BOUNDEDNESS OF MULTILINEAR PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES

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1 BOUNDEDNESS OF MULTILINEAR PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES SHAHLA MOLAHAJLOO, KASSO A. OKOUDJOU, AND GÖTZ E. PFANDER Abstract. Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.. Introduction and motivation Pseudodifferential operators have long been studied in the context of partial differential equations [39, 40, 42, 57, 59, 67, 69]. Among the most investigated topics on such operators are minimal smoothness and decay conditions on their symbols that guarantee their boundedness on function spaces of interest. In recent years, results from time-frequency analysis have been exploited to obtain boundedness results on so-called modulation spaces, which in turn yield boundedness on Bessel potential spaces, Sobolev spaces, and Lebesgue spaces via well established embedding results. In this paper, we develop time-frequency analysis based methods in order to establish boundedness of classes multilinear pseudodifferential operators on products of modulation spaces... Pseudodifferential operators. A pseudodiffrential operator is an operator T σ formally defined through its symbol σ by T σ fx) = σx, ξ) ˆfξ) e 2πix ξ dξ, R d where the Fourier transformation is formally given by Ff) ξ) = fξ) = R e 2πix ξ fx) dx. d Hörmander symbol classes are arguably the most used in investigating pseudodifferential operators. In particular, the class of smooth symbols with bounded derivatives was shown to yield bounded operator on L 2 in the celebrated work of Calderón and Vaillancourt []. More specifically, if σ S0,0 0, that is, for all non-negative integers α, β there exists C α,β with.) α x β ξ σx, ξ) C α,β, then T σ maps L 2 into itself..2. Time-frequency analysis of pseudodifferential operators. In [55], J. Sjöstrand defined a class of bounded operators on L 2 whose symbols do not have to satisfy a differentiability assumption and which contains those operators with symbol in S0,0 0. He proved that this class of symbols forms an algebra under the so-called twisted convolution [30, 34, 55, 56]. Incidentally, symbols of Sjöstrand s class operators are characterized by their membership in the modulation space M,, a space of tempered distributions introduced by Feitchinger via integrability and decay conditions on the distributions short-time Fourier transform [20]. Gröchenig and Heil Date: February, Mathematics Subject Classification. Primary 47G30; Secondary 35S99, 42A45, 42B5, 42B35.

2 2 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER then significantly extended Sjöstrands results by establishing the boundedness of his pseudodifferential operators on all modulation spaces [35]. These and similar results on pseudodifferential operators were recently extended by Molahajloo and Pfander through the introduction of ordered integrability conditions on the short-time Fourier transform of the operators symbols [49]. Similar approaches have been used to derive other boundedness results of pseudodifferential operators on modulation space like spaces [0]. The approach of varying integration orders of short-time Fourier transforms of, here, symbols of multilinear operators lies at the center of this paper. Today, the functional analytical tools developed to analyze pseudodifferential operators on modulation spaces form an integral part of time-frequency analysis. They are used, for example, to model time-varying filters prevalent in signal processing. By now, a robust body of work stemming from this point of view has been developed [8, 35, 36, 37, 54, 60, 6, 63, 66], and has lead to a number of applications to areas such as seismic imaging, and communication theory [47, 58]..3. Multilinear pseudodifferential operators. A multilinear pseudo-differential operator T σ with distributional symbol σ on R m+)d, is formally given by d.2) T σ f) x) = i= ξi) σx, ξ) f ξ ) f 2 ξ 2 )... f m ξ m ) dξ. R md e 2πix Here and in the following we use boldface characters as ξ = ξ,..., ξ m ) to denote products of m vectors ξ i R d, and it will not cause confusion to use the symbol f for both, a vector of m functions or distributions f = f,..., f m ), that is, a vector valued function or distribution on R d, and the rank one tensor f = f... f m, a function or distribution on R md. For example, we write fξ) = f ξ )... f m ξ m ), while fξ) = f ξ),..., f m ξ)). A trivial example of a multilinear operator is given by the constant symbol σ. Clearly, T σ f) is simply the product f x)f 2 x)... f m x). Thus, Hölder s inequality determines boundedness on products of Lebesgue spaces. On the other hand, when the symbol is independent of the space variable x, that is, when σx, ξ) τξ), the T σ = T τ is a multilinear Fourier multipliers. We refer to [2, 3, 7, 32, 48, 50] and the references therein for a small sample of the vast literature on multilinear pseudodiffrential operators. One of the questions that has been repeatedly investigated relates to minimal) conditions on the symbols σ that would guarantee the boundedness of.2) on products of certain function spaces, see [7, Theorem 34]. For example, one can ask if a multilinear version of.) exist. Bényi and Torres [2]) proved that unless additional conditions are added, there exist symbols which satisfy such multilinear estimates but for which the corresponding multilinear pseudodifferential operators are unbounded on products of certain Lebesgue spaces. Indeed, in the bilinear case, that is, when m = 2, the class of operators whose symbols satisfy for all non-negative integers α, β, γ,.3) α x β ξ γ η σx, ξ, η) C α,β,γ contains operators that do not map L 2 L 2 into L. Multilinear pseudodifferential operators in the context of their boundedness on modulation spaces, were first investigated in [6, 7]. Results obtained in this setting have been used to establish well posedness for a number of non-linear PDEs in these spaces [5, 9]. For example, and as opposed to the classical analysis of multilinear pseudodifferential operators, it was proved in [7] that symbols satisfying.3) yield boundedness from L 2 L 2 into the modulation space M,, a space that contains L. The current paper offers some new insights and results in this line of investigation.

3 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 3.4. Our contributions. Modulation spaces are defined by imposing integrability conditions on the short-time Fourier transform of the distribution at hand. Following ideas from Molahajloo and Pfander [49], we impose various ordered integrability conditions on the short-time Fourier transform of a tempered distribution σ on R m+)d which is a symbol of a multilinear pseudodifferential operator. By using this new setting, we establish new boundedness results for multilinear pseudodifferential operators on products of modulations spaces. For example, the following result follows from our main result, Theorem 4.. Theorem.. If p 0, p, p 2, q, q 2, q 3 satisfy p 0 p + p 2 and + q 3 q + q 2, and if for some Schwartz class function ϕ, the symbol short-time Fourier transform V ϕ σx, t, t 2, ξ, ξ 2, ν) = σ x, ξ, ξ 2 )ϕx x)ϕξ ξ )ϕξ 2 ξ 2 )) e 2πixν t ξ t 2 ξ 2 ) d x d ξ d ξ 2 satisfies.4) σ M,,);,,) = sup ξ,ξ 2 sup V ϕ σx, t, t 2, ξ, ξ 2, ν) dt dt 2 dν <, x then the pseudodifferential operator T σ initially defined on SR d ) SR d ) by T σ f, f 2 )x) = e 2πix ξ +ξ 2 ) σx, ξ, ξ 2 ) f ξ ) f 2 ξ 2 ) dξ 2 dξ extends to a bounded bilinear operator from M p,q M p 2,q 2 into M p 0,q 3. Moreover, there exists a constant C > 0 that only depends on d, the p i, and q i with T σ f, f 2 ) M p 0,q 3 C σ M,,);,,) f M p,q f 2 M p 2,q 2. We note that the classical modulation space M, R 3d ) can be continuously embedded into M,,),,,) R 3d ) implicitly defined by.4). Indeed, σ M,,);,,) = sup sup V ϕ σx, t, t 2, ξ, ξ 2, ν) dt dt 2 dν ξ,ξ 2 x sup V ϕ σx, t, t 2, ξ, ξ 2, ν) dt dt 2 dν = σ M ;. x,ξ,ξ 2 As a consequence Theorem. already extends the main result, Theorem 3., in [7]. The herein presented new approach allows us to investigate the boundedness of the bilinear Hilbert transform on products of modulation spaces. Indeed, in the one dimensional setting, d =, it can be shown that the symbol of the bilinear Hilbert transform σ H M,,r);,,) \ M,,);,,) for all r >. Hence, σ H M, and existing methods to investigate multilinear pseudodifferential operators on products of modulations spaces are not applicable. Using the techniques developed below, we obtain novel and wide reaching boundedness results for the bilinear Hilbert transform on the product of modulation spaces. For example, as a special case of our result, we prove that the bilinear Hilbert transform is bounded from L 2 L 2 into the modulation space M +ɛ, for any ɛ > 0. The results established here aim at generality and differ in technique from the ground breaking results about the bilinear Hilbert transformed as obtained by Lacey and Thiele [44, 43, 45, 46]. They are therefore not easily compared to those obtained using hard analysis techniques. Nonetheless, using our results and some embeddings of modulation spaces into Lebesgue space,

4 4 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER we discuss the relation of our results on the boundedness of the bilinear Hilbert transform to the known classical results. The herein given framework is flexible enough to allow an initial investigation of the trilinear Hilbert transform. Here we did not try to optimize our results but just show through some examples how one can tackle this more difficult operator in the context of modulation spaces..5. Outline. We introduce our new class of symbols based on a modification of the shorttime Fourier transform in Section 2. We then prove a number of technical results including some Young-type inequalities, that form the foundation of our main results. Section 3 contains most of the key results needed to establish our results. This naturally leads to our main results concerning the boundedness of multilinear pseudodifferential operators on product of modulation spaces. Section 4 is devoted to applications of our results. In Section 4. we specialize our results to the bilinear case, proving boundedness results of bilinear pseudodifferential operators on products of modulation spaces. We then consider as example the bilinear Hilbert transform in Section 4.2. In Section 4.3 we initiate an investigation of the boundedness of the trilinear Hilbert transform on products of modulation spaces. 2. Symbol classes for multilinear pseudodifferential operators 2.. Background on modulation spaces. Let r = r, r 2,..., r m ) where r i <, i =, 2,..., m. The mixed norm space L r R md ) is Banach space of measurable functions F on R md with finite norm [] ) F L r =... F x,..., x m ) r r2 /r ) rm/rm ) /rm. dx dx2... dxm R d R d R d Similarly, we define L r R md ) where r i = for some indices i. For a nonnegative measurable function w on R md wee define L r wr md ) to be the space all F on R md for which F w is in L r R md ), that is, F L r w = F w L r <. For the purpose of this paper, we define a mixed norm space depending on a permutation that determines the order of integration. For a permutation ρ on {, 2,..., n}, the weighted mixed norm space L r;ρ w R md ) is the set of all measurable functions F on R md for which F L r;ρ =... F x w, x 2,..., x n ) wx, x 2,..., x n ) r ρ) R d R d R d dx ρ) ) rρ2) /r ρ) dx ρ2) ) rρ3) /r ρ2)... dx ρn) ) /rρn) is finite. Let M ν denote modulation by ν R d, namely, M ν fx) = e 2πit ν fx), and let T t be translation by t R d, that is, T t fx) = fx t). The short-time Fourier transform V φ f of f S R d ) with respect to the Gaussian window φx) = e x 2 is given by V φ ft, ν) = F f T t φ ) ν) = f, M ν T t φ) = fx) e 2πixν φx t) dx. The modulation space M p,q R d ), p, q, is a Banach space consisting of those f S R d ) with f M p,q = V φ f L p,q = V φ ft, ν) p dt) q/p dν ) /q <, with usual adjustment of the mixed norm space if p = and/or q =. We refer to [20, 34] for background on modulation spaces.

5 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 5 In the sequel we consider weight functions w on R 2m+)d. We assume that w is continuous and sub-multiplicative, that is, wx + y) Cwx)wy). Associated to w will be a family of w- moderate weight functions v. That is v is positive, continuous and satisfies vx+y) Cwx)vy) A new class of symbols. The commonly used short-time Fourier transform analyzes functions in time ; as symbols have time and frequency variables, we base the herein used shorttime Fourier transform on a Fourier transform that takes Fourier transforms in time variables and inverse Fourier transforms in frequency variables. We then order the variables, first time, then frequency. That is, we follow the idea of symplectic Fourier transforms F s on phase space, F s F t, ν) = F x, ξ) e 2πiξt xν) dξdx. R m+)d For F S R m+)d ) and φ SR m+)d ), we define the symbol short-time Fourier transform V φ F of F with respect to φ by V φ F x, t, ξ, ν) = F s F Tx,ξ) φ ) t, ν) = F, M ν,t) T x,ξ) φ = e 2πi xν t ξ) F x, ξ, )φ x x, ξ ξ) d x d ξ R md R d where x, ν R d, and t, ξ R md. Note that the symbol short-time Fourier transform is related to the ordinary short-time Fourier transform by V φ F x, t, ξ, ν) = V φ F x, ξ, ν, t). Modulation spaces for symbols of multilinear operators are then defined by requiring the symbol short-time Fourier transform of an operator to be in certain weighted L p spaces. To describe these, we fix decay parameters p 0, p,..., p m, q, q 2,..., q m, q m+, and permutations κ on {0,,..., m} and ρ on {,..., m, m + }. The latter indicate the integration order of the time, respectively frequency, variables. Put, p = p, p 2,..., p m ), q = q, q 2,..., q m ) and let w be a weight function on R 2m+)d. Then L p 0,p),κ;q,q m+ ),ρ w R 2m+)d ) is the mixed norm space consisting of those measurable functions F for which the norm F p L 0,p),κ;q,q m+ ),ρ w = R d R d R d R d R d R d wt 0, t,..., t m, ξ,..., ξ m, ξ m+ ) F t 0, t,..., t m, ξ,..., ξ m, ξ m+ ) p κ0) dt κ0) ) pκ) /p κ0) dt κ) ) pκ2) /p κ)... dt κm) ) qρ) /p κm) dξ ρ) ) qρ2) /q ρ)... dξ ρm+) ) /qρm+) is finite. The weighted symbol modulation space M p 0,p),κ;q,q m+ ),ρ w R m+)d ) is composed of those F S R m+)d ) with F M p 0,p),κ;q,q m+ ),ρ w = V φ F L p 0,p),κ;q,q m+ ),ρ w <. When κ and ρ are identity permutations, then we denote L p 0,p),κ;q,q m+ ),ρ w R 2m+)d ) and w R 2m+)d ) and M p 0,p);q,q 0 ) w R 2m+)d ), respectively. M p 0,p),κ;q,q m+ ),ρ w R 2m+)d ) by L p 0,p);q,q 0 ) The dependence of the norm on the choice of κ, ρ, as well as the advantage of choosing a particular order will be discussed in Section 2.4. For clarity, we always refer to the variables x,y,t as time variables, even though a physical interpretation of time necessitates d =. Alternatively, one can consider multivariate x,y,t as spatial variables.

6 6 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER For simplicity of notation, we set Sξ) = m i= ξ i. For functions g and components of f in SR d ), the Rihaczek transform Rf, g) of f and g is defined by R f, g) x, ξ) = e 2πix ξ +...+ξ m) f ξ )... f m ξ m )gx) = e 2πix Sξ) fξ)gx). Multilinear pseudo-differential operators are related to Rihaczek transforms by T σ f, g = σ, Rf, g) a-priori for all functions f i and g in SR d ) and symbols σ SR m+)d ). With x ± t = x ± t,..., t m ) = x ± t,..., x ± t m ), it can be easily seen that where R f, g) x, ξ) = F t ξ f + x)) gx) F t ξ f + x)) ξ) = e 2πit ξ ft + x) dt. R md Lemma 2.. For ϕ real-valued, ϕ = ϕ,..., ϕ), f = f, f 2,..., f m ) SR d ) m, and g SR d ), V TA ϕ ϕ)t A f g)x, ξ, t, ν) = V ϕ f x t, ξ )... V ϕ f m x t m, ξ m ) V ϕ gx, ν Sξ)). Moreover, ) V Rϕ,ϕ) Rf, g) x, ξ, ν, t) = e 2πiξt V TA ϕ ϕ)t A f g) ) x, t, ν, ξ), and in particular, ) V Rϕ,ϕ) Rf, g) x, ξ, ν, t) = V TA ϕ ϕ)t A f g)x, ξ, t, ν) Proof. We compute VTA ϕ ϕ)t A f g) ) x, ξ, t, ν) = e 2πi xν+ tξ) ) T A f g x, t)t A ϕ ϕ) x x, t t) d x d t R md R d ) = e 2πi tξ f x t)ϕ x x t + t) d t e 2πi xν g x)ϕ x x) d x R d R md = fs)g x)e 2πiν x+ξ x s)) ϕs x t))ϕ x x) d x ds R d R { md } { } = e 2πiξs fs)ϕs x t)) ds e 2πiν+Sξ)) x g x)ϕ x x) d x R md R d = V ϕ f) x t, ξ) V ϕ g) x, ν + Sξ)). Further, ) V Rϕ,ϕ) Rf, g) x, ξ, ν, t) = e 2πiν x+t ξ) Rf, g) x, ξ)rϕ, ϕ) x x, ξ ξ) d x d ξ R md R d = e 2πiν x+t ξ) ) F t ξ f x ) g x)f t ξ ξ ϕ x x ))ϕ x x) d x d ξ R md R d ) = f x ) g x)f t ξ ξ ϕ x x ))ϕ x x) d x d ξ. R md R d e 2πiν x+t ξ) F t ξ

7 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 7 On the other hand, by using Parseval identity we have VTA ϕ ϕ)t A f g) ) x, t, ν, ξ) = e 2πi xν+ tξ) ) T A f g x, t)t A ϕ ϕ) x x, t t) d x d t R d R md ) = e 2πi xν g x)ϕ x x) d x = R d R d e 2πi tξ f x t)ϕ x x t + t) d t R md ) f x ) F t ξ R md F t ξ e 2πi tξ ϕ x x + t ) ) e 2πi xν g x)ϕ x x) d ξ d x. But, ) F e 2πi tξ ϕ x x + t ) = e 2πitξ ξ) F t ξ γ ξ ξ ϕ x x )), therefore, VTA ϕ ϕ)t A f g) ) x, t, ν, ξ) = e 2πitξ ) f x ) F t ξ ξ ϕ x x )) g x)ϕ x x) d x d ξ. R md R d e 2πit ξ v x) F t ξ 2.3. Young type results. The following results are consequences of Young s inequality and will be central in proving our main results. We use the convention that summation over the empty set is equal to 0. Lemma 2.2. Suppose that p k, r k for k = 0,,..., m and A) p k r k, k =,..., m; k A2), k = 0,..., m ; p l r l r 0 p k+ l= A3) = ; p l r l r 0 p 0 l= then F x, t) = fx t)gx) satisfies F L r 0,r) g L p 0 f L p. Proof. For simplicity, we use capital letters for the reciprocals of p k, r k, that is, P k = /p k, R k = /r k, k = 0,..., m. Recalling that summation over the empty set is defined as 0, our assumptions A) A3) are simply A) P k R k, k =,..., m; k A2) R 0 P k+ P l R l, k = 0,..., m ; A3) R l = l=0 l= P l. l=0 Define /b = B = R 0 + R P, and for k = 2,..., m, /b k = B k = B k + R k P k k = R 0 + R l P l. l=

8 8 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER The first application of Young s inequality below requires that p /r 0, r /r 0, b /r 0 and /p /r 0 ) + /b /r 0 ) = + /r /r 0 ). This translates to R 0 R, B, P and P + B = R 0 + R which is equivalent to R 0 R, P, R 0 + R P. But, condition A) of the hypothesis implies that P R. Thus we have, R 0 R, P and P R, that is, R 0 P R. Similarly, the successive applications of Young s inequality follow by replacing p, r, b, r 0 by p k, r k, b k, b k, respectively. That is, we require B k R k, B k + R k P k, P k which is equivalent to B k P k R k which follows from A). We shall also use the standard fact that for 0 < α, β, γ, δ <, and set fx) = f x). We compute f α γ L β = f αδ γ δ L β/δ, F rm L r 0,r =... f x t )... f m x t m ) gx) r 0 dx R d R d R d R d =... f t x) T t2 f2 x)... T tm fm x) gx) ) r 0 dx R d R d R d R d =... f ) r r 0 T t2 f2... T tm fm g r ) r 2 ) 0 r0 r rmdtm t ) dt... R d R d R d =... f r r 2 ) r 0 T t2 f2... T tm fm g r 0 r 3 ) r 0 r r2 rmdtm dt 2... R d R d R d L r /r 0... f r 2 r 2 r 0 r 0 T R d R d R d L p /r 0 t 2 f2... T tm fm g r 0 r 0 dt L b /r 0 2 =... f r 2 ) r3 ) r 2 L p R d R d R d T t2 f 2... T tm f m g b b r2 rmdtm dt L 2... = f rm L p... f 2 x t 2 )... f m x t m ) gx) b dx R d R d R d R d... ) rm f rm L p... f m rm L p m f m x t m ) gx) b m b dx m dt m R d R d = f rm L p... f m rm L p m f m b m g b m f rm L p... f m rm L p m f m rm = f rm L p... f m rm L pm g rm L p 0, L g p rm 0 p0 pm L rm b m rm b L m ) r ) r 2 ) r0 r rmdtm dt... ) r ) r 2 ) r0 r rmdtm dt... ) r 3 ) r2 rmdtm... ) r 2 ) r 3 ) b r2 rmdtm dt 2... where each inequality stems from an application of Young s inequality for convolutions. In the final step, we used b m = p 0 which follows by combining the definition of b m with hypothesis A3).

9 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 9 Remark 2.3. Observe that if we would add the condition p 0 r 0 in hypothesis A) of Lemma 2.2, then A) and A3) would combine to imply p k = r k for k = 0,..., m. Indeed, the strength of Lemma 2.2 lies in the fact that p 0 r 0 and p k = r k for k = 0,..., m are not implied by the hypotheses. Setting k = p k r k for k = 0,..., m, A) in Lemma 2.2 is,..., m 0 and condition A3) becomes 0 + m k= k = 0, a condition that allows 0 to be negative, that is p 0 > r 0. In short, all k > 0 contribute to compensate for 0 = r 0 p 0 being negative. Let us now briefly discuss condition A2) in Lemma 2.2. For k = 0, we have 0 r 0 p. To satisfy condition A2) for k =, we increase the left hand side by = p r 0, add to the right hand side the possibly negative term p p 2, and require that the sum on the left remains bounded above by the sum on the right. For k = 2, we increase the left hand side by 2 = p 2 r 2 0 and add to the right hand side p 2 p 3, maintaining that the right hand side dominates the left hand side. This is illustrated in Figure below. In the case m =, the conditions 0 and 0 + = 0 from Lemma 2.2 are amended by the requirement r 0 p, and, for example, if r 0 =, p 0 = 2, then Lemma 2.2 is applicable whenever p = r + 2, that is, if p = 2r r +2. If m = 2, then, 2 0 and = 0 from Lemma 2.2 are combined with the condition r 0 p and r 0 p 2. It is crucial in what follows to observe that these conditions are sensitive to the order of the p k and the r k. For example, the parameters r 0 =, p 0 = 2, r = = p, p 2 =, r 2 = 2 satisfy the hypothesis, while r 0 =, p 0 = 2, r 2 = = p 2, p =, r = 2 do not. Indeed, if for some k, k = p k r k is much smaller than p k p k+, then we would profit more from this if k is a small index, that is, the respective summands play a role early on in the summation. Below, we shall use this idea and reorder the indices. This allows us to first choose κ) = k {,..., d} with κ) = p κ) r κ) small, and then κ2) = k 2 so that p κ) p κ2) large. Clearly, the feasibility of κ2) also depends on the size of κ2) = p κ2) r κ2), so finding an optimal order cannot be achieved with a greedy algorithm. Moreover, note that the spaces M p 0,p),κ;q,q m+ ),σ and M p 0,p),id;q,q m+ ),id are not identical, hence, we cannot choose κ and ρ arbitrarily. is Figure. Depiction of condition A2) in Lemma 2.2. After adding a pair of colored fields, the top row must always exceed the lower row, with the lower row finally catching up in the last step, see Remark 2.3. Remark 2.4. Note that conditions A) and A3) follow from but are not equivalent to) the simpler condition A4) r 0 p r p 2... r m p m r m. Equality A3) can then be satisfied by choosing an appropriate p 0. The inequalities in A) imply that the LHS of A2) is positive and, hence, always r 0 p k r k for all k. Also, A) and A2) necessitate p k r k p m+.

10 0 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER Similarly to Lemma 2.2, we show the following. Lemma 2.5. Suppose that q k, s k for k =,..., m + and B) q k s k, k =,..., m; B2), k =,..., m; q l s l s m+ q k l=k+ B3) = ; q l s l s m+ q m+ l= then for Gt, x) = ft)gx + St)) we have G L s,s m+ f L q g L q m+. Proof. As before, our computations involve the introduction of an auxiliary parameter b k. We start with a formal computation, namely, G s m+ L r,s m+ ) s =... f t )... f m t m ) gx + t t m ) s 2 ) s m+ s dt... dt sm m dx R d R d R d = f m t m ) sm... f 2 t 2 ) s 2 f ) s t )gx t t 2... t m ) s 2 ) s 3 s s2 dt dt 2... R d R d R d R d = f m t m ) sm... f 2 t 2 ) s 2 f s g s x t 2 t 3... t m ) ) ) s2 s 3 ) s 4 s2 s3 s dt 2 dt 3... R d R d R d... = f m sm f m s m... f 2 s 2 f s gx) s ) ) s 2 s 3 ) sm ) s m+ s2 s s... m sm dx = R d f m sm f s m+ m sm sm = f m s m+ L qm f m s m... f 2 s 2 f s g s ) ) s 2 s 3 ) sm s2 s... L qm/sm fm s m f m s m... = f m s m+ L... f qm 2 s m+ L q 2 f s g s s f m s m+ L... f qm 2 s m+ L q 2 f m+ s s... f 2 s 2 f s g s ) ) s 2 s 3 s... f 2 s 2 f s g s ) ) s 2 s 3 ) s m s2 s... s m+ s L b 2 /s L q /s g s s m+ s L q m+ /s = f m s m+ L... f qm s m+ L q g s m+ L q m+. To justify the first application of Young s inequality, we require q m + b m = + s m s m Using reciprocals, this is equivalent to that is, s m+, s m sm s m+ L sm ) sm s m+ s2 s... m L bm/sm s m+ s m 2 s m L bm/s m s m s m+ q m s m, b m s m, s m+ s m. Q m + B m = S m + S m+, S m Q m, B m, S m+, B m = S m Q m + S m+, S m Q m, B m, S m+. ) s m+ sm dx ) s m+ sm dx

11 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS The subsequent application of Young s inequality requires q m + b m = + s m s m Using reciprocals, this is equivalent to b m, s m B m = S m Q m + B m = S m+ + In general, for k =,..., m 2, we require B m k = S m k Q m k + B m k+ = S m+ + l=m l=m k q m s m, b m s m, b m s m. S l Q l, S m Q m, B m, B m. S l Q l, S m k Q m k, B m k, B m k+, and finally, for the last application of Young s inequality, we require Q m+ = S Q + B 2 = S m+ + S l Q l, S Q, Q m+, B 2. Now, S k Q k for k =,..., m implies l=m k 0 S m+ B m B m... B 3 B 2 Q m+, hence, it suffices to postulate aside of S k Q k for k =,..., m the conditions S k B k for k = 2,..., m and S Q m+, B 2. For k = 2,..., m, we use that m+ l= S l Q l = 0 implies m l=k S l Q l = S m+ + Q m+ k l= S l Q l in order to rewrite S k B k in form of k S k B k = S m+ + S l Q l = Q m+ S l Q l which is l=k k Q m+ S k S l Q l. For k =, the above covers the condition Q m+ S. In summary, for k =,..., m + we obtained the sufficient conditions B ) q k s k, k =,..., m B2 ) B3 ) k l= l= l=, k = 0,..., m ; s l q l q m+ s k+ s l q l = q m+ s m+. Forming the difference of B3 ) and B2 ) gives B2 ), k = 0,..., m. s l q l s k+ s m+ l=k+ Reindexing leads to B2 ), k =,..., m, s l q l s k s m+ l=k and adding q k s k l= to both sides, and then multiplying both sides by - gives

12 2 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER B2) l=k+, k =,..., m. q l s l s m+ q k Remark 2.6. The conditions B) B3) are similar to those in A) A3). Indeed, a change of variable k m + k, that is, renaming q k = q m+ k and s k = s m+ k, k =,..., m +, turns B2) into l=k+ q m+ l s m+ l s m+ m+) q m+ k = s 0 We have m k = q m+ l s m+ l q l=k+ l l = hence, we obtain for k = m k the conditions k l = q l s l q m+ k, k =,..., m. s, l s, k = 0,..., m. 0 q k + We conclude that difference between the conditions in Lemma 2.2 and in Lemma 2.5 lies aside of naming the decay parameters simply in replacing in A) and A2) by in B) and B2). Hence, it comes to no surprise that B) and B2) follow from, but are not equivalent to B4) s q s 2... q m s m q m s m+. Moreover, B) implies l=k+ q l s l 0, and, hence, q m+ q k for k =,..., m Young type results with permutations. As observed in Remark 2.3, condition A2) in Lemma 2.2 and, similarly, B2) in Lemma 2.5 are sensitive to the order of the p k, r k, q k, and s k. To obtain a bound for operators as desired, we may have to reorder the parameters. This motivates the introduction of permutations κ and ρ. In addition to the flexibility obtained at cost of notational complexity, we observe that the permutation of the integration order will allow us to pull out integration with respect to some variables. In fact, setting t 0 = x and choosing j = κ 0), we arrive at F r κm) L r 0,r);κ =... R d R d =... R d R d R d R d f t 0 t )... f m t 0 t m ) gt 0 ) r κ0) dt κ0) R d m ) r κ) ) r κ2) f κl) t 0 t κl) )gt 0 ) r r κ0) dt κ0) r κ0) dt κ) κ)... R d l=0 = f κ0) r κm) L p f κ0) κ) r κm) L p... f κ) κj ) r κm) L p κj )... f κj+) x t κj+) )... f κm) x t κm) ) gx) r κj) dx ) r κ) ) r κ2) ) rκm) r κ0) r dt κ) κ)... dt κm) ) r κj+) r κj) We can then apply Lemma 2.2 to the iterated integral on the right hand side. This observation leads us to the following result. ) rκm) dt κm) ) r κj+2) ) rκm) r dt κj+) κj+)... dt κm).

13 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 3 Lemma 2.7. Let κ be a permutation on {0,,..., m}, z = κ 0), and let p k, r k, k = 0,,..., m, satisfy A0) p κl) = r κl), l = 0,..., z ; A) p κl) r κl), l = z,..., m; A2) A3) k l=z+ l=z+, k = z,..., m ; p κl) r κl) r 0 p κk+) =. p κl) r κl) r 0 p 0 Then for F x, t) = fx t)gx) it holds F L r 0,r)κ g L p 0 f L p. Remark 2.8. Loosely speaking, the decay of a function F x, t,..., t d ) in the variables x, t,..., t d ) is given by the parameters p 0, p,..., p d ), that is, L p 0 -decay in x, L p -decay in t,..., L p d-decay in t d. As we then use the flexibility of order of integration, it is worth noting that Minkowski s inequality for integrals implies that integrating with respect to variables with large exponents last, increases the size of the space. For example, if q p, we have F L p,q);0,) = F x, t ) p dx) q/pdt ) /q F x, t ) q dx) p/qdt ) /p = F L p,q);,0), which implies L p,q);0,) L p,q);,0) if q p, for example, L, );0,) L, );,0). This inclusion is strict in general, for example, choose F x, t ) = gx t ) L, );,0) \ L, );0,) for any function g L. Similarly to Lemma 2.7, we formulate the following. Lemma 2.9. Let ρ be a permutation on {,..., m + }, w = ρ m + ), and q k, s k be k =,..., m + satisfy B0) q κl) = s κl), l = w,..., m; B) q ρk) s ρk), k =,..., w ; B2) B3) w l=k+ w l=, k =,..., w q ρl) s ρl) s m+ q ρk) =. q ρl) s ρl) s m+ q m+ Then Gξ, ν) = fξ)gν + Sξ)) satisfies G L s,s m+ ),ρ f L q g L q m+. 3. Boundedness on modulation spaces When applying Lemmas 2.2, 2.5, 2.7, and 2.9 in the context of modulation spaces, we can use the property that M p,q embeds continuously in M p 2,q 2 if p p 2 and q q 2. To exploit this in full, the introduction of auxiliary parameters p and s is required as illustrated by Example 3.2 below. Proposition 3.. Given p 0, p, p, q, q, q m+, r 0, r, s, s m+ with p p r and s, q q. Let κ be a permutation on {0,..., m} and let z = κ 0). Similarly, let ρ be a permutation on {, 2,..., m + } and w = ρ m + ). Assume

14 4 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER ) 2) 3) 4) k l=z+ l=z+ w l=k+ w l= p κl) r κl) r 0 ; p κl) r κl) r 0 p 0 p κk+), k = z,..., m ; q, k =,..., w ; q ρl) s ρl) s m+ k q ρl) s ρl) s m+ q m+. Let v be a weight function on R 2m+)d and assume that w 0, w,..., w m are weights on R 2d such that 3.) vx, t, ξ, ν) w 0 x, ν + Sξ))w x t, ξ )... w m x t m, ξ m ). For ϕ SR d ) real valued, f Mw p,κ;q,ρ R md ), and g M p 0,q m+ R d ), we have V TA ϕ ϕ)t A f g) L r 0,r) κ,s,s m+ ) ρ v R 2m+)d ) with 3.2) V TA ϕ ϕ)t A f g) L r 0,r),κ;s,s m+ ),ρ v w 0 C f M p,q w where the LHS is defined by integrating the variables in the index order In particular, κ0), κ),..., κm), ρ),..., ρm), ρm + ). T A f g) M r 0,r)κ,s,s m+ )ρ v C f M p,q w Note that C depends only on the parameters p i, r i, q i, s i and d.... f m M pm,qm g p wm M 0,q m+, w 0... f m M pm,qm g p wm M 0,q m+. w 0 Proof. For simplicity we assume ρ = κ = id and use Lemma 2.2 and Lemma 2.5. The general case follows as Lemma 2.7 and Lemma 2.9 followed from Lemma 2.2 and Lemma 2.5. Let f = f, f 2,..., f m ), φ = φ, φ,..., φ) and w = w.w w m ). Then and by Lemma 2., we have V φ f = V φ f V φ f 2 V φ f m, V TA ϕ ϕ)t A f g)x, ξ, t, ν) = V ϕ fx t, ξ)v ϕ gx, ν Sξ)), where x, ν R d, and t, ξ R md. So, if 3.) and conditions 2) and 4) above hold with equality, then A) A3) in Lemma 2.2 and B) B3) in Lemma 2.5 will hold. Then vx, t, ξ, ν)v TA ϕ ϕ)t A f g)x, t, ξ, ν) L r 0,r),s,s m+ ) x,t,ξ,ν) wx t, ξ) V ϕ f) x t, ξ)w 0 x, ν + Sξ)) V ϕ g) x, ν + Sξ)) L r 0,r x,t) L s,s m+ ξ,ν) wt, ξ) V ϕ f) t, ξ) L p t) w 0 x, ν + Sξ)) V ϕ g) x, ν + Sξ)) L p L 0 x) s,s m+ ξ,ν) wt, ξ) V ϕ f) t, ξ) L p t) L q ξ) w0 x, ν + Sξ)) V ϕ g) x, ν + Sξ)) L p L 0 x) q m+ ν) = V ϕ f L p,q w V ϕg L p 0 q m+ w 0.

15 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 5 We now use that p p and q q implies f M p, q f M p,q, a property that clearly carries through to the class of weighted modulation spaces considered in this paper. If hypotheses 2) and 4) hold with strict inequalities, then we can increase p 0 to appropriate p 0 and q m+ to appropriate q m+ so that 2) and 4) will hold with equalities. The resulting inequalities involving p 0 and q m+ then again implies the weaker inequalities involving p 0 and q m+. Example 3.2. The conditions r 0 p k r k, k =,..., m, and m l= /p l /r l /r 0 /p 0 do not guarantee the existence of a permutation κ so that also k l= /p κl) /r κl) /r 0 /p κk+), k = 0,..., m. Indeed, consider for m = 2 the case r 0 =, p = p 2 = 0/9, r = r 2 = 2, and p 0 = 5. It is easy to see that no κ exist that allows us to apply Proposition 3. to obtain for these parameters 3.2). Using r 0 =, p = p 2 = 0/9, r = r 2 = 2, r = r 2 = 2, we can choose κ0) =, κ) = 0, κ2) = 2, to obtain 3.2) for p 0 5/3. Unfortunately, this is again not the best we can do. In fact, we can replace p 2 by p 2 = 5/9 [0/9, 8/9] = [p 2, r 2 ]. This choice allows us to choose for κ the identity which leads to sufficiency for p 0 2, which by inclusion also gives boundedness with r 0 =, p = p 2 = 0/9, r = r 2 = 2, r = r 2 = 2. Remark 3.3. Observe those k with r 0 > r k must satisfy κ k) < z; possibly there are also k with r 0 r k and κ k) < z. Importantly, only those k with p k < r k and κ k) > z contribute to filling the gap between p 0 and r 0, see Remark 2.3 As immediate consequence, we obtain our first main result. Theorem 3.4. Given p 0, p, p, q, q, q m+, r 0, r, s, s m+ with p p r and q, s q. Let κ be a permutation on {0,..., m} and let z = κ 0). Similarly, let ρ be a permutation on {, 2,..., m + } and w = ρ m + ) and ) + + r 0 p κk+) 2) r 0 + 3) 4) l=z+ + + s m+ q ρk) k l=z+ + k z+, k = z,..., m ; p κl) r κl) + m z + ; p κl) r κl) p 0 w l=k+ + w k, k =,..., w ; q ρl) s ρl) w + + w +. s m+ q l= ρl) s ρl) q m+ Let v be a weight function on R 2m+)d and assume that w 0, w,..., w m are weights on R 2d such that vx, t, ξ, ν) w 0 x, ν + Sξ)) w x t, ξ )... w m x t m, ξ m ). Assume that σ M r 0,r),κ;s,s m+ ),ρ v. Then the multilinear pseudodifferential operator T σ defined initially for f k SR d ) for k =, 2,..., m by.2) extends to a bounded multilinear operator from M p,q w M p 2,q 2 w 2... Mw pm,qm m into M p 0,q m+ w 0. Moreover, there exists a constant C so that for all f, we have T σ f M p 0 q m+ w 0 C σ r M 0,r),κ;s,s m+ ),ρ f p M,q v w... f m M pm,qm. wm

16 6 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER Proof. Let f k M p k,q k w k, k =,... m, ϕ SR d ), and denote ϕ = ϕ,..., ϕ). Note that sup{, g, g M p 0,q m+ /w 0 } defines a norm which is equivalent to M p 0 q m+ w 0 for p 0, q m+ [, ] see, for example, [65, Proposition.23)]). Hence, to complete our result on the basis of Lemma 2., we estimate for g M p 0,q m+ /w 0 as follows T σ f, g = σ, Rf, g) = V Rϕ,ϕ) σ, V Rϕ,ϕ) Rf, g) σ M r 0,r),κ;s,s m+ ),ρ v Rf, g) M r 0,r )κ,s,s. m+ )ρ Using the conjugate indices r 0, r κ, s m+, s ρ, it is easy to see that the conditions on the indices ) 4) are equivalent to those in Proposition 3.. Therefore, Rf, g) M r 0,r )κ,s,s m+ )ρ /w C f M p,q w /w... f m M pm,qm g. wm M p 0,q m+ w 0 Note that the criteria on time and frequency are separated. Even when it comes to order of integration, we do not link these, that is, the permutations κ and ρ are not necessarily identical. Corollary 3.5. If and r 0 p r p 2... r m p m r m ; s q s 2 q 2... q m s m q m s m+ ; r 0 + s m+ + l= l= p l + r l m + p 0 ; + m + ; q l s l q m+ then the conclusion of Theorem 3.4 for any symbol σ M r 0,r) κ,s,s 0 ) ρ v, where κ, ρ are the identity permutations. Proof. Note that since κ, ρ are the identity permutations, then z = 0 and ω = m +. ) + k + + k +, k = 0,..., m ; r 0 p k+ p l r l l= 2) m k +, k =,..., m, s m+ q k q l s l l=k+ follow from the monotonicity conditions. 4. Applications In Section 4. we simplify the conditions of Theorem 3.4 in case of bilinear operators, that is, m = 2. The focus of Section 4.2 lies on establishing boundedness of the bilinear Hilbert transform on products of modulation spaces. We stress that these results are beyond the reach of existing methods of time-frequency analysis of bilinear pseudodifferential operators as developed in [7, 6, 8, 9]. Finally, in Section 4.3 we consider the trilinear Hilbert transform.

17 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS Bilinear pseudodifferential operators. A bilinear pseudodifferential operator with symbol σ is formally defined by 4.) T σ f, g)x) = σx, ξ, ξ 2 ) ˆfξ )ĝξ 2 ) dξ dξ 2. R R For m = 2, Theorem 3.4 simplifies to the following. Theorem 4.. Let p 0, p, p 2, q, q 2, q 3, r 0, r, r 2, s, s 2, s 3. If and one of the following ) 2) 3) 4) 5) /p + /r, /p 2 + /r 2 p 0 r 0, using κ =, 2, 0) or 2,, 0)); + p 0 r 0 + r + p, r p 0, r 0, κ = 2, 0, )); + p 0 r 0 + r 2 + p 2, r 2 p 0, r 0, κ =, 0, 2)); p 0 r 0 r r 2 max{p, r 0 } +, r 2 p 0, r, r 2 r 0, κ = 0,, 2)); p p 0 r 0 + r + r 2 + as well as one of ) 2) 3) 4) 5) max{p 2, r 0 } + p, r p 0, r, r 2 r 0, κ = 0, 2, )); q 3 s 3, using ρ = 3,, 2) or 3, 2, )); + q 3 q + s + s 3, s 3 q 3, s, q, ρ =, 3, 2)); + q 3 q 2 + s 2 + s 3, s 3 q 3, s 2, q 2, ρ = 2, 3, )); q q 3 max{q, s } + max{q 2, s 2 } + +, s 3 q s 2 s 2, s 2, 3 max{q, s } + max{q 2, s 2 } + + +, ρ =, 2, 3)); s s 2 s 3 max{q, s } + max{q 2, s 2 } + +, s 3 q s s, s, 3 max{q, s } + max{q 2, s 2 } + + +, ρ =, 3, 2)), s s 2 s 3 hold. Assume that w 0, w, w 2, and v are weight functions satisfying vx, t, t 2, ξ, ξ 2, ν) w 0 x, ν + ξ + ξ 2 ) w x t, ξ ) w 2 x t 2, ξ 2 ). If σ M r 0,r,r 2 ),κ;s,s 2,s 3 ),ρ, the bilinear pseudodifferential operator T σ initially defined on SR d ) SR d ) by 4.) extends to a bounded bilinear operator from M p,q w M p 2,q 2 w 2 into M p 0,q 3 w 0. Moreover, there exists a constant C > 0, such that we have T σ f, f 2 ) M p 0,q 3 w 0 C σ M r 0,r,r 2 ),κ;s,s 2,s 3 ),ρ f M p,q w f 2 M p 2,q 2 w 2 with appropriately chosen order of integration κ, ρ.

18 8 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER Proof. This result is derived from Theorem 3.4 for m = 2, namely, we establish conditions on the p 0, p, p 2, r 0, r, r 2, q, q 2, q 3, s, s 2, s 3 for the existence of p, p 2, q, q 2 satisfying the conditions of Theorem 3.4. If κ = 2 0) or κ = 2 0), then z = 2 in Theorem 3.4 and we require in addition only r 0 p 0. For the remaining cases, we have to show that the conditions above imply the existence of p p, p 2 p 2 which allow for the application of Theorem 3.4. If κ = 0 2), we have z =, and we seek, with notation as before, P and P 2 with that is, 4.2) P P ; P2 P 2 ; P + R ; P2 + R 2 ; R 0 + P 2 ; R 0 + P 2 + R 2 + P 0 ; R P P ; R 0 R 2 + P 0, R 2 P 2 P 2, R 0 ; which defines a non empty set if and only if P + R, P 2 + R 2, R 2 P 0, R 0, + P 0 R 0 + R + P 2. For κ = 0 2) we have z = 0 in Theorem 3.4 and we require that some P and P 2 satisfy P P ; P2 P 2 ; P + R ; P2 + R 2 ; that is, 4.3) 4.4) 4.5) R 0 + P 2 ; R 0 + P 2 + P + R 2; R 0 + P 2 + R 2 + P + R 2 + P 0 ; R P P, R 0 ; R 2 P 2 P 2 ; 2 + P 0 R 0 R R 2 P + P 2 2 R 0 R. Note that 4.3) defines a vertical strip in the P, P 2 ) plane which is non-empty if and only if P + R and R 0 R. Similarly, 4.4) defines a horizontal strip which is not empty if we assume P 2 + R 2. Lastly, the diagonal strip given by 4.5) is nonempty if and only if P 0 R 2. To obtain a boundedness result, we still need to establish that the diagonal strip meats the rectangle given by the intersection of horizontal and vertical strips. This is the case if the upper right hand corner of the rectangle is above the lower diagonal given by P + P 2 = 2 + P 0 R 0 R R 2, that is, if min{p, R 0 } + P P 0 R 0 R R 2, and if the lower left corner of the rectangle lies below the upper diagonal, that is, if R + R 2 2 R 0 R,

19 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 9 which holds if R 0 R 2. Let us now turn to the frequency side. If ρ = 3 2 ) or ρ = 3 2), we have w = and an application Theorem 3.4 requires the single but strong assumption Q 3 S 3. For ρ = 3 2) we have w = 2 in Theorem 3.4. To satifiy the conditions, we need to establish the existence of Q and Q 2 satisfy Q Q ; Q2 Q 2 ; Q + S ; Q2 + S 2 ; S 3 + Q ; S 3 + Q + S + Q 3. The existence of such Q 2 is trivial, so we are left with + Q 3 S S 3, S 3 Q Q, S,. Note that this inequality is exactly 4.2) with S 3 replacing R 2, S replacing R 0 Q 3 replacing P 0, and Q, Q in place of P 2, P 2. We conclude that for the existence of Q, we require S 3 S, Q 3, Q, and + Q 3 Q + S + S 3. For ρ = 2 3) we have w = 3 in Theorem 3.4. We need to establish the existence of Q and Q 2 satisfy that is, choosing Q Q ; Q2 Q 2 ; Q + S ; Q2 + S 2 ; S 3 + Q 2 ; S 3 + Q 2 + Q + S 2 2; S 3 + Q + Q 2 + S 2 + S 2 + Q 3 ; Q = min{q, S }, and Q2 = min{q 2, S 2 }, we require min{q 2, S 2 } + S 3 ; 2 min{q, S } + min{q 2, S 2 } + S 2 + S 3 ; 2 + Q 3 min{q, S } + min{q 2, S 2 } + S + S 2 + S 3. Proof. Proof of Theoremm. Theorem. now follows from choosing κ and ρ to be the identity permutations, and r 0 = s = s 2 =, r = r 2 = s 3 =. Note that this result covers and extends Theorem 3. in [7]. Remark 4.2. Using Remark 2.8, we observe that M, R 3d ) M,,),,,) R 3d ). Indeed, in both cases we have the same decay parameters, but different integration orders, namely M, x, ξ, ξ 2, ν, t, t 2 ; M,,),,,) x, t, t 2, ξ, ξ 2, ν. Inclusion follows from the fact that we always moved a large exponent to the right of a small exponent. Note that for any r M, R) \ M, R), for example, a chirped signal rξ) = e 2πiξ2 uξ) with uξ) L 2 \ L, we have σx, ξ, ξ 2 ) = rξ ) M,,),,,) \ M,.

20 20 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER Example 4.3. With κ and ρ are the identity, that is, κ = 0,, 2) and ρ =, 2, 3), we illustrate the applicability of Theorem 4. for maps on L 2 L 2 = M 2,2 M 2,2, that is, p = p 2 = q = q 2 = 2. On the time side, we require r, r 2 2, r 0 and r 2 p 0 as well as p 0 r 0 r r 2 max{2, r 0 }. Our goal is to obtain results for r 0 large, hence, we assume r 0 2. In case of r 0 2, the last inequality above does not depend on r 0, and we can improve the result by fixing r 0 = 2.) We obtain the range of applicability r, r 2 2 r 0, and r 2 p 0, and + p 0 r 0 + r + r 2. On the frequency side, we have to satisfy the conditions s 3 2, s 2, q 3 max{2, s } + max{2, s 2 } + +, s 2 s 3 max{2, s } + max{2, s 2 } s s 2 s 3 Let us assume s 2 s 2, then we have the range of applicability s, s 3 2 s 2, 2 + s, 2 + q 3 s 2 + s 3. The range of applicability gives exponents that guarantee that a bilinear pseudodifferential operator maps boundedly L 2 L 2 into M p 0,q 3 if σ M r 0,r,r 2 ),s,s 2,s 3 ). In particular, when σ M,,),2,2,) we can take p 0 = q 3 =. So we get that T σ maps L 2 L 2 into M, M, The bilinear Hilbert transform. We now consider boundedness properties of the bilinear Hilbert transform on modulation spaces. Recall that this operator is defined for f, g SR) by BHf, g)x) = lim fx + y)gx y) ɛ 0 y >ɛ dy y. Equivalently, this operator can be written as a Fourier multiplier, that is, a bilinear pseudodifferential operator whose symbol is independent of the space variable, with symbol σ BH x, ξ, ξ 2 ) = σξ ξ 2 ), where σx) = πisign x), x 0. Our first goal is to identify which of the unweighted) spaces M r 0,r,r 2 ),κ;s,s 2,s 0 ),ρ the symbol σ BH belongs to. To this end consider the window function Ψx, ξ, ξ 2 ) = ψx)ψξ 2 )ψξ ξ 2 ), where ψ SR) such that ψx) = ψ x) ψ x) with ψ SR), 0 ψ x) for all x R. In addition, we require that the support of ψ is strictly included in 0, ). Then V Ψ σ BH x, t, t 2, ξ, ξ 2, ν) = V ψ x, ν) V ψ σξ ξ 2, t ) V ψ ξ 2, t + t 2 ) Assume that the two permutations κ of {0,, 2}, and ρ of {, 2, 3} are identities. Moreover, suppose that all the weights are identically equal to. Proposition 4.4. For r >, we have that σ BH M,,r),,,). Proof. Let r >. We shall integrate V Ψ σ BH x, t, ξ, ν) = V ψ x, ν) V ψ σξ ξ 2, t ) V ψ ξ 2, t + t 2 )

21 in the order SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 2 x r 0 = t r = t 2 r 2 = r > ξ s = ξ 2 s 2 = ν s 0 =. We estimate σ BH M,,),,,r) = = sup R ξ,ξ 2 sup R ξ,ξ 2 = ˆψ L sup ξ,ξ 2 R R R R R ) r ) /r sup V Ψ σ BH x, t, ξ, ν) dt dt 2 dν x ) r ) /r sup V ψ x, ν) V ψ σξ ξ 2, t ) V ψ ξ 2, t + t 2 ) dt dt 2 dν x ) r ) /r V ψ σξ ξ 2, t ) V ψ ξ 2, t + t 2 ) dt dt 2 R ˆψ L sup ξ,ξ 2 V ψ σξ ξ 2, ) V ψ ξ 2, ) L r ˆψ L sup ξ,ξ 2 V ψ σξ ξ 2, ) L r V ψ ξ 2, ) L = ˆψ 2 L sup ξ V ψ σξ ) L r, where we have repeatedly used the fact that V ψ x, ν) = e 2πixν ˆψν), and Vψ L x)l ν), that is sup V ψ x, ν) dν = ˆψ L <. x Thus, we are left to estimate sup V ψ σξ ) L r. ξ Recall that ψx) = ψ x) ψ x), hence, we have R V ψ σξ, t) = e 2πiξt [ ξ e 2πiyt ψy)dy + ξ ] e 2πity ψy)dy. A series of straightforward calculations yields ˆψ t) ˆψ t) if ξ V ψ σξ, t) = ˆψ t) χ [0, ξ] ˆψ t) + χ [ ξ,] ˆψ t) if ξ 0 ˆψ t) χ [ξ,] ˆψ t) + χ [0,ξ] ˆψ t) if 0 ξ, where χ [a,b] denotes the characteristic function of [a, b]. We note that that χ [0, ξ], χ [ ξ,], χ [ξ,], χ [0,ξ] L r uniformly for ξ for each r >. For ξ, we have V ψ σξ, ) L q 2 ˆψ L q for any q. Now consider ξ 0, then V ψ σξ, ) L r ˆψ L r + χ [0, ξ] ˆψ L r + χ [ ξ,] ˆψ L r ˆψ L r + ˆψ L χ [0, ξ] L r + χ [ ξ,] L r) ˆψ L r + C ˆψ L where C > 0 is a constant that depends only on r. Using a similar estimate for 0 ξ, we conclude that sup V ψ σξ, ) L r C < ξ

22 22 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER where C depends only on ψ and r. Observe that σ BH M,,r),,,) R 3 ) \ M,,),,,) R 3 ) for all r >. Consequently, to obtain a boundedness result for the bilinear Hilbert transform, we cannot apply any of the existing results on bilinear pseudodifferential operators. However, using the symbol classes introduced we obtain the following result: Theorem 4.5. Let p 0, p, p 2, q, q 2, q 3 satisfy p + p 2 > p 0 and that q + q 2 + q 3. Then the bilinear Hilbert transform extends to a bounded bilinear operator from M p,q M p 2,q 2 into M p 0,q 3. Moreover, there exists a constant C > 0 such that BHf, g) M p 0,q 3 C f M p,q g M p 2,q 2. In particular, for any p, q, and ɛ > 0, the BH continuously maps M p,q M p,q M +ɛ, and we have BHf, g) M +ɛ, C f M p,q g M p,q. into Proof. Since the symbol σ BH of BH satisfies σ BH M,,r),,,), the proof follows from Theorem 4.. Indeed, on the time side, all simple inequalities hold and we are left to check which is with r = ɛ On the frequency side, the conditions q 3 are clearly satisfied whenever 2 + p 0 r 0 + r + r p ɛ + max{p, r 0 } + p 2, max{p, } + p 2. max{q, s } + max{q 2, s 2 } + +, s 3 q s 2 s 2, s 2, 3 max{q, s } + max{q 2, s 2 } + + +, s s 2 s q 3 max{q, } + max{q 2, } Remark 4.6. It was proved in [44, 45] that the bilinear Hilbert transform BH continuously maps L p L p 2 into L p where p = p + p 2, p, p 2 and 2/3 < p. Our results give that if < p, q, p < then H maps continuously M p,q M p,q into M p,. One can use embeddings between modulation spaces and Lebesgue spaces to get some mixed boundedness results. For example, assume that q 2 and q p q, then it is known that see [60, Proposition.7]) L p M p,q and M p,q L p. Consequently, it follows from Theorem 4.5 that BH continuously maps L p M p,q into M p, L p.

23 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS The trilinear Hilbert transform. In this final section we consider the trilinear Hilbert transform TH given formally by THf, g, h)x) = lim fx t)gx + t)hx + 2t) ɛ 0 t >ɛ dt t. The trilinear Hilbert transform can be written as a trilinear pseudodifferential operator, or more specifically as a trilinear Fourier multiplier given by THf, g, h)x) = R R R σ TH x, ξ, ξ 2, ξ 3 ) ˆfξ )ĝξ 2 )ĥξ 3)e 2πixξ +ξ 2 +ξ 3 ) dξ dξ 2 dξ 3 where σ TH x, ξ, ξ 2, ξ 3 ) = σξ ξ 2 2ξ 3 ) = πisignξ ξ 2 2ξ 3 ). Recall from Section 4.2 that ψ SR) is chosen such that ψx) = ψ x) ψ x) with ψ SR), 0 ψ x) for all x R. Next we define Ψx, ξ, ξ 2, ξ 3 ) = ψx)ψξ 2 )ψξ 3 )ψξ ξ 2 2ξ 3 ). We can now compute the symbol window Fourier transform V Ψ σ TH of σ TH with respect to Ψ and obtain V Ψ σ TH x, t, ξ, ν) = V ψ x, ν)v ψ ξ 2, t t 2 )V ψ ξ 2, 2t t 3 )V ψ σξ ξ 2 2ξ 3, t ). Observe that V g x, η) = ĝη). Hence, V Ψ σ TH x, t, ξ, ν) = ˆψν) ˆψ t t 2 ) ˆψ 2t t 3 ) V ψ σξ ξ 2 2ξ 3, t ). But by the choice of ψ we see that ˆψ η) = ˆψη). Proposition 4.7. For r >, we have σ TH M,,r,r),,,,). In particular, this conclusion holds when r = + ɛ for all ɛ > 0. Proof. Let r >. We proceed as in the proof of Proposition 4.4, and integrate V Ψ σ TH x, t, ξ, ν) in the following order: x r 0 =, t r =, t 2 r 2 = r >, t 3 r 3 = r >, ξ s =, ξ 2 s 2 =, ξ 3 s 3 =, ν s 0 =.

24 24 S. MOLAHAJLOO, K. A. OKOUDJOU, G. E. PFANDER In particular, we estimate σ TH M,,r,r),,,,) = = R R dν dν sup ξ,ξ 2,ξ 3 sup ξ,ξ 2,ξ 3 dt 3 dt 2 R R R dt 3 dt 2 R R R ) /r dt sup V Ψ σ H x, t, ξ, ν) r x dt sup ˆψν) r ˆψ t t 2 ) r x ˆψ 2t t 3 ) r V ψ σξ ξ 2 2ξ 3, t ) r ) /r = ˆψ sup ξ,ξ 2,ξ 3 dt 3 dt 2 dt ˆψ t t 2 ) r ˆψ 2t t 3 ) r R R R V ψ σξ ξ 2 2ξ 3, t ) r ) /r = ˆψ sup ξ,ξ 2,ξ 3 dt 3 dt 2 dt ˆψt 2 + t ) r ˆψt 3 + 2t ) r R R R V ψ σξ ξ 2 2ξ 3, t ) r ) /r = ˆψ sup ξ,ξ 2,ξ 3 dt 3 dt 2 dt ˆψt 2 t ) r ˆψ2t 3 t )) r R R R V ψ σξ ξ 2 2ξ 3, t ) r ) /r = ˆψ sup ξ,ξ 2,ξ 3 dt 3 dt 2 dt ˆψt ) r ˆψ2t 2 t 3 t )) r R R R V ψ σξ ξ 2 2ξ 3, t 2 t ) r ) /r ) /r = ˆψ sup dt 3 dt 2 ˆψ r T t3 ˆψ2 r Ṽψσξ ξ 2 2ξ 3, ) r )t 2 ) ξ,ξ 2,ξ 3 R R where ˆψ 2 ξ) = ˆψ2ξ), and Ṽψσξ ξ 2 2ξ 3, η) = V ψ σξ ξ 2 2ξ 3, η). Consequently, σ TH M,,r,r),,,,) ˆψ ˆψ r sup ξ,ξ 2,ξ 3 = ˆψ ˆψ r sup ξ,ξ 2,ξ 3 ˆψ ˆψ r ˆψ 2 r dt 3 R R ) /r dt 2 ˆψ 2 t 2 t 3 ) r Ṽψσξ ξ 2 2ξ 3, t 2 ) r ) dt 3 ˆψ 2 r Ṽψσξ ξ 2 2ξ 3, ) r t 3 ) R sup ) /r dt 3 Ṽψσξ ξ 2 2ξ 3, t 3 ) r R ξ,ξ 2,ξ 3 = ˆψ ˆψ r ˆψ 2 r sup ξ,ξ 2,ξ 3 ) /r dt 3 V ψ σξ ξ 2 2ξ 3, t 3 ) r. R ) /r

25 SYMBOL CLASSES FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERTORS 25 The proof is complete by observing that the proof of Proposition 4.4 implies that ) /r sup dt 3 V ψ σξ ξ 2 2ξ 3, t 3 ) r <. ξ,ξ 2,ξ 3 R Using this result and Theorem 3.4 for m = 3 we can give the following initial result on the boundedness of TH on product of modulation spaces. Theorem 4.8. For p, p 0, p, ) and q, the trilinear Hilbert transform TH is bounded from M p, M p,q M p,q into M p 0, and we have the following estimate: THf, g, h) M p 0, C f M p, g M p,q h M p,q for all f, g, h SR), where the constant C > 0 is independent of f, g, h. Remark 4.9. Before proving this result we point out that the strongest results are obtained by choosing p 0 as close to as possible and p as close to as possible. As special case, we see that TH boundedly maps for every r < and ɛ > 0. M r, L 2 L 2 M +ɛ, Proof. We set r = min{p 0, p, p, p } >. The symbol of TH is in the symbol modulation space with decay parameters r 0 =, r =, r 2 = r 3 = r > as used in Theorem 3.4. Note that here, κ is the identity permutation, so z = 0. The boundedness conditions in Theorem 3.4 now read k = 0 : 0 + p ; k = : 0 + p 2 + p + 2; k = 2 : 0 + p 3 + p + + p 2 + r 3; k = 3 : 0 + p + + p 2 + r + p 3 + r 3 + p 0 ; where 4.6) p p r =, p 2 p 2 r, p 3 p 3 r. The four conditions above reduce to k = : + p ; p 2 k = 2 : p + p 2 + p 3 2 r ; k = 3 : p + p 2 + p r + p 0 ; For simplicity, we now set p 2 = p 2 = p 3 = p 3 [r, r ] and obtain k = : p ; p 3 k = 2 : p r ; k = 3 : p 2 r + p 0 ; that is k = : p p 3 ; k = 2 : p r ; k = 3 : p 2 r + p 0 ;