Boundedness of some multi-parameter fiber-wise multiplier operators
aa r X i v : . [ m a t h . C A ] J u l BOUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISEMULTIPLIER OPERATORS
FR´ED´ERIC BERNICOT AND POLONA DURCIK
Abstract.
We prove L p estimates for various multi-parameter bi- and trilinear operators withsymbols acting on fibers of the two-dimensional functions. In particular, this yields estimatesfor the general bi-parameter form of the twisted paraproduct studied in [14]. Introduction
The classical Coifman-Meyer theorem [4, 5] is concerned with bilinear operators of the form T m ( F , F )( x ) = Z R n c F ( ξ ) c F ( η ) m ( ξ, η ) e πix · ( ξ + η ) dξdη, defined for test functions F , F : R n → C and m a bounded function on R n . The Coifman-Meyer theorem states that if m , in addition, satisfies | ∂ αξ ∂ βη m ( ξ, η ) | ≤ C | ( ξ, η ) | −| α |−| β | (1.1)for all multi-indices α, β ∈ N n up to a sufficently large finite order and all ( ξ, η ) = 0, with0 ≤ C < ∞ , then the operator T m maps L p × L p to L p whenever 1 < p , p ≤ ∞ , 1 / < p < ∞ ,and 1 /p + 1 /p = 1 /p . A notable application of this result is to the fractional Leibniz rule byKato and Ponce [12], which has further applications to nonlinear PDE; see for instance the workby Christ and Weinstein [3].Multi-parameter variants of the Coifman-Meyer theory arise by considering multilinear opera-tors with symbols behaving as tensor products of symbols (1.1). A simple bi-parameter examplecan be obtained by considering the operator T m with m ( ξ, η ) = m ( ξ ) m ( η ) , (1.2)where m and m 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 Calder´on-Zygmundoperators. It can be observed that the symbol m in (1.2) satisfies | ∂ αξ ∂ βη 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 operatorsby Grafakos and Kalton [10] 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 obtainingvarious fractional Leibniz-type rules such as in the works by Muscalu, Pipher, Tao, Thiele [22, 23].In particular, these papers show bounds for the operators T m with symbols satisfying | ∂ α ( ξ ,η ) ∂ β ( ξ ,η ) m ( ξ, η ) | ≤ C | ( ξ , η ) | −| α | | ( ξ , η ) | −| β | , (1.4)where ξ = ( ξ , ξ ) , η = ( η , η ) ∈ R n × R n . More general operators with so-called flag singularitieswere studied by Muscalu [20, 21]. Some recent works in the area include the one by Muscalu Date : July 7, 2020.1991
Mathematics Subject Classification.
Primary 42B15; Secondary 42B20. and Zhai [27], who investigate a certain trilinear operator which falls under the class of singularBrascamp-Lieb integrals with non-H¨older scaling, and some related flag paraproducts studiedby Lu, Pipher, and Zhang [17].In the aforementioned papers, the symbols in question generalize products of Coifman-Meyersymbols (1.1), each symbol acting on one or several fibers of the input functions. For instance, thefiber-wise action in (1.4) means that the first factor on the right-hand side of (1.4) concerns only F ( · , ξ ) and F ( · , η ), while the second term concerns only the complementary fibers F ( ξ , · )and F ( η , · ). Several recent developments in the theory of singular integrals include the study ofmultilinear 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 ( ξ , · ) and F ( · , η ), or F ( · , ξ ) and F ( η , · ). A fiber-wise action of this kind was first studied by Kovaˇc [14] and the first author[2] in the one-parameter setting (1.1) and dimension n = 2. In this paper we address such asituation in the multi-parameter setting.We follow customary practice to model symbols by the convolution-type P − Q operators, seefor instance [14] or [24]. For k ∈ Z and 1 ≤ i ≤
2, let ϕ i,k and ψ i,k be smooth functions adaptedin the interval [ − k +1 , k +1 ], and let ψ i,k vanish on [ − k − , k − ]. A function ρ adapted toan interval I ⊆ R means a function supported in I and satisfying k ∂ α ρ k ∞ ≤ | I | − α for all multi-indices α up to order N for some large N ; see [29]. Let P i,k and Q i,k denote theone-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 ( R ). When we apply such operators to one-dimensional fibers of a two-dimensionalfunction we use a superscript to denote the fiber on which the action takes place. For instance, P (1) i,k F ( x , x ) = (cid:0) F ( · , x ) ∗ q ϕ i,k (cid:1) ( x ) , P (2) i,k F ( x , x ) = (cid:0) F ( x , · ) ∗ q ϕ i,k (cid:1) ( x ) , and similarly for Q (1) i,k F and Q (2) i,k F . The central objects of this paper are the two-dimensionalbi-parameter bilinear operators T ( F , F )( x ) = X ( k,l ) ∈ Z : k ≤ l ( Q (1)1 ,k P (2)2 ,l F )( x ) ( Q (1)2 ,l P (2)1 ,k F )( x ) and T ( F , F )( x ) = X ( k,l ) ∈ Z : k ≤ l ( P (1)1 ,k P (2)2 ,l F )( x ) ( Q (1)2 ,l Q (2)1 ,k F )( x ) , defined for test functions F , F : R → C . We prove the following bounds in Section 2. Theorem 1.
The operators T and T are bounded from L p ( R ) × L p ( R ) to L p ′ ( R ) whenever < p , p < ∞ , < p ′ < , and p + p = p ′ . Passing to the Fourier side, the operators T and T can be viewed as multiplier operatorswhich map the tuple ( F , F ) to the two-dimensional function defined by x Z R c F ( ξ ) c F ( η ) m ( ξ, η ) e πix · ( ξ + η ) dξdη (1.5)for a suitable bounded function m on R . In the case of T and T , the function m satisfies | ∂ α ( ξ ,η ) ∂ β ( ξ ,η ) m ( ξ, η ) | ≤ C | ( ξ , η ) | −| α | | ( ξ , η ) | −| β | (1.6)for all α, β ∈ N up to order N and all ( ξ, η ) = 0. In general, the estimates (1.6) alone are notsufficient for boundedness of the multiplier operator (1.5). We elaborate on this in Section 2.4by reducing a special case of (1.5) to the aforementioned counterexample from [10]. OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 3
Here and in the sequel, the notion of bi- and multi-parameter is related to the number ofparameters of frequency scales. For instance, each factor on the right-hand side of (1.6) givesrise to one parameter. Given the constraint k ≤ l , the symbols of T , T do not split into thetensor products of two symbols m ( ξ , η ) and m ( ξ , η ) . However, bounds for T , T implybounds on the multiplier operator (1.5) in the case when m is indeed of tensor type, i.e. m ( ξ, η ) = m ( ξ , η ) m ( ξ , η ) , (1.7)where m , m satisfy the estimates | ∂ α ( ζ ,ζ ) m i ( ζ , ζ ) | ≤ C | ( ζ , ζ ) | − α (1.8)for all α ∈ N up to order N and all 0 = ( ζ , ζ ) ∈ R . This can be seen by the classical conedecomposition. Corollary 2.
Let m be given as in (1.7) . Then the associated operator (1.5) is bounded from L p × L p to L p ′ whenever < p , p < ∞ , < p ′ < , and p + p = p ′ . The multiplier operator (1.5) with the symbol (1.7) was suggested by Camil Muscalu. Thecase when m and m are localized to cones in the frequency plane is sometimes called twistedparaproduct. A special case when one of the symbols m i is constantly equal to one, i.e. x Z R c F ( ξ ) c F ( η ) m ( ξ , η ) e πix · ( ξ + η ) dξdη, (1.9)has been previously studied by Kovaˇc [14] and the first author [2]. It is a degenerate case ofthe two-dimensional bilinear Hilbert transform investigated by Demeter and Thiele [6]. Theone-parameter operator (1.9) is known to map L p × L p → L p ′ in a larger range than stated inCorollary 2, namely whenever the exponents satisfy1 < p , p < ∞ , < p ′ < , and 1 p + 1 p = 1 p ′ . Indeed, the techniques developed in [14] yield bounds whenever 2 < p , p < ∞ , < p ′ < m is of thetensor type m ( ζ , ζ , ζ , ζ ) = m (( ζ a ) a ∈ S ) m (( ζ a ) a ∈ S ) m (( ζ a ) a ∈ S ) m (( ζ a ) a ∈ S ) , (1.10)where S i ⊆ { , , , } and m i are symbols on R | S i | , each of them satisfying the estimates analo-gous to (1.8). Then (1.7) is a special case with S = S = ∅ . For the purpose of this paper, letus restrict ourselves to the case when the sets S i are pairwise disjoint and consider (1.5) with FR´ED´ERIC BERNICOT AND POLONA DURCIK such a symbol. If S = { } , then due to boundedness of the one-dimensional Calder´on-Zygmundoperators we may replace the function b F in (1.5) by m ( ξ ) b F ( ξ ) . (1.11)Performing the analogous step for any singleton S i , we may reduce (1.5) to the case where eachnon-empty set S i satisfies | S i | ≥
2. Up to symmetries, mapping properties of the cases that ariseare discussed in Table 1 below.
Table 1.
Structure of m and the range of boundednessStructure of m ( ξ, η ) Known range of boundedness1 m ( ξ , η ) 1 < p , p ≤ ∞ , < p ′ < ∞ m ( ξ , ξ ) 1 < p , p < ∞ , < p ′ < ∞ m ( ξ , η ) 1 < p , p < ∞ , < p ′ < m ( ξ , ξ ) m ( η , η ) 1 < p , p < ∞ , < p ′ < ∞ m ( ξ , η ) m ( ξ , η ) 1 < p , p ≤ ∞ , < p ′ < ∞ m ( ξ , η ) m ( ξ , η ) 1 < p , p < ∞ , < p ′ < m ( ξ , ξ , η ) 1 < p , p < ∞ , < p ′ < ∞ m ( ξ , ξ , η , η ) 1 < p , p ≤ ∞ , < p ′ < ∞ The operator corresponding to Case 1 is a classical Coifman-Meyer multiplier [4, 5] acting onthe first fibers of the functions F , F . Case 2 is a pointwise product of a linear Fourier multiplierwith the identity operator and can be treated analogously as (1.11). The estimates for Case 2stated in Table 1 are obtained by H¨older’s inequality. Case 3 is the operator (1.9), which hasbeen studied in [14]. Case 4 is a pointwise product of two linear operators and the analogousreduction as in (1.11) applies. Case 5 is a bi-parameter version of a Coifman-Meyer multiplierand its boundedness follows by [22]. As remarked earlier, in [22] the authors are able to handlemultipliers m which are not necessarily of tensor type. This is in contrast with Case 6, which isthe content of Corollary 2. It remains to consider Case 8, which is the classical Coifman-Meyermultiplier, and Case 7. We prove the following bounds for Case 7 in Section 3. Theorem 3. If m ( ξ, η ) = m ( ξ , ξ , η ) , then the associated operator (1.5) is bounded from L p × L p to L p ′ whenever < p , p < ∞ , < p ′ < ∞ , and p + p = p ′ . We emphasize that extending the range of exponents for p ′ in Cases 3 and 6 remains open.Dualizing the operators in (1.5) with symbols of the form (1.10), the corresponding trilinearforms are particular examples of singular Brascamp-Lieb integrals with several singular kernels,see also the survey [9]. Indeed, they can be represented by the trilinear forms Z R F ( x + A s + B t ) F ( x + A s + B t ) F ( x ) K ( s, t ) dxdsdt (1.12)for suitable matrices A , B , A , B ∈ M ( R ) and a kernel K , whose Fourier transform b K is ofthe form (1.10). In particular, the operator with (1.7) is associated with a tensor product of two OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 5
Calder´on-Zygmund kernels K ( s, t ) = K ( s ) K ( t ) , where K = q m , K = q m . In this case, if one of the kernels K , K in (1.12) specializesto the Dirac delta, then the object reduces to a one-parameter family of the two-dimensionalbilinear Hilbert transforms [6]. Studying (1.12) for an arbitrary choice of matrices A i , B i andany Calder´on-Zygmund kernels K , K remains an open problem.Furthermore, the operators T and T can be viewed as particular fiber-wise versions of theparaproducts studied by Muscalu, Tao, and Thiele in [24]. Following the setup from [24], let n ≥ n ∈ Z n be a convex polytope of the formΩ n = { ( k , . . . , k n ) ∈ Z n : k i α ≥ k i ′ α for all 1 ≤ α ≤ K } , where i α , i ′ α ∈ { , . . . , n } and K ≥ n -tuple( F i ) ni =1 of test functions on R to the two-dimensional function x X ( k ,...,k n ) ∈ Ω n n Y 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 adaptedin [ − k +1 , k +1 ] and vanishes on [ − k − , k − ]. When Ω n is of the form Ω n × Z n , boundsfor (1.13) follow from [24]. When n = 1, (1.13) reduces to a classical linear Calder´on-Zygmundoperator. When n = 2, one can classify the cases similarly as in Table 1. Indeed, this can be seenby summing over Ω in at least two parameters k i in (1.13). Using the fact that P k ∈ Z Q i,k arelinear Calder´on-Zygmund operators and hence satisfy the desired bounds, the problem is reducedto objects with the summation over Ω , such as T , T , and other bi-parameter analogues in [22].This yields L p × L p to L p ′ bounds whenever 1 < p , p < ∞ and 1 < p ′ <
2, but the knownrange may, in particular cases, be larger. However, as it is the case for T and T , the symbolsof (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-valuedestimates for the operator (1.9) with one constant symbol. The key step is in observing thatthe operator is localized in frequency due to the frequency supports of the fibers of the inputfunctions. This can be seen in sharp contrast with (1.9) itself, where such localization doesnot 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 applicationsof H¨older’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 Calder´on-Zygmund decompositionfrom [2]. This decomposition does not seem applicable in the context of Theorem 1, as thesymbols act on both fibers of the input functions. Extending the range of exponents in Theorem1 remains an open problem.Theorem 3 is proven in Section 4 and in the Banach case it relies on the bounds for theoperator (1.9). In Theorem 4, the operator acts on only one fiber of the function F ; in thiscase we are able to use the fiber-wise Calder´on-Zygmund decomposition to prove quasi-Banachestimates as stated in Theorem 3. Multilinear and multi-parameter generalizations.
At present there is only partial un-derstanding of the multilinear generalizations of Theorem 1. In the case of (1.7), multilinearoperators with only one non-constant symbol can be described in the language of bipartite graphsand were studied by Kovaˇc in [13] in a dyadic model.In this paper, we discuss a particular tri-parameter trilinear example, which can be seen as anatural generalization of (1.7). Let m , m , m be symbols satisfying (1.8) for 1 ≤ i ≤
3. Define
FR´ED´ERIC BERNICOT AND POLONA DURCIK the trilinear operator which maps a triple ( F , F , F ) of functions on R to the two-dimensionalfunction given by x Z R b F ( ξ ) b F ( η ) b F ( τ ) m ( ξ , η ) m ( η , τ ) m ( τ , ξ ) e πix · ( ξ + η + τ ) dξdηdτ, (1.14)where ξ = ( ξ , ξ ), η = ( η , η ), τ = ( τ , τ ). We prove the following bounds. Theorem 4.
The operator (1.14) is bounded from L p × L p × L p to L p ′ when < p , p , p < ∞ , < p < ∞ , P i =1 1 p = 1 , and p + p > , p + p > , p + p > . Note that the range in Theorem 4 is non-empty. For example, it contains exponents in vicinityof 3 < p = p = p <
4. The proof of Theorem 4 is detailed in Section 4 and can be seen as aniteration of the steps used in the proof of Theorem 1 and its corollary, by gradually reducing tothe vector-valued estimates for the operators with one or more constant symbols. Iterating thisprocedure and applying estimates for one- and two-parameter operators, which hold in restrictedranges 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 multilineari-ties. Beside facing the issues with the exponent range, objects with constant symbols which arisein 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. Ob-taining 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 twistedparaproduct which maps a tuple ( F , F ) of functions on R to the two-dimensional function x sup N> (cid:12)(cid:12)(cid:12) X | k | For two non-negative quantities A, B we write A . B if there exists an absolute con-stant C such that A ≤ CB . If C depends on the parameters P , . . . , P n , we write A . P ,...,P n B . Acknowledgments. The authors thank Cristina Benea, Vjekoslav Kovaˇc, Camil Muscalu andChristoph Thiele for several motivating discussions. The first author is supported by ERCproject FAnFArE no.637510. The second author acknowledges the hospitality of Universit´e deNantes while this research was performed.2. The bi-parameter bilinear operators T and T This section is devoted to the proof of Theorem 1. Throughout this section we will use theshorthand notation k ≪ l to denote k < l − k ≫ l will denote k > l + 200 and k ∼ l will mean l − ≤ k ≤ l + 200. OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 7 It will be evident from the proof that the argument will not rely on the particular choice ofthe bump functions ϕ i,k , ψ i,k as long as they satisfy the assumptions in the definition of T and T . For simplicity of notation, we shall therefore only discuss the case ϕ ,k = ϕ ,k = ϕ k and ψ ,k = ψ ,k = ψ k for each k ∈ Z . We shall also write P ,k = P ,k = P k and Q ,k = Q ,k = Q k .2.1. Boundedness of T . First we split the summation over k ≤ l into the regimes where k ∼ l and k ≪ l respectively, i.e. T ( F, G ) = X ( k,l ) ∈ Z l − ≤ k ≤ l ( Q (1) k P (2) l F ) ( Q (1) l P (2) k F ) + X ( k,l ) ∈ Z k ≪ l ( Q (1) k P (2) l F ) ( Q (1) l P (2) k F ) . (2.1)Bounds for the sum over l − ≤ k ≤ l follow by Cauchy-Schwarz. Indeed, using Cauchy-Schwarz in l ∈ Z we pointwise bound this term as (cid:12)(cid:12)(cid:12) X − ≤ s ≤ X l ∈ Z ( Q (1) l + s P (2) l F ) ( Q (1) l P (2) l + s F ) (cid:12)(cid:12)(cid:12) ≤ X − ≤ s ≤ k Q (1) l + s P (2) l F k ℓ l ( Z ) k Q (1) l P (2) l + s F k ℓ l ( Z ) . We consider the product of terms on the right-hand side for each fixed − ≤ s ≤ 0. To estimatethe L p ′ norm of the product we apply H¨older’s inequality and use bounds for the classical squarefunction. We obtain kk Q (1) l + s P (2) l F k ℓ l ( Z ) k Q (1) l P (2) l + s F k ℓ l ( Z ) k L p ′ ( R ) ≤ k Q (1) l + s P (2) l F k L p ( ℓ l ) k Q (1) l P (2) l + s F k L p ( ℓ l ) . p ,p k F k L p ( R ) k F k L p ( R ) whenever 1 /p ′ = 1 /p + 1 /p and 1 < p , p < ∞ . In the end it remains to sum the individualcontributions of these finitely many terms.It remains to consider the case k ≪ l in (2.1). By duality it suffices to study the correspondingtrilinear form and show (cid:12)(cid:12)(cid:12) X ( k,l ) ∈ Z : k ≪ l Z R ( Q (1) k P (2) l F ) ( Q (1) l P (2) k F ) F (cid:12)(cid:12)(cid:12) . p ,p ,p k F k L p ( R ) k F k L p ( R ) k F k L p ( R ) for any choice of exponents 1 < p , p < ∞ , 2 < p < ∞ with 1 /p + 1 /p + 1 /p = 1. By thefrequency supports of F , F , the form on the left-hand side can be written up to a constant as X ( k,l ) ∈ Z : k ≪ l Z R ( Q (1) k P (2) l F ) ( Q (1) l P (2) k F ) ( Q (1) l P (2) l F ) , (2.2)where P l and Q l are Fourier multipliers with symbols adapted to [ − l +3 , l +3 ], the symbol of P l is constant on [ − l +2 , l +2 ], and the symbol of Q l vanishes on [ − l − , l − ].Then we write P l = ϕ l (0) I + ( P l − ϕ l (0) I ) where I is the identity operator, and plug thisdecomposition into the form (2.2). This yields(2.2) = X ( k,l ) ∈ Z : k ≪ l M k,l + X ( k,l ) ∈ Z : k ≪ l E k,l , (2.3)where we have defined M k,l = Z R ( Q (1) k F ) ( Q (1) l P (2) k F ) ( ϕ l (0) Q (1) l P (2) l F ) , E k,l = Z R ( Q (1) k ( P (2) l − ϕ l (0) I (2) ) F ) ( Q (1) l P (2) k F ) ( Q (1) l P (2) l F ) . FR´ED´ERIC BERNICOT AND POLONA DURCIK First we consider the term involving M k,l . Since | ϕ l (0) | ≤ 1, up to redefining P l we mayassume that ϕ l (0) = 1 for each l . We split the summation as X ( k,l ) ∈ Z : k ≪ l M k,l = X ( k,l ) ∈ Z M k,l − X ( k,l ) ∈ Z : k ∼ l M k,l − X ( k,l ) ∈ Z : 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 H¨older’s inequality in l andin the integration, yielding (cid:12)(cid:12)(cid:12) X l ∈ Z X s ∼ M l + s,s (cid:12)(cid:12)(cid:12) ≤ X s ∼ k Q (1) l + s F k L p ( ℓ ∞ l ) k Q (1) l P (2) l + s F k L p ( ℓ l ) kQ (1) l P (2) l F k L p ( ℓ 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 . ByCauchy-Schwarz in l and H¨older’s inequality in the integration we estimate (cid:12)(cid:12)(cid:12) X ( k,l ) ∈ Z M k,l (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) X k ∈ Z ( Q (1) k F ) ( Q (1) l P (2) k F ) (cid:13)(cid:13)(cid:13) L p ′ ( ℓ l ) kQ (1) l P (2) l F k L p ( ℓ l ) , (2.5)where 1 /p + 1 /p ′ = 1. The second term on the right-hand side is a classical square function. Tobound the first term we use L p ( R ) × L p ( ℓ ) → L p ′ ( ℓ ) vector-valued estimates for the operator(1.9), which hold whenever 1 < p ′ < < p , p < ∞ . These vector-valued estimates areobtained by freezing the function F ∈ L p and using the linear inequalities of Marcinkiewiczand Zygmund [18], together with scalar-valued boundedness of (1.9). We obtain (cid:13)(cid:13)(cid:13) X k ∈ Z ( Q (1) k F ) ( Q (1) l P (2) k F ) (cid:13)(cid:13)(cid:13) L p ′ ( ℓ l ) . p k F k L p ( R ) k Q (1) l F k L p ( ℓ l ) . p ,p k F k L p ( R ) k F k L p ( R ) , where we have used the Littlewood-Paley inequality for the last bound. This yields the desiredestimate for the first term in (2.3).It remains to estimate the second term in (2.3). By frequency consideration in the secondfiber, one has X ( k,l ) ∈ Z : k ≪ l E k,l = c X ( k,l ) ∈ Z : k ≪ l Z R ( Q (1) k e Q (2) l F ) ( Q (1) l P (2) k F ) ( Q (1) l P (2) l F ) , (2.6)where c is an absolute constant and e Q l is a Fourier multiplier with symbol adapted to [ − l +3 , 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 , k ≪ l and k ≫ l . Bythe analogous considerations as in the paragraphs from (2.4) to (2.5) we note that it suffices toinstead consider the case when the sum is unconstrained, i.e. X ( k,l ) ∈ Z Z R ( Q (1) k e Q (2) l F ) ( Q (1) l P (2) k F ) ( Q (1) l P (2) l F ) . By the Cauchy-Schwarz inequality in l and H¨older’s inequality in the integration we bound thelast display by (cid:13)(cid:13)(cid:13) X k ∈ Z ( Q (1) k e Q l F ) ( Q (1) l P (2) k F ) (cid:13)(cid:13)(cid:13) L p ′ ( ℓ l ) kQ (1) l P (2) l F k L p ( ℓ l ) . The second term is a square function. Bounds for the first term follow from L p ( ℓ ) × L q ( ℓ ) → L r ( ℓ ) vector-valued estimates for the twisted paraproduct (1.9) and two applications of theLittlewood-Paley inequality. The vector valued estimates which we need in this case are a OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 9 consequence of scalar boundedness of the operator (1.9) and a result by Grafakos and Martell[11, Theorem 9.1]. This yields the desired bound for the second term in (2.3) and in turnestablishes the claim for T .2.2. Boundedness of T . The proof for T proceeds in the analogous way as the proof for T and we only sketch the necessary ingredients. By duality it suffices to bound (cid:12)(cid:12)(cid:12) X ( k,l ) ∈ Z : k ≤ l Z R ( P (1) k P (2) l F ) ( Q (1) l Q (2) k F ) F (cid:12)(cid:12)(cid:12) . p ,p ,p k F k L p ( R ) k F k L p ( R ) k F k L p ( R ) for any choice of exponents 1 < p , p < ∞ , 2 < p < ∞ with 1 /p + 1 /p + 1 /p = 1. The casewhen l − ≤ k ≤ l is bounded by H¨older’s inequality, so it suffices to consider k ≪ l .By frequency considerations, the form on the left-hand side is a constant multiple of X ( k,l ) ∈ Z : k ≪ l Z R ( P (1) k P (2) l F ) ( Q (1) l Q (2) k F ) ( Q (1) l P (2) l F )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 ). Byconsiderations as in the discussion after (2.3) it remains to estimate the analogue of the termassociated with M k,l , i.e. X ( k,l ) ∈ Z Z R ( P (1) k F ) ( Q (1) l Q (2) k F ) ( ϕ (0) Q (1) l P (2) l F ) , and the analogue of the term associated with E k,l , i.e. X ( k,l ) ∈ Z Z R ( P (1) k e Q (2) l F ) ( Q (1) l Q (2) k F ) ( Q (1) l P (2) l F ) , where e 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 is to deduce them from the bounds for T via a telescoping identity, which ”swaps” the P - and Q -type operators. Indeed, this can beachieved in a special case when T and T are related by a condition on the bump functions asin Proposition 8 in Section 4 below. Then, one can deduce a general case of T from the specialcase by an averaging argument. We perform such arguments in a trilinear tri-parameter settingin Section 4 below.2.3. Proof of Corollary 2. This corollary can be deduced from Theorem 1 by a classical conedecomposition, as performed for instance in [19]. For completeness we outline the relevant steps.Let m be a function on R \ { } satisfying | ∂ α m ( ζ ) | . | ζ | −| α | for all α up to a large finite order and all ζ = 0 in R . Let ϕ be a smooth function supportedin [ − , 2] and constantly equal to 1 on [ − , ψ = ϕ − ϕ (2 · ). Then P k ∈ Z ψ (2 − k τ ) = 0 foreach τ = 0. We can write m ( ζ ) = X ( k,l ) ∈ Z m ( ζ ) ψ (2 − k ζ ) ψ (2 − l ζ ) . Splitting the sum into regions when k ≤ l and k > l , respectively, and summing in the smallerparameter we obtain m ( ζ ) = X k ∈ Z m ( ζ ) ϕ (2 − k ζ ) ψ (2 − k ζ ) + X k ∈ Z m ( ζ ) ψ (2 − k ζ ) ϕ (2 − ( k − ζ ) . We consider the first sum; the second is treated analogously.Note that for each k ∈ Z , the summand is supported in { ζ ∈ R : | ζ | ≤ k +1 , − k − ≤ | ζ | ≤ k +1 } . The smooth restriction of m to that region can be decomposed into a double Fourier series,which yields X k ∈ Z m ( ζ ) ϕ (2 − k ζ ) ψ (2 − k ζ ) = X ( n ,n ) ∈ Z X k ∈ Z C kn ,n ϕ (2 − k ζ ) e cπi − k ζ n ψ (2 − k ζ ) e cπi − k ζ n where c is a fixed constant and the Fourier coefficients C kn ,n satisfy | C kn ,n | . (1 + | n | ) − N (1 + | n | ) − N for any N > k ∈ Z . For details we refer to [19], Chapter 2.13.Normalizing the bump functions and the coefficients, the last display can be written as anabsolute constant times X ( n ,n ) ∈ Z (1 + | n | ) − (1 + | n | ) − X k ∈ Z e C kn ,n ϕ k,n ( ζ ) ψ k,n ( ζ ) , where ϕ k,n ( ζ ) = c ϕ ( ζ ) e cπiζ n , ψ k,n ( ζ ) = c ψ ( ζ ) e cπiζ n , and the constants c , c are such that both functions are adapted to [ − k +1 , k +1 ]. Moreover, ϕ k,n (0) is the same constant for each k ∈ Z , ψ k,n vanishes in [ − k − , k − ], and | e C kn ,n | ≤ . This reduces the matters to considering symbols for each fixed n , n with bounds uniform in n , n . Note that the coefficients e C kn ,n can assumed to be equal to 1 by subsuming them intothe definitions of ψ k,n .We perform this decomposition for the symbols m and m in (1.7). Then the bounds for theassociated operator follow from bounds on T and T .2.4. A counterexample. In this section we show that the estimates (1.6) alone are not sufficientfor boundedness of the operator (1.5). We restrict ourselves to the Banach regime. Moreprecisely, we show that there exists a bounded function m on R satisfying | ∂ α ( ξ ,η ) ∂ β ( ξ ,η ) m ( ξ, η ) | . α,β | ( ξ , η ) | −| α | | ( ξ , η ) | −| β | for all multi-indices α, β ∈ N such that the associated operator (1.5) does not satisfy L p × L p to L p ′ estimates for any exponents 1 < p , p , p < ∞ .To see this we first recall a result from [10]; see the remark at the end of Section 3 in [10].Existence is shown of a symbol m on R \ { } satisfying | ∂ αξ ∂ βη m ( ξ , η ) | . α,β | ξ | − α | η | − β for all multi-indices α, β ∈ N , such that the one-dimensional operator mapping a tuple ( f , f )of functions on R to a one-dimensional function x Z R b f ( ξ ) b f ( η ) m ( ξ , η ) e πix ( ξ + η ) dξ dη , (2.7)does not satisfy any L p × L p to L p ′ 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 , ξ , η , η ) = m ( ξ , η ) e m ( ξ , η ) e m ( η , ξ ) , OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 11 where e m is a smooth symbol on R \ { } supported in the cone { ( ζ , ζ ) : | ζ | . | ζ |} , satisfying e m ( ζ , 0) = 1 whenever | ζ | 6 = 0, and | ∂ α e m ( ζ , ζ ) | . α | ( ζ , ζ ) | −| α | for all α ∈ N and all ( ζ , ζ ) = 0. Then m satisfies (1.6) and we have m ( ξ , , η , 0) = m ( ξ , η ).We dualize the operator (1.5) associated with this multiplier and consider the correspondingtrilinear form, which reads Z R c F ( ξ ) c F ( η ) c F ( ξ + η ) m ( ξ, η ) dξdη. (2.8)Let λ > < p , p , p < ∞ be such that 1 /p + 1 /p + 1 /p = 1. For 1 ≤ i ≤ F i ( x , x ) = f i ( x ) λ − /p i ϕ ( λ − x ) , where f i and ϕ are one-dimensional smooth compactly supported functions and b ϕ ≥ 0. Pluggingthese particular functions F i into the trilinear form (2.8) we obtain λ − Z R b f ( ξ ) λ b ϕ ( λξ ) b f ( η ) λ b ϕ ( λη ) b f ( ξ + η ) λ b ϕ ( λ ( ξ + η )) m ( ξ , ξ , η , η ) dξ dξ dη dη . (2.9)By rescaling in ξ and η it suffices to consider Z R b f ( ξ ) b ϕ ( ξ ) b f ( η ) b ϕ ( η ) b f ( ξ + η ) b ϕ ( ξ + η ) m ( ξ , λ − ξ , η , λ − η ) dξ dξ dη dη . Letting λ → ∞ and then integrating in ξ , η we obtain a non-zero constant multiple of Z R b f ( ξ ) b f ( η ) b f ( ξ + η ) m ( ξ , η ) dξ dη , which can be recognized as (2.7) paired with a function f . Therefore, the form in the lastdisplay does not satisfy any L p estimates. Since k F i k L pi ( R ) equals k f i k L pi ( R ) k ϕ k L pi ( R ) , it followsthat (2.9) does not satisfy any L p estimates as well.3. A one-parameter bilinear operator This section is devoted to the proof of Theorem 3. By a three-dimensional analogue of thecone decomposition outlined in Section 2.3 it suffices to estimate the operators mapping ( F , F )to two-dimensional functions given by x X k ∈ Z ( Q (1) k P (2)1 ,k F )( x ) ( P (1)2 ,k F )( x ) , (3.1) x X k ∈ Z ( P (1)1 ,k P (2)2 ,k F )( x ) ( Q (1) k F )( x ) , (3.2)and x X k ∈ Z ( P (1)1 ,k Q (2) k F )( x ) ( P (1)2 ,k F )( x ) , (3.3)where P i,k , Q k are Fourier multipliers with symbols ϕ i,k and ψ k , respectively, which are adaptedto [ − k +1 , k +1 ], and in addition ψ k vanishes on [ − k − , k − ].We will first prove bounds for each of them in the open Banach range. Then we will extendthe range with a fiber-wise Calder´on-Zygmund decomposition from [2]. 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 × L p to L p ′ whenever 1 < p , p , p < ∞ and1 /p + 1 /p + 1 /p = 1.To bound the first operator (3.1), we first reduce to the case when the symbol of P ,k issupported in [ − k − , k − ]. Indeed, if this is not the case we split P ,k = e P ,k + Q ,k , where the symbols of e P ,k and Q ,k are supported in [ − k − , k − ] and [ − k +1 , − k − ] ∪ [2 k − , k +1 ], respectively. We split the operator accordingly and note that the term with Q ,k immediately reduces to two square functions by Cauchy-Schwarz.Now we dualize and consider the corresponding trilinear form. By the frequency localizationin the first fibers of the functions, it suffices to show (cid:12)(cid:12)(cid:12) X k ∈ Z Z R ( Q (1) k P (2)1 ,k F ) ( e P (1)2 ,k F ) ( Q (1) k F ) (cid:12)(cid:12)(cid:12) . p ,p ,p k F k L p ( R ) k F k L p ( R ) k F k L p ( R ) for any choice of exponents 1 < p , p , p < ∞ and 1 /p + 1 /p + 1 /p = 1. Here Q k is adaptedto [ − k +3 , k +3 ] and the corresponding symbol vanishes at the origin. This estimate followsimmediately by H¨older’s inequality in k and in the integration, together with bounds on thesquare and maximal function.For the second operator (3.2) we proceed in the same way; this time reducing to the case whenthe symbol of P ,k is supported in [ − k − , k − ].It remains to consider (3.3). By considerations as above, we may assume that P ,k is supportedin [ − k − , k − ]. By duality it suffices to study the corresponding trilinear form X k ∈ Z Z R ( P (1)1 ,k Q (2) k F ) ( P (1)2 ,k F ) F = c X k ∈ Z Z R ( P (1)1 ,k Q (2) k F ) Q (2) k (cid:16) ( P (1)2 ,k F ) F (cid:17) , (3.4)where c is a constant and Q k satisfies the properties as in the previous display. By Cauchy-Schwarz in k and H¨older’s inequality in the integration, we bound the last display by k P (1)1 ,k Q (2) k F k L p ( ℓ k ) (cid:13)(cid:13)(cid:13) Q (2) k (cid:16) ( P (1)2 ,k F ) F (cid:17)(cid:13)(cid:13)(cid:13) L p ′ ( ℓ k ) . The first term is a square function bounded in the full range. For the second term, we notethat by Fatou’s lemma, it suffices to restrict the ℓ k norm to | k | ≤ K for some K > K . By Kintchine’s inequality, this term reduces to having to estimate (cid:13)(cid:13)(cid:13) X | k |≤ K a k Q (2) k (cid:16) ( P (1)2 ,k F ) F (cid:17)(cid:13)(cid:13)(cid:13) L p ′ ( R ) for uniformly bounded coefficients a k . Dualizing with G ∈ L p ( R ) it suffices to show (cid:12)(cid:12)(cid:12) X | k |≤ K Z R a k ( Q (2) k G ) ( P (1)2 ,k F ) F (cid:12)(cid:12)(cid:12) . p ,p ,p k G k L p ( R ) k F k L p ( R ) k F k L p ( R ) . This estimate holds in the range 1 < p , p < ∞ , 2 < p < ∞ , and H¨older scaling, since theleft-hand side is an operator of the form (1.9) paired with the function F .To obtain bounds for (3.3) in the full range, it suffices to show estimates with 1 < p , p < ∞ and 2 < p < ∞ . The left-hand side of (3.4) can be written as X k ∈ Z Z R ( P (1)1 ,k Q (2) k F ) ( P (1)2 ,k F ) P (1) k F for a multiplier P k with symbol adapted to [ − k +3 , k +3 ]. Writing P ,k = ϕ ,k (0) I + ( P ,k − ϕ ,k (0) I ) we reduce the last display to X k ∈ Z Z R ( P (1)1 ,k Q (2) k F ) F ( ϕ ,k (0) P (1) k F ) + X k ∈ Z Z R ( P (1)1 ,k Q (2) k F ) ( Q (1) k F ) ( P (1) k F )for a suitably defined Q k whose symbol vanishes at the origin. The first term is analogous tothe left-hand side of (3.4), satisfying the estimates with 1 < p , p < ∞ and 2 < p < ∞ . Thesecond term has two Q -type operators and bounds in the full range follow by H¨older’s inequalityand bounds on the maximal and square functions.3.2. Fiber-wise Calder´on-Zygmund decomposition. In this section we show that the op-erators (3.1), (3.2) and (3.3) are bounded from L p × L p to L p ′ whenever 1 < p , p < ∞ ,1 / < p ′ < ∞ and 1 /p + 1 /p = 1 /p ′ .By real bilinear interpolation, it suffices to prove weak-type bounds with p ′ < T ( F , F )( x ) = X k ∈ Z ( P (1)1 ,k P (2)3 ,k F )( x ) ( P (1)2 ,k F )( x ) , where P i,k are Fourier multipliers with symbols adapted to [ − k +1 , k +1 ]. We are going to applythe fiberwise Calder´on-Zygmund from [2] to the function F .It suffices to show that for any 1 < p , p < ∞ and 1 / < p ′ < 1, the operator T satisfies theweak L p × L p → L p ′ , ∞ estimates. Fix F ∈ L p ( R ) and F ∈ L p ( R ). By homogeneity wemay assume k F k L p ( R ) = k F k L p ( R ) = 1 . Our goal is to show that for every λ > |{ x ∈ R : | T ( F , F )( x ) | > λ }| . p ,p λ − p ′ . We write x = ( x , x ). Fix λ > F ( · , x ) for fixed x ∈ R at level λ p ′ /p . This yields functions g x and atoms a i,x supported ondisjoint dyadic intervals intervals I i,x such that F ( · , x ) = g x + P i a i,x and for all x k g x k L p ( R ) ≤ k F ( · , x ) k L p ( R ) and k g x k L ∞ ( R ) . λ p ′ /p . Moreover, for all i , the atom a i,x has vanishing mean on I i,x , k a i,x k L p ( I i,x ) . λ p ′ /p | I i,x | /p , and the intervals satisfy X i | I i,x | . λ − p ′ k F ( · , x ) k p L p ( R ) . (3.5)First we consider the good part T ( F , g ), where g = ( x , x ) g x ( x ). We have k g k L p ( R ) ≤ k F k L p ( R ) = 1 and k g k L ∞ ( R ) . λ p ′ /p . Therefore, for q > p we get k g k L q ( R ) . p λ p ′ (1 /p − /q ) . By the boundedness in the Banach range obtained in the previous section we obtain |{ x ∈ R : | T ( F , g )( x ) | > λ }| ≤ λ − s k T ( F , g ) k s L s ( R ) . p ,p λ − s k F k s L p ( R ) k g k s L q ( R ) . λ − s + sp ′ (1 /p − /q ) = λ − p ′ , where 1 < q , s < ∞ are any exponents that satisfy 1 /p + 1 /q = 1 /s , q > p and ( p , q , 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 , b ), where b = ( x , x ) P i a i,x ( x ). Denote E = [ x ∈ R [ i (2 I i,x × { x } ) , where 2 I i,x is the interval with the same center as I i,x but twice the length. Note that | E | . Z R | ∪ i I i,x | dx . λ − p ′ Z R k F ( · , x ) k p L p ( R ) dx . λ − p ′ k F k p L p ( R ) = λ − p ′ , where we have used (3.5). Therefore, it remains to estimate |{ x E : | T ( F , b )( x ) | > λ }| . Bythe definition we have P (1)2 ,k b ( x ) = X i P ,k [ a i,x ]( x ) . Fix x ∈ R and take x ∈ R \ ∪ i I i,x . Denote by c i,x the center of the interval I i,x . Since P ,k is a convolution operator with some smooth function ρ k (with its Fourier transform adapted to[ − k +1 , k +1 ] up to a large order), using the mean zero property of a i,x , we obtain | P ,k [ a i,x ]( x ) | ≤ Z I i,x | a i,x ( w ) || ρ k ( x − w ) − ρ k ( x − c i,x ) | dw . Z I i,x | a i,x ( w ) | k | I i,x | (cid:0) k | x − c i,x | (cid:1) − dw . λ p ′ /p k | I i,x | (cid:0) k | x − c i,x | (cid:1) − . Here we used k a i,x k L ( R ) ≤ | I i,x | /p ′ k a i,x k L p ( R ) . λ p ′ /p | I i,x | . Therefore, since | x − c i,x | ≥ | I i,x | / X k ∈ Z k | I i,x | (cid:16) | x − c i,y | − k (cid:17) − . (cid:16) | x − c i,y || I i,x | (cid:17) − . Therefore, we have showed that for x ∈ R \ ∪ i I i,x it holds | T ( F , b )( x ) | . λ p ′ /p M ( F )( x ) H ( x ) , where M is the Hardy-Littlewood maximal function and H ( x ) = X i (cid:16) | x − c i,y || I i,y | (cid:17) − . Next we use boundedness of the Marcinkiewicz functions associated with a disjoint collection ofthe intervals ( I i,x ) i , i.e. (cid:13)(cid:13)(cid:13) X i (cid:16) | x − c i,x || I i,x | (cid:17) − (cid:13)(cid:13)(cid:13) L p x ( R ) . p (cid:16) X i | I i,x | (cid:17) /p . (See [28] or Grafakos, Exercise 4.6.6.) This yields k H k L p ( R ) . p (cid:13)(cid:13)(cid:13)(cid:16) X i | I i,x | (cid:17) /p (cid:13)(cid:13)(cid:13) L p x ( R ) . λ − p ′ /p kk F ( x , x ) k L p x ( R ) k L p x ( R ) . λ − p ′ /p . OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 15 Therefore, |{ x E : | T ( F , b )( x ) | > λ }| . |{ x ∈ R : λ p ′ /p M ( F )( x ) H ( x ) & λ }| = |{ x ∈ R : M ( F )( x ) H ( x ) & λ − p ′ /p }| . λ − p ′ +( p ′ ) /p kM ( F ) k p ′ L p ( R ) k H k p ′ L p ( R ) . p ,p λ − p ′ +( p ′ ) /p − ( p ′ ) /p . λ − p . Summarizing, we have shown that the operator T satisfies the weak inequality L p × L p → L p ′ , ∞ . 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 Calder´on-Zygmund inthis case (and not in the case of Theorem 1) because the operators (3.1), (3.2) and (3.3) acton only fiber of the function F . Alternatively, one could apply the classical two-dimensionalCalder´on-Zygmund decomposition by decomposing the function F . Remark 7. An alternative approach to prove quasi-Banach estimates for (3.1) and (3.2), whichavoids the use of fiberwise Calder´on-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 ′ ⊆ L p ′ , is thefollowing. Note that the operators (3.1) and (3.2) are localized in frequency in the first coordinateat scale k . So we can use the argument from [1] about the inequality k f k p . k S ( f ) k p for0 < p ≤ 1, where S is the Littlewood-Paley square function. More precisely, following the scalarcase of [1, Theorem 3.1], one obtains for 0 < p ′ ≤ k X k ∈ Z ( Q (1) k P (2)1 ,k F ) ( P (1)2 ,k F ) k L p ′ ( R ) . k ( Q (1) k P (2)1 ,k F ) ( P (1)2 ,k F ) k L p ′ ( ℓ k ) . To apply the arguments from [1] it is important is to obtain the localized estimates which arereduced through duality by factorizing a Q k operator on the dual function, which is possible inthis situation. Then the square function on the right-hand side is easily estimated by the productof a maximal and square function, both of them which are bounded. A similar reasoning canbe done for the second operator of type (3.2). In that way we recover the L p ′ –boundedness ofoperators (3.1) and (3.2) and we also prove boundedness in the Hardy space H p ′ .4. A tri-parameter trilinear operator The goal of this section it to prove Theorem 4. Throughout this section we will use theshorthand notation k ≪ l to denote k < l − 50. Similarly, k ≫ l will denote k > l + 50 and k ∼ l will mean l − ≤ k ≤ l + 50.For k ∈ Z and 1 ≤ i ≤ Q i,k be Fourier multipliers with symbols ψ i,k . For k ∈ Z and4 ≤ j ≤ P j,k be Fourier multipliers with symbols ϕ j,k . Here ψ i,k and ϕ j,k are smoothone-dimensional functions. In what follows we will consider various classes of symbols and wewill apply more assumptions on them as we proceed. We define the trilinear operators U and U acting on two-dimensional functions F , F , F : R → C by U ( F , F , F )( x ) = X ( k,l,m ) ∈ Z ( Q (1)1 ,k P (2)4 ,m F )( x ) ( Q (1)2 ,l P (2)5 ,k F )( x ) ( Q (1)3 ,m P (2)6 ,l F )( x ) ,U ( F , F , F )( x ) = X ( k,l,m ) ∈ Z ( P (1)5 ,k P (2)4 ,m F )( x ) ( Q (1)2 ,l Q (2)1 ,k F )( x ) ( Q (1)3 ,m P (2)6 ,l F )( x ) . We keep in mind that they depend on the particular choice of the functions ψ i,k , ϕ j,k , but wehave suppressed that in the notation. Let the exponents p , p , p , and p ′ satisfy the assumptions stated in Theorem 4. The first stepin the proof of Theorem 4 is to decompose the symbols m , m , m as in Section 2.3. This givesthat it suffices to prove L p × L p × L p to L p ′ estimates for U and U under the assumptionsthat for 1 ≤ i ≤ ψ i,k is adapted to [ − k +1 , k +1 ], and for 4 ≤ j ≤ ϕ j,k is adapted to[ − k +1 , k +1 ]. Moreover, each ψ i,k vanishes on [ − k − , k − ] and ϕ j,k (0) = ϕ j, (0) for each k .Indeed, note that while cone decomposition gives 8 terms, all other terms that are of the formof U and U with P - and Q -type operators interchanged can be deduced from the bounds on U and U 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 and U resulting after a cone decomposition will be deduced from Lemmas 8-10.Lemma 8 concerns a particular case of U . Lemma 8. Let p , p , p and p ′ be exponents as in Theorem 4. For ≤ i ≤ and k ∈ Z let ψ i,k be a smooth function supported in [ − k +3 , k +3 ] , which vanishes on [ − k − , k − ] . Assume thatfor each ≤ i ≤ , ψ i,k is of the form ψ i,k = ϕ i,k − ϕ i,k − , (4.1) where ϕ i,k is a bump function adapted to [ − k +3 , k +3 ] . For ≤ j ≤ and k ∈ Z let ϕ j,k be adapted to [ − k +4 , k +4 ] and assume that the function ψ j,k = ϕ j,k − ϕ j,k − is supported in [ − k +4 , k +4 ] and vanishes on [ − k − , k − ] .Under these assumptions on ψ i,k and ϕ j,k , the associated operator U is bounded from L p × L p × L p to L p ′ . Next we state Lemmas 9 and 10, which concern operators of the form (1.14) with one constantsymbol. In particular, these results will be used in the proof of Lemma 8. Lemma 9. For k ∈ Z let Q ,k , Q ,k , P ,k and P ,k be Fourier multipliers with symbols adapted to [ − k +10 , k +10 ] . Further, assume that the symbols of Q ,k , Q ,k vanish on [ − k − , k − ] . Thenthe trilinear operator which maps ( F , F , F ) to the two-dimensional function given by x X ( k,l ) ∈ Z : k ≪ l ( Q (1)1 ,k F )( x ) ( Q (1)2 ,l P (2)5 ,k F )( x ) ( P (2)6 ,l F )( x ) is bounded from L p × L p × L p to L p ′ in the range < p , p , p < ∞ , < p < ∞ , P i =1 1 p i = 1 ,and p + p > .Proof of Lemma 9. It will be clear from the proof that the argument will not depend on theparticular choice of the frequency projections satisfying the requirements from the lemma, so forsimplicity of the notation, we only discuss the special cases P ,k = P ,k = P k , Q ,k = Q ,k = Q k .By duality it suffices to consider the corresponding quadrilinear form X ( k,l ) ∈ Z : k ≪ l Z R ( Q (1) k F ) ( Q (1) l P (2) k F ) ( P (2) l F ) F . By the frequency localization of the functions F i in the first fibers, to prove estimates for thequadrilinear form it suffices to study X ( k,l ) ∈ Z : k ≪ l M k,l = X ( k,l ) ∈ Z M k,l − X ( k,l ) ∈ Z : k ∼ l M k,l − X ( k,l ) ∈ Z : k ≫ l M k,l , (4.2)where we have defined M k,l = Z R ( Q (1) k F ) ( Q (1) l P (2) k F ) Q (1) l (cid:16) ( P (2) l F ) F (cid:17) . OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 17 Here Q l is a Fourier multiplier with a symbol which is adapted to [ − l +12 , l +12 ] and vanisheson [ − l − , l − ].Note that the sum over k ≫ l in (4.2) vanishes due to frequency supports. To bound the sumwith k ∼ l we use H¨older’s inequality to obtain (cid:12)(cid:12)(cid:12) X l ∈ Z X s ∼ M l + s,l (cid:12)(cid:12)(cid:12) ≤ X s ∼ k Q (1) l + s F k L p ( ℓ ∞ l ) k Q (1) l P (2) l + s F k L p ( ℓ l ) (cid:13)(cid:13)(cid:13) Q (1) l (cid:16) ( P (2) l F ) F (cid:17)(cid:13)(cid:13)(cid:13) L r ( ℓ l ) whenever 1 /p + 1 /p + 1 /r = 1 and 1 < p , p , r < ∞ . For a fixed s , bounds for the first twoterms follow by boundedness of the maximal and square functions. The third term satisfies (cid:13)(cid:13)(cid:13) Q (1) l (cid:16) ( P (2) l F ) F (cid:17)(cid:13)(cid:13)(cid:13) L r ( ℓ l ) . p ,p k F k L p ( R ) k F k L p ( R ) (4.3)whenever 1 < p , r < ∞ , 2 < p < ∞ and 1 /r = 1 /p + 1 /p . Indeed, this follows by Kintchine’sinequality to linearize the square-sum and then use bounds for (1.9). In the end, it remains tosum in s ∼ k, l ) ∈ Z . In this case we note thatthe third factor in M k,l does not depend on k . This allows to apply Cauchy-Schwarz in l andH¨older’s inequality in the integration, yielding (cid:12)(cid:12)(cid:12) X ( k,l ) ∈ Z M k,l (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) X k ∈ Z ( Q (1) k F ) ( Q (1) l P (2) k F ) (cid:13)(cid:13)(cid:13) L s ( ℓ l ) (cid:13)(cid:13)(cid:13) Q (1) l (cid:16) ( P (2) l F ) F (cid:17)(cid:13)(cid:13)(cid:13) L r ( ℓ 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 + 1 /p = 1 /r and 1 < p , r < ∞ ,2 < p < ∞ . The first factor maps L p ( R ) × L p ( ℓ ) → L s ( ℓ ) whenever 1 /p + 1 /p = 1 /s and1 < p , p < ∞ , 1 /s > / 2. This follows from vector-valued estimates for (1.9) as discussed inthe display following (2.5). (cid:3) Lemma 10. For k ∈ Z , let Q ,k , Q ,k be Fourier multipliers with symbols σ ,k , σ ,k which arefunctions supported in [ − k +10 , k +10 ] and vanish on [ − k − , k − ] . For k ∈ Z let P ,k , P ,k beFourier multipliers with symbols ρ ,k , ρ ,k , which are functions adapted to [ − k +10 , k +10 ] .Assume that for each i ∈ { , , , } , these functions are of the form σ i,k = ρ i,k − ρ i,k − , where σ ,k , σ ,k are further functions supported in [ − k +10 , k +10 ] , which vanish on [ − k − , k − ] ,while ρ ,k , ρ ,k are functions adapted to [ − k +10 , k +10 ] . Then the trilinear operator which maps ( F , F , F ) to the two-dimensional function given by x X ( k,l ) ∈ Z : k ≫ l ( Q (1)1 ,k F )( x ) ( Q (1)2 ,l P (2)5 ,k F )( x ) ( P (2)6 ,l F )( x ) (4.4) is bounded from L p × L p × L p to L p ′ in the range < p , p , p < ∞ , < p < ∞ , P i =1 1 p i = 1 ,and p + p > .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 ∈ { , , , } .Observe that because of the condition on the bump functions we have the telescoping identity k X k = k P i,k f Q j,k g + k X k = k Q i,k f P j,k − g = P i,k f P j,k g − P i,k − f P j,k − g (4.5) for any f, g ∈ L ( R ) and i, j ∈ { , , , } . Also note that letting k → −∞ and k → ∞ , theright-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 P k ∈ Z P l ∈ Z : l ≪ k and use (4.5) in l . This gives(4.4) = − X ( k,l ) ∈ Z : k ≫ l ( Q (1)1 ,k F ) ( P (1)2 ,l − P (2)5 ,k F ) ( Q (2)6 ,l F ) (4.6)+ X k ∈ Z ( Q (1)1 ,k F ) ( P (1)2 ,k − P (2)5 ,k F ) ( P (2)6 ,k − F ) (4.7)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 X k ∈ Z Z R Q (1) k (cid:16) ( Q (1)1 ,k F ) ( P (1)2 ,k − P (2)5 ,k F ) (cid:17) ( P (2)6 ,k − F ) F , where Q k has a symbol adapted to [ − k +12 , k +12 ] which vanishes on [ − k − , k − ]. Note thatby the support assumption on σ ,k we necessary have that ρ ,k (0) is the same constant c ∈ R for each k ∈ Z . Then we write P ,k − = c I + ( P ,k − − c I ) and split the operator accordingly.By the frequency support information in the first fibers it suffices to bound the terms X k ∈ Z Z R c Q (1) k (cid:16) ( Q (1)1 ,k F ) ( P (2)5 ,k F ) (cid:17) ( P (2)6 ,k − F ) F and (4.8) X k ∈ Z Z R Q (1) k (cid:16) ( Q (1)1 ,k F ) ( e Q (1) k P (2)5 ,k F ) (cid:17) ( P (2)6 ,k − F ) F , (4.9)where the symbol of e Q k is adapted in [ − k +100 , k +100 ] and vanishes at the origin. The desiredbounds for the first term (4.8) follow from (cid:12)(cid:12)(cid:12) X k ∈ Z Z R ( Q (1)1 ,k F ) ( P (2)5 ,k F ) Q (1) k (cid:16) ( P (2)6 ,k − F ) F (cid:17)(cid:12)(cid:12)(cid:12) ≤ k ( Q (1)1 ,k F ) ( P (2)5 ,k F ) k L r ( ℓ k ) (cid:13)(cid:13)(cid:13) Q (1) k (cid:16) ( P (2)6 ,k − F ) F (cid:17)(cid:13)(cid:13)(cid:13) L s ( ℓ k ) (4.10)whenever 1 /r + 1 /s = 1 and 1 < r, s < ∞ . For the first term in (4.10) we use H¨older’s inequalityto obtain k ( Q (1)1 ,k F ) ( P (2)5 ,k F ) k L r ( ℓ k ) ≤ k Q (1)1 ,k F k L p ( ℓ k ) k P (2)5 ,k F k L p ( ℓ ∞ k ) . p ,p k F k L p ( R ) k F k L p ( R ) . whenever 1 /r = 1 /p + 1 /p , 1 < p , p < ∞ . For the last inequality we have used bounds on themaximal and square functions. For the second term in (4.10) we use Kintchine’s inequality tolinearize the square-sum. Then we apply bounds for the corresponding operator (1.9), yielding (cid:13)(cid:13)(cid:13) Q (1) k (cid:16) ( P (2)6 ,k − F ) F (cid:17)(cid:13)(cid:13)(cid:13) L s ( ℓ k ) . p ,p k F k L p ( R ) k F k L p ( R ) whenever 1 /s = 1 /p + 1 /p , 1 < p < ∞ , and 2 < p < ∞ . This yields the desired estimate.For (4.9) we proceed in the analogous way, this time estimating e Q (1) k P (2)5 ,k F by the Hardy-Littlewood maximal function. OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 19 It remains to estimate (4.6), for which we still want to switch the projections in k . Now wewrite the double sum over k ≫ l as P l ∈ Z P k ∈ Z : k ≫ l and telescope (4.6) in k , giving(4.6) = X ( k,l ) ∈ Z : k ≫ l ( P (1)1 ,k − F ) ( P (1)2 ,l − Q (2)5 ,k F ) ( Q (2)6 ,l F ) , (4.11)+ X l ∈ Z ( P (1)1 ,l +51 F ) ( P (1)2 ,l − P (2)5 ,l +51 F ) ( Q (2)6 ,l F ) , (4.12) − X l ∈ Z ( F ) ( P (1)2 ,l − F ) ( Q (2)6 ,l F ) . (4.13)The term (4.13) is the boundary term at k = ∞ , which is just a pointwise product of F with an operator of the form (1.9). We obtain L p × L p × L p → L p ′ estimates in the range1 < p , p , p , p < ∞ , 1 /p + 1 /p > / 2, whenever the exponents satisfy the H¨older scaling.The term (4.11) follows from Lemma 9 and symmetry considerations, i.e. after interchangingthe role of F and F and the role of the first and second fibers. We obtain estimates in therange 1 < p , p , p < ∞ , 2 < p < ∞ , 1 /p + 1 /p > / 2. Finally, note that(4.12) = X l ∈ Z ( P (1)1 ,l +51 F ) ( P (1)2 ,l − P (2)5 ,l − F ) ( Q (2)6 ,l F ) (4.14)+ X l ∈ Z ( P (1)1 ,l +51 F ) ( P (1)2 ,l − Q (2) l F ) ( Q (2)6 ,l F ) , (4.15)where the symbol of Q l is up to a constant adapted in [ − l +51 , l +51 ] and it vanishes near theorigin. The term (4.15) is bounded by H¨older’s inequality and bounds on the maximal and squarefunctions in the full range. For (4.14) one can proceed analogously as for (4.7), with the rolesof the first and second fiber interchanged. This gives bounds in the range 1 < p , p , p < ∞ ,2 < p < ∞ and H¨older scaling. (cid:3) Now we are ready to prove Lemma 8. Proof of Lemma 8. For simplicity of notation, we shall assume that P ,k = P ,k = P k and Q ,k = Q ,k = Q k , but as it will be clear from the proof, the arguments will work for general Fouriermultiplier operators with symbols satisfying the above assumptions.The proof will be based on analyzing the ordering of the parameters k, l , and m . Note thatby symmetry we may assume k ≤ l ≤ m . If, in addition, it holds that k ∼ l and l ∼ m , thenall frequencies are comparable up to an absolute constant and the claim follows by H¨older’sinequality.Consider now the case with two dominating frequencies when k ≪ l and l ∼ m , i.e. l = m + s for some − ≤ s ≤ 0. Then we need to estimate X m ∈ Z X − ≤ s ≤ X k ∈ Z : k ≪ m + s ( Q (1) k P (2) m F ) ( Q (1) m + s P (2) k F ) ( Q (1) m P (2) m + s F ) . By Cauchy-Schwarz in m and H¨older’s inequality, its L p ′ norm is bounded by X − ≤ s ≤ (cid:13)(cid:13)(cid:13) X k ∈ Z : k ≪ m + s ( Q (1) k P (2) m F )( Q (1) m + s P (2) k F ) (cid:13)(cid:13)(cid:13) L p ′ ( ℓ m ) k Q (1) m P (2) m + s F k L p ( ℓ m ) whenever 1 < p < ∞ and 1 /p ′ = 1 /p + 1 /p ′ . For a fixed s , the second term is a squarefunction, bounded in the full range. For the first term we use Kintchine’s inequality, which reduces to showing (cid:13)(cid:13)(cid:13) X | m |≤ M a m X k ∈ Z : k ≪ m + s ( Q (1) k P (2) m F )( Q (1) m + s P (2) k F ) (cid:13)(cid:13)(cid:13) L p ′ ( R ) . p ,p k F k L p ( R ) k F k L p ( R ) uniformly in M > 0, where | a m | ≤ 1. This estimate holds when 1 < p , p , p , p < ∞ , 1 /p +1 /p = 1 /p ′ , 1 /p + 1 /p > / 2. Indeed, this follows from Theorem 1 after normalizing thesymbols of P m and Q m + s . In the end, it remains to sum in − ≤ s ≤ k ≤ l ≪ m with one dominating frequency. We dualize and consider thecorresponding trilinear form. By the frequency localizations it suffices to bound X ( k,l,m ) ∈ Z : k ≤ l ≪ m Z R ( Q (1) k P (2) m F ) ( Q (1) l P (2) k F ) ( Q (1) m P (2) l F ) ( Q (1) m P (2) m F ) , where Q m , P m are adapted to [ − m +6 , m +6 ] and Q m vanishes in [ − m − , m − ]. As in the proofof Lemma 10, we note that ϕ m (0) = c is a constant with | c | ≤ m ∈ Z . We write P m = c I + ( P m − c I ) for the projection acting on the second fiber of F . Then this reduces to X ( k,l,m ) ∈ Z : k ≤ l ≪ m M k,l,m + X ( k,l,m ) ∈ Z : k ≤ l ≪ m E k,l,m , where we have set M k,l,m = Z R ( Q (1) k F ) ( Q (1) l P (2) k F ) ( Q (1) m P (2) l F ) ( Q (1) m P (2) m F ) , (4.16) E k,l,m = Z R ( Q (1) k e Q (2) m F ) ( Q (1) l P (2) k F ) ( Q (1) m P (2) l F ) ( Q (1) m P (2) m F ) . (4.17)Here e Q m is associated with a bump function, which is up to a constant multiple, adapted to[ − m +10 , m +10 ]. Moreover, the bump function vanishes at the origin. We have redefined P m bysubsuming c into its definition.First we consider (4.16). We write the sum over ( k, l, m ) ∈ Z with k ≤ l ≪ m as X ( k,l,m ) ∈ Z : k ≤ l ≪ m M k,l,m = (cid:16) X ( k,l,m ) ∈ Z : k ≤ l ≪ m M k,l,m − X ( k,l,m ) ∈ Z : k ≤ l M k,l,m (cid:17) (4.18)+ X ( k,l,m ) ∈ Z : 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 ), applyCauchy-Schwarz in m and H¨older’s inequality in the integration. This yields (cid:12)(cid:12)(cid:12) X ( k,l,m ) ∈ Z : k ≤ l M k,l,m (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) X ( k,l ) ∈ Z : k ≤ l ( Q (1) k F ) ( Q (1) l P (2) k F ) ( Q (1) m P (2) l F ) (cid:13)(cid:13)(cid:13) L p ′ ( ℓ m ) kQ (1) m P (2) m F k L p ( ℓ m ) . (4.20)Since the second term is a square function, it remains to prove L p ( R ) × L p ( R ) × L p ( ℓ ) → L p ′ ( ℓ ) vector-valued estimates for the trilinear operator X ( k,l ) ∈ Z : k ≤ l ( Q (1) k F ) ( Q (1) l P (2) k F ) ( P (2) l F ) . (4.21)These estimates will again follow by freezing the functions F and F and using Marcinkiewicz-Zygmund inequalities, provided we show scalar-valued boundedness of (4.21). To show this, wesplit the summation in (4.21) into regions where k ∼ l , k ≪ l , or k ≫ l . The case k ∼ l is bounded OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 21 by H¨older’s inequality. The remaining two cases follow by Lemmas 9 and 10 above. Summarizing,we obtain estimates for (4.20) when 1 < p , p , p < ∞ , 2 < p < ∞ , 1 /p + 1 /p > / /p + 1 /p > 1, and P i =1 /p i = 1.It remains to estimate (4.18), which splits further into several terms. We fix k ≤ l andthen we consider several cases depending on the size of m relative to k and l . If for any triple( j , j , j ) ∈ { k, l, m } with all three distinct entries it holds j ∼ j and j ∼ j , then the claimfollows by H¨older’s inequality. If l ≫ max( k, m ), then the corresponding term is zero due tofrequency 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 eachparameter. Bounds are deduced by two applications of H¨older’s inequality as follows. Takingthe triangle inquality, enlarging the sum in m to be over the whole Z and applying H¨older’sinequality in l , we obtain (cid:12)(cid:12)(cid:12) X − ≤ s ≤ X l ∈ Z X m ∈ Z : m ≪ l + s M l + s,l,m (cid:12)(cid:12)(cid:12) ≤ X − ≤ s ≤ Z R X m ∈ Z k Q (1) l + s F k ℓ l k Q (1) l P (2) l + s F k ℓ l k Q (1) m P (2) l F k ℓ ∞ l |Q (1) m P (2) m F | By Cauchy-Schwarz in m and H¨older in the integration, we bound the term for a fixed s by k Q (1) l + s F k L p ( ℓ l ) k Q (1) l P (2) l + s F k L p ( ℓ l ) k Q (1) m P (2) l F k L p ( ℓ m ( ℓ ∞ l )) kQ (1) m P (2) m F k L p ( ℓ m ) . (4.22)Each of the terms is bounded in full range. The third term, square-maximal function term isbounded in by the Fefferman-Stein inequality, see [22]. The remaining terms are square functions.Next we consider the case k ≪ l and m ∼ l . We split X ( k,l,m ) ∈ Z : k ≪ l,m ∼ l M k,l,m = X ( k,l,m ) ∈ Z : m ∼ l M k,l,m − X ( k,l,m ) ∈ Z : l − ≤ k ≤ l +100 ,m ∼ l M k,l,m − X ( k,l,m ) ∈ Z : k ≫ l +50 ,m ∼ l M k,l,m . Note that bounds for the second sum follow by H¨older’s inequality, while the third sum is zerodue to frequency supports. Thus, it remains to consider the case m ∼ l . We again split the sumover ( k, l ) ∈ Z and write m = l + s , s ∼ 0. Now we use H¨older’s inequality in l to bound X s ∼ (cid:12)(cid:12)(cid:12) X ( k,l ) ∈ Z M k,l,l + s (cid:12)(cid:12)(cid:12) ≤ X s ∼ (cid:13)(cid:13)(cid:13) X k ∈ Z ( Q (1) k F ) ( Q (1) l P (2) k F ) (cid:13)(cid:13)(cid:13) L r ( ℓ l ) k Q (1) l + s P (2) l F k L p ( ℓ l ) k Q (1) l + s P (2) l + s F k L p ( ℓ ∞ l ) (4.23)where 1 /r + 1 /p + 1 /p = 1. The second and third term are maximal and square func-tions, respectively, while bounds for the first term follow by L p ( R ) × L q ( ℓ ) → L r ( ℓ ) vector-valued estimates for the operator (1.9). Summing in s , we obtain estimates in the range1 < p , p , p , p < ∞ , 1 /p + 1 /p > / 2, and P i =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 ( ℓ ) × L p ( R ) × L p ( ℓ ) → L p ( ℓ ) vector-valued estimates for that operator and inthe display analogous to (4.22) we now have the maximal-square function k Q (1) l + s e Q (2) m F k L p ( ℓ ∞ m ( ℓ l )) , which is also bounded in the full range by the Fefferman-Stein inequality. The analogue for thevector-valued estimates used in (4.23) are now L p ( ℓ ) × L q ( ℓ ) → L r ( ℓ ) vector-valued estimatesfor the operator of the form (1.9). (cid:3) Finally, we are ready to tackle the operators that result after the cone decomposition. Recallthat we need to prove bounds for U and U under the assumptions that for 1 ≤ i ≤ ψ i,k isadapted to [ − k +1 , k +1 ], and for 4 ≤ j ≤ ϕ j,k is adapted to [ − k +1 , k +1 ]. Moreover, each ψ i,k vanishes on [ − k − , k − ] and ϕ j,k (0) = ϕ j, (0).4.1. Completing the proof of Theorem 4. Let φ k be a function adapted in [ − k +1 , k +1 ],which satisfies φ k (0) = φ (0) for each k ∈ Z . We write φ k = c − ( c φ k − e φ k ) + c − e φ k , (4.24)where e φ k is constantly equal to c φ (0) on [ − k − , k − ], and | c | ≤ e φ k is a functionadapted in [ − k +1 , k +1 ]. Note that the function c φ k − e φ k vanishes at the origin.Consider U and U which resulted from the cone decomposition. Writing each of the symbolsof the P -type operators as in (4.24), it suffices to bound the operators under the followingassumptions. For each 1 ≤ i ≤ ψ i,k is adapted in [ − k +1 , k +1 ] and vanishes on [ − k − , k ].For each 4 ≤ j ≤ ϕ j,k is supported in [ − k +1 , 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) on [ − k − , k − ].(2) There is an index 4 ≤ j ≤ 6, such that ϕ j ,k (0) = 0 for each k . Moreover, for j = j , ϕ j,k is constantly equal to ϕ j, (0) on [ − k − , k − ].(3) There is an index 4 ≤ j ≤ 6, such that ϕ j ,k is constantly equal to ϕ j , (0) on[ − k − , k − ]. Moreover, for j = j , 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 and U and we start with U . Our first aim is to reduce considerations to the case when the symbols ψ i,k are supportedin [ − k +3 , k +3 ], vanish on [ − k − , k − ], and are of the form ϕ i,k − ϕ i,k − for a function ϕ i,k supported in [ − k +3 , k +3 ]. The argument that follows is along the lines of an argument used inSection 6 of [14] 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 . Asbefore 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 [ − − . , − . ] and such that φ is constantlyequal to one on [ − − . , − . ]. For a ∈ R define ϑ a and ρ a by ϑ a ( ξ ) = φ (2 − a − ξ ) − φ (2 − a ξ ) ,ρ a ( ξ ) = φ (2 − a − . ξ ) − φ (2 − a − . ) . Note that ϑ a is supported in [ − a +0 . , a +0 . ] and vanishes on [ − a − . , a − . ]. Moreover, ϑ a is constantly equal to one on [2 a − . , a +0 . ] and [ − a +0 . , − a − . ]. Finally, ρ a is supported in[ − a +0 . , a +0 . ] and vanishes on [ − a − . , a − . ]. In particular, we have ϑ a = 1 on the supportof ρ a . Moreover, for k ∈ Z and 1 ≤ i ≤ P l = − ρ k +0 . l = 1 on supp( ψ i,k ) , (4.25) P l = − ρ k +0 . l = 0 on supp( ψ i,k ′ ) if | k ′ − k | ≥ . (4.26)Let n ∈ Z , 0 ≤ s ≤ 9. Note that due to (4.26) and (4.25) we may write for ξ, η ∈ R X k ∈ Z ψ i,k ( ξ ) ϕ j,k ( η ) = X s =0 X n ∈ Z ψ i, n + s ( ξ ) ϕ j, n + s ( η )= X s =0 20 X l = − X n ∈ Z ρ n + s +0 . l ( ξ )Ψ i,s ( ξ ) ϕ j, n + s ( η ) , OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 23 where Ψ i,s = P k ∈ Z ψ i, k + s .Let us now consider U with bump functions satisfying (Q) and (P). Applying the aboveconsiderations in each parameter k, l, m , it suffices to prove bounds for X ( n ,n ,n ) ∈ Z (cid:16) ( P (1) ρ n s . l P (2) ϕ , n s G ,s ) ( P (1) ρ n s . l P (2) ϕ , n s G ,s )( P (1) ρ n s . l P (2) ϕ , n s G ,s ) (cid:17) for fixed 0 ≤ s , s , s ≤ − ≤ l , l , l ≤ 20, where G i,s = P (1)Ψ i,s F i . The Mikhlin-H¨ormander theorem in one variable gives k G i,s k L pi ( R ) = k P (1)Ψ i,s F i k L pi ( R ) . s k F i k L pi ( R ) . Now, for b ∈ R and 0 ≤ s ≤ G i,s,b = P n ∈ Z P (1) ρ n + s + b G i,s . By support considerations, P (1) ϑ k + b G i,s,b = P (1) ρ n + s + b G i,s if k = 10 n + s ∈ Z + s and P (1) ϑ k + b G i,s,b = 0 if k n + s . The Littlewood-Paley inequality in one variable gives k G i,s,b k L pi ( R ) . p i ,s,b k G i,s k L pi ( R ) . Thus, it suffices to bound X ( k ,k ,k ) ∈ Z ( P (1) ϑ k b P (2) ϕ ,k G ,s ,b ) ( P (1) ϑ k b P (2) ϕ ,k G ,s ,b ) ( P (1) ϑ k b P (2) ϕ ,k G ,s ,b ) (4.27)for each fixed b i = 0 . l i , − ≤ l i ≤ 20 and 0 ≤ s i ≤ 9, 1 ≤ i ≤ ϑ b is supported in [ − , ] and vanishes on [ − − , − ]. Moreover, ( ϑ b ) k equals( φ b ) k − ( φ b ) k − , where φ b = φ (2 − b − · ) is supported in [ − , ].Now we are ready to distinguish further cases depending on the form of ϕ j,k . Case (1) of U . Bounds for (4.27) follow from Lemma 8, applied with ψ i,k being a constantmultiple of ϑ k + b i for 1 ≤ i ≤ Case (2) of U . Without loss of generality, we may assume j = 4. Then we dualize andwrite the quadrilinear form in question up to a constant as Z R X ( k,m ) ∈ Z ( Q (1)1 ,k P (2)4 ,m F )( x ) Q (1)1 ,k (cid:16) X l ∈ Z ( Q (1)2 ,l P (2)5 ,k F )( x ) ( Q (1)3 ,m P (2)6 ,l F )( x ) F ( x ) (cid:17) dx, where Q ,k is a constant multiple of a function adapted in [ − , ], which vanishes on [ − − , − ].By Cauchy-Schwarz in k, m , and H¨older’s inequality in the integration, it suffices to bound afiber-wise square function and (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z (cid:12)(cid:12)(cid:12) Q (1)1 ,k X l ∈ Z ( Q (1)2 ,l P (2)5 ,k F )( x ) ( Q (1)3 ,m P (2)6 ,l F )( x ) F ( x ) (cid:12)(cid:12)(cid:12) (cid:17) / (cid:13)(cid:13)(cid:13) L p ′ ( ℓ m ) . Dualizing with a function e F and using Kintchine’s inequality we see that it suffices to proveL p × L p × L p → L p ′ bounds for the operator X ( k,l ) ∈ Z a k ( Q (1)1 ,k e F )( x )( Q (1)2 ,l P (2)5 ,k F )( x ) ( P (2)6 ,l F )( x ) . Now we perform the analogous averaging argument as the one that led to (4.27). Then it sufficesto consider the case when a k = 1 and Q ,k has a symbol supported in [ − , ], which vanisheson [ − − , − ], and is of the form θ k − θ k − for a suitable function θ . The desired bounds thenfollow by Lemmas 9 and 10. Case (3) of U . Without loss of generality, we may assume j = 6. Then we write theoperator in question as X ( k,m ) ∈ Z ( Q (1)1 ,k P (2)4 ,m F )( x ) (cid:16) X l ∈ Z ( Q (1)2 ,l P (2)5 ,k F )( x ) ( Q (1)3 ,m P (2)6 ,l F )( x ) (cid:17) . By the Cauchy-Schwarz inequality in k and m , this reduces to vector-valued estimates for theoperator (1.9) and a fiber-wise square function. We obtain L p × L p × L p → L p ′ estimates withH¨older scaling and in the range 1 < p , p , p , p < ∞ , 1 /p + 1 /p > / Case (4) of U . This case easily follows from three applications of H¨older’s inequality, re-ducing to fiber-wise square functions.It remains to treat U . By the analogous argument as leading to (4.27), it suffices to con-sider the case when for 1 ≤ i ≤ 3, the symbols ψ i,k are supported in [ − k +3 , k +3 ], vanish on[ − k − , k − ], and are of the form ϕ i,k − ϕ i,k − for a function ϕ i,k supported in [ − k +3 , k +3 ].Then we distinguish further cases depending on the form of ϕ j,k . Case (1) of U . Let e Q ,k and e P ,k be associated with e ψ ,k and e ϕ ,k , respectively, where e ψ ,k = ϕ ,k − ϕ ,k − and e ϕ ,k = ϕ ,k − (with ϕ ,k defined in (4.1)). Observe that e ψ ,k is supported in [ − k +3 , k +3 ] andvanishes on [ − k − , k − ]. Moreover, e ϕ ,k is supported in [ − k +4 , k +4 ] and observe that e ϕ ,k − e ϕ ,k − = ϕ ,k − − ϕ ,k − = ψ ,k − is supported in [ − k +4 , k +4 ] and vanishes on [ − k − , k − ].An application of the telescoping identity (4.5) in k yields that U then equals − X ( k,l,m ) ∈ Z ( e Q (1)5 ,k P (2)4 ,m F ) ( Q (1)2 ,l e P (2)1 ,k F ) ( Q (1)3 ,m P (2)6 ,l F ) (4.28)+ X ( l,m ) ∈ Z ( P (2)4 ,m F ) ( Q (1)2 ,l F ) ( Q (1)3 ,m P (2)6 ,l F ) (4.29)Bounds for the first term follow from Lemma 8, applied with ψ ,k being e ψ ,k and ϕ ,k = e ϕ ,k .The desired bounds for the second term follow from Lemmas 9 and 10, and by H¨older’s inequalityused for the portion of the sum when l ∼ m . Case (2) of U . Let j = 6. Performing the analogous steps as in Case (2) of U , we see thatit suffices to show estimates for X ( k,m ) ∈ Z a m ( P (1)5 ,k P (2)4 ,m F ) ( Q (2)1 ,k F ) ( Q (1)3 ,m e F ) , where Q ,m is as in Case (2) of U . By an averaging argument (as leading to (4.27)) it sufficesto consider the case when a k = 1 and Q ,m has a symbol supported in [ − , ], which vanisheson [ − − , − ], and is of the form θ k − θ k − for a suitable function θ . The clam then follows bythe telescoping identity (4.5) in m , Lemmas 9 and 10, and bounds for (1.9). If j = 5, we canproceed as in Case (2) of U . Then we need to show estimates for X ( l,m ) ∈ Z a l ( P (2)4 ,m F ) ( Q (1)2 ,l e F ) ( Q (1)3 ,m P (2)6 ,l F ) . If j = 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-valuedestimates for the operator (1.9). For (4.28), we can proceed as in Case (2) of U , up to obviousmodifications. Cases (3) and (4) of U . These two cases are analogous to Cases (3) and (4) of U . (cid:3) OUNDEDNESS OF SOME MULTI-PARAMETER FIBER-WISE MULTIPLIER OPERATORS 25 References [1] C. Benea and C. 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