MAXIMAL FOURIER INTEGRALS AND MULTILINEAR MULTIPLIER OPERATORS

Transcription

1 MAXIMAL FOURIER INTEGRALS AND MULTILINEAR MULTIPLIER OPERATORS A Dissertation presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by HANH VAN NGUYEN Dr. Loukas Grafakos, Dissertation Supervisor July 206

2 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: MAXIMAL FOURIER INTEGRALS AND MULTILINEAR MULTIPLIER OPERATORS presented by Hanh Van Nguyen, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Dr. Loukas Grafakos Dr. Steven Hofmann Dr. David G. Retzloff Dr. Igor E. Verbitsky

3 ACKNOWLEDGMENTS I owe my deepest gratitude to Dr. Loukas Grafakos for introducing me to Harmonic Analysis and for tirelessly guiding me and inspiring me to push my research forward. I had a hard time in the summer 20 when my mother unexpectedly passed away after an unsuccessful brain tumor surgery. At that time, it was really hard for me to keep my concentration on research and to be productive. Luckily, Dr. Grafakos was enormously sympathetic and encouraged me to return to Columbia and to continue my research. I am indebted to him for responsively answering all my questions and for spending hours to explore my ideas. I am also grateful to him for being my supervisor during my graduate study at Mizzou. I would like to express my gratitude to the faculty and staff of the Department of Mathematics at the University of Missouri, Columbia for running interesting courses, and for constantly helping international students with their special needs. Many thanks, in particular, to Dr. Peter Casazza for introducing me Frame Theory and letting me be a part of his research group, to Dr. Steven Hofmann, Dr. David G. Retzloff, and Dr. Igor E. Verbitsky for being members of my committee. I owe thanks to the Vietnamese Ministry of Education and Training and the Vietnam International Education Development which supported me for the first two years of my doctoral program. I am so thankful to the Department of Mathematics at the University of Missouri Columbia and to the Department of Mathematics of College of Education at Hue University in Vietnam for their vital support. Finally, I cannot thank enough my family for understanding my choice to leave Vietnam and to pursue a doctoral degree in the USA. Especially, I would like to express my deep gratitude to my wife Phuong Truong, who has endlessly supported my research work and has assisted me to complete this dissertation timely, and to my son Hayden and my little daughter Hannah who made this process more enjoyable. ii

4 TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT ii v CHAPTER Introduction Maximal Fourier Integrals Historical overview Maximal Fourier integral operator The spherical maximal operator on the sphere The proof of Theorem Multilinear Multiplier Operators Historical overview Preliminaries and known results Regularization of the multiplier Minimality of conditions Endpoint estimate Discretization of the multiplier The proof of the main result The first case: 0 < p i, i m The second case: 0 < p i or p i = The third case: 0 < p i or 2 p i The last case: 0 < p i, i m Proofs of technical lemmas iii

5 3.8. Proof of Lemma Proof of Lemma Proof of Lemma APPENDIX A Interpolation illustration in R A. The smoothness condition in the unit cube A.. Near the origin A..2 Three wedges A..3 The lower local tetrahedron A..4 Three side local tetrahedrons A..5 Three face-out local tetrahedrons A..6 Two upper local tetrahedrons BIBLIOGRAPHY VITA iv

6 ABSTRACT The first topic of this dissertation is concerned with the L 2 boundedness of a maximal Fourier integral operator which arises by transferring the spherical maximal operator on the sphere S n to a Euclidean space of the same dimension. Thus, we obtain a new proof of the boundedness of the spherical maximal function on S n. In the second part, we obtain boundedness for m-linear multiplier operators from a product of Lebesgue (or Hardy spaces) on R n to a Lebesgue space on R n, with indices ranging from zero to infinity. The multipliers lie in an L 2 -based Sobolev space on R mn uniformly over all annuli, just as in Hörmander s classical multiplier condition. Moreover, via proofs or counterexamples, we find the optimal range of indices for which the boundedness holds within this class of multilinear Fourier multipliers. v

7 Chapter Introduction Fourier singular integral operators naturally appear in the study of linear partial differential equations and have proved to be useful tools in the solution of linear partial differential equations with variable coefficients. These operators generalize the Fourier transform, or general Fourier multiplier operators, and pseudo-differential operators, such as those that arise in the study of the wave equation. The fundamental L 2 -estimate for Fourier integral operators was established by Hörmander [2]; this was extended by Seeger, Sogge, and Stein [34] to L p for < p <. In this dissertation we consider maximal Fourier singular integral operators on R n that arise from natural process of a pullback of a maximal operator on the unit sphere S n in R n+. Such maximal Fourier integral operators are associated with symbols of three variable σ(t, ξ, x), where the supremum is taken over t > 0, x is the space variable, and ξ is the frequency variable. Under certain nondegeneracy conditions on σ and on the phase, we establish the L 2 boundedness of such maximal Fourier integral operators via an extension of the Rubio de Francia s technique [6]. As an application we obtain an alternative direct proof of Sogge s result [35] about establishing the boundedness of the spherical maximal operator on the unit sphere S n for n 3. In the second part of this dissertation, we address the following question: What

8 is the minimal regularity of a multilinear multiplier so that the associated multilinear operator is bounded from a product of Hardy spaces to another Lebesgue space? Inspired by the work of Calderón and Torchinsky [3], Grafakos and Kalton [3], and Miyachi and Tomita [27], we found an answer to the question. We provide necessary and sufficient conditions on multipliers which lie uniformly over all annuli in an L 2 - based Sobolev space for the associated operators to be bounded from a product of Lebesgue (or Hardy) spaces to another Lebesgue space for the entire range of indices possible. The essential results in this work are organized as follows: Chapter 2 contains the L 2 boundedness of a maximal Fourier integral and a short proof of the boundedness of the spherical maximal function on the unit sphere S n. The study of minimal regularity of multilinear multipliers is given in Chapter 3. 2

9 Chapter 2 Maximal Fourier Integrals 2. Historical overview The spherical maximal operator on R n was introduced by Stein in [39] as the supremum of spherical averages of a function. Precisely, this is defined as the operator Mf(x) = sup r>0 σ r (S(x, r)) S(x,r) f(y)dσ r (y), x Rn (2.) where σ r is the induced measure on the sphere S(x, r) = {y R n : y x = r}, and f is any function on R n, initially taken to be in the Schwartz class. The study of this operator has provided the impetus for the subsequent development of Fourier integral operators and their maximal counterparts. Stein [39] obtained the L p (R n ) boundedness of M whenever for p > n n and n 3. The two-dimensional case n = 2 turned out to be more complicated and was completed about a decade later by Bourgain [2]. An alternative proof of these results was given by Mockenhaupt, Seeger and Sogge [28]. The study of the operator M in (2.) has stimulated significant new research. Greenleaf [9] has considered the case where the sphere is replaced by a surface with 3

10 non-zero principal curvature. Sogge and Stein [36] have studied the case where the Gaussian curvature of the surface is nonvanishing to infinite order at any point and later extended this result to variable coefficient maximal operators associated with a family of surfaces of finite order [37]; the same authors have also considered the case of non-vanishing rotational curvature in [38]. In this work, Sogge and Stein studied the maximal operator with respect to dilations of (n )-dimensional compact surfaces embedded in an n-dimensional manifold. Using intricate estimates for oscillatory integrals, Isosevich and Sawyer [23] provided a necessary and sufficient condition for the maximal function associated with surface measure on a smooth convex surface having finite order of contact with its tangent lines, to be bounded on L p (R n ) for p > 2. The spherical maximal function on non-euclidean ambient spaces has also been considered. In this situation, this operator is defined as the supremum of all averages over geodesic spheres. Kohen [24] has obtained the boundedness for this spherical maximal operator on hyperboloids. Ionescu [22] extended this result for noncompact symmetric spaces of real rank one. In [30] Narayanan and Thangavelu showed the L p -boundedness of the spherical maximal operator on the Heisenberg group H n for p > 2n/(2n ) and n 2. Müller and Seeger [29] have proved the boundedness of the spherical maximal function on 2-step nilpotent Lie groups; since S 3 is a 2-step nilpotent Lie group, this result covers the spherical maximal function on S 3. Fischer [8] has considered the case of free 2-step nilpotent Lie groups. Nevo and Ratnakumar [3] have proved endpoint restricted weak type ( n, n ) estimates for the spherical n n maximal operator acting on radial functions defined on n dimensional symmetric spaces of constant curvature; these fail for general functions. Sogge [35] has employed the powerful machinery of Fourier integral operators to deduce the boundedness of the spherical maximal operator on compact manifolds without boundary and positive injectivity radius. 4

11 Rubio de Francia [6] studied the L p (R n ) boundedness of maximal operators formed by dilations of a fixed singular multiplier transformation. In the present chapter, we provide an extension of the result in [6], obtaining the boundedness for a maximal (over t > 0) Fourier integral operator containing a symbol of three variable σ(t, ξ, x), under certain nondegeneracy conditions on σ and on the phase. As an application we obtain an alternative proof of the boundedness of the spherical maximal operator on the n-dimensional sphere S n for n 3, which is transferred to R n via the stereographic projection. 2.2 Maximal Fourier integral operator The main result of this section is the boundedness of a maximal Fourier integral operator on a domain possessing a special geometrical property, called the cone property; this property is crucial in proving the Sobolev embedding theorem; see []. An open set Ω in R n has the cone property if there exists a finite truncated cone C whose vertex is the origin such that for each point x Ω there exists a rigid motion R of R n such that x + R[C] is contained in Ω. We have the following theorem concerning maximal Fourier integral operators on such domains. Theorem 2.. Let Ω be an open set in R n having the cone property. Fix 0 < λ and let σ : [0, λ) R n Ω C be a C k function, for some k n. Let ϕ : [0, λ) R n R n be a C -function. For a Schwartz function f supported inside Ω, define the maximal Fourier integral operator T f(x) = sup f(ξ)σ(t, ξ, x)e 2πiϕ(t,x) ξ dξ, x Ω. 0<t<λ R n Suppose that there exist constants A, A 2, r > 0 and a > 2 such that σ(t, ξ, x) is supported in the set {(t, ξ, x) [0, λ) R n Ω : tξ r} and such that 5

12 . α x σ(t, ξ, x) + t α x σ(t,ξ,x) ξ tξ a ψ α (x) with ψ α L (Ω) for all α k, ξ 0; 2. A det((j x ϕ)(t, x)) and t ϕ(t, x) A 2 for all x Ω and 0 t < λ. Then, there exists a finite constant C(Ω, a, n, k, A, A 2, ψ) such that T f L 2 (Ω) C(Ω, a, n, k, A, A 2, ψ)r 2 a f L 2 (Ω) for all Schwartz functions f. Thus T extends to a bounded map from L 2 (Ω) to L 2 (Ω) with constant depending only on a, n, k, A, A 2, r, ψ, and on Ω. Proof. We suppose first that σ(t, ξ, x) is supported in the set s tξ 4s for some s > 0. For 0 t < λ, denote T t f(x) = R n f(ξ)σ(t, ξ, x)e 2πiϕ(t,x) ξ dξ. Then T t f(x) ( λ 2 2 T t f(x) 2 dt 0 t ) /2 ( λ 0 t d dt ( T t f(x) ) ) /2 2 dt. t We have d ( T t f(x) ) = dt + R n f(ξ) t σ(t, ξ, x)e 2πiϕ(t,x) ξ dξ n l= R n f(ξ)σ(t, ξ, x) t ϕ l (t, x)ξ l e 2πiϕ(t,x) ξ dξ, in which ϕ l is the lth component function of ϕ. Denote ( λ G σ f(x) = T t f(x) ) /2 ( 2 dt λ = t 0 0 R n f(ξ)σ(t, ξ, x)e 2πiϕ(t,x) ξ dξ ) /2 2 dt. t Since Ω has the cone property, the Sobolev embedding theorem for Ω ([]) says that 6

13 for k n there is a constant C = C(Ω, n, k) such that sup z Ω R n f(ξ)σ(t, ξ, z)e 2πiϕ(t,x) ξ dξ C α k Ω R n f(ξ) α z σ(t, ξ, z)e 2πiϕ(t,x) ξ dξ dz. It follows that G σ f(x) C ( λ α k Ω 0 R n α f(ξ) z σ(t, ξ, z)e 2πiϕ(t,x) ξ dξ ) /2 2 dt dz. t Taking the L 2 -norm, the above inequality gives Gσ f C L 2 (Ω) α k Ω ( λ 0 Ω R n α f(ξ) z σ(t, ξ, z)e 2πiϕ(t,x) ξ dξ 2 dx dt ) /2 dz. t Now letting y = ϕ(t, x) and noting that A det (J x ϕ)(t, x), we have G σ f L 2 (Ω) A C α k A C α k Ω Ω ( λ 0 ( λ 0 R n R n R n α f(ξ) z σ(t, ξ, z)e 2πiy ξ dξ f(ξ) z α σ(t, ξ, z) 2 dξ dt ) /2 dz. t 2 ) dy dt /2 dz t By the hypothesis x α σ(t, ξ, x) tξ a ψ α (x), we deduce G σ f L 2 (Ω) ln(4)a Cs a f L 2 (Ω) ψ α L (Ω). (2.2) α k Denote σ (t, ξ, x) = t t σ(t, ξ, x) and σ l (t, ξ, x) = tξ lσ(t, ξ, x) (l =,..., n). Then, it is easy to see that ( T f(x) 2 G σ f(x) G σ f(x) + A 2 n l= ) G σ f(x) l, x Ω. 7

14 Taking the integral of both sides of the preceding inequality, we obtain T f 2 L 2 (Ω) G σf L 2 (Ω) Applying estimate (2.2) to G σ, G σ, and G σ l ( G σ f L 2 (Ω) + A 2 n l= we conclude that T f L 2 (Ω) C(Ω, n, k, A, A 2 )s 2 a f L 2 (Ω) ) G σ f L 2 (Ω). l ψ α L (Ω). (2.3) α k Assume now that σ is supported in the set tξ r. Fix functions γ 0, γ C 0 such that γ is supported in r 2 ξ 2r and γ 0 (ξ) + j 0 γ(2 j ξ) =, ξ 0, where γ 0 is supported in ξ r. Since σ vanishes on tξ < r, we have σ(t, ξ, x) = j 0 σ(t, ξ, x)γ(2 j tξ) for all t [0, λ), ξ 0 and x Ω. Set σ j (t, ξ, x) = σ(t, ξ, x)γ(2 j tξ), j 0 and T j f(x) = sup 0<t<λ R n f(ξ)σj (t, ξ, x)e 2πiϕ(t,x) ξ dξ. Notice that σ j enjoys the same properties as σ and is supported in 2 j r tξ 2 j+ r. It is easy to see that T f(x) j 0 T j f(x), x Ω. 8

15 Combining the above inequality with estimate (2.3) for each σ j we obtain T f L 2 (Ω) C(Ω, n, A, A 2 ) 2 (j )( 2 a) r 2 a f L 2 (Ω) ψ α L (Ω). j 0 α k Thus, we have just derived the inequality T f L 2 (Ω) C(Ω, a, n, A, A 2 )r 2 a f L 2 (Ω) ψ α L (Ω) α k for all Schwartz functions on R n supported in Ω. This result can be extended to L p (Ω) functions via the following lemma whose easy proof is omitted. Lemma 2.2. Let X be a normed space and let T : Y R be a sublinear map initially defined on a dense subspace Y of X such that T (y) X C y X for all y Y. Then T can be extended to a bounded map on X with the same norm. 2.3 The spherical maximal operator on the sphere We now apply our theorem to derive the boundedness of the spherical maximal operator on the sphere. Let S n be the n-sphere on R n+. For a point x S n and r > 0 we define the (n )-sphere Γ(x, r) = {y S n : x y = 2r} and the spherical cap (x, r) = {y S n : x y < 2r}. 9

16 For brevity, we set (x) = (x, /4) for all x S n. In analogy with the Euclidean setting, the spherical maximal operator is defined in the spherical-geometry setting as follows: Given a function g L (S n ), we define M S (g)(x) = sup 0<r</8 r n Γ(x,r) g(y)dh n (y), (2.4) where H n is the induced (n )-Hausdorff measure of R n+ on Γ(x, r). The main result of this section is to show an alternative proof of the following [35]: Theorem 2.3 ([35]). For n 3 and p > that n, there is a constant C = C(n, p) such n M S (g) L p (S n,σ n) C g L p (S n,σ n), (2.5) holds for all smooth functions g on S n, where σ n = H n is the spherical measure on S n and the constant C depending only on p and n. Therefore, M S can be uniquely extended to a bounded operator on L p (S n, σ n ) with the same constant. The main idea of the proof is to apply the stereographic projection to transfer M S to the Euclidean space where one has many available tools, such as the Fourier transform, maximal Fourier integral operators and the Hardy-Littlewood maximal operator. Now estimate (2.5) can be localized because of the compactness of the sphere. In fact, we can find a positive integer N 0 such that S n N 0 j= (x j, /8). For g L (S n ), using partition of unity ρ j subordinated to (x j, /8) to split the function g into several pieces, g = N 0 j= gρ j, where 0 ρ j and supp(ρ j ) (x j, /8). The sublinearity of the operator in (2.4) yields immediately the inequality N 0 M S (g) M S (g j ), j= with g j = gρ j, j N 0. Note that if function g is supported in (x 0, 8 ), then 0

17 M S (g) is supported in (x 0 ) = (x 0, ). Thus, to prove the estimate (2.5), it is 4 enough to show that (x j ) M S (g j ) p dσ n C(n, p) g j p dσ n, (x j ) where g j is supported in (x j, 8 ). Let e n+ = (0,..., 0, ) R n+. Since M S (g R )(Rx) = M S (g)(x) for all rotations R O(n + ). A suitable choice of rotations, it suffices to prove the inequality ( e n+ ) M S (g) p (x)dσ n (x) C(n, p) g(x) p dσ n (x) (2.6) ( e n+ ) for every integrable function g supported in ( e n+, 8 ). Now the stereographic projection Π : R n S n, given by Π(x) = 2 + x 2 x + x 2 + x 2 e n+, will allow us to transfer the problem to the Euclidean setting. Fix x B(0, ) R n and denote Σ(x, r) = Π (Γ(Π(x), r)), (0 < r < /8). Note that Σ(x, r) is an (n )- sphere S n (c x, ρ x ) R n. Let µ n = (Π ) (H n ) be the pushforward measure of H n associated with the map Π. Then M S (g)(π(x)) = sup 0<r</8 r n = sup 0<r</8 r n Γ(Π(x),r) Σ(x,r) g(y) dh n (y) (g Π)(z) dµ n (z). Since the measure µ n is comparable with the normal surface measure on Σ(x, r) =

18 S n (c x, ρ x ), we have M S (g)(π(x)) sup 0<r</8 r n S n (c x,ρ x) (g Π)(y) dσ n (y). (The notation means that the quantities are comparable.) Now change variables y = x r 2 ( + x 2 ) r r2 ( + x 2 ) r 2 ( + x 2 ) z, then M S (g)(π(x)) is comparable to ( ) x r sup a(r, x) n r2 ( + x (g Π) 0<r</8 S 2 )z dσ r n 2 ( + x 2 n (z) ), r2 ( + x 2 ) in which the quantity a(r, x) = is comparable to a positive constant r 2 ( + x 2 ) for all 0 < r < /8 and x <. Set s = r 2 ( + x 2 ), (0 < s < /32). Then M S (g)(π(x)) can be controlled by sup 0<s< 32 ( x s + (g Π) S s x 2 s ) z dσ n (z) s. n This sequence of calculations leads us to define an operator acting on every locally integrable function h on R n by setting N(h)(x) = sup 0<s< 32 S n h ( x s + s x 2 s ) z dσ n (z) s. The required inequality (2.6) will be established if the operator N is bounded from L p to L p when acting on functions supported in the unit ball. The following result is the main ingredient needed to prove Theorem

19 Lemma 2.4. For n 3 and p > n, the operator N satisfies the estimate n N(h) L p (B(0,2)) C h L p (B(0,2)) (2.7) for all smooth functions h supported in the ball B(0, 2). The proof of Lemma 2.4 will be provided in the next section. Now we examine certain examples that provide us information about the behavior of this operator. Fix x 0 R n and δ > 0. Acting N on χ B(x0,δ) yields N(χ B(x0,δ))(x) = sup 0<s< 32 S n χ B (0, ( s)δ s + x 2 s ( ) ) x ( s)x 0 z dσ n (z). s + x 2 s Since s + x 2 s > /2 for all 0 < s < /32 and x <, N(χ B(x0,δ))(x) sup 0<s< 32 χ ( ( s)x0 x )(z) S n B s + x 2 s, δ dσ n (z). s Denote G x0 = 0<s< 32 { x R n : x x 0 2 = s + s x 0 2}. For 0 < s < 32, let x s = ( s)x 0 x s + x 2 s Sn. Then we have ( δ s ) n N(χ B(x0,δ))(x) χ B(xs, δ S n s ) (z) dσ n (z) b n cn δ n. Thus, we have just established the lower bound N(χ B(x0,δ))(x) c n δ n χ Gx0. This inequality tells us that N does not map L p to L p for p < n. n More interestingly, note that G x0 is arbitrarily large whenever x 0 near infinity; 3

20 consequently, the operator N fails to map L p (R n ) to L p (R n ) globally for all 0 < p <. However, for p > next section. n n To exclude the case p = the local boundedness of the operator will be established in the n, we use the inequality n ( ) N(χ B(0,) )(x) c x 2 n, n x 0 < x < The proof of Theorem 2.3 To complete the proof of Theorem 2.3, we fix a smooth function h on R n supported in some ball and we prove (2.7) for this h. The passage to general h is achieved via Lemma 2.2. Before turning to the proof of Theorem 2.3, let us recall the operator N(h)(x) = sup 0<s< 32 S n h ( x s + s x 2 s ) z dσ n (z) s, for which we need to obtain L p bounds. Expressing the smooth function h in terms of its Fourier transform ĥ, we may write: N(h)(x) = sup 0<s< 32 ( ĥ(ξ)m s + x 2 R n s s ξ ) e 2πi s x ξ dξ, where m(ξ) = dσ(ξ) is the Fourier transform of the spherical measure. It is easy to see that if h is supported in the unit ball then N(h) is supported in B(0, 2). This operator is similar to the one that was introduced by Rubio de Francia [6]. However, the method that was used in [6] is based on Plancherel s theorem and cannot be applied directly because the multiplier here is a function of two variables x and ξ. To avoid this issue, we need to modify slightly the parameter of the operator. 4

21 In fact, the substitution t = s s s + x 2 yields a bigger maximal operator T h(x) = sup 0<t</8 ĥ(ξ)m(tξ)e 2πiα(t, x 2 )x ξ dξ, R n in which α : (0, ) [0, ) (0, ) is determined by α(t, s) = 2( + t 2 ) s2 + 2s( + 2t 2 ) + + s. First, we have the following elementary properties of the function α:. α(t, s) 2 for all (t, s) (0, ) [0, ). 2. 2s α s (t, s) + α(t, s) for all (t, s) (0, ) [0, ) For fixed t (0, 2 ), the map x ϕ t(x) = α(t, x 2 )x is one-to-one on R n. 4. The Jacobian matrix of ϕ t (x) is precisely represented as the sum of two matrices J(ϕ t (x)) = α(t, x 2 )I n + 2 α s (t, x 2 )B, when B is the matrix with entries b ij = x i x j. 5. The determinant of the Jacobian matrix is bounded below by a uniform constant ( det(j(ϕ t (x))) =α(t, x 2 ) n + 2 α(t, s x 2 ) ) α(t, x 2 ) x 2 ( =α(t, x 2 ) n 2 x 2 α ) s (t, x 2 ) + α(t, x 2 ) 2, t (0, 2 ). 6. α (t, s) 6 for all t (0, ) and s 0. t Our goal is to establish the L p local boundedness for the operator T. To do this, we first fix functions ψ 0, ψ C (R n ) such that ψ is smooth and supported in 5

22 2 ξ 2 and that ψ0 (ξ) + j ψ(2 j ξ) =, where ψ 0 is a smooth function supported inside the ball ξ 2. Now using the idea of the Littlewood-Paley decomposition to break the function h into frequencies, i.e. h = (ψ 0 ) t h + j ψ 2 j t h, where for a function g, define the dilation g t (x) = t n g(x/t), t > 0, x R n. We have the following simple estimate T h(x) Tj h(x), x R n, j=0 with and T j h(x) = T 0 h(x) = sup 0<t</8 sup 0<t</8 ĥ(ξ)m(tξ) ψ 0 (tξ)e 2πiα(t, x 2 )x ξ dξ, R n ĥ(ξ)m(tξ) ψ(2 j tξ)e 2πiα(t, x 2 )x ξ dξ, j. R n Next, we will show that T 0 is dominated by the Hardy-Littlewood maximal operator for all x 2; consequently, T 0 L p (B(0,2)) L p (B(0,2)) <, < p <. Indeed, we have ĥ(ξ)m(tξ) ψ 0 (tξ)e 2πiα(t, x 2 )x ξ dξ R n = h(y) m(tξ) ψ 0 (tξ)e 2πi(α(t, x 2 )x y) ξ dξ dy R n R n = h(y) m(ξ) ψ t R n 0 (ξ)e 2πi (α(t, x 2 )x y) t ξ dξ dy n R n 6

23 = ( h(y)(ψ t R n 0 dσ n ) β(t, x 2 )x + x y ) dy, t n where β(t, s) = 2tα(t, s) s2 + 2s( + 2t 2 ) + + s + + 2t 2. Notice that 0 < β(t, s) for all (t, s) (0, ) [0, ). 2 Since the smooth function ψ 0 has compact support, ψ 0 is a Schwartz function. Applying Lemma 2.5 to ψ 0 with j = 0 deduces (ψ 0 dσ n ) ( β(t, x 2 )x + x y ) C(n) (3 + β(t, x 2 )x + x y ) n t t ( C(n) + x y n. ) t Thus, T 0 is majorized by Hardy-Littlewood maximal function on B(0, 2). It remains to show that j Tj h L p (B(0,2)) C(n, p) h L p (B(0,2)), p > n/(n ). (2.8) For j, in view of the fact that m(tξ) + t (m(tξ)) / ξ C tξ n 2, ξ 0, Theorem 2. implies the estimate T j h L 2 (B(0,2)) C2 ( n 2 )j h L 2 B(0,2). (2.9) Next, we show that T j h is pointwise controlled by a multiple of the Hardy- Littlewood maximal function with a constant that grows like a multiple of 2 j. Indeed, 7

24 we will show that T j h(x) C(n)2 j Mh(x), x B(0, 2). (2.0) Using the weak type (, ) boundedness of the Hardy-Littlewood maximal function and (2.0), we deduce T j h L, (B(0,2)) C(n)2 j h L (B(0,2)) (2.) for our fixed function h supported inside the ball B(0, 2). Now interpolating between inequalities (2.) and (2.9) yields: T j h L p (B(0,2)) C(n, p)2 j( n p + n) h L p (B(0,2)), ( < p < 2). (2.2) Summing up all inequalities from (2.2) and noting that p > is established for n n < p < 2; hence, Lemma 2.4 is proved for n n, the inequality (2.8) n n < p < 2. Since the operator N maps L to L, estimate (2.7) is also true for p 2 via an interpolation between L p (for some n < p < 2) and n L. Thus we are only left with establishing inequality (2.0). The following lemma, whose proof can be found in [0, p ], can be used to prove (2.0). Lemma 2.5. Support φ is a Schwartz function in R n. For every M > n, the convolution of φ 2 j with the spherical measure dσ n can be estimated pointwise by φ 2 j dσ n (x) C(M, n)2j (3 + x ) M, j 0, where C(M, n) is a finite constant depending only on M and n. Now, the same argument as in proving the pointwise estimate for T 0 h(x) can be used together with the above lemma to deduce the inequality (2.0). 8

25 So far we have been working with a smooth function g with compact support. To extend the boundedness for M S to all functions in L p (S n, σ n ), we use Lemma

26 Chapter 3 Multilinear Multiplier Operators 3. Historical overview Let σ be a bounded function on R n. We denote by T σ the linear Fourier multiplier operator, whose action on Schwartz functions is given by T σ (f)(x) = σ(ξ) f(ξ)e 2πixξ dξ. (3.) R n Mikhlin s classical result [26] states that the T σ admits an L p -bounded extension for < p <, whenever α ξ σ(ξ) C α ξ α, ξ 0 (3.2) for all multi-indices α with α [ n ] +. This result was refined by Hörmander [20] 2 who proved that (3.2) can be replaced by the Sobolev-norm condition sup σ(2 j ( )) ψ W s <, (3.3) 20

27 for some s > n, where ψ is a smooth function supported in ξ 2 that satisfies 2 2 ψ(2 j ξ) = for all 0 ξ R n. Here g W s = (I ) s/2 g L 2, where I is the identity operator and = n j= 2 j is the Laplacian on R n. Calderón and Torchinsky [3] showed that the Fourier multiplier operator in (3.) admits a bounded extension from the Hardy space H p to H p with 0 < p if sup σ(t ) ψ W s < t>0 and s > n p n 2. Here the index s = n p n 2 is critical in the sense that the boundedness of T σ on H p does not hold if s n n. This was pointed out later by Miyachi and p 2 Tomita [27]. The multilinear counterpart of the Fourier multiplier theory has been rather similar in the formulation of results, but substantially more complicated in its proofs. The theory of multilinear operators, and in particular that of multilinear multiplier operators, originated in the work of Coifman and Meyer [4], [5], [25] and resurfaced in the work of Grafakos and Torres [8]. Multilinear Fourier multipliers are bounded functions σ on R mn = R n R n associated with the m-linear Fourier multiplier operator in the following way T σ (f,..., f m )(x) = e 2πix (ξ + +ξ m) σ(ξ,..., ξ m ) f (ξ ) f m (ξ m ) dξ, R mn (3.4) where f j are in the Schwartz space of R n and dξ = dξ dξ m. Tomita [40] obtained L p L pm L p boundedness ( < p,..., p m, p < ) for multilinear multiplier operators under a condition analogous to (3.3). Grafakos and Si [7] extended Tomita s results to the case p by using L r -based Sobolev 2

28 norms for σ with < r 2. Fujita and Tomita [9] provided weighted extensions of these results but also noticed that the Sobolev space W s in (3.3) can be replaced by a product-type Sobolev space W (s,...,s m) when p > 2. Grafakos, Miyachi, and Tomita [5] extended the range of p in [9] to p > and obtained boundedness even in the endpoint case where all but one indices p j are equal to infinity. Miyachi and Tomita [27] provided extensions of the Calderón and Torchinsky results [3] for Hardy spaces in the bilinear case; note that in [27] it was pointed out that the conditions on the indices are sharp, even in the linear case, i.e., in the Calderón and Torchinsky theorem. Following this stream of work, we provide extensions of the result of Calderón and Torchinsky [3] (m = ) and of Miyachi and Tomita [27] (m = 2) to the case m 3. Our work is inspired by that of Calderón and Torchinsky [3], Grafakos and Kalton [3], and certainly of Miyachi and Tomita [27]. As in [27], we find necessary and sufficient conditions, which coincide with those in [27] when m = 2, that imply boundedness for multilinear multiplier operators on a products of Hardy or Lebesgue spaces on the entire range of indices 0 < p. One important aspect of this work is an appropriate regularization of the multilinear multiplier operator which allows the interchange of its action with infinite sums of H p j atoms (see Section 3.3). Also, we point out that the complexity of the problem increases significantly as m increases. In fact, the main difficulty concerns the case where < p j < 2, in which the boundedness holds exactly in the interior of a convex simplex in R m. This simplex has m2 m + vertices but it is not enough to obtain the corresponding estimates for the vertices of the simplex, since interpolation between the vertices does not yield minimal smoothness in the interior. We overcome this difficulty by establishing estimates for all the points inside the simplex being arbitrarily close to those m2 m + points without losing smoothness. We introduce the Sobolev spaces that will be used throughout this chapter. First, 22

29 for x R n we set x = + x 2. For s,..., s m > 0, we denote by W (s,...,s m) the Sobolev space (of product type) consisting all functions f L 2 (R mn ) such that ( ) f W (s,...,sm) := f(y,..., y m ) y s y m sm 2 2 dy dy m <. R mn Throughout this work, let ψ be a smooth function on R mn whose Fourier transform ψ is supported in 2 ξ 2 and satisfies ψ(2 j ξ) =, ξ 0. Also, H p (R n ) denotes the real-variable Hardy space of Fefferman and Stein [7], for 0 < p. This space coincides with the Lebesgue space L p (R n ) when < p. The following is the main result of this chapter. Theorem 3.. Let 0 < p,..., p m, 0 < p, p + + p m = p, s,..., s m > n/2, and suppose k J ( sk n p k ) > 2 for every nonempty subset J {, 2,..., m}. If σ satisfies (3.5) A := sup σ(2 j ) ψ W (s,...,sm) <, (3.6) then we have T σ H p H pm L p A, (3.7) where L p should be replaced by BMO if p =. Moreover, this result is optimal in the sense that if (3.6) and (3.7) are valid then we must necessarily have s j n/2 for all j m, and k J ( sk n p k ) 2 23

30 for every nonempty subset J of {, 2,..., m}. Remark 3.2. When 0 < p j for all j m, conditions (3.5) imply that s j > n 2. Moreover, the condition in (3.6) is sufficient to guarantee that σ lies in L (R mn ). Indeed, suppose that σ is a function on R mn that satisfies (3.6). It is easy to see that ψ( 2 x) + ψ(x) + ψ(2x) = for all x 2. Now we want to verify that σ(2 j 0 x) is uniformly bounded in j 0 Z for a.e. x 2. Applying the Cauchy-Schwarz inequality and using the conditions s k > n, we write 2 σ(2 j 0 x) = σ(2 j 0 x) ψ(2 l x) ( σ(2 j 0 l ) ψ ) (ξ)e 2l+πixξ dξ l l R mn m ( + ξ k t 2 ) s k 2 m ( + tξ k t 2 ) s k ( 2 σ(2 j 0 l ) ψ ) (ξ,..., ξ m ) dξ dξ m l R mn k= k= C(s,..., s m, n) σ ψ j0 l W (s,...,sm) 3C(s,..., s m, n) sup σ j ψ W (s,...,sm), l for almost all x satisfying x 2. Here we set σ j ( ξ ) = σ(2 j ξ ). Thus σ L (R mn ) 3C(s,..., s m, n) sup σ j ψ W (s,...,sm) <. Throughout this chapter, we use the notation A B to indicate that A CB, where the constant C is independent of any essential parameters, and A B if both A B and B A hold simultaneously. 24

31 3.2 Preliminaries and known results Now fix 0 < p and a Schwartz function Φ with Φ(0) 0. Then the Hardy space H p contains all tempered distributions f on R n such that f H p := sup Φ t f L p <. 0<t< It is well known that the definition of the Hardy space does not depend on the choice of the function Φ. Note that H p = L p for all p >. When 0 < p, one of nice features of Hardy spaces is the atomic decomposition. More precisely, any function f H p (0 < p ) can be decomposed as f = k λ ka k, where a k s are L -atoms for H p supported in cubes Q k such that a k L Q k p and x γ a k (x)dx = 0 for all γ < N, and the coefficients λ k satisfy k λ k p 2 p f p Hp. The order N of the moment condition can be taken arbitrarily large. A fundamental L 2 estimate for T σ is given in the following theorem. Theorem 3.3 ([5]). If s,..., s m > n/2, then T σ L 2 L L L 2 C sup σ(2 j ) ψ W (s,...,sm). The following two lemmas are essentially contained in [27], modulo a few minor modifications. Lemma 3.4 ([27]). Let m be a positive integer, σ be a function defined on R mn, and K = σ, the inverse Fourier transform of σ. Suppose that σ is supported in the ball { y R mn : y 2 } and suppose l n, s i 0 for i m and p q. Then for each multi-index α there exists a constant C α such that y s y l s l α y K(y) L q (R ml, dy dy l ) C α y s y l s l K(y) L p (R ml, dy dy l ), 25

32 where y = (y,..., y m ) with y j R n. Proof. Denote y = (y, y ), where y = (y,..., y l ) and y = (y l+,..., y m ). Take ϕ a Schwartz function on R ln such that ϕ(y ) = for all y R ln, y 2. It is easy to see that σ(y, y ) = σ(y, y ) ϕ(y ) for all y R ln and y R (m l)n. Using the inverse Fourier transform we have ( ) K(y, y ) = K (ϕ δ 0 ) (y, y ) = K(y u, y u )ϕ(u )δ 0 (u )du du R ln R (m l)n = K(y u, y )ϕ(u )du, R ln where δ 0 is the Dirac distribution. Therefore, y s y l s l K(y, y ) = y s y l s l K(y u, y )ϕ(u )du R ( ln l ) y j u j s j K(y u, y ) u s u l s l ϕ(u ) du R ln j= C y s y l s l K(y, y ) L p (R ln,dy ) u s u l s l ϕ(u ) L p (R ln,du ) C 2 y s y l s l K(y, y ) L p (R ln,dy ). Here we used Hölder s inequality in the second to last line. Thus, we have proved y s y l s l α y K(y) L (R ml, dy dy l ) y s y l s l K(y) L p (R ml, dy dy l ). Now interpolate between the last inequality above with the trivial identity y s y l s l α y K(y) L p (R ml, dy dy l ) = y s y l s l K(y) L p (R ml, dy dy l ), we obtain the required inequality in the lemma. Lemma 3.5 ([27]). Let s i > n 2 for i m, and let ζ be a smooth function which is 26

33 supported in an annulus centered at zero. Suppose that Φ is a smooth function away from zero that satisfies the estimates α ξ Φ(ξ) Cα ξ α for all ξ R mn, ξ 0, and for all multi-indices α. Then there exists a constant C such that sup σ(2 j )Φ(2 j ) ζ W (s,...,sm) C sup σ(2 j ) ψ W (s,...,sm). Adapting the Calderón and Torchinsky interpolation techniques in the multilinear setting (for details on this we refer to [5, p. 38]) allows us to interpolate between two estimates for multilinear multiplier operators from a product of some Hardy spaces or Lebesgue spaces to Lebesgue spaces. Theorem 3.6 ([5]). Let 0 < p i, p i,k and s i,k > 0 for i = 0, and k m. For 0 < θ <, set p = θ p 0 + θ p, p k = θ p 0,k + θ p,k, and s k = ( θ)s 0,k + θs,k. Assume that the multilinear operator T σ defined in (3.4) satisfies the estimates T σ H p i, H p i,m L p i C i sup σ(2 j ) ψ W (s i,,...,s i,m ), i = 0,, where L p i should be replaced by BMO if p i =. Then T σ H p H pm L p C sup σ(2 j ) ψ W (s,...,sm), where L p should be replaced by BMO if p =. The following result is due to Fujita and Tomita [9] for 2 < p <, while the extension to p > and the endpoint case where all but one indices are equal to infinity is due to Grafakos, Miyachi and Tomita [5]. 27

34 Theorem 3.7 ([9, 5]). Let < p,..., p m, < p < and p + + p m = p. If σ satisfies (3.6), then T σ is bounded from L p L pm most a multiple of A. to L p with constant at We also use the following lemmas. Lemma 3.8 ([5, Lemma 3.3]). Let s > n/2, max{, n/s} < q < 2, and ζ j (x) = 2 jn ( + 2 j x ) sq, j Z, x R n. Suppose σ W (s,...,s) (R mn ) and supp σ { ξ 2 j+ } for some j Z. Then there exists a constant C > 0 depending only on m, n, s, and q such that T σ (f,..., f m )(x) C σ(2 j ) W (s,...,s)(ζ j f q )(x) /q... (ζ j f m q )(x) /q for all x R n. Lemma 3.9 ([5, Lemma 3.2]). Let ϕ S(R n ) be such that ϕ(0) = 0, and set j f(x) = e 2πix ξ ϕ(2 j ξ) f(ξ) dξ, j Z. R n Let ε > 0 and ζ j (x) = 2 jn ( + 2 j x ) n ε, j Z, x R n. Then the following inequalities hold for each 0 < q < 2: j f(x) 2 dx C f 2 L2, (3.8) R n (ζ j f q )(x) 2/q (ζ j j g q )(x) 2/q dx C q f 2 L 2 g 2 BMO. (3.9) R n Lemma 3.0. Suppose {F j } S (R n ) and suppose there exists a constant B > such that supp F j {ζ R n B 2 j ζ B2 j } for all j Z. Then, for each 28

35 0 < p <, j F H ( j p ) /2 F j 2 L p. j The preceding lemma is well known in the Littlewood-Paley theory, see for example [4, 5.2.4] and [, Lemma 7.5.2]. 3.3 Regularization of the multiplier In this section, we show that the operator defined in (3.) with enough smoothness of the multiplier can be approximated by a family of very nice operators. For a Schwartz function K, we denote the multilinear operator of convolution type associated with the kernel K by T K (f,..., f m )(x) = K(x y,..., x y m )f (y ) f m (y m )dy dy m. R mn Theorem 3.. Let σ be a function on R mn and s k > n 2 (3.6). Then there exists a family of functions ( σ ε) 0<ε< 2 for k m satisfying such that K ε := ( σ ε) is smooth and compactly supported for every 0 < ε < 2 ; also sup 0<ε< 2 sup σ ε (2 j ) ψ W (s,...,sm) sup σ(2 j ) ψ W (s,...,sm), (3.0) and lim T ε(f,..., f m ) T σ (f,..., f m ) L 2 = 0 (3.) ε 0 for all functions f k L 2m, k m, where T ε are multilinear singular integral operators of convolution type associated to K ε. The following lemma is the first step in constructing such a family of functions σ ε as stated in Theorem

36 Lemma 3.2. Let ϕ be a Schwartz function. Suppose σ is a function on R mn satisfying (3.6) for s k > n. Then we have 2 sup ε>0 sup [(ϕ ε σ)(2 j )] ψ W (s,...,sm) sup σ(2 j ) ψ W (s,...,sm), where ϕ ε (x,..., x m ) = ε mn ϕ(ε x,..., ε x m ) for all x k R n, k m. Proof. For ρ Z, ρ 2 denote F ρ = { y R mn : 2 ρ 2 y 2 ρ+ + 2 }. Fix x = (x,..., x m ) R mn. Then we have (ϕ ε σ)(2 j x) ψ(x) ={ ε mn ϕ ( ε y ) σ(2 j x y)dy} ψ(x) ={ ε mn 2 jmn ϕ ( ε 2 j y ) σ(2 j (x y))dy} ψ(x) = ρ Z { ϕ ε2 j(y)σ(2 j (x y)) ψ(2 ρ (x y))dy} ψ(x) = ρ 3 + ρ 2 { ϕ ε2 j(x y)σ(2 j y) ψ(2 } ρ y)dy ψ(x) (3.2) ϕ ε2 j(y)σ(2 j (x y)) ψ(2 ρ (x y)) ψ(x)dy (3.3) + ρ 3 { ϕ ε2 j(x y)σ(2 j y) ψ(2 ρ y)dy} ψ(x). (3.4) The W (s,...,s m) norm of term (3.3) can be estimated easily by ρ 2 ϕ ε2 j(y) σ(2 j ( y)) ψ(2 ρ ( y)) ψ W (s,...,sm)dy ρ 2 ϕ ε2 j(y) σ(2 j ( y)) ψ(2 ρ ( y)) W (s,...,sm) ψ W (s,...,sm)dy σ(2 j+ρ ) ψ W (s,...,sm) ρ 2 ϕ ε2 j(y) dy sup σ(2 j ) ψ W (s,...,sm), 30

37 in which the second last inequality follows from the fact [9, Proposition A.2] that fg W (s,...,sm) f W (s,...,sm) g W (s,...,sm), when f, g W (s,...,s m) for s,..., s m > n 2. Now fix integer numbers N k s k, ( k m) and set N = N + + N m. Since f W (s,...,sm) f W N, the Sobolev norm of the term in (3.2) is bounded by { ϕ ε2 j( y)σ(2 j y) ψ(2 y)dy} ρ ψ ρ 3 ρ 3 α + β N W N { (ε2 j ) α ( α ϕ) ε2 j( y)σ(2 j y) ψ(2 } ρ y)dy β ψ = { ( α ϕ) ε2 j(y) σ(2 j ( y)) ρ 3 α + β N 4 y 9 (ε2 j ) ψ(2 } ρ ( y))dy β ψ α 4 ( y ) α ( α ϕ) ε2 j ε2 j(y) σ(2 j ( y)) ψ(2 ρ ( y)) dy L 2 ρ 3 α N 2 ρn ( y ) α 2 σ(2 j+ρ ) ψ ( α ϕ) L 2 ε2 j ε2 j(y) dy ρ 3 α N 2 ρn 2 σ(2 j+ρ ) ψ W s y α ( α ϕ)(y) dy ρ 3 sup σ(2 j ) ψ W (s,...,sm). α N L 2 L 2 Finally, we deal with term (3.4). We have { ϕ ε2 j( y)σ(2 j y) ψ(2 y)dy} ρ ψ ρ 3 { ϕ ε2 j( y)σ(2 j y) ψ(2 y)dy} ρ ψ ρ 3 ρ 3 α + β N = ρ 3 α + β N W (s,...,sm) W N { (ε2 j ) α ( α ϕ) ε2 j( y)σ(2 j y) ψ(2 } ρ y)dy β ψ { (ε2 j ) α ( α ϕ) ε2 j(y)σ(2 j ( y)) ψ(2 } ρ ( y))dy β ψ F ρ L 2 L 2 3

38 ρ 3 α + β N = ρ 3 α + β N ρ 3 α + β N ρ 3 α N α N ρ 3 α N ρ 3 ( y ) α ( α ϕ) F ρ 2 ρ ε2 j ε2 j(y) σ(2 j ( y)) ψ(2 ρ ( y)) β ψ L 2dy ( y ) α ( α ϕ) F ρ 2 ρ ε2 j ε2 j(y) σ(2 j ) ψ(2 ρ )( β L ψ)( + y) 2dy ( y ) α ( α ϕ) ε2 j ε2 j(y) σ(2 j+ρ ) ψ( β ψ)(2ρ +y) L 2dy 2 ρn 2 F ρ σ(2 σ(2 2 ρn 2 F ρ j+ρ ) ψ j+ρ ) ψ sup σ(2 j+ρ ) ψ W (s,...,sm) ρ Z sup σ(2 j ) ψ W (s,...,sm). ( y ) α ( α ϕ) ε2 j ε2 j(y) σ(2 j+ρ ) ψ L 2 (B( 2 ρ y,2 ))dy ρ ( y ) α ( α ϕ) L F ρ ε2 j ε2 j(y) dy ( y ) α W (s,...,sm) ( α ϕ) F ρ ε2 j ε2 j(y) dy ( y ) α ( α ϕ) ε2 j ε2 j(y) dy α N ρ 3 F ρ The proof of the lemma is now complete. Proof of Theorem 3.. Fix 0 < ε <. Choose a smooth function ϕ such that ϕ 2 is supported in the unit ball and ϕ(0) =. Denote by σ ε = ϕ ε ( σφ ε), where φ ε = θ(ε ) θ(ε ), and θ is a smooth function satisfying θ(x) = 0 for all x and θ(x) = for all x 2. We note that these functions are suitable regularized versions of the multiplier in Theorem 3.. Indeed, let K ε = ( σ ε) ( = σφ ε ) ϕ(ε( )); then, K ε are smooth functions with compact support for all 0 < ε <. 2 Using the fact that α φ ε (ξ) C α,θ ξ α, ξ 0, 0 < ε < 2, Lemma 3.2 applied to the function σφ ε combined with Lemma 3.5 gives sup 0<ε< 2 sup σ ε (2 j ) ψ W (s,...,sm) sup sup σ(2 j )φ ε (2 j ) ψ W (s,...,sm) 0<ε< 2 32

39 sup σ(2 j ) ψ W (s,...,sm), which yields (3.0). Thus, we are left with establishing (3.). For ε > 0, now recall T ε (f,..., f m )(x) = = K ε (x y,..., x y m )f (y ) f m (y m )dy σ ε (ξ,..., ξ m ) f (ξ ) f m (ξ m )e 2πix(ξ + +ξ m) dξ. Involve estimate (3.0) with Theorem 3.7, we can see that T σ and T ε are uniformly bounded from L 2m L 2m L 2 for all 0 < ε <. By density, it suffices to 2 verify (3.) for all functions in the Schwartz class. Now fix Schwartz functions f k, for k m. The Fourier transform of T σ (f,..., f m ) can be written by R n(m ) σ ( ξ,..., ξ m, ξ m l= m ) ( ξ l f (ξ ) f m (ξ m ) f m ξ Similarly, the Fourier transform of T ε (f,..., f m ) is R n(m ) σ ε( ξ,..., ξ m, ξ m l= l= m ) ( ξ l f (ξ ) f m (ξ m ) f m ξ l= ξ l ) dξ dξ m. ξ l ) dξ dξ m. We now claim that σ ε converges pointwise to σ. Take this claim for granted, we have ( ( T ε (f,..., f m ) ) (ξ) T σ (f,..., f m )) (ξ), ε 0 (3.5) for a.e. ξ R n. Notice that T ε (f,..., f m ) T σ (f,..., f m ) L 2 = ( T ε (f,..., f m ) ) ( T σ (f,..., f m ) ) L 2. Since σ ε L σ L < for all ε > 0, Lebesgue s dominated convergence theorem 33

40 implies that ( Tε (f,..., f m ) ) ( T σ (f,..., f m ) ) as ε 0 in L 2, and this establishes (3.). It remains to prove (3.5) about the pointwise convergence of σ ε as ε 0. Now fix j 0 Z, we want to show that σ ε (x) σ(x) for a.e. a.e. 2 j 0 x 2 j 0+. Indeed, let 0 < ε < min { 2 2j 0 2, 2 j 0 2 } be an arbitrarily small positive number. Then we have σ ε (x) σ(x) y ε + + y ε y > ε ϕ ε (y) σ(x y) sup 2 j 0 x 2 j 0 + φ ε (x y) dy ϕ ε (y) σ(x y) σ(x) dy ϕ ε (y) σ(x y)φ ε (x y) σ(x) dy. The first integral vanishes since φ ε (x) = for all 2ε x. To estimate the second ε integral, we denote Ψ(x) = l 2 ψ(2 l x). Then Ψ(x) = for all 4 x 4. Therefore we have Ψ(2 j 0 (x y)) = Ψ(2 j 0 x) = for all 2 j 0 x 2 j 0+ and y 2 j 0. Now recall σ j (x) = σ(2 j x) and estimate y ε = ϕ ε (y) σ(x y) σ(x) dy y ε ϕ L ϕ ε (y) σ(x y) Ψ(2 j 0 (x y)) σ(x) Ψ(2 j 0 x) dy sup σ( y) Ψ(2 j 0 ( y)) σ Ψ(2 j0 ) y L ε 34

41 ϕ L = ϕ L We would like to show j 0 +2 sup j=j 0 2 y ε j 0 +2 sup j=j 0 2 y 2 j ε (σ j ψ)( 2 j y) (σ j ψ) L (σ j ψ)( y) (σj ψ) L. lim sup ε 0 y 2 j ε (σ j ψ)( y) (σj ψ) L = 0. The preceding limit apparently converges to 0 as ε 0 because σ j ψ W (s,...,s m) for s k > n, k m. The last term is majorized by 2 C σ L y ε ϕ(y) dy, which tends to 0 when ε 0. Thus σ ε (x) σ(x) as ε 0 for a.e. 2 j 0 x 2 j 0+. Hence, σ ε converges to σ pointwise on R mn. Also σ ε L (R mn ) σ L (R mn ) uniformly for all ε > 0. The proof of Theorem 3. is now complete. Proposition 3.3. Let K be a smooth function on R mn with compact support. Then we have T K H p H pm L p C K < for all 0 < p,..., p m, p < and p = p + + p m, where T K is the multilinear singular integral operator of convolution type associated with the kernel K. Proof. The boundedness of the operator T K can be deduced from [2, Lemma 4.2], which provides the estimate (for some sufficiently large integer N) T K (f,..., f m )(x) 35 m M N (f k )(x), (3.6) k=

42 for all f k L 2 H p k, in which M N (f)(x) = sup sup sup ϕ F N t>0 y B(x,t) (ϕ t f)(y) is the grand maximal function with respect to N, and F N := { ϕ S(R n ) : ( + x ) N R n α N+ α ϕ(x) dx }. Taking the L p quasinorm, applying Holder s inequality to (3.6), and using the quasinorm equivalence of some maximal functions [, Theorem 6.4.4] yields m m T K (f,..., f m ) L p M N (f k ) L p k C K f k H p k. k= k= Working with smooth kernels K with compact support comes handy when dealing with infinite sums of atoms, since we are able to freely interchange the action of T K with infinite sums of atoms. Precisely, a consequence of the boundedness of T K, given in Proposition 3.3, is the following result. Proposition 3.4. Let 0 < p,..., p m and let K be a smooth function with compact support. Then for every f k H p k with atomic representation fk = j k λ k,jk a k,jk, where a k,jk are L -atoms for H p k and j k λ k,jk p k 2 p k fk p k H p k for k m. Then T K (f,..., f m )(x) = j λ,j λ m,jm T K (a,j,..., a m,jm )(x) j m for a.e. x R n. Proof. Let 0 < p < be number such that p = p + + p m. For any positive 36

43 integers N,..., N m, Proposition 3.3 gives the estimate T K (f,..., f m ) C K N j = m k= f k N m j m= N k j k = λ,j λ m,jm T K (a,j,..., a m,jm ) L p λ k,jk a k,jk H p k f l H p l. l k Now passing to the limit, we obtain T K (f,..., f m )(x) = λ,j λ m,jm T K (a,j,..., a m,jm )(x) j = j m= for a.e. x R n. Similarly, we can obtain the following result. Proposition 3.5. Let 0 < p,..., p l and < p l+,..., p m. Let K be a smooth function with compact support. Suppose f i H p i, i l, has atomic representation f i = k i λ i,ki a i,ki, where a i,ki are L -atoms for H p i and k i λ i,ki p i 2 p i f i p i H p i. Suppose f i L p i for l + i m. Then T K (f,..., f m )(x) = k λ,k λ l,kl T K (a,k,..., a l,kl, f l+,..., f m )(x) k l for almost all x R n. 3.4 Minimality of conditions In this section we will show that conditions (3.5) and s k > n 2 are minimal in general that guarantee boundedness for multilinear multiplier operators. We fix a smooth function ψ whose Fourier transform is supported in {2 3 4 ξ }, it satisfies 37

44 ψ(ξ) = for all 2 4 ξ 2 4, and for some nonzero constant c ψ(2 j ξ) = c, ξ 0. Now we have the following theorem: Theorem 3.6. Let 0 < p k, 0 < p <, and s k > 0 for k m. Suppose that the estimate T σ (f,..., f m ) L p sup σ(2 j ) ψ W (s,...,sm) m f k H p k k= holds for all f k H p k and σ L such that sup σ(2 j ) ψ W (s,...,sm) <. The following conditions are then necessary: s k n, k m, (3.7) 2 and k J ( sk n p k ) 2 (3.8) for every nonempty subset J {,..., m}. The following lemma is obvious by changing variables, so its proof is omitted. Lemma 3.7. Let ϕ be a nontrivial Schwartz function and s > 0. Then ( ϕ(εy) 2( + y 2) s dy ) 2 ε n 2 s for all 0 < ε. Proof of Theorem 3.6. We show first the necessary conditions (3.7) for k m. Without loss of generality, we will show s n. To establish this inequality, we need 2 38

45 to construct some functions σ ε, (0 < ε ), and f k H p k such that fk H p k = for all k m, and T σ ε(f,..., f m ) L p, and further that sup σ ε (2 j ) ψ W (s,...,sm) ε n 2 s. Once these functions are constructed, one observes that T σ ε(f,..., f m ) L p sup σ ε (2 j ) ψ W (s,...,sm) for all 0 < ε. Therefore we must have s n 2. m f k H p k ε n 2 s Let ϕ be a nontrivial Schwartz function such that ϕ is supported in the unit ball, k= and let φ 2 = = φ m be Schwartz functions whose Fourier transforms, φ2, is supported in an annulus ξ, and identical to on ξ. 7m 3m 6m 4m Similarly, fix a Schwartz function φ m with φ m { ξ R n : 2 3 on an annulus For 0 < ε < 240m, set ξ Take a, b Rn with a = 5m ( σ ε ξ a ) (ξ,..., ξ m ) = ϕ φ2 (ξ 2 ) ε φ m (ξ m ). ξ 4 3} and φm and b =. It is easy to check that supp σ ε { 2 4 ξ 2 4 } ; hence, σ ε (2 j ) ψ = σ ε for j = 0 and σ ε (2 j ) ψ = 0 for j 0. This directly implies that sup σ ε (2 j ) ψ W (s,...,sm) = σ ε W (s,...,sm). Taking the inverse Fourier transform of σ ε gives (σ ε ) (x,..., x m ) = ε n e 2πia x ϕ(εx )φ 2 (x 2 ) φ m (x m ). 39

46 Now apply Lemma 3.7 to have σ ε W (s,...,sm) ε n 2 s. Thus sup σ ε (2 j ) ψ W (s,...,sm) ε n 2 s. Now choose f k (ξ) = ε n n p k ϕ( ξ a ) for k m, and f ε m (ξ) = ε n we will show that these functions are what we needed to construct. pm n ϕ( ξ b ε ). Then In the following estimates, we will use the fact, its proof can be done by using the Littlewood-Paley characterization for Hardy spaces, that if f is a function whose Fourier transform is supported in a fixed annulus centered at the origin, then f H p f L p for 0 < p <, (cf. [9, Remark 7.]). Indeed, using the above fact and checking that each f k is supported in an annulus centered at zero and not depending on ε allow us to estimate H p -norms via L p -norms, namely f k H p k f k L p k =, ( k m). Thus, we are left with showing that T σ (f,..., f m ) L p. Notice that φ k (ξ) = on the support of f k for 2 k m. Therefore, ( ( a ) T σ ε(f,..., f m )(x) = ϕ ε n n ( p a )) (x) ( ) (x) ) (x) ϕ φ2 f2 ( φm fm ε ε ( ( a ) = ϕ ε n n ( p a )) (x) ( ) (x) ) (x) ϕ f2 ( fm ε ε = ε n p + + n pm e 2πi[(m )a+b] x (ϕ ϕ)(εx)[ϕ(εx)] m = ε n p e 2πi[(m )a+b] x (ϕ ϕ)(εx)[ϕ(εx)] m, which obviously gives T σ ε(f,..., f m ) L p. So far, we have proved that s n 2 ; hence, by symmetry, we have s k n 2 for all k m. 40