2 k-100 , 2 k+1 ], respectively. We split the operator accordingly and note that the term with Q 2,k immediately reduces to two square functions by Cauchy-Schwarz. Now we dualize and consider the corresponding trilinear form. By the frequency localization in the first fibers of the functions, it suffices to show

k∈Z 2 (Q (1) k P (2) 1,k F 1 ) ( P (1) 2,k F 2 ) (Q (1) k F 3 ) p 1 ,p 2 ,p 3 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 ) F 3 L p 3 (R 2 )
for any choice of exponents 1 < p 1 , p 2 , p 3 < ∞ and 1/p 1 + 1/p 2 + 1/p 3 = 1. Here Q k is adapted to [-2 k+3 , 2 k+3 ] and the corresponding symbol vanishes at the origin. This estimate follows immediately by Hölder's inequality in k and in the integration, together with bounds on the square and maximal function.

For the second operator (3.2) we proceed in the same way; this time reducing to the case when the symbol of

P 1,k is supported in [-2 k-99 , 2 k-99 ].
It remains to consider (3.3). By considerations as above, we may assume that P 1,k is supported in [-2 k-99 , 2 k-99 ]. By duality it suffices to study the corresponding trilinear form k∈Z R 2 (P

(1) 1,k Q (2) k F 1 ) (P (1) 2,k F 2 ) F 3 = c k∈Z R 2 (P (1) 1,k Q (2) k F 1 ) Q (2) k (P (1) 2,k F 2 ) F 3 , (3.4)
where c is a constant and Q k satisfies the properties as in the previous display. By Cauchy-Schwarz in k and Hölder's inequality in the integration, we bound the last display by

P (1) 1,k Q (2) k F 1 L p 1 ( 2 k ) Q (2) k (P (1) 2,k F 2 ) F 3 L p 1 ( 2 k )
.

The first term is a square function bounded in the full range. For the second term, we note that by Fatou's lemma, it suffices to restrict the 2 k norm to |k| ≤ K for some K > 0 and prove bounds uniform in K. By Kintchine's inequality, this term reduces to having to estimate

|k|≤K a k Q (2) k (P (1) 2,k F 2 )F 3 L p 1 (R 2 )
for uniformly bounded coefficients a k . Dualizing with

G ∈ L p 1 (R 2 ) it suffices to show |k|≤K R 2 a k (Q (2) k G) (P (1) 2,k F 2 ) F 3 p 1 ,p 2 ,p 3 G L p 1 (R 2 ) F 2 L p 2 (R 2 ) F 3 L p 3 (R 2 ) .
This estimate holds in the range 1 < p 1 , p 2 < ∞, 2 < p 3 < ∞, and Hölder scaling, since the left-hand side is an operator of the form (1.9) paired with the function F 3 .

To obtain bounds for (3.3) in the full range, it suffices to show estimates with 1 < p 1 , p 3 < ∞ and 2 < p 2 < ∞. The left-hand side of (3.4) can be written as

k∈Z R 2 (P (1) 1,k Q (2) k F 1 ) (P (1) 2,k F 2 ) P (1) k F 3 for a multiplier P k with symbol adapted to [-2 k+3 , 2 k+3 ]. Writing P 2,k = ϕ 2,k (0)I + (P 2,k - ϕ 2,k (0)I) we reduce the last display to k∈Z R 2 (P (1) 1,k Q (2) k F 1 ) F 2 (ϕ 2,k (0)P (1) k F 3 ) + k∈Z R 2 (P (1) 1,k Q (2) k F 1 ) (Q (1) k F 2 ) (P (1) k F 3 )
for a suitably defined Q k whose symbol vanishes at the origin. The first term is analogous to the left-hand side of (3.4), satisfying the estimates with 1 < p 1 , p 3 < ∞ and 2 < p 2 < ∞. The second term has two Q-type operators and bounds in the full range follow by Hölder's inequality and bounds on the maximal and square functions.

3.2. Fiber-wise Calderón-Zygmund decomposition. In this section we show that the operators (3.1), (3.2) and (3.3) are bounded from L p 1 × L p 2 to L p 3 whenever 1 < p 1 , p 2 < ∞, 1/2 < p 3 < ∞ and 1/p 1 + 1/p 2 = 1/p 3 .

By real bilinear interpolation, it suffices to prove weak-type bounds with p 3 < 1 for the operators (3.1), (3.2) and (3.3). Each of them is of the form

T (F 1 , F 2 )(x) = k∈Z (P (1) 1,k P (2) 3,k F 1 )(x) (P (1) 2,k F 2 )(x),
where P i,k are Fourier multipliers with symbols adapted to [-2 k+1 , 2 k+1 ]. We are going to apply the fiberwise Calderón-Zygmund from [START_REF] Bernicot | Fiber-wise Calderón-Zygmund decoposition and application to a bi-dimensional paraproduct[END_REF] to the function F 2 .

It suffices to show that for any 1 < p 1 , p 2 < ∞ and 1/2 < p 3 < 1, the operator T satisfies the weak L

p 1 × L p 2 → L p 3 ,∞ estimates. Fix F 1 ∈ L p 1 (R 2 ) and F 2 ∈ L p 2 (R 2 )
. By homogeneity we may assume

F 1 L p 1 (R 2 ) = F 2 L p 2 (R 2 ) = 1. Our goal is to show that for every λ > 0 |{x ∈ R 2 : |T (F 1 , F 2 )(x)| > λ}| p 1 ,p 2 λ -p 3 .
We write x = (x 1 , x 2 ). Fix λ > 0 and perform a fiber-wise Calderón-Zygmund decomposition of F 2 (•, x 2 ) for fixed x 2 ∈ R at level λ p 3 /p 2 . This yields functions g x 2 and atoms a i,x 2 supported on disjoint dyadic intervals intervals

I i,x 2 such that F 2 (•, x 2 ) = g x 2 + i a i,x 2 and for all x 2 g x 2 L p 2 (R) ≤ F 2 (•, x 2 ) L p 2 (R) and g x 2 L ∞ (R) λ p 3 /p 2 .
Moreover, for all i, the atom a i,x 2 has vanishing mean on I i,x 2 ,

a i,x 2 L p 2 (I i,x 2 ) λ p 3 /p 2 |I i,x 2 | 1/p 2 ,
and the intervals satisfy

i |I i,x 2 | λ -p 3 F 2 (•, x 2 ) p 2 L p 2 (R) . (3.5)
First we consider the good part T (F 1 , g), where g = (x 1 , x 2 ) → g x 2 (x 1 ). We have

g L p 2 (R 2 ) ≤ F 2 L p 2 (R 2 ) = 1 and g L ∞ (R 2 ) λ p 3 /p 2 .
Therefore, for q 2 > p 2 we get g L q 2 (R 2 ) p 2 λ p 3 (1/p 2 -1/q 2 ) . By the boundedness in the Banach range obtained in the previous section we obtain

|{x ∈ R 2 : |T (F 1 , g)(x)| > λ}| ≤ λ -s T (F 1 , g) s L s (R 2 ) p 1 ,p 2 λ -s F 1 s L p 1 (R 2 ) g s L q 2 (R 2 ) λ -s+sp 3 (1/p 2 -1/q 2 ) = λ -p 3 ,
where 1 < q 2 , s < ∞ are any exponents that satisfy 1/p 1 + 1/q 2 = 1/s, q 2 > p 2 and (p 1 , q 2 , s) belong to the open Banach range. This is the desired estimate on the level set.

It remains to consider the bad part T

(F 1 , b), where b = (x 1 , x 2 ) → i a i,x 2 (x 1 ). Denote E = x 2 ∈R i (2I i,x 2 × {x 2 }),
where 2I i,x 2 is the interval with the same center as I i,x 2 but twice the length. Note that

|E| R | ∪ i 2I i,x 2 |dx 2 λ -p 3 R F 2 (•, x 2 ) p 2 L p 2 (R) dx 2 λ -p 3 F 2 p 2 L p 2 (R 2 ) = λ -p 3 ,
where we have used (3.5). Therefore, it remains to estimate |{x ∈ E :

|T (F 1 , b)(x)| > λ}|.
By the definition we have

P (1) 2,k b(x) = i P 2,k [a i,x 2 ](x 1 ). Fix x 2 ∈ R and take x 1 ∈ R \ ∪ i 2I i,x 2 .
Denote by c i,x 2 the center of the interval I i,x 2 . Since P 2,k is a convolution operator with some smooth function ρ k (with its Fourier transform adapted to [-2 k+1 , 2 k+1 ] up to a large order), using the mean zero property of a i,x 2 , we obtain

|P 2,k [a i,x 2 ](x 1 )| ≤ I i,x 2 |a i,x 2 (w)||ρ k (x 1 -w) -ρ k (x 1 -c i,x 2 )|dw I i,x 2 |a i,x 2 (w)| 2 2k |I i,x 2 | 1 + 2 k |x 1 -c i,x 2 | -100 dw λ p 3 /p 2 2 2k |I i,x 2 | 2 1 + 2 k |x 1 -c i,x 2 | -100 .
Here we used

a i,x 2 L 1 (R) ≤ |I i,x 2 | 1/p 2 a i,x 2 L p 2 (R) λ p 3 /p 2 |I i,x 2 |. Therefore, since |x -c i,x 2 | ≥ |I i,x 2 |/2 we have k∈Z 2 2k |I i,x 2 | 2 1 + |x 2 -c i,y | 2 -k -100 1 + |x 2 -c i,y | |I i,x 2 | -2
.

Therefore, we have showed that for

x 1 ∈ R \ ∪ i 2I i,x 2 it holds |T (F 1 , b)(x)| λ p 3 /p 2 M(F 1 )(x)H(x),
where M is the Hardy-Littlewood maximal function and

H(x) = i 1 + |x 1 -c i,y | |I i,y | -2
.

Next we use boundedness of the Marcinkiewicz functions associated with a disjoint collection of the intervals (I i,x 2 ) i , i.e.

i 1 + |x 1 -c i,x 2 | |I i,x 2 | -2 L p 2
x 1 (R)

p 2 i |I i,x 2 | 1/p 2 .
(See [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF] or Grafakos, Exercise 4.6.6.) This yields

H L p 2 (R 2 ) p 2 i |I i,x 2 | 1/p 2 L p 2 x 2 (R) λ -p 3 /p 2 F 2 (x 1 , x 2 ) L p 2 x 1 (R) L p 2 x 2 (R) λ -p 3 /p 2 .
Therefore,

|{x ∈ E : |T (F 1 , b)(x)| > λ}| |{x ∈ R 2 : λ p 3 /p 2 M(F 1 )(x)H(x) λ}| = |{x ∈ R 2 : M(F 1 )(x)H(x) λ 1-p 3 /p 2 }| λ -p 3 +(p 3 ) 2 /p 2 M(F 1 ) p 3 L p 1 (R 2 ) H p 3 L p 2 (R 2 )
p 1 ,p 2 λ -p 3 +(p 3 ) 2 /p 2 -(p 3 ) 2 /p 2 λ -p 3 . Summarizing, we have shown that the operator T satisfies the weak inequality L p 1 × L p 2 → L p 3 ,∞ . By interpolation it is bounded in the range claimed by Theorem 4. Remark 6. We emphasize that we are able to make use of the fiberwise Calderón-Zygmund in this case (and not in the case of Theorem 1) because the operators

(Q (1) k P (2) 1,k F 1 ) (P (1) 2,k F 2 ) L p 3 (R 2 ) (Q (1)
k P

(2)

1,k F 1 ) (P (1) 2,k F 2 ) L p 3 ( 2 k ) .
To apply the arguments from [START_REF] Benea | Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions[END_REF] it is important is to obtain the localized estimates which are reduced through duality by factorizing a Q k operator on the dual function, which is possible in this situation. Then the square function on the right-hand side is easily estimated by the product of a maximal and square function, both of them which are bounded. A similar reasoning can be done for the second operator of type (3.2). In that way we recover the L p 3 -boundedness of operators (3.1) and (3.2) and we also prove boundedness in the Hardy space H p 3 .

A tri-parameter trilinear operator

The goal of this section it to prove Theorem 4. Throughout this section we will use the shorthand notation k l to denote k < l -50. Similarly, k l will denote k > l + 50 and k ∼ l will mean l -50 ≤ k ≤ l + 50.

For k ∈ Z and 1 ≤ i ≤ 3 let Q i,k be Fourier multipliers with symbols ψ i,k . For k ∈ Z and 4 ≤ j ≤ 6 let P j,k be Fourier multipliers with symbols ϕ j,k . Here ψ i,k and ϕ j,k are smooth one-dimensional functions. In what follows we will consider various classes of symbols and we will apply more assumptions on them as we proceed. We define the trilinear operators U 1 and U 2 acting on two-dimensional functions F 1 , F 2 , F 3 : R 2 → C by

U 1 (F 1 , F 2 , F 3 )(x) = (k,l,m)∈Z 3 (Q (1) 1,k P (2) 4,m F 1 )(x) (Q (1) 2,l P (2) 5,k F 2 )(x) (Q (1) 3,m P (2) 6,l F 3 )(x), U 2 (F 1 , F 2 , F 3 )(x) = (k,l,m)∈Z 3 (P (1) 5,k P (2) 4,m F 1 )(x) (Q (1) 2,l Q (2) 1,k F 2 )(x) (Q (1) 3,m P (2) 6,l F 3 )(x).
We keep in mind that they depend on the particular choice of the functions ψ i,k , ϕ j,k , but we have suppressed that in the notation.

Let the exponents p 1 , p 2 , p 3 , and p 4 satisfy the assumptions stated in Theorem 4. The first step in the proof of Theorem 4 is to decompose the symbols m 1 , m 2 , m 3 as in Section 2.3. This gives that it suffices to prove L p 1 × L p 2 × L p 3 to L p 4 estimates for U 1 and U 2 under the assumptions that for 1 ≤ i ≤ 3, ψ i,k is adapted to [-2 k+1 , 2 k+1 ], and for 4 ≤ j ≤ 6, ϕ j,k is adapted to [-2 k+1 , 2 k+1 ]. Moreover, each ψ i,k vanishes on [-2 k-1 , 2 k-1 ] and ϕ j,k (0) = ϕ j,0 (0) for each k. Indeed, note that while cone decomposition gives 8 terms, all other terms that are of the form of U 1 and U 2 with P -and Q-type operators interchanged can be deduced from the bounds on U 1 and U 2 by symmetry considerations, that is, up to interchanging the role of the functions F i and considering their transposes (x, y) → F i (y, x).

Bounds for U 1 and U 2 resulting after a cone decomposition will be deduced from Lemmas 8-10. Lemma 8 concerns a particular case of U 1 .

Lemma 8. Let p 1 , p 2 , p 3 and p 4 be exponents as in Theorem 4. For

1 ≤ i ≤ 3 and k ∈ Z let ψ i,k be a smooth function supported in [-2 k+3 , 2 k+3 ], which vanishes on [-2 k-3 , 2 k-3 ]. Assume that for each 1 ≤ i ≤ 3, ψ i,k is of the form ψ i,k = ϕ i,k -ϕ i,k-1 , (4.1)
where ϕ i,k is a bump function adapted to [-2 k+3 , 2 k+3 ]. For 4 ≤ j ≤ 6 and k ∈ Z let ϕ j,k be adapted to [-2 k+4 , 2 k+4 ] and assume that the function

ψ j,k = ϕ j,k -ϕ j,k-1 is supported in [-2 k+4 , 2 k+4 ] and vanishes on [-2 k-4 , 2 k-4 ].
Under these assumptions on ψ i,k and ϕ j,k , the associated operator

U 1 is bounded from L p 1 × L p 2 × L p 3 to L p 4 .
Next we state Lemmas 9 and 10, which concern operators of the form (1.14) with one constant symbol. In particular, these results will be used in the proof of Lemma 8. Lemma 9. For k ∈ Z let Q 1,k , Q 2,k , P 5,k and P 6,k be Fourier multipliers with symbols adapted to [-2 k+10 , 2 k+10 ]. Further, assume that the symbols of Q 1,k , Q 2,k vanish on [-2 k-10 , 2 k-10 ]. Then the trilinear operator which maps (F 1 , F 2 , F 3 ) to the two-dimensional function given by

x → (k,l)∈Z 2 :k l (Q (1) 1,k F 1 )(x) (Q (1) 2,l P (2) 5,k F 2 )(x) (P (2) 6,l F 3 )(x) is bounded from L p 1 ×L p 2 ×L p 3 to L p 4 in the range 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞, 4 i=1 1 p i = 1, and 1 p 1 + 1 p 2 > 1 2
. Proof of Lemma 9. It will be clear from the proof that the argument will not depend on the particular choice of the frequency projections satisfying the requirements from the lemma, so for simplicity of the notation, we only discuss the special cases

P 5,k = P 6,k = P k , Q 1,k = Q 2,k = Q k .
By duality it suffices to consider the corresponding quadrilinear form

(k,l)∈Z 2 :k l R 2 (Q (1)
k F 1 ) (Q (1) l P
(2)

k F 2 ) (P (2)
l F 3 ) F 4 .
By the frequency localization of the functions F i in the first fibers, to prove estimates for the quadrilinear form it suffices to study

(k,l)∈Z 2 :k l M k,l = (k,l)∈Z 2 M k,l - (k,l)∈Z 2 :k∼l M k,l - (k,l)∈Z 2 :k l M k,l , (4.2)
where we have defined

M k,l = R 2 (Q (1)
k F 1 ) (Q (1) l P
(2)

k F 2 ) Q (1) l (P (2)
l F 3 ) F 4 .
Here Q l is a Fourier multiplier with a symbol which is adapted to [-2 l+12 , 2 l+12 ] and vanishes on [-2 l-12 , 2 l-12 ]. Note that the sum over k l in (4.2) vanishes due to frequency supports. To bound the sum with k ∼ l we use Hölder's inequality to obtain

l∈Z s∼0 M l+s,l ≤ s∼0 Q (1) l+s F 1 L p 1 ( ∞ l ) Q (1) l P
(2)

l+s F 2 L p 2 ( 2 l ) Q (1) l (P (2)
l F 3 ) F 4 L r ( 2 l )
whenever 1/p 1 + 1/p 2 + 1/r = 1 and 1 < p 1 , p 2 , r < ∞. For a fixed s, bounds for the first two terms follow by boundedness of the maximal and square functions. The third term satisfies

Q (1) l (P (2)
l F 3 ) F 4 L r ( 2 l ) p 3 ,p 4 F 3 L p 3 (R 2 ) F 4 L p 4 (R 2 ) (4.3)
whenever 1 < p 3 , r < ∞, 2 < p 4 < ∞ and 1/r = 1/p 3 + 1/p 4 . Indeed, this follows by Kintchine's inequality to linearize the square-sum and then use bounds for (1.9). In the end, it remains to sum in s ∼ 0. Therefore, it suffices to bound the form in (4.2) with (k, l) ∈ Z 2 . In this case we note that the third factor in M k,l does not depend on k. This allows to apply Cauchy-Schwarz in l and Hölder's inequality in the integration, yielding

(k,l)∈Z 2 M k,l ≤ k∈Z (Q (1) k F 1 ) (Q (1) l P (2) k F 2 ) L s ( 2 l ) Q (1) l (P (2)
l F 3 ) F 4 L r ( 2 l )
whenever 1/r + 1/s = 1, 1 < r, s < ∞. The second factor on the right-hand side equals (4.3).

From the known bounds, we obtain the condition 1/p 3 + 1/p 4 = 1/r and 1 < p 3 , r < ∞, 2 < p 4 < ∞. The first factor maps L p 1 (R 2 ) × L p 2 ( 2 ) → L s ( 2 ) whenever 1/p 1 + 1/p 2 = 1/s and 1 < p 1 , p 2 < ∞, 1/s > 1/2. This follows from vector-valued estimates for (1.9) as discussed in the display following (2.5).

Lemma 10. For k ∈ Z, let Q 1,k , Q 2,k be Fourier multipliers with symbols σ 1,k , σ 2,k which are functions supported in [-2 k+10 , 2 k+10 ] and vanish on [-2 k-10 , 2 k-10 ]. For k ∈ Z let P 5,k , P 6,k be Fourier multipliers with symbols ρ 5,k , ρ 6,k , which are functions adapted to [-2 k+10 , 2 k+10 ].

Assume that for each i ∈ {1, 2, 5, 6}, these functions are of the form

σ i,k = ρ i,k -ρ i,k-1 ,
where σ 4,k , σ 5,k are further functions supported in [-2 k+10 , 2 k+10 ], which vanish on [-2 k-10 , 2 k-10 ], while ρ 1,k , ρ 2,k are functions adapted to [-2 k+10 , 2 k+10 ]. Then the trilinear operator which maps (F 1 , F 2 , F 3 ) to the two-dimensional function given by

x → (k,l)∈Z 2 :k l (Q (1) 1,k F 1 )(x) (Q (1) 2,l P (2)
5,k F 2 )(x) (P (2) 6,l F 3 )(x) (4.4) is bounded from L p 1 ×L p 2 ×L p 3 to L p 4 in the range 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞, 4 i=1
1 p i = 1, and 1 p 2 + 1 p 3 > 1 2
. Proof of Lemma 10. Let us augment the definitions in the statement of the lemma by setting Q i,k and P i,k to be Fourier multipliers with the respective symbols σ i,k and ρ i,k for any i ∈ {1, 2, 5, 6}. Observe that because of the condition on the bump functions we have the telescoping identity

k 1 k=k 0 P i,k f Q j,k g + k 1 k=k 0 Q i,k f P j,k-1 g = P i,k 1 f P j,k 1 g -P i,k 0 -1 f P j,k 0 -1 g (4.5)
for any f, g ∈ L 1 loc (R) and i, j ∈ {1, 2, 5, 6}. Also note that letting k 0 → -∞ and k 1 → ∞, the right-hand side becomes the pointwise product f g.

Our goal is to deduce the claim from Lemma 9 by two applications of the telescoping identity (4.5). We write the sum over k l in (4.4) as k∈Z l∈Z:l k and use (4.5) in l. This gives

(4.4) = - (k,l)∈Z 2 :k l (Q (1) 1,k F 1 ) (P (1) 2,l-1 P
(2)

5,k F 2 ) (Q (2)
6,l F 3 ) (

+ k∈Z (Q (1) 1,k F 1 ) (P 4.6)
2,k-49 P

5,k F 2 ) (P

6,k-49 F 3 ) (

Indeed, (4.7) is the boundary term at l = k -49, while the term at -∞ vanishes.

Let us consider (4.7). We dualize it and write the corresponding form up to a constant as

k∈Z R 2 Q (1) k (Q (1) 1,k F 1 ) (P (1) 2,k-49 P
(2)

5,k F 2 ) (P (2) 6,k-49 F 3 ) F 4 ,
where

Q k has a symbol adapted to [-2 k+12 , 2 k+12 ] which vanishes on [-2 k-12 , 2 k-12
]. Note that by the support assumption on σ 2,k we necessary have that ρ 2,k (0) is the same constant c 0 ∈ R for each k ∈ Z. Then we write P 2,k-49 = c 0 I + (P 2,k-49 -c 0 I) and split the operator accordingly. By the frequency support information in the first fibers it suffices to bound the terms

k∈Z R 2 c 0 Q (1) k (Q (1)
1,k F 1 ) (P

5,k F 2 ) (P

6,k-9 F 3 ) F 4 and (4.8)

k∈Z R 2 Q (1) k (Q (1) 1,k F 1 ) ( Q (1)
k P

(2)

5,k F 2 ) (P (2) 6,k-9 F 3 ) F 4 , (4.9)
where the symbol of Q k is adapted in [-2 k+100 , 2 k+100 ] and vanishes at the origin. The desired bounds for the first term (4.8) follow from

k∈Z R 2 (Q (1) 1,k F 1 ) (P (2) 5,k F 2 ) Q (1) k (P (2) 6,k-49 F 3 ) F 4 ≤ (Q (1) 1,k F 1 ) (P (2) 5,k F 2 ) L r ( 2 k ) Q (1) k (P (2) 6,k-49 F 3 ) F 4 L s ( 2 k ) (4.10)
whenever 1/r + 1/s = 1 and 1 < r, s < ∞. For the first term in (4.10) we use Hölder's inequality to obtain (Q

(1) 1,k F 1 ) (P (2) 5,k F 2 ) L r ( 2 k ) ≤ Q (1) 1,k F 1 L p 1 ( 2 k ) P (2) 5,k F 2 L p 2 ( ∞ k ) p 1 ,p 2 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 ) .
whenever 1/r = 1/p 1 + 1/p 2 , 1 < p 1 , p 2 < ∞. For the last inequality we have used bounds on the maximal and square functions. For the second term in (4.10) we use Kintchine's inequality to linearize the square-sum. Then we apply bounds for the corresponding operator (1.9), yielding

Q (1) k (P (2) 6,k-49 F 3 ) F 4 L s ( 2 k ) p 3 ,p 4 F 3 L p 3 (R 2 ) F 4 L p 4 (R 2 )
whenever 1/s = 1/p 3 + 1/p 4 , 1 < p 3 < ∞, and 2 < p 4 < ∞. This yields the desired estimate. For (4.9) we proceed in the analogous way, this time estimating Q

1,k-1 F 1 ) (P

(1) 2,l-1 Q (2) 5,k F 2 ) (Q (2)
6,l F 3 ), (4.11)

+ l∈Z (P (1)
1,l+51 F 1 ) (P

2,l-1 P

(2)

5,l+51 F 2 ) (Q (2)
6,l F 3 ), (4.12)

- l∈Z (F 1 ) (P (1) 2,l-1 F 2 ) (Q (2)
6,l F 3 ). (4.13)

The term (4.13) is the boundary term at k = ∞, which is just a pointwise product of F 1 with an operator of the form (1.9). We obtain L

1,l+51 F 1 ) (P

(2)

l F 2 ) (Q (2)
6,l F 3 ), (

where the symbol of Q l is up to a constant adapted in [-2 l+51 , 2 l+51 ] and it vanishes near the origin. The term (4.15) is bounded by Hölder's inequality and bounds on the maximal and square functions in the full range. For (4.14) one can proceed analogously as for (4.7), with the roles of the first and second fiber interchanged. This gives bounds in the range 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞ and Hölder scaling. Now we are ready to prove Lemma 8.

Proof of Lemma 8. For simplicity of notation, we shall assume that P 1,k = P 2,k = P k and Q 1,k = Q 2,k = Q k , but as it will be clear from the proof, the arguments will work for general Fourier multiplier operators with symbols satisfying the above assumptions. The proof will be based on analyzing the ordering of the parameters k, l, and m. Note that by symmetry we may assume k ≤ l ≤ m. If, in addition, it holds that k ∼ l and l ∼ m, then all frequencies are comparable up to an absolute constant and the claim follows by Hölder's inequality.

Consider now the case with two dominating frequencies when k l and l ∼ m, i.e. l = m + s for some -50 ≤ s ≤ 0. Then we need to estimate m∈Z -50≤s≤0 k∈Z: k m+s (Q

(1) k P (2) m F 1 ) (Q (1)
m+s P

(2)

k F 2 ) (Q (1) m P (2)
m+s F 3 ).

By Cauchy-Schwarz in m and Hölder's inequality, its L p 4 norm is bounded by

-50≤s≤0 k∈Z: k m+s (Q (1) k P (2) m F 1 )(Q (1) m+s P
(2)

k F 2 ) L p 3 ( 2 m )
Q (1) m P

(2) k P (2) m F 1 )(Q

m+s F 3 L p 3 ( 2
m+s P

(2)

k F 2 ) L p 3 (R 2 ) p 1 ,p 2 F 1 L p 1 (R 2 ) F 2 L p 2 (R 2 )
uniformly in M > 0, where |a m | ≤ 1. This estimate holds when 1 < p 1 , p 2 , p 3 , p 4 < ∞, 1/p 1 + 1/p 2 = 1/p 3 , 1/p 1 + 1/p 2 > 1/2. Indeed, this follows from Theorem 1 after normalizing the symbols of P m and Q m+s . In the end, it remains to sum in -50 ≤ s ≤ 0. Now we consider the k ≤ l m with one dominating frequency. We dualize and consider the corresponding trilinear form. By the frequency localizations it suffices to bound

(k,l,m)∈Z 3 :k≤l m R 2 (Q (1) k P (2) m F 1 ) (Q (1)
l P

(2)

k F 2 ) (Q (1) m P (2) l F 3 ) (Q (1) m P (2) m F 4 ),
where Q m , P m are adapted to [-2 m+6 , 2 m+6 ] and Q m vanishes in [-2 m-6 , 2 m-6 ]. As in the proof of Lemma 10, we note that ϕ m (0) = c 0 is a constant with |c 0 | ≤ 1 for each m ∈ Z. We write P m = c 0 I + (P m -c 0 I) for the projection acting on the second fiber of F 1 . Then this reduces to

(k,l,m)∈Z 3 :k≤l m M k,l,m + (k,l,m)∈Z 3 :k≤l m E k,l,m ,
where we have set

M k,l,m = R 2 (Q (1)
k F 1 ) (Q

(2)

k F 2 ) (Q (1) m P (2) l F 3 ) (Q (1) m P (2) m F 4 ), (4.16)
E k,l,m = R 2 (Q (1) k Q (2) m F 1 ) (Q (1) l P
(2)

k F 2 ) (Q (1) m P (2) l F 3 ) (Q (1) m P (2) m F 4 ). (4.17)
Here Q m is associated with a bump function, which is up to a constant multiple, adapted to [-2 m+10 , 2 m+10 ]. Moreover, the bump function vanishes at the origin. We have redefined P m by subsuming c 0 into its definition. First we consider (4.16). We write the sum over (k, l, m) ∈ Z 3 with k ≤ l m as

(k,l,m)∈Z 3 :k≤l m M k,l,m = (k,l,m)∈Z 3 :k≤l m M k,l,m - (k,l,m)∈Z 3 :k≤l M k,l,m (4.18)
+ (k,l,m)∈Z 3 :k≤l M k,l,m . (4.19)
Let us first focus on (4.19). We split the summation into the sums over m and (k, l), apply Cauchy-Schwarz in m and Hölder's inequality in the integration. This yields

(k,l,m)∈Z 3 :k≤l M k,l,m ≤ (k,l)∈Z 2 :k≤l (Q (1) k F 1 ) (Q (1) l P (2) k F 2 ) (Q (1) m P (2) l F 3 ) L p 4 ( 2 m ) Q (1) m P (2) m F 4 L p 4 ( 2 m ) . (4.20)
Since the second term is a square function, it remains to prove L

p 1 (R 2 ) × L p 2 (R 2 ) × L p 3 ( 2 ) → L p 4 ( 2 )
vector-valued estimates for the trilinear operator (4.18), which splits further into several terms. We fix k ≤ l and then we consider several cases depending on the size of m relative to k and l. If for any triple (j 1 , j 2 , j 3 ) ∈ {k, l, m} 3 with all three distinct entries it holds j 1 ∼ j 2 and j 2 ∼ j 3 , then the claim follows by Hölder's inequality. If l max(k, m), then the corresponding term is zero due to frequency supports. Therefore, the remaining two cases with k ≤ l are: k ∼ l and m k, or k l and m ∼ l. Consider first the case when k ∼ l and m k. Then we have two Q-type projections in each parameter. Bounds are deduced by two applications of Hölder's inequality as follows. Taking the triangle inquality, enlarging the sum in m to be over the whole Z and applying Hölder's inequality in l, we obtain

(k,l)∈Z 2 :k≤l (Q (1) k F 1 ) (Q (1) l P (2) k F 2 ) (P (2) l F 3 ). ( 4
-50≤s≤0 l∈Z m∈Z:m l+s M l+s,l,m ≤ -50≤s≤0 R 2 m∈Z Q (1) l+s F 1 2 l Q (1) l P (2) l+s F 2 2 l Q (1) m P (2) l F 3 ∞ l |Q (1) m P (2) m F 4 |
By Cauchy-Schwarz in m and Hölder in the integration, we bound the term for a fixed s by

Q (1) l+s F 1 L p 1 ( 2 l ) Q (1) l P
(2)

l+s F 2 L p 2 ( 2 l ) Q (1) m P (2) l F 3 L p 3 ( 2 m ( ∞ l )) Q (1) m P (2) m F 4 L p 4 ( 2 m ) . (4.22)
Each of the terms is bounded in full range. The third term, square-maximal function term is bounded in by the Fefferman-Stein inequality, see [START_REF] Muscalu | Bi-parameter paraproducts[END_REF]. The remaining terms are square functions. Next we consider the case k l and m ∼ l. We split

(k,l,m)∈Z 3 : k l,m∼l M k,l,m = (k,l,m)∈Z 3 : m∼l M k,l,m - (k,l,m)∈Z 3 : l-50≤k≤l+100,m∼l M k,l,m - (k,l,m)∈Z 3 : k l+50,m∼l M k,l,m .
Note that bounds for the second sum follow by Hölder's inequality, while the third sum is zero due to frequency supports. Thus, it remains to consider the case m ∼ l. We again split the sum over (k, l) ∈ Z 2 and write m = l + s, s ∼ 0. Now we use Hölder's inequality in l to bound

s∼0 (k,l)∈Z 2 M k,l,l+s ≤ s∼0 k∈Z (Q (1) k F 1 ) (Q (1) l P (2) k F 2 ) L r ( 2 l ) Q (1) l+s P (2) l F 3 L p 3 ( 2 l ) Q (1)
l+s P

(2)

l+s F 3 L p 4 ( ∞ l ) (4.23)
where 1/r + 1/p 3 + 1/p 4 = 1. The second and third term are maximal and square functions, respectively, while bounds for the first term follow by L p (R 2 ) × L q ( 2 ) → L r ( 2 ) vectorvalued estimates for the operator (1.9). Summing in s, we obtain estimates in the range 1 < p 1 , p 2 , p 3 , p 4 < ∞, 1/p 1 + 1/p 2 > 1/2, and 4 i=1 1/p i = 1. To bound (4.17), we proceed analogously as for (4.16), except that in the analogue of (4.21) we use L p 1 ( 2 ) × L p 2 (R 2 ) × L p 3 ( 2 ) → L p 4 ( 2 ) vector-valued estimates for that operator and in the display analogous to (4.22) we now have the maximal-square function

Q (1) l+s Q (2) m F 1 L p 1 ( ∞ m ( 2 l ))
, which is also bounded in the full range by the Fefferman-Stein inequality. The analogue for the vector-valued estimates used in (4.23) are now L p ( 2 ) × L q ( 2 ) → L r ( 2 ) vector-valued estimates for the operator of the form (1.9). Finally, we are ready to tackle the operators that result after the cone decomposition. Recall that we need to prove bounds for U 1 and U 2 under the assumptions that for 1 ≤ i ≤ 3, ψ i,k is adapted to [-2 k+1 , 2 k+1 ], and for 4 ≤ j ≤ 6, ϕ j,k is adapted to [-2 k+1 , 2 k+1 ]. Moreover, each ψ i,k vanishes on [-2 k-1 , 2 k-1 ] and ϕ j,k (0) = ϕ j,0 (0). 4.1. Completing the proof of Theorem 4. Let φ k be a function adapted in [-2 k+1 , 2 k+1 ], which satisfies φ k (0) = φ 0 (0) for each k ∈ Z. We write

φ k = c -1 0 (c 0 φ k -φ k ) + c -1 0 φ k , (4.24)
where φ k is constantly equal to c 0 φ 0 (0) on [-2 k-1 , 2 k-1 ], and

|c 0 | ≤ 1 is such that φ k is a function adapted in [-2 k+1 , 2 k+1
]. Note that the function c 0 φ k -φ k vanishes at the origin.

Consider U 1 and U 2 which resulted from the cone decomposition. Writing each of the symbols of the P -type operators as in (4.24), it suffices to bound the operators under the following assumptions. For each 1 ≤ i ≤ 3, ψ i,k is adapted in [-2 k+1 , 2 k+1 ] and vanishes on [-2 k-1 , 2 k1 ]. For each 4 ≤ j ≤ 5, ϕ j,k is supported in [-2 k+1 , 2 k+1 ] and exactly one of the following holds:

(1) For each 4 ≤ j ≤ 6, the function ϕ j,k is constantly equal to ϕ j,0 (0) on [-2 k-1 , 2 k-1 ].

(2) There is an index 4 ≤ j 0 ≤ 6, such that ϕ j 0 ,k (0) = 0 for each k. Moreover, for j = j 0 , ϕ j,k is constantly equal to ϕ j,0 (0) on [-2 k-1 , 2 k-1 ]. (3) There is an index 4 ≤ j 0 ≤ 6, such that ϕ j 0 ,k is constantly equal to ϕ j 0 ,0 (0) on [-2 k-1 , 2 k-1 ]. Moreover, for j = j 0 , it holds ϕ j,k (0) = 0 for each k. (4) For each 4 ≤ j ≤ 6, it holds ϕ j,k (0) = 0 for each k.

We will analyze each of these cases for the operators of the form U 1 and U 2 and we start with U 1 . Our first aim is to reduce considerations to the case when the symbols ψ i,k are supported in [-2 k+3 , 2 k+3 ], vanish on [-2 k-3 , 2 k-3 ], and are of the form ϕ i,k -ϕ i,k-1 for a function ϕ i,k supported in [-2 k+3 , 2 k+3 ]. The argument that follows is along the lines of an argument used in Section 6 of [START_REF] Kovač | Boundedness of the twisted paraproduct[END_REF] when transitioning from the dyadic to the continuous setting.

Let us denote by P ϕ the one-dimensional Fourier multiplier with symbol ϕ, i.e. P ϕ f = f * q ϕ. As before we shall denote its fiber-wise action on a two-dimensional function with a superscript. Let φ be a non-negative smooth function supported in [-2 -0.4 , 2 -0.4 ] and such that φ is constantly equal to one on [-2 -0.6 , 2 -0.6 ]. For a ∈ R define ϑ a and ρ a by ϑ a (ξ) = φ(2 -a-1 ξ) -φ(2 -a ξ), ρ a (ξ) = φ(2 -a-0.6 ξ) -φ(2 -a-0.5 ).

Note that ϑ a is supported in [-2 a+0.6 , 2 a+0.6 ] and vanishes on [-2 a-0.6 , 2 a-0.6 ]. Moreover, ϑ a is constantly equal to one on [2 a-0.4 , 2 a+0.4 ] and [-2 a+0.4 , -2 a-0.4 ]. Finally, ρ a is supported in [-2 a+0.2 , 2 a+0.2 ] and vanishes on [-2 a-0.1 , 2 a-0.1 ]. In particular, we have ϑ a = 1 on the support of ρ a . Moreover, for k ∈ Z and 1 ≤ i ≤ 3 we have By the Cauchy-Schwarz inequality in k and m, this reduces to vector-valued estimates for the operator (1.9) and a fiber-wise square function. We obtain L p 1 × L p 2 × L p 3 → L p 4 estimates with Hölder scaling and in the range 1 < p 1 , p 2 , p 3 , p 4 < ∞, 1/p 2 + 1/p 3 > 1/2.

Case (4) of U 1 . This case easily follows from three applications of Hölder's inequality, reducing to fiber-wise square functions.

It remains to treat U 2 . By the analogous argument as leading to (4.27), it suffices to consider the case when for 1 ≤ i ≤ 3, the symbols ψ i,k are supported in [-2 k+3 , 2 k+3 ], vanish on [-2 k-3 , 2 k-3 ], and are of the form ϕ i,k -ϕ i,k-1 for a function ϕ i,k supported in [-2 k+3 , 2 k+3 ]. Then we distinguish further cases depending on the form of ϕ j,k .

Case (1) of U 2 . Let Q 5,k and P 1,k be associated with ψ 5,k and ϕ 1,k , respectively, where

4,m F 1 ) (Q

2,l P

1,k F 2 ) (Q

3,m P If j 0 = 4, we first use the telescoping identity (4.5) in k, giving terms of the form (4.28) and (4.29). For (4.29), we apply the Cauchy-Schwarz inequality in m, which leads to vector-valued estimates for the operator (1.9). For (4.28), we can proceed as in Case (2) of U 1 , up to obvious modifications.

Cases (3) and (4) of U 2 . These two cases are analogous to Cases (3) and (4) of U 1 .

  (3.1), (3.2) and (3.3) act on only fiber of the function F 2 . Alternatively, one could apply the classical two-dimensional Calderón-Zygmund decomposition by decomposing the function F 1 . Remark 7. An alternative approach to prove quasi-Banach estimates for (3.1) and (3.2), which avoids the use of fiberwise Calderón-Zygmund decomposition and in fact proves a stronger claim, namely, that (3.1) and (3.2) are bounded with values in the Hardy space H p 3 ⊆ L p 3 , is the following. Note that the operators (3.1) and (3.2) are localized in frequency in the first coordinate at scale k. So we can use the argument from [1] about the inequality f p S(f ) p for 0 < p ≤ 1, where S is the Littlewood-Paley square function. More precisely, following the scalar case of [1, Theorem 3.1], one obtains for 0 < p 3 ≤ 1 the bound k∈Z

  F 2 by the Hardy-Littlewood maximal function. It remains to estimate (4.6), for which we still want to switch the projections in k. Now we write the double sum over k l as l∈Z k∈Z:k l and telescope (4.6) in k, giving (4.6) = (k,l)∈Z 2 :k l (P

m ) whenever 1

 1 < p 3 < ∞ and 1/p 4 = 1/p 3 + 1/p 3 . For a fixed s, the second term is a square function, bounded in the full range. For the first term we use Kintchine's inequality, which

20 l=- 20 ρ 9 s=0 20 l=- 20

 202092020 k+0.1l = 1 on supp(ψ i,k ), (4.25) 20 l=-20 ρ k+0.1l = 0 on supp(ψ i,k ) if |k -k| ≥ 10. (4.26) Let n ∈ Z, 0 ≤ s ≤ 9. Note that due to (4.26) and (4.25) we may write for ξ, η ∈ R k∈Z ψ i,k (ξ)ϕ j,k (η) = 9 s=0 n∈Zψ i,10n+s (ξ)ϕ j,10n+s (η) = n∈Z ρ 10n+s+0.1l (ξ)Ψ i,s (ξ)ϕ j,10n+s(η), Case (3)of U 1 . Without loss of generality, we may assume j 0 = 6. Then we write the operator in question as (k,m)∈Z 2 F 3 )(x) .

ψ 5 ,

 5 k = ϕ 5,k -ϕ 5,k-1 and ϕ 1,k = ϕ 1,k-1 (with ϕ 1,k defined in (4.1)). Observe that ψ 5,k is supported in [-2 k+3 , 2 k+3 ] and vanishes on [-2 k-3 , 2 k-3 ]. Moreover, ϕ 1,k is supported in [-2 k+4 , 2 k+4 ] and observe that ϕ 1,kϕ 1,k-1 = ϕ 1,k-1 -ϕ 1,k-2 = ψ 1,k-1 is supported in [-2 k+4 , 2 k+4 ] and vanishes on [-2 k-4 , 2 k-4]. An application of the telescoping identity (4.5) in k yields that U 2 then equals -

29 ) 2 a 2 a

 2922 Bounds for the first term follow from Lemma 8, applied with ψ 1,k being ψ 5,k and ϕ 5,k = ϕ 1,k . The desired bounds for the second term follow from Lemmas 9 and 10, and by Hölder's inequality used for the portion of the sum when l ∼ m.Case (2) of U 2 . Let j 0 = 6. Performing the analogous steps as in Case (2) of U 1 , we see that it suffices to show estimates for(k,m)∈Z F 3 ),where Q 3,m is as in Case (2) of U 1 . By an averaging argument (as leading to (4.27)) it suffices to consider the case when a k = 1 and Q 3,m has a symbol supported in [-2 9 , 2 9 ], which vanishes on [-2 -9 , 2 -9 ], and is of the form θ k -θ k-1 for a suitable function θ. The clam then follows by the telescoping identity (4.5) in m, Lemmas 9 and 10, and bounds for (1.9). If j 0 = 5, we can proceed as in Case (2) of U 1 . Then we need to show estimates for (l,m)∈Z

Table 1 .

 1 Structure of m and the range of boundedness Structure of m(ξ, η) Known range of boundedness

  p 1 × L p 2 × L p 3 → L p 4 estimates in the range 1 < p 1 , p 2 , p 3 , p 4 < ∞, 1/p 2 + 1/p 3 > 1/2,whenever the exponents satisfy the Hölder scaling. The term (4.11) follows from Lemma 9 and symmetry considerations, i.e. after interchanging the role of F 1 and F 3 and the role of the first and second fibers. We obtain estimates in the range 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞, 1/p 2 + 1/p 3 > 1/2. Finally, note that

	(4.12) =	(P	(1) 1,l+51 F 1 ) (P	(1) 2,l-1 P	(2) 5,l-49 F 2 ) (Q (2) 6,l F 3 )	(4.14)
	l∈Z
	+	(P
	l∈Z

  The case k ∼ l is bounded by Hölder's inequality. The remaining two cases follow by Lemmas 9 and 10 above. Summarizing, we obtain estimates for (4.20) when 1 < p 1 , p 2 , p 3 < ∞, 2 < p 4 < ∞, 1/p 1 + 1/p 2 > 1/2, 1/p 2 + 1/p 3 > 1, and 4 i=1 1/p i = 1. It remains to estimate

.21)

These estimates will again follow by freezing the functions F 1 and F 2 and using Marcinkiewicz-Zygmund inequalities, provided we show scalar-valued boundedness of (4.21). To show this, we split the summation in (4.21) into regions where k ∼ l, k l, or k l.

where Ψ i,s = k∈Z ψ i,10k+s .

Let us now consider U 1 with bump functions satisfying (Q) and (P). Applying the above considerations in each parameter k, l, m, it suffices to prove bounds for (n 1 ,n 2 ,n 3 )∈Z 3 (P (1) ρ 10n 1 +s 1 +0.1l 1 P (2) ϕ 4,10n 3 +s 3 G 1,s 1 ) (P (1) ρ 10n 2 +s 2 +0.1l 2 P (2) ϕ 5,10n 1 +s 1 G 2,s 2 ) (P (1) ρ 10n 3 +s 3 +0.1l 3

for fixed 0 ≤ s 1 , s 2 , s 3 ≤ 9 and -20 ≤ l 1 , l 2 , l 3 ≤ 20, where G i,s = P (1) Ψ i,s F i . The Mikhlin-Hörmander theorem in one variable gives

Thus, it suffices to bound

Now we are ready to distinguish further cases depending on the form of ϕ j,k .

Case (1) of U 1 . Bounds for (4.27) follow from Lemma 8, applied with ψ i,k being a constant multiple of ϑ k+b i for 1 ≤ i ≤ 3.

Case (2) of U 1 . Without loss of generality, we may assume j 0 = 4. Then we dualize and write the quadrilinear form in question up to a constant as

By Cauchy-Schwarz in k, m, and Hölder's inequality in the integration, it suffices to bound a fiber-wise square function and

2,l P

(2)

3,m P

6,l F 3 )(x)F 4 (x)

.

Dualizing with a function F 1 and using Kintchine's inequality we see that it suffices to prove L p 1 × L p 2 × L p 3 → L p 4 bounds for the operator

6,l F 3 )(x). Now we perform the analogous averaging argument as the one that led to (4.27). Then it suffices to consider the case when a k = 1 and Q 1,k has a symbol supported in [-2 9 , 2 9 ], which vanishes on [-2 -9 , 2 -9 ], and is of the form θ k -θ k-1 for a suitable function θ. The desired bounds then follow by Lemmas 9 and 10.


Petr Brož
email: petr.broz@ig.cas.cz

Ondřej Krýza

Ernst Hauber

S J Conway

J Raack

M R Patel

M R Balme

A Mazzini

M E Sylvest

Violent Mud Propagation on Mars: Evidence From Laboratory Simulations

Here we present the results of experiments performed inside a low pressure chamber to investigate how mud would propagate over a 'warm' (~295 K) unconsolidated sandy surface under Martian atmospheric pressure conditions (~7 mbar). The results show that, flowing mud is capable of eroding down into the substrate. The gas released by boiling allows the mud to propagate into the subsurface and to form a subsurface flow which acts as a platform for further mud propagation over the surface. Escaping gasses can cause pockets of mud to levitate for a limited period of time (similar to [1,2]) and hence cause faster and further propagation than would be possible on Earth.

Introduction

Even though most of the Martian surface is cold today, locally warm surface temperatures can be achieved. Therefore sedimentary volcanism, if present on Mars [4][5][6][7], could represent a source of erupted mud in such warm regions. The extrusion of mud on cold Martian surfaces (i.e. sedimentary volcanism [3][4][5][6]) induces rapid freezing and the formation of a protective frozen crust on top of the mud flow, leading to a behavior similar to pahoehoe lava on Earth (see the EPSC abstract #122 for details). On the other hand, warm (i.e. non-freezing) surface temperatures (which can locally occur [7]) preclude freezing and in such conditions the mud propagation should be different. As the physical instability of water under current Martian atmospheric pressure leads to boiling [e.g., 1,2,8,9], this suggests that the propagation of a muddy mixture would also be different from our terrestrial experience.

Experimental setup

We used the Mars Simulation Chamber at the Open University (UK) into which we inserted a 0.9 × 0.4 m aluminum tray filled with a ~2 cm deep sediment (natural sand, ~200 µm) bed together with a reservoir containing 500 ml of low viscosity mud hanging ~5 cm above the tray. Mud and sand were at room temperature (~20°C). The tray was inclined by 5° or 10° to force the mud to move under gravity once poured on the surface. The mud was released from the container under reduced (~7 mbar) pressure and the movement of the mixture was observed and recorded by four cameras from different angles. Each experimental run was performed in triplicate to confirm the reproducibility of the results; comparative experiments under terrestrial pressure were also performed.

Observations

Once the atmospheric pressure is reduced, the mud in the container starts to boil. The boiling intensifies as the pressure gets closer to the 12-14 mbar and continues all the way down to 7 mbar. When a pressure of 7 mbar is reached, the mud is manually released by tipping the container, letting it flow over the 'hot' (20°C) sandy surface. The contact of the mud with the 'hot' surface triggers explosive activity, which causes ejection of sandy grains to a height of several centimetres. The particles land both on the mud and on the surrounding sand. The deposition of the sand grains forms a small raised rim around the contact area resulting in a crater-like depression (Fig. 1). The explosive activity decreases with time. At the beginning the mud is not visible inside the crater area as it gets covered by a layer of loose sand which is repeatedly disturbed by bubbling (Fig. 1,8 s). Within seconds, mud can be observed on the surfacenot necessarily at the site where it was directly poured from the containerpropagating inside the crater (Fig. 1). At the boundary between the mud and the sand layer, a large amount of millimetre-scale explosion pits formed, from which gas continued to eject particles for several minutes. This enabled a progressive expansion of the rim. Continued mud supply causes the flow to breach the sandy rim and a new lobe of mud advances over the warm sand (Fig. 1, 22 s). This flow front triggers new explosions as the mud propagates. The escape of gas at the bottom of the mud flow causes the lobe to vibrate vertically and to quickly propagate over the first few centimetres (Fig. 2a). Then the lobe stalls and small millimetre-scale explosions occur around its edge causing the formation of small ridges. Simultaneously fresh mud outpouring from the crater starts to propagate over the lobe's surface and accumulate at the front of the flow. Once enough material has accumulated to overcome the small ridges at the edges a new lobe forms and the process repeats until the supply of new mud is exhausted.

The movement of mud through the lobes creates a trough with a curvy and irregular shape. This internal structure is supported by a hardened mixture of mud and sand. We also observed that the bottom of the trough is covered by fine-grained clay. Here, holes formed as the result of repetitive explosions caused by escaping gases were located above small subsurface pockets infilled by mud (see Fig. 2b). After several tens of minutes the chamber was decompressed and we inspected the interior of the mud flows by breaking them apart. Liquid mud was still present in the subsurface covered by sand. This implies that the sand partly acts as a protective layer insulating the mud from the surrounding desiccating environment.

Conclusions

Our experiments show that a warm and unconsolidated surface has a profound effect on the behaviour of flowing mud in a low pressure environment, because of boiling. This causes levitation of the mud over the surface for a short period of time as well as the erosion of the unconsolidated sandy substrate. Both mechanisms alter the mud propagation in a low pressure environment. Moreover, as Mars has a lower gravitational acceleration than Earth, we expect that these processes would be even more effective on Mars, because gravity does not change boiling rate, but the sediments can be more easily entrained [1,2]. The gas released should levitate mud for a more extended period of time, as also similarly suggested for wet sand [1,2], hence allowing the mud to propagate over larger distances than on Earth. Our work shows that the behaviour of mud and its propagation in a low pressure environment is strongly dependent on the surface temperature as freezing [see the EPSC abstract #122 for details] or rapid boiling would significantly change the final morphologies of resulting surface flow features

Figure 1 :

 1 Figure 1: A sequence of images capturing the propagation of mud under an atmospheric pressure of 7 mbar and over a non-freezing 10° inclined surface. See the text for details.

Figure 2 :

 2 Figure 2: A sequence of images showing the levitation of the mud caused by boiling (a) and the sandy crust and repetitive explosions associated with small mud pockets (b).

Acknowledgements:

The access to the Large Mars Chamber at the Open University was provided by Europlanet 2020 RI which has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 654208.


Candice Hansen
email: cjhansen@psi.edu

Susan Conway

Ganna Portyankina

Nicolas Thomas

Alfred Mcewen

Jason Perry

Antoine Pommerol

Camila Cesar

Searching for Seasonal Jets on Mars in CaSSIS and HiRISE Images

Spring is an active time on Mars as the CO 2 seasonal cap sublimates. Stereo images from HiRISE and CaSSIS have been collected to search for the gas jets that produce seasonal fans visible every spring.

Introduction

Every year in the spring fans appear on Mars' seasonal polar cap. The generally accepted Kieffer model [START_REF] Kieffer | Cold jets in the Martian polar caps[END_REF] is that sunlight penetrates the translucent layer of CO 2 ice, and warms the surface, which causes basal sublimation of the layer of seasonal ice. Gas trapped below the ice builds up pressure until the overlying ice ruptures and the gas escapes, entraining material from the surface. The particles are carried out, get blown downwind, and fall to the surface in fan-shaped deposits (see Figure 1).

HiRISE Search for Jets

Every spring tens of thousands of fans appear on the seasonal cap. Ideally a stereo pair of HiRISE images would detect the jet of particles shooting above the surface. It turns out to be surprisingly difficult to achieve this seemingly simple objective. The amount of gas released is limited by the supply, and might last just minutes to hours. The closest spacing for HiRISE images is on adjacent orbits, with ~2 hr separation, but only at latitudes poleward of 80 o . Given other constraints on sequencing it is typical to get just one of these stereo pairs per week. At the most active time in the spring that means that only 2 -5 stereo pairs may be scheduled. In the last 7 Mars years of MRO operation only ~23 pairs have been returned.

While the stereo data shows very interesting detail on surface morphology no jets have been detected.

Another limitation is that the MRO orbit is fixed at a mean local solar time of ~3 pm. It is possible that the jets are more likely to erupt in the morning or midday. If that is the case HiRISE will never be able to image them. seasonal layer of ice every winter, and show typical spring activity with fans and blotches. But is this an example of gas being released and carrying material out in the manner hypothesized in the Kieffer model? Without stereo data to pin down the source and vent structure it is difficult to be certain.

CaSSIS Opportunities

The CaSSIS camera [START_REF] Thomas | The Colour and Stereo Surface Imaging Systemfor the ExoMars Trace Gas Orbiter[END_REF] has the ability to complete a stereo pair within minutes.

The most poleward latitude is limited by the inclination of the TGO orbit to 74 o , but the full range of latitudes equatorward of this limit can be imaged in stereo quickly. Furthermore, the TGO orbit evolves through local time, giving us the opportunity to image over the full range of a martian day.

A particularly opportune time took place from May to July of 2018, when high latitudes were visible at local morning between 6 and 10 local time. the date of this writing 16 stereo pairs have already been analyzed to look for plumes. As of yet none have been identified, but the mere fact that we have as many stereo pairs to look at as in the entire 7 Mars years of HiRISE observations bodes well.

One limitation that works against CaSSIS is its lower resolution (~5m) relative to HiRISE (~1m). Fans are typically shorter at the latitudes CaSSIS observes, which may be related to the gas supply and the thickness of the seasonal ice layer. If this is related to the height the jets achieve, which is likely [START_REF] Thomas | HiRISE observations of gas sublimation-driven activity in Mars' southern polar regions: IV. Fluid dynamics models of CO2 jets[END_REF], then it will be more difficult to identify a jet with certainty in the CaSSIS stereo images. Estimates for jet height are on the order of 70m [START_REF] Thomas | HiRISE observations of gas sublimation-driven activity in Mars' southern polar regions: IV. Fluid dynamics models of CO2 jets[END_REF], but this is relevant to the higher latitudes and longer fans.

Summary and Conclusions

A fair amount of luck is required to catch an eruption underway. The combination of HiRISE and CaSSIS means that almost all latitudes are accessible with stereo.

HiRISE has the advantage of higher resolution and larger fans to observe than CaSSIS, but CaSSIS offers more opportunities.

It is also possible that jets will never be imaged if the dust load is not opaque enough to be visible in images acquired at relatively low phase angles. The opacity of the jet may just not be enough to be detectable, but we will continue to look.

Figure 1 :

 1 Figure 1: Fan-shaped deposits appear on top of the seasonal layer of ice every spring. This image, ESP_011671_0935, was taken at 86.4S / 99.0E, at L s 196 in a region informally known as Manhattan.

  The most promising HiRISE image to-date is shown in Figure2, in Russell Crater dunes. Dust appears to be billowing from one of the linear gullies. At ~54 o south latitude these dunes are covered with a EPSC Abstracts Vol. 13, EPSC-DPS2019-388-2, 2019 EPSC-DPS Joint Meeting 2019 c Author(s) 2019. CC Attribution 4.0 license.

Figure 1 :

 1 Figure 1: This image of Russell crater dunes, ESP_047078_1255, was taken at 54.3S / 12.9E, L s 202. A thin cloud of dust might be coming from a rupture in the ice.

Acknowledgements

We would like to thank our funding source for this work: NASA and the MRO project.


Y Tao
email: yu.tao@ucl.ac.uk

J-P Muller
email: j.muller@ucl.ac.uk

S J Conway
email: susan.conway@univ-nantes.fr

3D multi-resolution mapping of RSLs at Valles Marineris

We demonstrate techniques and results of multiresolution 3D mapping of the whole of the Valles Marineris (VM) area of Mars using stereo images from the Mars Express High Resolution Camera (HRSC), Mars Reconnaissance Orbiter (MRO) Context Camera (CTX), High Resolution Imaging Science Experiment (HiRISE) and the ESA ExoMars Trace Gas Orbiter (TGO) Colour and Stereo Surface Imaging System (CaSSIS). Following this, we demonstrate the methods employed for the automated detection and tracking of the Recurring Slope Lineae (RSL) features over the mapped region using CTX and repeat HiRISE images along with their associated 3D information.

Introduction

RSLs are metre-to decametre-wide dark streaks found on steep slopes, which grow during the warmest times of the year, fading during the cooler periods and reappearing again (but not necessarily in exactly the same place). The origin of these features is strongly contested, with some authors suggesting they are formed by surface water [1] or brine and others suggesting they are completely dry processes [2]. The implications of each of these formation mechanisms is fundamental to constraining Mars' water budget and habitability.

The VM area is where the highest concentration of RSLs is found on Mars as well as being the sole location where the triple point of water can be reached during the Martian summertime. This study focuses on multi-resolution 3D mapping of the whole VM area using the cascaded HRSC-CTX-CaSSIS-HiRISE dataset, and automated RSL detection and tracking using the CTX and repeat HiRISE images. We focus on techniques and will publish the results of the multiresolution 3D mapping and RSL tracking.

Multi-resolution 3D mapping

Previously, within the completed EU FP-7 iMars (http://www.i-mars.eu) project, a fully automated multiresolution Digital Terrain Model (DTM) processing chain was developed at UCL for NASA CTX and HiRISE stereo-pairs, called the Co-registration ASP-Gotcha Optimised (CASP-GO), based on the open source NASA Ames Stereo Pipeline (ASP) [3], tiepoint based multi-resolution image co-registration [4], and the Gotcha [5] sub-pixel refinement method. The CASP-GO system guarantees global geo-referencing congruence with respect to the aerographic coordinate system defined by HRSC, level-4 products and thence to the MOLA, providing much higher resolution stereo derived DTMs. By mid 2018, CASP-GO has been used to process ~5,300 planet-wide CTX DTMs [6] which are now being published through the ESAC Guest Storage Facilities (GSF) [7].

In this study, we refined and updated the CTX 3D mapping results for the whole VM area using a hybrid processing approach combining the CASP-GO method and the new ASP MGM method [3]. In addition, multi-resolution HRSC level 2 stereo images are being processed at UCL using CASP-GO to merge with the existing DLR HRSC level 4 DTMs to create a mosaiced base map of the VM area. A few TGO CaSSIS stereo images are currently being experimented with to form the cascaded HRSC-CTX-CaSSIS-HiRISE 3D dataset, which will be completed by mid June, 2019.

Super-resolution restoration

Within the completed EU FP-7 Planetary Robotics Vision Data Exploitation (PRoViDE) project (http://provide-space.eu), we developed a novel superresolution restoration (SRR) algorithm called Gotcha Partial Differential Equation based Total Variation (GPT) SRR [8] to restore distorted features from multi-angle observations using advanced feature and area matcher and regularization approaches, achieving a factor of 2x-5x enhancement in resolution for repeatpass HiRISE images [START_REF] Tao | ISPRS 2016 Commission IV[END_REF].

More recently funded by the UKSA CEOI (SuperRes-EO project), we have further developed the SRR system using advanced machine learning algorithms, applied to Earth Observation (EO) data. The new Multi-Angle GPT with Generative Adversarial Network refinement (MAGiGAN) SRR system [START_REF] Tao | [END_REF]11] not only retrieves subpixel information from multiangle distorted features from the original GPT SRR algorithm, but also uses the losses calculated from feature maps of the GAN network to replace the pixel wise difference based content loss of the original GPT algorithm in order to retrieve high texture detail.

We have ported most of the MAGiGAN SRR system to a GPU array and are about to form a training dataset from CTX and HiRISE images to apply this back to CTX and HiRISE images.

RSL detection and tracking

Studying the transport and formation processes on the Martian surface requires accurate measurements of dynamic features and the underlying 3D static surface. Tracking of such dynamic features has never been achieved automatically before due to the fact that detection and classification methods usually require a static reference frame and do not perform well when the feature itself is changing all the time. Previously, we demonstrated a new approach to detect and track the dynamic features by extracting the "static part" of the Martian surface through SRR using repeat-pass HiRISE observations. Due to the ability of the SRR technique to extract super-resolution for static features, we are able to restore matched (unchanged) features and meanwhile automatically track the unmatched (dynamic) pixels to characterize and measure the "change". Combining with a feature classifier, the detected dynamic changes can be further classified to known dynamic features, such as RSLs.

Previously, we reported on our initial RSL detection and tracking results of one of the RSL sites (centre coordinates: 41.6°S, 202.3°E) at Palikir Crater using the GPT SRR dynamic feature masking and SVM based classification from 8 repeat-pass HiRISE images [12].

In this study, we show modifications of the RSL detection and tracking system that involves a deep image network for better classification results at the VM area. The detection will be demonstrated using both CTX and HiRISE (where detection from CTX is positive) images. Tracking of the RSL features will be demonstrated using repeat HiRISE images.

RSL modelling

In the final stage, adding the 3D information derived from the multi-resolution DTMs, detected and tracked RSL features will be associated with their slopes and orientations, providing a more comprehensive interpretation of the RSL formation processes in 3D.

Future work

In the longer term, we aim to provide a regional map of RSL occurrence, with associated growth rates, timings (including inter-annual variability) and topographic information (including slopes and orientation) as well as uncertainties associated with the detection results.

Acknowledgements

The research leading to these results is receiving funding from the UKSA Aurora programme (2018-2021) under grant no. ST/S001891/1. The research leading to these results also received partial funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under iMars grant agreement n˚ 607379. Part of this research has also received funding from the UK Space Agency Centre for Earth Observation Instrumentation under SuperRes-EO project grant number RP10G0435A05 and OVERPaSS project grant number RP10G0435C206. SJC is grateful for the financial support of CNES in support of her CaSSIS work. The authors thank the spacecraft and instrument engineering teams for the successful completion and operation of CaSSIS. CaSSIS is a project of the University of Bern funded through the Swiss Space Office via ESA's PRODEX programme. The instrument hardware development was also supported by the Italian Space Agency (ASI) (ASI-INAF agreement no.I/018/12/0), INAF/Astronomical Observatory of Padova, and the Space Research Center (CBK) in Warsaw. Support from SGF (Budapest), the University of Arizona (LPL) and NASA are also gratefully acknowledged.


S J Conway
email: susan.conway@univ-nantes.fr

A Pommerol

N Thomas

J Raack

M Philippe

G Cremonese

Seasonal ices and gullies on Mars studied with CaSSIS

Gullies on Mars are kilometer-scale downslope mass wasting systems that resemble water-worn gullies on Earth. Recent observations have revealed that they are active today and that this activity is linked is linked with the seasonal frost cycle of carbon dioxide. In this study we use images from the Colour and Stereo Surface Imaging System (CaSSIS) aboard ESA's Trace Gas Orbiter (TGO) to study the relationship between surface frosts and gullies.

Introduction

Gullies on Mars were first reported in 2000 [1] and were hailed as evidence of recent liquid water flows on Mars. Since that time monitoring of gullies has revealed they are active today at times of year when the martian surface is at its coldest and when CO 2 is condensed on the surface [2,3]. To the first order the distribution of CO 2 frost [4,5] and the geographic locations of gullies [6] match -between latitudes of 30° and 45° on pole-facing steep slopes and 45° to the poles on slopes with any orientation. Detailed observations of gullies on martian sand dunes has hinted at a complex interplay between frosted and non-frosted surfaces [7] and recent gully-deposits have been observed to be emplaced atop frosted gully-fans [3]. In order to explore the relationship between surface frosts and gully-activity further we focus on Sisyphi Cavi near the south pole of Mars, where gully-activity has been studied [8] and CaSSIS obtained a dense temporal coverage of in 2018.

Approach

We use CaSSIS colour images in order to study the relationship between surface frosts and gullies in Sisyphi Cavi. CaSSIS has four colour filters: BLU, PAN, RED and NIR (centred on 500, 675, 836 and 937 nm respectively; see [9]); where the BLU filter is particularly useful for picking up surface frosts [10]. the timing of surface defrosting is latitude dependent. Although the images we use in this study are not from the same latitude we are able to place them into "defrosting" order by using surface temperature measurements from the Thermal Emission Spectrometer (TES) aboard Mars Global Surveyor. We assume that the surface materials vary little over this constricted region and hence thermal inertia variations are less important than latitude in controlling defrosting-timing at a regional scale.

Figure 1: Map of CaSSIS footprints over Sisyphi Cavi on a MOLA colour-coded elevation map with shading from CTX. Gullies are mapped in red and

CaSSIS PAN filter footprints are in black.

Results and discussion

In winter frost covers all surfaces and dark spots and fans can be seen across the slopes with gullies and preferentially around the gully channels (Figure 2a). The dark spots seem to form preferentially on sandy surfaces; such as the dunes at the bottom of Figure 2a. This is consistent with previous observations and has been interpreted to be the surface expression of gasjets generated by the sublimation of CO 2 underneath a continuous slab of CO 2 ice on the surface [11]. The jets occur when the pressure fractures the slab ice. As the surface temperature increases towards 200K the top of the slopes are the first to defrost (Figure 2b) followed by sun facing parts of the alcoves and channels (Figure 2c). Notice in both Figures 2b and2c that the base of the alcove remains frost covered.

As the surface temperature approaches and exceeds 250K and the surrounding terrain is completely defrosted, the last parts of the gully to remain frost covered are the fans (Figure 2d). We interpret this to be a result of the fans having slightly lower thermal inertia than the surrounding materials. This lower thermal inertia could be because the fans have a lower content of water ice (i.e. a thicker lag on top of the ice-table), because of recent depositional events. It is at this time of year when gullies are active [8].

Hence we infer that for gully activity to occur there needs to be both frosted and defrosted surfaces available to drive vigorous sublimation of the CO 2 ice [3,7]. Finally, once defrosting has almost fully completed and surface temperatures have reached their seasonal maximum of ~270K the only remaining surface frosts are in pole-facing niches at the base of gully-alcoves (Figure 2e).

Conclusions

 Ices and patterns of defrosting are easy monitor with CaSSIS.  Gully-alcoves defrost before the fans and gullies defrost later than surrounding terrainsuggests activity is driven by the availability of "hot" sediment to trigger more efficient sublimation.  Further work will examine whether frost patterns differ where activity occurs.

Figure 2 :

 2 Figure 2: Defrosting of gullies in Sisyphi Cavi as seen by CaSSIS. (a) MY34_002397_317_1, Ls 189° at 07:55 and 73°S. (b) MY34_003751_266_0, Ls 257° at 03:23 and 73°S. (c) MY34_003454_281_0 Ls 241° at 12:38 and 70°S. (d) MY34_003464_256_2, Ls 242° at 07:22and 68°S. (e) MY34_004349_276_2, Ls 288° at 18:01and 73°S.

Acknowledgments

SJC is grateful for the financial support of CNES in support of her CaSSIS work. The authors thank the spacecraft and instrument engineering teams for the successful completion and operation of CaSSIS. CaSSIS is a project of the University of Bern funded through the Swiss Space Office via ESA's PRODEX programme. The instrument hardware development was also supported by the Italian Space Agency (ASI) (ASI-INAF agreement no. I/018/12/0), INAF/ Astronomical Observatory of Padova, and the Space Research Center (CBK) in Warsaw. Support from SGF (Budapest), the University of Arizona (LPL) and NASA are also gratefully acknowledged.