MULTILINEAR HARMONIC ANALYSIS. 1. Introduction

Transcription

1 MULTILINEAR HARMONIC ANALYSIS LOUKAS GRAFAKOS Abstract. This article contains an expanded version of the material covered by the author in two 90-minute lectures at the 9th international school on Nonlinear Analysis, Function spaces, and Applications, held in Třešt, Czech Republic during the period September to September 7, Introduction In memory of Nigel Kalton An operator acting on function spaces may not only depend on a main variable but also on several other function-variables that are often treated as parameters. Examples of such operators are ubiquitous in harmonic analysis: multiplier operators, homogeneous singular integrals associated with functions on the sphere, Littlewood- Paley operators, the Calderón commutators, and the Cauchy integral along Lipschitz curves. Of the aforementioned examples, we discuss the latter: The Cauchy integral along a Lipschitz curve Γ is given by C Γ h)z) = 2πi p.v. Γ hζ) ζ z dζ, where h is a function on Γ, which is taken to be the graph of a Lipschitz function A : R R, and z is a point on the curve Γ. A. Calderón wrote ) C Γ h)z) = i) m C m f; A)x), 2πi m=0 where z = x + iax), fy) = hy + iay)) + ia y)), and ) m Ax) Ay) fy) C m f; A)x) = p.v. x y x y dy. R The operators C m f; A) are called the mth Calderón commutators and they provide examples of singular integrals whose action on the function has inspired the fundamental work on the T theorem []. Identity ) reduces the boundedness of C Γ h) to that of the operators C m f; A) recall fy) = hy + iay)) + ia y))); certainly for this approach to bear fruit, one Date: November 2, Mathematics Subject Classification. Primary 42B20. Secondary 42B5, 46B70, 47G30. Key words and phrases. Calderón-Zygmund singular integrals, multilinear operators, multipliers, interpolation, T Theorem. Research partially supported by the NSF under grant DMS

2 2 LOUKAS GRAFAKOS would also need to know that the operators C m f; A) are bounded with norms having moderate growth in m. At this point, it seems that we reduced the boundedness of a linear operator to another operator that contains powers of the function A and thus it is nonlinear in it. To adopt a truly multilinear point of view, we introduce the m + )-linear operator E m+ f; A,..., A m )x) = p.v. R A x) A y) x y )... Am x) A m y) x y ) fy) x y dy and seek estimates for it. Any estimate for E m+ from a product of function spaces Z Z 2 Z m+, where Z 2 = = Z m+ gives yield to an estimate for C m f; A) in terms of f and A. This point of view leads to the following result: Theorem. [5]) Let 0 < /p = m+ j= /p j. Then the m + )-linear operator E m+ maps L p R) L p m+ R) to L p, R) whenever p,..., p m+ and it also maps L p R) L p m+ R) to L p R) when < p j < for all j. In particular it maps L R) L R) to L /m+), R). The endpoint conclusion L R) L R) to L /m+), R) of Theorem has been obtained by C. Calderón [2] when m = and Coifman and Meyer [6] when m =, 2 while the case m 3 was completed by Duong, Grafakos, and Yan [5]. The underlying idea in the proof of Theorem is the simultaneous Calderón- Zymgund decomposition on all functions that E m+ acts on. This decomposition resembles the classical linear Calderón-Zymgund decomposition, but is more complicated due to the presence of several tuples of combinations of good and bad functions. This decomposition is discussed in Section 3. However, the proof contained in Section 3 does not directly apply to Theorem ; the latter requires a more flexible version of the Calderón-Zymgund decomposition, since the kernel of E m+ does not obey the standard multilinear Calderón-Zymgund kernel conditions, see Section 3. Indeed the kernel of E m+ is the function of m + )-variables Ky 0,..., y m+ ) = )ey m+ y 0 )m y 0 y m+ ) m+ χ yl ) miny 0,y m+ ),maxy 0,y m+ )) which contains characteristic functions. The proof of Theorem is achieved in [5] and is modeled after the approach devised by Duong and A. McIntosh [6] for linear operators. Another class of operators closely related to the commutators of Calderón is the family H α,α 2 f, f 2 )x) = p.v. f x α t)f 2 x α 2 t) dt R t, α, α 2, x R, called today the bilinear Hilbert transforms. These were also introduced by A. Calderón in an attempt to show that the commutator C f; A) is bounded on L 2 R) when At) is a function on the line with bounded derivative. The idea of this approach is that the linear operator f C f; A) can be expressed as the average 2) C f; A)x) = 0 l= H,α f, A )x) dα,

3 MULTILINEAR HARMONIC ANALYSIS 3 and thus the boundedness of C f; A) can be reduced to the uniform in α) boundedness of H,α. Naturally, the estimates for H,α should depend linearly) on both functions f and A. This operator is discussed in Section 7 The previous discussion leads to the conclusion that treating the function A as a frozen parameter provides limited results in terms of its smoothness. If we have estimates in terms of a few function space norms of both f and A, we may use the power of multilinear interpolation, to deduce boundedness of C f; A) on various function spaces, of ranging degree of regularity. Certainly this fact is not only pertinent to Calderón s first commutator C, but all multilinear operators. In summary, we advocate the following point of view in the study of multivariable operators: unfreeze the functions serving the roles of a parameter and treat them as input variables. This approach often yields sharper results in terms of the regularity of the input functions. In these notes we pursue this idea in a systematic way. We present certain fundamental results concerning linear or sublinear) operators of several variables, henceforth called multilinear or multisublinear), that contain challenges that appear in their study, despite the great resemblances with their linear analogues. The proofs given in the next sections contain most necessary details but references are provided for the sake of completeness in the exposition. 2. Examples of multivariable operators We embark on the study of multilinear harmonic analysis with the class of operators that extends the concept of Calderón-Zygmund operators in the multilinear setting. These operators have kernels that satisfy standard estimates and possess boundedness properties analogous to those of the classical linear ones. This class of operators has been previously studied by Coifman and Meyer [6], [7], [8], [9], [34], assuming sufficient smoothness on their symbols and kernels. If an m-linear operator T commutes with translations in the sense that 3) T f,..., f m )x + t) = T f + t),..., f m + t))x) for all t, x R n, then it incorporates a certain amount of homogeneity. Indeed, if it maps L p L pm to L p, then one must necessarily have /p + + /p m /p; this was proved in [25] for compactly supported kernels but extended for general kernels in [3]. We use the following definition for the Fourier transform in n-dimensional Euclidean space fξ) = fx)e 2πix ξ dx, R n while f ξ) = f ξ) denotes the inverse Fourier transform. Multilinear operators that commute with translations as in 3) are exactly the multilinear multiplier operators that have the form 4) T f,..., f m )x) = σξ,..., ξ m ) f ξ ) f m ξ m )e 2πix ξ + +ξ m) dξ... dξ m R n ) m for some bounded function σ.

4 4 LOUKAS GRAFAKOS Endpoint estimates for linear singular integrals are usually estimates of the form L L or L L,. The analogous m-linear estimates are L L L /m,. Although one expects some similarities with the linear case, there exist some differences as well. For example, if a linear translation-invariant operator has a positive kernel and it maps L L,, then it must have an integrable kernel and thus it actually maps L to L. In the multilinear case, it is still true that if a multilinear translation-invariant operator has a positive kernel and maps L L to L /m,, then it must have an integrable kernel, but having an integrable positive kernel does not necessarily imply that the corresponding operator maps L L to L /m. Results of this type have been obtained in [23]. We provide a few examples of mutlivariable multilinear and multisublinear) operators: Example : The identity operator in the m-linear setting is the product operator T f,..., f m )x) = f x) f m x). By Hölder s inequality T maps L p L pm L p whenever /p + +/p m = /p. Example 2: The action of a linear operator L on the product f f m gives rise to a more general degenerate m-linear operator T 2 f,..., f m )x) = Lf f m )x). that still maps L p L pm L p whenever /p + + /p m = /p, provided L is a bounded operator on L p. Example 3: The previous example captures the majority of interesting m-linear operators. Let L 0 be a linear operator acting on functions defined on R mn. We define T 3 f,..., f m )x) = L 0 f f m )x). Here f f m is the tensor product of these functions, defined as a function on R mn as follows: f f m )x,..., x m ) = f x )... f m x m ). In particular, L 0 could be a singular integral acting on functions on R mn. The boundedness of T 3 from L p L pm L p whenever /p + + /p m = /p may not always be an easy task. It often requires a delicate study aspects of which are investigated in this article for certain classes of linear and also sublinear) operators L 0. The situation where /p + + /p m = /p will be referred to as the singular integral case. This is because, it needs to be distinguished from the fractional integral case in which /p < /p + +/p m. This name is due to the fact that most examples of multilinear operators bounded in this case have fractional integral homogeneity, such as these: f,..., f m ) f x y )... f m x y m ) x y + + x y m ) mn+α dy... dy m. R mn Example 4: Taking L 0 to be a linear multiplier operator on R n ) m with symbol σ, we obtain a multilinear multiplier operator of the form 4) Then σ is called the symbol or multiplier of the m-linear multiplier.

5 MULTILINEAR HARMONIC ANALYSIS 5 Multilinear multipliers operators arise in many situations. For instance, to prove the Kato-Ponce inequality [28] Leibniz rule for fractional derivatives D α, α > 0) D α fg) [ D L r C α p,q,r f L p g L q + f L p D α g ] L q where /p + /q = /r, one would have to study bilinear multiplier operators with symbols 5) ξ + η [ α Ψ2 j ξ)φ2 j η) + Φ2 j ξ)ψ2 j η) + ] Ψ2 j ξ)ψ2 k η). j j k 2 Here Ψ and Φ are smooth functions supported in an annulus and in small disjoint ball, both centered at the origin, respectively. The main idea is in the first term in 5) we have ξ+η ξ and so ξ+η α could be replaced by ξ α via some multiplier theorem. This would yield the term D α f L p g L q in L r norm. An analogous estimate with the roles of f and g interchanged holds for the second term in 5), while the third term is easier. Such a study requires a multiplier theory for multilinear operators. The topic of multilinear multipliers will be addressed in Section 5. Example 5: The maximal function Mf,..., f m )x) = sup Q x Q m Q )... f y )... f m y m ) dy... dy m Q where the supremum is taken over all cubes in R n with sides parallel to the axes. This was introduced in the work of Lerner, Ombrosi, Pérez, Torres and Trujillo-González [32] and plays an important role in the characterization of the class of multiple A p weights. Example 6: A larger operator is the strong multilinear maximal function. It is defined for x R n as M R f,..., f m )x) = sup R x )... f R m y )... f m y m ) dy... dy m, R R where the supremum is taken over all rectangles R in R n with sides parallel to the axes. When m =, this operator reduces to the strong maximal function on R n. 3. Multilinear Calderón-Zygmund operators In this section we set up the background of the theory of multilinear Calderón- Zygmund operators. We will be working on n-dimensional space R n. We denote by S R n ) the space of all Schwartz functions on R n and by S R n ) its dual space, the set of all tempered distributions on R n. An m-linear operator T : S R n ) S R n ) S R n ) is linear in every entry and consequently it has m formal transposes. The first transpose T of T is defined via T f, f 2,..., f m ), h = T h, f 2,..., f m ), f, for all f, f 2,..., f m, h in S R n ). Analogously one defines T j, for j 2 and we also set T 0 = T.

6 6 LOUKAS GRAFAKOS Let Kx, y,..., y m ) be a locally integrable function defined away from the diagonal x = y = = y m in R n ) m+, which satisfies the size estimate 6) Kx, y,..., y m ) A x y + + x y m ) mn for some A > 0 and all x, y,..., y m ) R n ) m+ with x y j for some j. Furthermore, assume that for some ε > 0 we have the smoothness estimates 7) Kx, y,..., y m ) Kx, y,..., y m ) A x x ε x y + + x y m ) mn+ε whenever x x 2 max x y,..., x y m ) and also that 8) Kx, y, y 2,..., y m ) Kx, y, y 2,..., y m ) A y j y j ε x y + + x y m ) mn+ε whenever y y max x y 2,..., x y m ) as well as a similar estimate with the roles of y and y j reversed. Kernels satisfying these conditions are called multilinear Calderón-Zygmund kernels and are denoted by m-czka, ε). A multilinear operator T is said to be associated with K if T f,..., f m )x) = Kx, y,..., y m )f y )... f m y m ) dy... dy m, R n ) m whenever f,..., f m are smooth functions with compact support and x does not lie in the intersection of the support of f j. Certain homogeneous distributions of order mn are examples of kernels in the class m-czka, ε). For this reason, boundedness properties of operators T with kernels in m-czka, ε) from a product L p L pm into another L p space can only hold when + + = p p m p as dictated by homogeneity. If such boundedness holds for a certain triple of Lebesgue spaces, then the corresponding operator is called multilinear Calderón-Zygmund. A fundamental result concerning these operators is the multilinear extension of the classical Calderón-Zygmund [3]; the linear result states that if an operator with smooth enough kernel is bounded on a certain L r space, then it is of weak type, ) and is also bounded on all L p spaces for < p <. A version of this theorem for operators with kernels in the class m-czka, ε) has been obtained by Grafakos and Torres [25]. A special case of this result was also obtained by Kenig and Stein [29]; both approaches build on previous work by Coifman and Meyer [6]. Theorem 2. [25]) Let T be a multilinear operator with kernel K in m-czka, ε). Assume that for some q,..., q m and some 0 < q < with + + = q q m q, T maps L q L qm L q,. Then T can be extended to a bounded operator from L L into L /m,. Moreover, for some constant C n that depends only on the

7 parameters indicated) we have that MULTILINEAR HARMONIC ANALYSIS 7 9) T L L L /m, C n A + T L q L qm L q, ). Proof. Set B = T L q L qm L q,. Fix an α > 0 and consider functions f j L for j m. Without loss of generality we may assume that f L = = f m L =. Set E α = {x : T f,..., f m )x) > α}. We need to show that there is a constant C = C m,n such that 0) E α CA + B) /m α /m. Once 0) has been established for f j s with norm one, the general case follows by replacing each f j by f j / f j L. Let γ be a positive real number to be determined later. Apply the Calderón-Zygmund decomposition to the function f j at height αγ) /m to obtain good and bad functions g j and b j, and families of cubes {Q j,k } k with disjoint interiors such that and f j = g j + b j b j = k b j,k where supportb j,k ) Q j,k b j,k x)dx = 0 b j,k x) dx Cαγ) /m Q j,k k Q j,k Cαγ) /m b j L C g j L s Cαγ) /ms for all j =, 2,..., m and any s. Define the sets E ={x : T g, g 2,..., g m )x) > α/2 m } E 2 ={x : T b, g 2,..., g m )x) > α/2 m } E 3 ={x : T g, b 2,..., g m )x) > α/2 m }... E 2 m ={x : T b, b 2,..., b m )x) > α/2 m }, where each set E s has the form {x : T h, h 2,..., h m )x) > α/2 m } with h j {g j, b j } and all the sets E s are distinct. Since {x : T f,..., f m )x) > α} 2 m s= E s, it will suffice to prove estimate 0) for each one of the 2 m sets E s.

8 8 LOUKAS GRAFAKOS Chebychev s inequality and the L q L qm L q, boundedness give ) E 2m B) q g α q q L q... g m q L CBq qm α q = C B q α q αγ) m q ) q m = C B q α m γ q m. αγ) j= q mq j Consider now a set E s defined above with 2 s 2 m. Suppose that for some l m we have l bad functions and m l good functions appearing in T h,..., h m ), where h j {g j, b j } and assume that the bad functions appear at the entries j,..., j l. We will show that 2) E s Cα /m γ /m + γ /m Aγ) /l). Let lq) denote the side-length of a cube Q and let Q be a certain dimensional dilate of Q with the same center. Fix an x / m j= k Q j,k ). Also fix for the moment the cubes Q j,k,..., Q jl,k l and without loss of generality suppose that Q j,k has the smallest size among them. Let c j,k be the center of Q j,k. For fixed y j2,..., y jl R n, the mean value property of the function b j,k gives Kx, y,..., y j,..., y m )b j,k y j ) dy j Q j,k = Kx, y,..., y j,..., y m ) Kx, y,..., c j,k,..., y m ) ) b j,k y j ) dy j Q j,k A y j c j,k b j,k y j ) ε Q j,k x y + + x y m ) dy mn+ε j C A lq j,k b j,k y j ) ) ε Q j,k x y + + x y m ) dy mn+ε j, where the previous to last inequality above is due to the fact that y j c j,k c n lq j,k ) 2 x y j 2 max j m x y j. Multiplying the just derived inequality Kx, y)b j,k y j ) dy j Q j C A b j,k y j ) lq j,k ),k Qj,k ε x y + + x y m ) dy mn+ε j

9 by i/ {j,...,j l } estimate 3) MULTILINEAR HARMONIC ANALYSIS 9 g i y i ) and integrating over all y i with i / {j,..., j l }, we obtain the R n ) m l i/ {j,...,j l } C A C A g i y i ) Kx, y)b j,k y j ) dy j dy i/ {j,...,j l } Q j i,k i/ {j,...,j l } A C lq j,k g i L b j,k y j ) ) ε Q j,k l j= x y j ) dy mn m l)n+ε j lq j,k g i L b j,k ) ε L l j= lq i,k i ) + x c i,ki ) ) nl+ε i/ {j,...,j l } i/ {j,...,j l } g i L b j,k L l lq ji,k i ) ε l. lq i,ki ) + x c i,ki ) n+ ε l i= The inequality before the last one is due to the fact that for x / m j= k Q j,k ) and y j Q j,k we have that x y j lq j,kj ) + x c j,kj, while the last inequality is due to our assumption that the cube Q j,k has the smallest side length. It is now a simple consequence of 3) that for x / m j= k Q j,k ) we have T h,..., h m )x) CA g i y i ) R n ) m CA g i L CA i/ {j,...,j l } i/ {j,...,j l } C Aαγ) m l m where i/ {j,...,j l } i= l i= l b ji,k i y ji ) ) k i i=2 lq ji,k i ) ε l lq i,ki ) + x c i,ki ) n+ ε l k i Kx, y ) b j,k y j ) dy j Q j,k R n ) l i=2 l b g i ji,ki lq ) ε ) L L l ji,ki lq i=2 k i,ki ) + x c i,ki ) n+ ε l i l αγ)/m lq ji,ki )n+ ε ) l = C A α γ lq i,ki ) + x c i,ki ) n+ ε l M i,ε/l x) = k i lq ji,k i ) n+ ε l i j dy i l b ji,k i y ji ) ) dy i2... dy il k i lq i,ki ) + x c i,ki ) n+ ε l l M i,ε/l x), is the Marcinkiewicz function associated with the union of the cubes {Q i,ki } k. It is a known fact see for instance [36]) that R n M i,ε/l x) dx C ki Q i,ki C αγ) /m. Now, since m j= k Q j,k ) Cαγ) /m, i=

10 0 LOUKAS GRAFAKOS inequality 2) will be a consequence of the estimate 4) {x / m j= k Q j,k ) : T h,..., h m )x) > α/2 m } Cαγ) /m Aγ) /l. We prove 4) using an L /l estimate outside m j= k Q j,k ) ; recall here that we are considering the situation where l is not zero. Using the size estimate derived above for T h,..., h m )x) outside the exceptional set, we obtain {x / m j= k Q j,k ) : T h,..., h m )x) > α/2 m } Cα /l R n \ mj= kq j,k ) αγa M,ε/l x)... M l,ε/l x) ) /l dx /l ) /l CγA) R /l M,ε/l x)dx)... M l,ε/l x)dx n R n C γa) /l αγ) /m... αγ) /m) /l = C α /m Aγ) /l γ /m, which proves 4) and thus 2). We have now proved 2) for any γ > 0. Selecting γ = A + B) in both ) and 2) we obtain that all the sets E s satisfy 0). Summing over all s 2 m we obtain the conclusion of the theorem. Example Let R be the bilinear Riesz transform in the first variable x y R f, f 2 )x) = p.v. x y, x y 2 ) f y 3 )f 2 y 2 ) dy dy 2. R R Using an m-linear T theorem, it was shown in [25]) that R maps L p R) L p 2 R) to L p R) for /p + /p 2 = /p, < p, p <, /2 < p <. Thus by Theorem 2 it also maps L L to L /2,. However, it does not map L L to any Lorentz space L /2,q for q <. In fact, letting f = f 2 = χ [0,], an easy computation shows that R f, f 2 )x) behaves at infinity like x 2. This fact indicates that in Theorem 2 the space L /2, is best possible and cannot be replaced by any smaller space, in particular, it cannot be replaced by L /2. 4. Endpoint estimates and interpolation for multilinear Calderón-Zygmund operators The theory of multilinear interpolation according to the real method is significantly more complicated than the linear one. Early versions appeared in the work of Janson [27] and Strichartz [37]. In this exposition we will use a version of real multilinear interpolation appearing in [8]. This makes use of the notion of multilinear restricted weak type p,... p m, p) estimates. These are estimates of the form sup λ { x : T χ A,..., χ Am )x) > λ } /p M A /p... A m /pm λ>0 and have a wonderful interpolation property: if an operator T satisfies restricted weak type p,..., p m, p) and q,..., q m, q) estimates with constants M 0 and M,

11 MULTILINEAR HARMONIC ANALYSIS respectively, then it also satisfies a restricted weak type r,..., r m, r) estimate with constant M0 θ M θ, where,...,, ) = θ),...,, ) + θ,...,, ). r r m r p p m p q q m q More refined ideas can be employed to obtain the following multilinear interpolation result; for a precise formulation and a proof see [8]. Theorem 3. Let 0 < p ij, p i, i =,..., m +, j =,..., m, and suppose that the points /p,..., /p m ), /p 2,..., /p m2 ), /p m+),..., /p m+)m ) satisfy a certain nondegeneracy condition. Let /q,..., /q m ) be in the interior of the convex hull of these m + points. Suppose that a multilinear operator T satisfies restricted weak type p i,..., p im, p i ) estimates for i =,..., m +. Then T has a bounded extension from L q L qm L q whenever /q /q + + /q m. There is also an interpolation theorem saying that if a linear operator that satisfies a mild assumption) and its transpose are of restricted weak type, ), then the operator is L 2 bounded. We prove here a multilinear analogue of this result due to Grafakos and Tao [24]: Theorem 4. [24]) Let < p,..., p m < be such that /p + +/p m = /p <. Suppose that an m-linear operator has the property that 5) sup A 0 /p A /p... A m /pm T χ A,..., χ Am ) dx A 0,A,...,A m A < 0 where the supremum is taken over all subsets A 0, A,..., A m of finite measure. Also suppose that T j, j = 0,,..., m are of restricted weak type,,..., /m); this means that these operators map L L to L /m, when restricted to characteristic functions with constants B 0, B,..., B m, respectively. Then there is a constant C p,...,p m such that T maps the product of Lorentz spaces L p, L pm, to weak L p when restricted to characteristic functions with norm at most C p,...,p m B /2p) 0 B /2p )... B /2p m) m. Proof. We will make use of the following characterization of weak L p due to Tao): 6) g L p, sup inf p gt) dt E E E E E < E 2 E The easy proof of 6) is omitted. Let M be the supremum in 5). We consider the following two cases: Case : Suppose that A 0 B0 max A B,..., Am ) Bm. Since T maps L L to weak L /m when restricted to characteristic functions, there exists a subset A 0 of A 0 of measure A 0 A 2 0 such that T χ A,..., χ Am )dx C B 0 A... A m A 0 /m A 0

12 2 LOUKAS GRAFAKOS for some constant C. Then T χ A,..., χ Am )dx A 0 A 0 T χ A,..., χ Am )dx + A 0 \A 0 T χ A,..., χ Am )dx C B 0 A... A m A 0 m+ + M A p... A 2 p 2 2 A 0 ) p CB 0 A B ) p p... A m B ) m p pm m m s= A 0 P p m+ s B0 B0 + M 2 p A p... A m pm A 0 p. It follows that M has to be less than or equal to B ) /p B ) /p C B 0 m m... + M 2 /p B0 B0 and consequently M C 2 /p B /2p) 0 B /2p )... B /2p m) m. Case 2: Suppose that A j max A 0 Bj B0,..., Am ) Bm for some j. To simplify notation, let us take j =. Here we use that T maps L L to weak L /m when restricted to characteristic functions. Then there exists a subset A of A of measure A A 2 such that T χ A0,..., χ Am )dx C B A 0 A 2... A m A m+ A for some constant C. Equivalently, we write this statement as T χ A, χ A2..., χ Am )dx C B A 0 A 2... A m A m+. A 0 by the definition of the first dual operator T. Therefore we obtain T χ A,... )dx T χ A, χ A2,... )dx A 0 A + T χ A \A 0 A, χ A 2,... )dx 0 m ) m CB A 0 A s A m+ + M A 0 p A s ) ps 2 A p s=2 CB A m++ p +P m s=2 p s By the definition of M, it follows that M s=2 B ) 0 m p A 0 p B s=2 A s B ) s p ps s B + M 2 /p A p A 2 p 2... A m pm A 0 p. C 2 /p B/2p) 0 B /2p )... B /2p m) m.

13 MULTILINEAR HARMONIC ANALYSIS 3 Then the statement of the theorem follows with C p,...,p m = C max 2,..., /p 2, ). /pm 2 /p Assumption 5) is not as restrictive as it looks. To apply this theorem for m-linear Calderón-Zygmund operators, one needs to consider the family of operators whose kernels are truncated near the origin, i.e., K δ x, y,..., y m ) = Kx, y,..., y m )ζ x y + + x y m )/δ ), where ζ is a smooth function that is equal to on [2, ) and vanishes on [0, ]. The kernels K δ are essentially in the same Calderón-Zygmund kernel class as K, that is if K lies in m-czka, ε), then K ε lie in m-czka, ε), where A is a multiple of A. Using Hölder s inequality with exponents p,..., p m, p, it is easy to see that for the operators T δ with kernels K δ, assumption 5) holds with constants depending on δ. Theorem 4 provides an interpolation machinery needed to pass from bounds at one point to bounds at every point for multilinear Calderón-Zygmund operators. An alternative interpolation technique was described in [25].) We have: Theorem 5. Suppose that an operator T with kernel in m-czka, δ) and all of its truncations T δ map L r L rm L r for a single tuple of indices r,..., r m, r satisfying /r + + /r m = /r and < r,..., r m, r < uniformly in δ. Then T is bounded from L p L pm L p for all indices p,..., p m, p satisfying /p + + /p m = /p and < p,..., p m <, /m < p <. Proof. Since T δ maps L r L rm L r and r >, duality gives that Tδ maps L r L r 2 L rm L r and likewise for the remaining adjoints uniformly in δ). It follows from Theorems 4 and 3 that T δ are bounded from L p L pm L p for all indices p,..., p m, p satisfying /p + + /p m = /p and < p,..., p m <, /m < p <. Passing to the limit, using Fatou s lemma, the same conclusion may be obtained for the non truncated operator T. This approach has the drawback that it uses the redundant assumption that if T is bounded from L r L r 2 L r, then so are all its truncations T δ uniformly in δ > 0). This is hardly a problem in concrete applications since the kernels of T and T δ satisfy equivalent estimates uniformly in δ > 0) and the method used in the proof of the boundedness of the former almost always applies for the latter. 5. The m-linear Mikhlin-Hörmander multiplier theorem In this section, we focus on an analogue of a classical linear multiplier theorem in the multilinear case. We first note that the Marcinkiewicz multiplier theorem fails for multilinear operators, see [9]. However, the Mikhlin -Hörmander multiplier theorem see [35], [26]) has a multilinear extension, which we discuss below.

14 4 LOUKAS GRAFAKOS The multilinear Fourier multiplier operator T σ associated with a symbol σ is defined by T σ f,..., f m )x) = e 2πix ξ + +ξ m) σξ,..., ξ m ) f ξ ) f m ξ m ) dξ dξ m R n ) m for f i S R n ), i =,, m. Coifman and Meyer [8] proved that if σ is a bounded function on R mn \ {0} that satisfies 7) α ξ αm ξ m σξ,..., ξ m ) C α ξ + + ξ m ) α + + α m ) away from the origin for all sufficiently large multiindices α j, then T σ is bounded from the product L p R n ) L pm R n ) to L p R n ) for all < p,..., p m, p < satisfying p + + p m =. Their proof is based on the idea of writing the Fourier p multiplier σ as a rapidly convergent sum of products of functions of the variables ξ j. The multiplier theorem of Coifman and Meyer was extended to indices p < and larger than /m by Grafakos and Torres [25] and Kenig and Stein [29] when m = 2). A different approach was taken by Tomita [38] who extended the proof of the Hörmander multiplier theorem in [4] to obtain the following result in the m-linear case: Theorem A. [38] Let σ L R mn ). Let Ψ be a Schwartz function whose Fourier transform is supported in the set { ξ R n ) m : /2 ξ 2} and satisfies 8) Ψ ξ/2 j ) = j Z for all ξ R n ) m \{0}. Suppose that for some s > mn/2, the function σ L R mn ) satisfies where for k Z, σ k is defined as sup σ k Ψ L 2 s <. k Z 9) σ k ξ,..., ξ m ) = σ2 k ξ,..., 2 k ξ m ). Then T σ is bounded from L p R n ) L pm R n ) to L p R n ), whenever < p, p 2,..., p m, p < and /p + + /p m = /p. Let S R d ) be the set of all Schwartz functions Ψ on R d, whose Fourier transform is supported in an annulus of the form {ξ : c < ξ < c 2 }, is nonvanishing in a smaller annulus {ξ : c ξ c 2} for some choice of constants 0 < c < c < c 2 < c 2 < ), and satisfies 20) Ψ2 j ξ) = constant, ξ R d \ {0}. j Z Theorem A has an extension to the case where the target space is L p for p :

15 MULTILINEAR HARMONIC ANALYSIS 5 Theorem 6. [22]) Let < r 2. Suppose that σ is a function on R nm and Ψ is a function in S R nm ) that satisfies for some γ > mn r 2) sup σ k Ψ L r γ R mn ) = K <, k Z where σ k is defined in 9). Then there is a number δ = δmn, γ, r) satisfying 0 < δ r, such that the m-linear operator T σ, associated with the multiplier σ, is bounded from L p R n ) L pm R n ) to L p R n ), whenever r δ < p j < for all j =,..., m, and p is given by 22) p = + +. p p m In the rest of this section, we prove Theorem Preliminary material. We develop some preliminary material needed in the proof of Theorem 6. For s R we denote by w s the weight w s x) = + 4π 2 x 2 ) s/2. Definition. For p <, the weighted Lebesgue space L p w s ) is defined as the set of all measurable functions f on R d such that ) /p f L p w s) = fx) p w s x) dx <. R d 23) We note that for < r 2 one has ĝ L r w s) = ) r w s dξ ĝ r R d = ĝ w s/r r R d = via the Hausdorff-Young inequality. ) r dξ R d [ I ) s 2r g ] r dξ R d I ) s 2r g r dx = g L r s/r, Lemma. Let p < q <. Then for every s 0 there exists a constant C = Cp, q, s, d) > 0 such that for all functions g supported in a ball of a fixed finite radius in R d we have g L p s R d ) C g L q s R d ). Proof. Since g is supported in a ball of finite fixed radius, then g = g ϕ for some compactly supported smooth function ϕ that is equal to one on the support of g. Pick r such that /p = /q + /r. ) r ) r

16 6 LOUKAS GRAFAKOS The Kato-Ponce rule [28] gives the estimate g L p s R d ) = I ) s/2 g ϕ) L p C [ I ) s/2 g L q ϕ L r + g L q I ) s/2 ϕ ] L r = C ϕ [ I ) s/2 g L q + g L q]. Now the Bessel potential operator J s = I ) s/2 is bounded from L q to itself for all s > 0. This implies that g L q C I ) s/2 g L q Combining this estimate with the one previously obtained, we deduce that g L p s R d ) 2 C ϕ C I ) s/2 g L q R d ) = C g L q sr d ). Lemma 2. Suppose that s 0 and < r <. Assume that ϕ lies in S R d ). Then there is a constant c ϕ such that for all g L r sr d ) we have g ϕ L r s c ϕ g L r s. Proof. We write I ) s/2 g ϕ) = ϕτ)i ) s/2 g e 2πiτ ) ) dτ. R d It will suffice to show that the L r norm of I ) s/2 g e 2πiτ ) ) is controlled by C M + τ ) M times the L r norm of I ) s/2 g, for some M > 0. This statement is equivalent to showing that the function + ξ τ 2 + ξ 2 is an L r Fourier multiplier with norm at most a multiple of + τ ) M. But this is an easy consequence of the Mihlin multiplier theorem. Lemma 3. Let k be the Littlewood-Paley operator given by k g) ξ) = ĝξ) Ψ2 k ξ), k Z, where Ψ is a Schwartz function whose Fourier transform is supported in the annulus {ξ : 2 b < ξ < 2 b }, for some b Z + and satisfies Ψ2 k Z k ξ) = c 0, for some constant c 0. Let 0 < p <. Then there is a constant c = cn, p, c 0, Ψ), such that for L p functions f we have f L p c ) /2 k f) 2 L. p k Z Proof. Let Φ be a Schwartz function with integral one. Then the following quantity provides a characterization of the H p norm: ) s 2 f H p sup f Φ t L p. t>0 It follows that for f in H p L 2, which is a dense subclass of H p, one has the estimate f sup f Φ t, t>0

17 MULTILINEAR HARMONIC ANALYSIS 7 since the family {Φ t } t>0 is an approximate identity. Thus f L p c f H p whenever f is a function in H p. Keeping this observation in mind we can write: f L p c f H p ) /2 j f) 2 L p j Z = c ) 2) /2 Lp j k f) j Z k Z c ) /2 k f) 2 L p k Z in view of the fact that j k = 0 unless j k b The proof of Theorem 6. Having disposed of the preliminary material, we now prove Theorem 6. Proof. For each j =,..., m, we let R j be the set of points ξ,..., ξ m ) in R n ) m such that ξ j = max{ ξ,..., ξ m }. For j =,..., m, we introduce nonnegative smooth functions φ j on [0, ) m that are supported in [0, 0 ]m such that m ξ = φ j ξ j,..., ξ j ξ j,..., ξ ) m ξ j j= for all ξ,..., ξ m ) 0, with the understanding that the variable with the tilde is missing. These functions introduce a partition of unity of R n ) m \ {0} subordinate to a conical neighborhood of the region R j. Each region R j can be written as the union of sets R j,k = { ξ,..., ξ m ) R j : ξ k ξ s for all s j } over k =,..., m. We need to work with a finer partition of unity, subordinate to each R j,k. To achieve this, for each j, we introduce smooth functions φ j,k on [0, ) m 2 supported in [0, 0 ]m 2 such that m ) = k= k j ξ φ j,k ξ k,..., ξ k ξ k,..., ξ j ξ k,..., ξ m ξ k for all ξ,..., ξ m ) in the support of φ j with ξ k 0 with missing kth and jth entries). We now have obtained the following partition of unity of R n ) m \ {0}: m m = φ j... ) φ j,k... ), j= k= k j where the dots indicate the variables of each function.

18 8 LOUKAS GRAFAKOS We introduce a nonnegative smooth bump ψ supported in [0m), 2] and equal to on the interval [5m), 2], and we decompose the identity on 0 Rn ) m \ {0} as follows m m [ ] = Φj,k + Ψ j,k, j= k= k j where ξk ) ) Φ j,k ξ,..., ξ m ) = φ j... ) φ j,k... ) ψ ξ j and ξk ) Ψ j,k ξ,..., ξ m ) = φ j... ) φ j,k... )ψ. ξ j This partition of unity induces the following decomposition of σ: m m [ ] σ = σ Φj,k + σ Ψ j,k. j= k= k j We will prove the required assertion for each piece of this decomposition, i.e., for the multipliers σ Φ j,k and σ Ψ j,k for each pair j, k) in the previous sum. In view of the symmetry of the decomposition, it suffices to consider the case of a fixed pair j, k) in the previous sum. To simplify notation, we fix the pair m, m ), thus, for the rest of the proof we fix j = m and k = m and we prove boundedness for the m-linear operators whose symbols are σ = σ Φ m,m and σ 2 = σ Ψ m,m. These correspond to the m-linear operators T σ and T σ2, respectively. The important thing to keep in mind is that σ is supported in the set where and σ 2 is supported in the set where max ξ,..., ξ m 2 ) 0 ξ m 0 5m ξ m max ξ,..., ξ m 2 ) 0 ξ m and ξ m 0m ξ m 2. We first consider T σ f,..., f m ), where f j are fixed Schwartz functions. We fix a Schwartz radial function η whose Fourier transform is supported in the annulus ξ 2 and satisfies 25 η2 j ξ) =, ξ R n \{0}. j Z Associated with η we define the Littlewood-Paley operator j f) = f η 2 j, where η t x) = t n ηt x) for t > 0. We decompose the function f m as j Z jf m ) and we note that the spectrum i.e. the Fourier transform) of T σ f,..., f m, j f m )) is contained in the set { ξ : ξ 3 2j 5m } + + { ξm : ξ m 3 2j 5m } + { ξm : j ξ m 2 2 j}

19 MULTILINEAR HARMONIC ANALYSIS 9 This algebraic sum of these sets is contained in the annulus {z R n : j z j }. We now introduce another bump that is equal to on the annulus {z R n 9 65 : z } and vanishes in the complement of the larger annulus {z R n 8 66 : < z < }. We call j the Littlewood-Paley operators associated with this bump and we note that j T σ f,..., j f m ))) = T σ f,..., j f m )). Finally, we define an operator S j by setting S j g) = g ζ 2 j, where ζ is a smooth function whose Fourier transform is equal to on the ball z < 3/5m and vanishes outside the double of this ball. Using this notation, we may write T σ f,..., f m, f m ) = T σ f,..., f m, j f m ) ) j = j = j T σ Sj f ),..., S j f m ), j f m ) ) j Tσ S j f ),..., S j f m ), j f m )) ). Since the Fourier transforms of j Tσ S j f ),..., S j f m ), j f m )) ) have bounded overlap, Lemma 3 yields that [ T σ f,...,, f m ) L p C Tσ Sj f ),..., S j f m ), j f m ) ) 2] 2 Obviously, we have j T σ Sj f ),..., S j f m ), j f m ) ) x) m = e 2πix ξ + +ξ m) σ ξ,..., ξ m ) Ŝ j f k )ξ k ) j f m )ξ m ) dξ dξ m. R n ) m k= A simple calculation yields that the support of the integrand in the previous integral is contained in the annulus { } ξ,..., ξ m ) R n ) m 7 : 0 2j < ξ,..., ξ m ) < 2 0 2j, so one may introduce in the previous integral the factor Ψ2 j ξ,..., 2 j ξ m ), where Ψ is a radial function in S R n ) m ) whose Fourier transform is supported in some annulus and is equal to on the annulus { } z,..., z m ) R n ) m 7 : z 0,..., z m ) 2. 0 L p

20 20 LOUKAS GRAFAKOS Inserting this factor and taking the inverse Fourier transform, we obtain that is equal to T σ Sj f ),..., S j f m ), j f m ) ) x) R n ) m 2 mnj σ j Ψ) 2 j x y ),..., 2 j x y m )) m i= S j f i )y i ) j f m )y m ) d y, where d y = dy... dy m, the check indicates the inverse Fourier transform in all variables, and σξ j, ξ 2,..., ξ m ) = σ 2 j ξ,..., 2 j ξ m ). We pick a ρ such that < ρ < r 2 and γ > mn/ρ. This is possible since γ > mn/r; for instance ρ = mn + mn r ) is a good choice if this number is γ 000 γ bigger than, otherwise we set ρ = +r. We define δ = r ρ. We now have: 2 T σ S j f ),..., S j f m ), j f m ))x) w γ 2 j x y ),..., 2 j x y m ) ) σ j Ψ) 2 j x y ),..., 2 j x y m )) R n ) m 2mnj S j f )y ) S j f m )y m ) j f m )y m ) w γ 2j x y ),..., 2 j x y m ) ) [ wγ σ j Ψ) ) 2 j x y ),..., 2 j x y m )) ] ρ ρ d y R n ) m 2 mnj S j f )y ) S j f m )y m ) j f m )y m ) ρ R n ) m w γρ 2j x y ),... 2 j x y m ) ) d y ) C w γρ y,..., y m ) σ j Ψ) y,..., y m ) ρ ρ d y R n ) m Rn)m 2 mnj S j f )y ) S j f m )y m ) j f m )y m ) ρ d y + 2 j x y ) γρ/m + 2 j x y m ) γρ/m σ j Ψ) L ρ w γρ ) m i= σ j Ψ) m L ρ w γρ ) cm/ρ where we used that R n Rn 2 jn S j f i )y i ) ρ i= a consequence of the fact that γρ/m > n. + 2 j x y i ) γρ/m dy i ) ρ Rn 2 jn j f m )y m ) ρ + 2 j x y m ) dy γρ/m m ) ρ MMf i ) ρ )x)) ρ M j f m ) ρ )x)) ρ, 2 jn hy) dy c Mh)x), + 2 j x y ) γρ/m d y ) ρ ) ρ

21 We now have the sequence of inequalities: MULTILINEAR HARMONIC ANALYSIS 2 σ j Ψ) L ρ w γρ ) σj Ψ L ρ γ C σ j Ψ L r γ C σ j Ψ L r γ < C K, justified by the result in the calculation 23) for the first, Lemma together with the facts that < ρ < r and σ j is supported in a ball of a fixed radius for the second inequality, Lemma 2 for the third, and the hypothesis of Theorem 6 for the last inequality. Thus we have obtained the estimate: T σ S j f ),..., S j f m ), j f m )) C K m i= MMfi ) ρ ) ) ρ M j f m ) ρ )) ρ. We now square the previous expression, we sum over j Z and we take square roots. Since r δ = ρ, the hypothesis p j > r δ implies p j > ρ, and thus each term MMf i ) ρ )) ρ is bounded on L p j R n ). We obtain Tσ f,..., f m, f m ) L p R n ) { C K } T σ S j f ),..., S j f m ), j f m )) 2 2 j C { K M j f m ) ρ ) 2 ρ j C { K M j f m ) ρ ) 2 ρ C K j f i L p ir n ) i= } 2 L pmr n ) i= } ρ 2 ρ L pm/ρ R n ) m L p R n ) M Mf i ) ρ)) ρ m i= f i L p ir n ) L p ir n ) in view of the Fefferman-Stein vector-valued inequality for the Hary-Littlewood maximal function [7] and the Littlewood-Paley theorem. Next we deal with σ 2. Using the notation introduced earlier, we write T σ2 f,..., f m, f m ) = j Z T σ2 f,..., f m, j f m )). The key observation in this case is that T σ2 f,..., f m, j f m )) = T σ2 S j f ),..., S jf m 2 ), jf m ), j f m ) ) for some other Littlewood-Paley operator j which is given on the Fourier transform by multiplication with a bump Θ2 j ξ), where Θ is equal to one on the annulus {ξ R n 24 : ξ 4} and vanishes on a larger annulus. Also, 25 0m S j is given by convolution with ζ 2, where ζ is a smooth function whose Fourier transform is j equal to on the ball z < 22 and vanishes outside the double of this ball. 0

22 22 LOUKAS GRAFAKOS As in the previous case, one has that in the support of the integral T σ2 S j f ),..., S jf m 2 ), jf m ), j f m ) ) x) = e 2πix ξ + +ξ m) σ 2 m 2 ξ ) Ŝ j f t)ξ t ) j f m )ξ m ) j f m )ξ m ) dξ R n ) m we have that t= ξ + + ξ m 2 j, thus one may insert in the integrand the factor Ψ2 j ξ,..., 2 j ξ m ), for some Ψ in S R n ) m ) that is equal to one on a sufficiently wide annulus. A calculation similar to the one in the case for σ yields the estimate T σ2 S jf ),..., S jf m 2 ), jf m ), j f m )) C K m 2 i= MMf i ) ρ )) ρ M j f m ) ρ ) ) ρ M j f m ) ρ )) ρ. Summing over j and taking L p norms yields T σ2 f,...,, f m, f m ) L p R n ) m 2 C K MMf i ) ρ )) ρ M j f m ) ρ)) ρ M j f m ) ρ )) ρ L p C K i= m 2 i= MMf i ) ρ )) ρ j Z { m i=m j Z M j f i ) ρ ) 2 ρ } 2 L p R n ) where the last step follows by the Cauchy-Schwarz inequality and we omitted the prime from the term with i = m for matters of simplicity. Applying Hölder s inequality and using that ρ < 2 and Lemma B we obtain the conclusion that the expression above is bounded by This concludes the proof of the theorem. C K f L p R n ) f m L pmr n ). 6. The multilinear strong maximal function In this section we study the maximal function M R introduced in Example 6 of Section 2. It turns out that this operator can be used to characterize the class of multiple A p weights introduced in [32] suitably modified for rectangles, see [2]. Here, we will be concerned with endpoint boundedness properties of M R. This will require a quick review of some facts from the theory of Orlicz spaces. A Young function is a continuous, convex, increasing function Φ : [0, ) [0, ) with Φ0) = 0 and such that Φt) as t. The properties of Φ easily imply that for 0 < ɛ < and t 0 24) Φɛ t) ɛ Φt).

23 MULTILINEAR HARMONIC ANALYSIS 23 The Φ-norm of a function f over a set E with finite measure is defined by { ) } fx) 25) f Φ,E = inf λ > 0 : Φ dx. E λ It follows from this definition that 26) f Φ,E > if and only if Φ fx) ) dx >. E E Associated with each Young function Φ, there is its complementary Young function 27) Φs) = sup {st Φt)} t>0 for s 0. Such Φ is also a Young function and has the property that [ 28) st C Φt) + Φs) ] for all s, t 0. Also the Φ-norms are related to the L Φ -norms via the the generalized Hölder inequality, namely 29) fx) gx) dx 2 f Φ,E g Φ,E. E E In this section we will work with the pair of Young functions Φ n t) := t[loge + t)] n and Φ n t) Ψ n t) := expt n ), t 0. It is the case that the pair Φ n, Ψ n satisfies 28), see the article by Bagby [], page 887. Observe that the above function Φ n is submultiplicative, that is, for s, t > 0 E Φ n st) c Φ n s) Φ n t). m times {}}{ We introduce the function Φ m) := Φ Φ Φ which is increasing with respect to the input variable and also with respect to m N. 6.. Some Lemmas. We begin by proving some useful general lemmas about averaging functions and Orlicz spaces. Lemma 4. Let Φ be any Young function, then for any f 0 and any measurable set E < f Φ,E f Φ,E Φfx)) dx. E Proof. Indeed, by homogeneity this is equivalent to f Φ,E, λ f,e where But this is the same as λ f,e = Φfx)) dx. E E fx) ) Φ dx E E λ f,e E

24 24 LOUKAS GRAFAKOS by definition of the norm 25). In view of Property 24), it would be enough to show that λ f,e = Φfx)) dx. E E But this is exactly the case in view of Property 26). Lemma 5. Let Φ be a submultiplicative Young function, let m N and let E be any set. Then there is a constant c such that whenever 30) < f i Φ,E holds, then 3) f i Φ,E c i= i= i= Φ m) f i x)) dx. E E Proof. a) The case m =. This is the content of Lemma 4. b) The case m = 2. Fix functions for which 30) holds: 2 < f i Φ,E. i= Without loss of generality we may assume that f Φ,E f 2 Φ,E. Observe that by 30) we must have f 2 Φ,E >. Suppose first that f Φ,E, then 3) follows from Lemma 4: 2 2 < f i Φ,E Φf i x)) dx E i= with m = and c =. Assume now f Φ,E f 2 Φ,E. Then we have by Lemma 4, submultiplicativity and Jensen s inequality 2 < f i Φ,E i= = f Φ,E f 2 Φ,E i= = f f 2 Φ,E Φ,E c Φf x) f 2 Φ,E ) dx E E c Φf x)) dx Φ f 2 Φ,E ) E E c Φf x)) dx Φc Φf 2 x)) dx ) E E E E E

25 MULTILINEAR HARMONIC ANALYSIS 25 c Φf x)) dx Φ 2) f 2 x)) dx E E E E 2 c Φ 2) f i x)) dx, E i= E which is exactly 3). c) The case m 3. By induction, assuming that the result holds for the integer m 2, we will prove it for m. Fix functions for which 30) holds: < f i Φ,E, i= and without loss of generality assume that f Φ,E f 2 Φ,E f m Φ,E. Observe that we must have f m Φ,E >. If we suppose that f Φ,E, then 3) follows directly from Lemma 4: < f i Φ,E Φf i x)) dx E i= with c = and Φ instead of Φ 2). Assume now that for some integer k {, 2,..., m } we have Since i= f Φ,E f 2 Φ,E f k Φ,E f k+ Φ,E f m Φ,E. < f i Φ,E = f Φ,E i= E m i=2 f i Φ,E, we must have m i=2 f i Φ,E >. Using the induction hypothesis we have m m 32) f Φ,E f i Φ,E c f Φ,E Φ m ) f i x)) dx = f R Φ,E, E i=2 i=2 E where R = m i=2 E E Φm ) f i x)) dx. Applying Lemma 4 to the function f R we obtain by submultiplicativity and Jensen s inequality f R Φ,E c Φf x) R) dx E E c Φf x)) dx ΦR) E c E c E E E E Φf x)) dx Φf x)) dx i=2 i=2 Φ E E E E ) Φ m ) f i x)) dx Φ m) f i x)) dx.

26 26 LOUKAS GRAFAKOS Combining this result with 32) we deduce f i Φ,E c E thus proving 3). i= i= E Φ m) f i x)) dx, 6.2. The main result. The previous lemmas are used in the proof of the following result due to Grafakos, Liu, Pérez, and Torres. Theorem 7. [2] There exists a positive constant C depending only on m and n such that for all λ > 0, 33) { {x R n : M R f } )x) > λ m m C i= Φ m) n R n ) fi x) dx λ } /m for all f i on R n and for all i =,..., m. Furthermore, the theorem is sharp in the sense that we cannot replace Φ m) n by Φ n k) for k m. Proof. By homogeneity, positivity of the operator, and the doubling property of Φ n, it is enough to prove 34) { { x R n : M R f } m /m )x) > C Φ m) n f j x)) dx}, R n for some constant C independent of the nonnegative functions f = f,, f m ). Let E = {x R n : M R f )x) > }, then by the continuity property of the Lebesgue measure we can find a compact set K such that K E and j= K E 2 K. Such a compact set K can be covered with a finite collection of rectangles {R j } N j= such that 35) f i y) dy >, j =,, N. R j R j i= We will use the following version of the Córdoba-Fefferman rectangle covering lemma [0] due to Bagby [] Theorem 4. C)): there are dimensional positive constants δ, c and a subfamily { R j } l j= of {R j } N j= satisfying and S l j= e R j exp N R j c δ j= l j= l j= R j, ) n l χ erj x) dx 2 j= R j.

27 MULTILINEAR HARMONIC ANALYSIS 27 Setting Ẽ = l R j= j and recalling that Ψ n t) = expt n ) the latter inequality is l ) Ẽ Ψ n δ χ erj x) dx ee which is equivalent to 36) by the definition of the norm. Now, since j= l χ erj Ψn, j= E e δ E 2 K C Ẽ we can use 35) and Hölder s inequality as follows Ẽ = l j= R j l R j j= l m ) m f i y) dy er j j= m i= l i= j= m i= m = i= S l j= e R j ee j= ) m f i y) dy er j l j= ) m χ erj y)f i y) dy l χ erj y)f i y) dy ) m. By this inequality and 29), we deduce l i= Ẽ χ erj y)f i y) dy ee j= l χ erj Ψn, f i Φn, e i= j= E e E δ f i Φn, E e = i= f i δ i= Φn, e E.

28 28 LOUKAS GRAFAKOS Finally, it is enough to apply Lemma 5 and that Φ m) n the proof of 34). Finally, we turn to the claimed sharpness that one cannot replace Φ m) n is submultiplicative to conclude k m in 33). In the case m = n = 2, we show that the estimate E) { x R 2 : M R f, g)x) > α 2} { ) fx) gx) C dx α α R 2 Φ 2 R 2 Φ 2 by Φ k) n ) } 2 dx cannot hold for α > 0 and functions f, g with a constant C independent of these parameters. For N =, 2,..., consider the functions f = χ [0,] 2 and g N = Nχ [0,] 2 and the parameter α =. Then the left hand side of E) reduces to 0 { x R 2 : M R f, g N )x) > } { = x R 2 : M R χ [0,] 2)x) > 00 0 } N N log N), where the last estimate is a simple calculation concerning the strong maximal function. However, the right hand side of E) is equal to CΦ 2 /α)) /2 Φ 2 N/α)) /2 = CΦ 2 0)) /2 Φ 2 0N)) /2 N log N and obviously it cannot control the left hand side of E) for N large. For general m, the vector f = f,..., f m ) with for also provides a counterexample. f = f 2 = = f m = χ [0,] 2 and f m = Nχ [0,] 2 7. The bilinear Hilbert transform and the method of rotations It is a classical result obtained by Caldeŕon and Zygmund [4] using the method of rotations, that homogeneous linear singular integrals with odd kernels are always L p bounded for < p <. We indicate what happens if the method of rotations is used in the multilinear setting. For an integrable function Ω on S 2n with vanishing integral, we consider the bilinear operator 37) T Ω f, f 2 )x) = R 2n Ωy, y 2 )/ y, y 2 ) ) y, y 2 ) 2n f x y )f 2 x y 2 ) dy dy 2. Suppose that Ω is an odd function on S 2n. Using polar coordinates in R 2n we express { + T Ω f, f 2 )x) = Ωθ, θ 2 ) f x tθ )f 2 x tθ 2 ) dt } dθ, θ 2 ). S 2n 0 t Replacing θ, θ 2 ) by θ, θ 2 ), changing variables, and using that Ω is odd we obtain { + T Ω f, f 2 )x) = Ωθ, θ 2 ) f x + tθ )f 2 x + tθ 2 ) dt } dθ, θ 2 ) S 2n 0 t

29 and averaging these identities we deduce that T Ω f, f 2 )x) = { + Ωθ, θ 2 ) 2 S 2n MULTILINEAR HARMONIC ANALYSIS 29 f x tθ )f 2 x tθ 2 ) dt t } dθ, θ 2 ). The method of rotations gives rise to the operator inside the curly brackets above and one would like to know that this operator is bounded from a product of two Lebesgue spaces into another Lebesgue space and preferably) uniformly bounded in θ, θ 2. Motivated by this calculation, for vectors u, v R n we introduce the family of operators + H u,v f, f 2 )x) = p.v. f x tu)f 2 x tv) dt t We call this operator the directional bilinear Hilbert transform in the direction indicated by the vector u, v) in R 2n ). In the special case n =, we use the notation H α,β f, g)x) = p.v. + fx αt)gx βt) dt t for the bilinear Hilbert transform defined for functions f, g on the line and x, α, β R. We mention results concerning the boundedness of these operators. The operator H α,β was first shown to be bounded by Lacey and Thiele [30], [3] in the range 38) < p, q, 2/3 < r <, /p + /q = /r. Uniform L r bounds in α, β) for H α,β were obtained by Grafakos and Li [20] in the local L 2 case, i.e the case when 2 < p, q, r < ) and extended by Li [33] in the hexagonal region 39) < p, q, r <, p < q 2, p < r 2, q < r 2. We use an idea similar to that Calderón used to express the first commutator C as an average of the bilinear Hilbert transforms as in 2), to obtain new bounds for a higher dimensional commutator introduced by Christ and Journé [5]. The n- dimensional commutator is defined as 40) C n) f, a)x) = p.v. Kx y) fy)a t)x + ty) dt dy R n 0 where Kx) is a Calderón-Zygmund kernel in dimension n and f, a are functions on R n. Christ and Journé [5] proved that C n) is bounded from L p R n ) L R n ) to L p R n ) for < p <. Here we discuss some off-diagonal bounds L p L q L r, whenever /p + /q = /r and < p, q, r <. As the operator C n) f, a) is n-dimensional, we will need to transfer H α,β in higher dimensions. To achieve this we use rotations. We have the following lemma: Lemma 6. Suppose that K is kernel in R 2n which may be a distribution) and let T K be the bilinear singular integral operator associated with K T K f, g)x) = Kx y, x z)fy)gz) dy dz.