MULTILINEAR CALDER ON-ZYGMUND THEORY

5 downloads 0 Views 409KB Size Report
A systematic treatment of multilinear Calder on-Zygmund operators is ... on products of Lp spaces in the spirit of the celebrated theorem of David and Journ e.
MULTILINEAR CALDERO N-ZYGMUND THEORY LOUKAS GRAFAKOS AND RODOLFO H. TORRES Abstract. A systematic treatment of multilinear Calderon-Zygmund operators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T 1 theorem, and a variety of results regarding multilinear multiplier operators.

1. Introduction The classical Calderon-Zygmund theory and its rami cations have proved to be a powerful tool in many aspects of harmonic analysis and partial di erential equations. The main thrust of the theory is provided by the Calderon-Zygmund decomposition, whose impact is deep and far-reaching. This decomposition is a crucial tool in obtaining weak type (1; 1) estimates and consequently Lp bounds for a variety of operators acting on function spaces on Rn and taking values in some Banach spaces. The Littlewood-Paley theory, with its incisive characterizations of function spaces on Rn, can also be obtained as a consequence of the Calderon-Zygmund theory. The realization of pseudodi erential operators in terms of singular integrals of CalderonZygmund type is yet another accomplishment of the theory that brings up intimate connections with operator theory and partial di erential equations. The study of multilinear operators is not motivated by a mere quest to generalize the theory of linear operators but rather by their natural appearance in analysis. Coifman and Meyer were one of the rst to adopt a multilinear point of view in their study of certain singular integral operators, such as the Calderon commutators, paraproducts, and pseudodi erential operators. The remarkable proof of the boundedness of the bilinear Hilbert transform by Lacey and Thiele [16], [17] provides, in our view, a further motivation for the systematic development and study of multilinear singular integrals. Within this framework, the bilinear Hilbert transform naturally arises in the context of the bilinear method of rotations, as described at the end of this article. In this work we prove a variety of theorems regarding what we call multilinear Calderon-Zygmund operators. Their name is justi ed by the fact that these operators have kernels which satisfy standard estimates and bear boundedness properties analogous to those of the classical linear ones. Particular examples of these operators 1991 Mathematics Subject Classi cation. Primary 42B20. Secondary 42B15, 46B70, 47G30. Key words and phrases. Calderon-Zygmund singular integrals, multilinear operators, multipliers, interpolation, T 1 Theorem, pseudodi erential operators. Grafakos' research partially supported by the NSF under grant DMS 9623120. Torres' research partially supported by the NSF under grant DMS 9696267. 1

2

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

have been previously studied by Coifman and Meyer [6], [7], [8], [9], [18], assuming sucient smoothness on their symbols and kernels. Our approach provides a systematic treatment of general m-linear Calderon-Zygmund operators on Rn under minimal smoothness assumptions on their kernels. Our rst result, Theorem 1, is concerned with the natural weak type endpoint estimate of Calderon-Zygmund operators on the m-fold product of L1 spaces. This theorem improves on results of Coifman and Meyer [6], [7], and Kenig and Stein [15]. A further re nement of Theorem 1, needed for interpolation purposes, is presented in Theorem 2. Next, we show that boundedness of multilinear Calderon-Zygmund operators on one product of Lp spaces implies boundedness on all suitable products of Lebesgue spaces. The precise statement of this result is given in Theorem 3. The situation is then in complete analogy with the linear case, where boundedness on just one Lp space implies boundedness on all Lp spaces for 1 < p < 1; the characteristic feature of Calderon-Zygmund operators. Other results we discuss include the multilinear version of the Peetre-Spanne-Stein theorem [19], [20], [21], on the action of singular integrals on L1 and the multilinear T 1 theorem, Theorem 4. This last result provides a powerful characterization of boundedness of multilinear singular integrals on products of Lp spaces in the spirit of the celebrated theorem of David and Journe [10]. Our characterization says that an m-linear Calderon-Zygmund singular integral operator T is bounded on products of Lebesgue spaces if and only if supn : : : sup n kT (e2i1 (  ); : : : ; e2im (  ) )kBMO < 1; 1 2R

m 2R

and similarly for the m transposes of T . A di erent formulation of a T 1 theorem for multilinear forms was given before by Christ and Journe [5]. We apply our version of the multilinear T 1 theorem to obtain some new continuity results for multilinear translation invariant operators and multilinear pseudodi erential operators. We devote the last section of this article to a further analysis of multilinear multipliers. 2. Notation and preliminaries We will be working on n-dimensional space Rn. We denote by S (Rn ) the space of all Schwartz functions on Rn and by S 0 (Rn) its dual space, the set of all tempered distributions on Rn. Similarly we denote by D(Rn) the set of all C 1 functions with compact support on Rn and by D0(Rn) the set of all distributions on Rn. We denote by Lp = Lp(Rn) the classical Lebesgue spaces of measurable functions whose modulus to the pth power is integrable, with the usual modi cation when p = 1. We also denote by Lp;q = Lp;q (Rn) the Lorentz spaces de ned by (1)

8Z 1  1  > q dt q 1  < t p f (t) t kf kLp;q = > 0 1 t p f (t) :sup t>0

if q < 1 and 0 < p  1, if q = 1 and 0 < p  1,

where f  is the nonincreasing rearrangement of f on (0; 1). Clearly Lp;p = Lp and Lp;1 = weak Lp. BMO = BMO(Rn) denotes the usual space of functions with

MULTILINEAR CALDERO N-ZYGMUND THEORY

3

bounded mean oscillation. We use the notation p0 = p=(p ? 1) for 1 < p < 1, 10 = 1, and 10 = 1. We will use the following de nition for the Fourier transform in n-dimensional euclidean space Z bf () = f (x)e?2ix dx; Rn

while f _( ) = fb(? ) will denote the inverse Fourier transform. The action of a distribution u on a test function f will be denoted by hu ; f i : Let T be an m-linear operator from S (Rn)   S (Rn) into S 0 (Rn) which is continuous with respect to the natural topologies of these spaces. A version of the Schwartz kernel theorem (c.f. [13]) gives that any such T has a kernel K , which is a tempered distribution on (Rn)m+1 , such that for all f1 ; : : : ; fm; g in S (Rn ) (2) hT (f1; : : : ; fm); gi = hK ; g f1    fm i: Here g f1    fm denotes the function (x; y1; : : : ; ym) ! g(x)f1(y1) : : : fm (ym): Conversely every tempered distribution K on (Rn)m+1 de nes a continuous m-linear map from S (Rn)      S (Rn) into S 0 (Rn) whose kernel is K . We will occasionally write K (x; y1; : : : ; ym) for the distribution K to indicate the variables on which it acts. In this work we study m-linear operators de ned on products of test functions and we seek conditions to extend them as bounded operators on certain products of Banach spaces. We will use the notation kT kX1Xm !X = sup kT (f1; : : : ; fm)kX kf kXj =1 1j m

to denote the norm of an m-linear operator T from a product of Banach spaces of functions X1      Xm into a quasi-Banach space X . We say that T is bounded from X1      Xm into X when the norm above is nite. An m-linear operator T : S (Rn)      S (Rn) ! S 0 (Rn) is linear in every entry and consequently it has m formal transposes. The j th transpose T j of T is de ned via hT j (f1 ; : : : ; fm ) ; hi = hT (f1; : : : ; fj?1; h; fj+1; : : : ; fm) ; fj i; for all f1; : : : ; fm ; g in S (Rn). It is easy to check that the kernel K j of T j is related to the kernel K of T via (3) K j (x; y1; : : : ; yj?1; yj ; yj+1; : : : ; ym) = K (yj ; y1; : : : ; yj?1; x; yj+1; : : : ; ym): Note that if a multilinear operator T maps a product of Banach spaces X1    Xm into another Banach space X , then the transpose T j maps the product of Banach spaces X1  : : : Xj?1  X   Xj+1    Xm into Xj. Moreover, the norms of T and T j are equal. It is sometimes customary to work with the adjoints of an m-linear operator T whose kernels are the complex conjugates of the kernels K j de ned above. In this

4

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

paper we choose to work with the transposes, as de ned above, to simplify the notation. This choice presents no di erences in the study of these operators. Let K (x; y1; : : : ; ym) be a locally integrable function de ned away from the diagonal x = y1 =    = ym in (Rn)m+1, which satis es the size estimate jK (x; y1; : : : ; ym)j  (jx ? y j +   A+ jx ? y j)nm (4) 1 m n m +1 for some A > 0 and all (x; y1; : : : ; ym) 2 (R ) with x 6= yj for some j . Furthermore, assume that for some " > 0 we have the smoothness estimates jK (x; y1; : : : ; yj ; : : : ; ym) ? K (x0 ; y1; : : : ; yj ; : : : ; ym)j 0" (5)  (jx ? y j +Ajx +? jxxj? y j)nm+" 1 m 1 whenever jx ? x0 j  2 max1jn jx ? yj j and also that for each j , jK (x; y1; : : : ; yj ; : : : ; ym) ? K (x; y1; : : : ; yj0 ; : : : ; ym)j (6) Ajy ? y0 j"  (jx ? y j +   j + jxj ? y j)nm+" 1 m 1 0 whenever jyj ? yj j  2 max1jn jx ? yj j. Note that condition (5) is a regularity condition for K j de ned in (3) in terms of K . For convenience in the notation, we will assume a more symmetric form of the above estimates. It is easy to see that with an appropriate constant cn;m > 0, condition (4) can also be written as A (7) jK (y0; y1; : : : ; ym)j  (Pm cn;m mn ; k;l=0 jyk ? yl j) while conditions (5) and (6) follow from the more concise estimate c Ajy ? y0 j" (8) jK (y0; : : : ; yj ; : : : ; ym) ? K (y0; : : : ; yj0 ; : : : ; ym)j  (Pmn;m jy j? y jj)mn+" ; l k;l=0 k whenever 0  j  m and jyj ? yj0 j  21 max0km jyj ? yk j. We also note that condition (8) with " = 1 is a consequence of jrK (y0; y1; : : : ; ym)j  (Pm jy A? y j)nm+1 ; l k;l=0 k where r denotes the gradient in all possible variables. We will reserve the letter A for the constant that appears in the size and regularity estimates of K . (If these numbers are di erent, we will take A to be the largest of all these constants.) In this article we study m-linear operators T : S (Rn )      S (Rn ) ! S 0 (Rn) for which there is a function K de ned away from the diagonal x = y1 =    = ym in (Rn)m+1 satisfying (7) and (8) and such that (9)

T (f1; : : : ; fm )(x) =

Z

(Rn )m

K (x; y1; : : : ; ym)f1(y1) : : : fm (ym) dy1 : : : dym;

MULTILINEAR CALDERO N-ZYGMUND THEORY

5

whenever f1; : : : ; fm 2 D(Rn) and x 2= \mj=1 supp fj . Let Ke be the Schwartz kernel of T . Note that if (7), (8), and (9) are satis ed, then for f1 ; : : : ; fm; g in D(Rn) with \mj=1supp fj \ supp g = ;, we have that

e g f1    fm i = hK;

Z Z

Rn (Rn )m

K (x; y1; : : : ; ym)f (y1) : : : fm(ym ) dy1 : : : dym dx;

as an absolutely convergent integral. For this reason, we will say that T is an m-linear operator with Calderon-Zygmund kernel K . The class of all functions satisfying (7) and (8) with parameters m, A, and " will be denoted by m-CZK (A; "). We plan to investigate boundedness properties of operators T with kernels in the class m-CZK (A; ") from a product of Lp spaces into another Lebesgue space. Since kernels satisfying condition (4) include certain distributions which are homogeneous of degree ?mn, if the corresponding operator maps Lp1      Lpm ! Lp, then the equation 1 ++ 1 = 1 p1 pm p must hold, as dictated by homogeneity. For notational convenience, we will occasionally write (y1; : : : ; ym) = ~y; K (x; y1; : : : ym) = K (x; ~y); dy1 : : : dym = d~y: As in the linear case, the smoothness assumption on the kernel allows us to extend the action of an operator T with kernel in m-CZK (A; ") to functions in (C 1 \ L1). To achieve this, let us x a C 1 function supported in the ball of radius two in Rn and satisfying 0  (x)  1 and (x) = 1 when 0  jxj  1. Let k (x) = (2?k x). We have the following. Lemma 1. Every multilinear operator T with kernel K in m-CZK (A; ") can be extended to (C 1 \ L1)      (C 1 \ L1 ) as an element of D0(Rn ) via ?T ( f ; : : : ; f ) + G( f ; : : : ; f ); T (f1 ; : : : ; fm ) = klim k 1 k m k 1 k m !1 where Z G( k f1 ; : : : ; k fm ) = ? K (0; ~y)( k f1)(y1) : : : ( k fm )(ym) d~y; jy j>1 1min j m j

and the limit above is taken in the weak -topology of D0(Rn ). Proof. Let fj 2 L1 \ C 1 for 1  j  m. Set Fk = T ( k f1; : : : ; k fm ) + G( k f1; : : : ; k fm): 0 n Since D (R ) is sequentially complete, it is enough to show that for each function g 2 D(Rn) the limk!1hFk ; gi exists. Let B (0; R) be the ball centered at zero of radius R > 0. Fix g 2 D(Rn), select a positive integer k0 so that supp g  B (0; 2k0 ), and write for k > k0 hFk ; gi = hFk0 ; gi + hFk ? Fk0 ; gi:

6

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

Any multilinear operator L satis es the identity (10)

L(f1 ; : : : ; fm) ? L(h1 ; : : : ; hm ) =

m X j =1

L(h1; : : : ; hj?1; fj ? hj ; fj+1; : : : ; fm);

with the obvious interpretations when j = 1 or j = m. Using (10) we can write hFk ? Fk0 ; gi = m X

(11)

j =1 m

hT ( k0 f1 ; : : : ;

XZ

j =1 Rn

k0 fj ?1 ; ( k ? k0 )fj ; k fj +1; : : : ; k fm ); g i +

G( k0 f1; : : : ;

k0 fj ?1; ( k ? k0 )fj ; k fj +1 ; : : : ; k fm )g (x) dx:

We need to show that each term in (11) has a limit as k ! 1. Note that for all k  k0 , \mj=1supp ( k ? k0 )fj \ supp g = ;; so using (9), we control the two terms for j = 1 in (11) by

Z Z

Z

Rn 2min jy j>1 jy1 j>2k0 lm l

+

j( Zk (y1) ?

Z Z

Rn 2min jy j1 jy1 j>2k0 lm l

j( k (y1) ?

jK (x; ~y) ? K (0; ~y)j k0 (y1 ))f1 (y1 ) k (y2 )f2 (y2 )    k (ym )fm (ym )g (x)j d~y dx

jK (x; ~y)j k0 (y1 ))f1 (y1 ) k (y2 )f2 (y2 )    k (ym )fm (ym )g (x)j d~ydx;

and similarly for j  2. Using the smoothness and size conditions on the kernel, it is an easy consequence of the Lebesgue dominated convergence theorem that (11) converges to m Z Z X

(12)

+

Z

?K (x; ~y) ? K (0; ~y)

jy j>1 jyj j>2k0 j =1 Rn 1min lm l l6=j (1 ? k0 (yj ))f1(y1)f2(y2)    fm(ym )g(x) d~y dx m

XZ Z

Z

K (x; ~y)

jy j>1 jyj j>2k0 j =1 Rn 1min lm l l6=j (1 ? k0 (yj ))f1(y1)f2(y2)    fm(ym )g(x) d~ydx;

as k ! 1. We conclude that hFk ; gi = hFk0 ; gi + hFk ? Fk0 ; gi has a limit as k ! 1.

MULTILINEAR CALDERO N-ZYGMUND THEORY

7

If the functions fj have compact support, then by choosing k0 large enough (12) becomes zero. Thus for f1 ; : : : ; fm in D(Rn) the actual value of T (f1 ; : : : ; fm ) is di erent from the value given in the above lemma by the constant (13)

G(f1 ; : : : ; fm) = ?

Z

jy j>1 1min j m j

K (0; ~y)f1(y1) : : : fm(ym ) d~y:

This slight discrepancy, however, will cause no ambiguities when T (f1; : : : ; fm) is seen as an element of BMO. We will later need to apply induction on the degree m of multilinearity of an operator T . We will then need to consider (m ? 1)-linear operators obtained by freezing one of the functions on which T acts. The following lemma will be useful for this purpose. Lemma 2. Let K be in m-CZK (A; "), let fm 2 L1, and for (x; y1; : : : ; ym?1) not in the diagonal of (Rn)m de ne (14)

Kfm (x; y1; : : : ; ym?1) =

Z

Rn

K (x; y1; : : : ; ym?1; ym)fm (ym) dym:

Then for some constant cn;m > 0 we have that K is in (m?1)-CZK (cn;mkfm kL1 A; "). Proof. Using estimate (4) we obtain

Z

(jx ? y1j +    + jx ? ymj)?nm dym jKfm (x; y1; : : : ; ym?1)j  k k Rn  cn;mkfmkL1 A(jx ? y1j +    + jx ? ym?1j)?n(m?1) ;

f m L1 A

which gives the size estimate for Kfm . To verify the smoothness conditions, set x = y0 and assume that for some 0  j  m ? 1 we have 1 max jy ? y j: jyj ? yj0 j  21 0max j y ? y j  j k km?1 2 0km j k Then, using (8), we obtain jKfm (y0; : : : ; yj ; : : : ; ym?1) ? Kfm (y0; : : : ; yj0 ; : : : ; ym?1)j Z 0" kfmkL1 n (Pcmn;mAjyjyj??yyjj)jmn+" dym ZR k;l=0 ckn;mAljyj ? yj0 j" kfmkL1 n ? Pm?1 jy ? y jmn+" dym R jy0 ? ym j + k;l l =0 k 0 " Ajyj ? yj j kfmkL1 ? Pm?cn;m (m?1)n+" : 1 k;l=0 jyk ? yl j

8

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

By symmetry, Lemma 2 is also true if we freeze any other variable in K instead of ym . Moreover, given an m{linear operator T and a xed function fj for some 1  j  m, we can construct the following (m ? 1){linear operator (15) Tfj (f1; : : : ; fj?1; fj+1; : : : ; fm) = T (f1; : : : ; fj?1; fj ; fj+1; : : : ; fm ): It is easy to check that the transposes of the operators de ned this way are (Tfj )k =(T k )fj ; when k = 1; : : : ; j ? 1; (16)  k  ( k +1) (Tfj ) =(T ) fj ; when k = j; : : : ; m ? 1: Lemma 3. Assume that T is a multilinear operator with kernel K in m-CZK (A; ") which extends to a bounded operator from Lp1      Lpm into Lp for some indices 1  p1; : : : ; pm; p < 1 and 1  pm  1. Fix a compactly supported bounded function fm and let Tfm be as in (15). Then Tfm is an (m ? 1)-linear operator with CalderonZygmund kernel Kfm given by (14). Proof. Let f1 ; : : : ; fm?1 in D and let fm be a compactly supported function in L1 . We need to show that Tfm (f1 ; : : : ; fm?1)(x) Z (17) = K (x; y ; : : : ; y )f (y ) : : : f (y ) dy : : : dy (Rn )m?1 \jm=1?1 supp fj .

fm

1

m?1 1 1

m?1 m?1

1

m?1

for x 2= We will ?prove (17) by testing against smooth functions h supported in the complement of \jm=1?1 supp fj . Duality gives (18) hT (f1; : : : ; fm?1; fm ); hi = hT m(f1; : : : ; fm?1 ; h); fmi; where T m (f1; : : : ; fm?1 ; h) is a well-de ned function in Lp0m , if pm < 1, and in L1 , if pm = 1. Moreover, since \jm=1?1 supp fj \ supp h = ;, this function is given by the absolutely convergent integral

z! for all z

Z

(Rn )m 2 Rn. It

Z Z

Rn (Rn )m?1

K (x; y1 ; : : : ; ym?1 ; z)f1(y1) : : : fm?1 (ym?1)h(x) dy1 : : : dym?1dx follows that (18) is given by the absolutely convergent integral

Kfm (x; y1; : : : ; ym?1)f1(y1) : : : fm?1 (ym?1) dy1 : : : dym?1 h(x) dx

which implies (17). Remark. If fm does not have compact support, then we cannot conclude from (9) that Tfm is associated to the Calderon-Zygmund kernel Kfm as de ned in (14) (though the function Kfm satis es the right estimates). We therefore choose to work with L1 functions with compact support when we consider the operators Tfm , to ensure that their kernels are indeed given by (14) and are a fortiori in the class (m ? 1)-CZK (kfmkL1 A; "). We also observe that if an m-linear operator with kernel in m-CZK (A; ") extends to a bounded operator on a product of Lpj spaces with pj < 1, then the integral

MULTILINEAR CALDERO N-ZYGMUND THEORY

9

representation (9) still holds for compactly supported and bounded functions fj . This last statement can be easily shown using an elementary limiting argument. 3. An endpoint weak type estimate The Calderon-Zygmund decomposition is the key tool used in obtaining weak type (1; 1) boundedness for classical linear singular integral operators. In this section we use this decomposition to obtain endpoint weak type results for the multilinear operators discussed in Section 1. Since L1 is a natural endpoint space for boundedness of singular integrals in the scale of Lp spaces (1 < p < 1), it is not surprising that the corresponding endpoint result for m-linear operators is attained on the m-fold product L1      L1 . By homogeneity this product should be mapped into L1=m;1 . In fact, for operators given by homogeneous kernels, such weak type estimates have been recently obtained by Kenig and Stein [15], building on previous work by Coifman and Meyer [6]. The theorem bellow is sharp in the sense that the space L1=m;1 cannot be replaced by L1=m , as indicated by an example given in Section 6. Theorem 1. Let T be a multilinear operator with kernel K in m-CZK (A; "). Assume that for some 1  q1 ; q2 ; : : : ; qm  1 and some 0 < q < 1 with 1 + 1 +    + 1 = 1; q1 q2 qm q T maps Lq1      Lqm into Lq;1. Then T can be extended to a bounded operator from the m-fold product L1    L1 into L1=m;1 . Moreover, for some constant Cn;m (that depends only on the parameters indicated) we have that  ? kT kL1L1!L1=m;1  Cn;m A + kT kLq1 Lqm !Lq;1 : (19) Proof. Set B = kT kLq1 Lqm !Lq;1 . Fix an > 0 and consider functions fj 2 L1 for 1  j  m. Without loss of generality we may assume that kf1kL1 =    = kfm kL1 = 1. Setting E = fx : jT (f1; : : : ; fm)(x)j > g, we need to show that for some constant C = Cm;n we have (20) jE j  C (A + B )1=m ?1=m : (Once (20) has been established for fj 's with norm one, the general case follows immediately by scaling.) Let be a positive real number to be determined later. Apply the Calderon-Zygmund decomposition to the function fj at height ( )1=m to obtain `good' and `bad' functions gj and bj , and families of cubes fQj;k gk with disjoint interiors such that fj = g j + b j and X bj = bj;k k

10

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

where support(bj;k )  Qj;k

Z

Z

bj;k (x)dx = 0

jbj;k (x)jdx  C ( )1=mjQj;k j j [k Qj;k j  C ( )?1=m kbj kL1  C kgj kLs  C ( )1=ms0 for all j = 1; 2; : : : ; m and any 1  s  1; (s0 is here the dual exponent of s). Now let

E1 =fx : jT (g1; g2; : : : ; gm)(x)j > =2mg E2 =fx : jT (b1; g2; : : : ; gm)(x)j > =2mg E3 =fx : jT (g1; b2 ; : : : ; gm)(x)j > =2mg ::: E2m =fx : jT (b1; b2 ; : : : ; bm)(x)j > =2mg; where each Es = fx : jT (h1; h2; : : : ; hm)(x)j > =2mg with hPj 2m fgj ; bj g and all the sets Es are distinct. Since jfx : jT (f1; : : : ; fm )(x)j > gj  2s=1 jEsj, it will suce to prove estimate (20) for each of the 2m sets Es. Let us start with set E1 which is the easiest. Chebychev's inequality and the Lq1      Lqm ! Lq;1 boundedness give

m q m B )q qY (2 CB q q jE1j  q kg1kLq1 : : : kgmkLqm  q ( ) mqj0 j =1 (21) 0 q q 1 1 1 = C Bq ( )(m? q ) m = C 0B q ? m q? m : Consider a set Es as above with 2  s  2m. Suppose that for some 1  l  m we have l bad functions and m ? l good functions appearing in T (h1 ; : : : ; hm ), where hj 2 fgj ; bj g and assume that the bad functions appear at the entries j1 ; : : : ; jl . We will show that

(22)

?



jEsj  C ?1=m ?1=m + ?1=m (A )1=l :

Let l(Q) 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 2= [mj=1 [k (Qj;k ) . Also x for the moment the cubes Qj1 ;k1 ; : : : ; Qjl;kl and without loss of generality suppose that Qj1;k1 has the smallest size among them. Let cj1;k1 be the center of Qj1;k1 . For xed yj2 ; : : : ; yjl 2 Rn,

MULTILINEAR CALDERO N-ZYGMUND THEORY

11

the mean value property of the function bj1 ;k1 gives

Z Qj1;k1 K (x; y1; : : : ; yj1 ; : : : ; ym)bj1;k1 (yj1 ) dyj1 Z ?  K (x; y1; : : : ; yj1 ; : : : ; ym) ? K (x; y1; : : : ; cj1;k1 ; : : : ; ym) bj1;k1 (yj1 ) dyj1 = Z Qj1;k1 A jyj ? cj ;k j"



jbj1;k1 (yj1 )j (jx ? y j +  1  + jx1 ?1 y j)mn+" dyj1 1 m ZQj1;k1 "  jbj1;k1 (yj1 )j (jx ? y j C+A l(+Qjj1x;k1?) y j)mn+" dyj1 ; m

1

Qj1 ;k1

where the previous to last inequality above is due to the fact that

jx ? yj j: jyj1 ? cj1;k1 j  cn l(Qj1 ;k1 )  21 jx ? yj1 j  21 1max j m Multiplying the just derived inequality

Z Z C A jbj1;k1 (yj1 )j l(Qj1;k1 )" dy  K ( x; ~ y ) b ( y ) dy j1 ;k1 j1 j1 j1 Qj1;k1 Qj1 ;k1 (jx ? y1 j +    + jx ? ym j)mn+" Q jg (y )j and integrating over all y with i 2= fj ; : : : ; j g, we obtain the

by i=2fj1 ;:::;jlg estimate

i i

1

l

Z Y dyi j gi(yi)j K (x; ~y)bj1 ;k1 (yj1 ) dyj1 (Rn )m?l i=2fj1 ;:::;jl g Qj1 ;k1 i=2fj1 ;:::;jlg Z " Y 1 ;k1 )  kgikL1 jbj1;k1 (yj1 )j Pl A C l(Qjmn ?(m?l)n+" dyj1 ( j x ? y j ) Q j j ;k j =1 1 1 i=2fj1 ;:::;jlg Y l(Qj1 ;k1 )" C A kgikL1 kbj1;k1 kL1 ? Pl nl+" ( l ( Q ) + j x ? c j ) i;k i;k i i i=2fj1 ;:::;jl g j =1 " l Y Y l(Qji;ki ) l Z

(23)

i

C A

Y

kgikL1 kbj1;k1 kL1

i=2fj1 ;:::;jl g

i=1

(l(Qi;ki ) + jx ? ci;ki j)n+ "l

:

The penultimate inequality above is due to the fact that for x 2= [mj=1 [k (Qj;k ) and yj 2 Qj;k we have that jx ? yj j  l(Qj;kj ) + jx ? cj;kj j, while the last inequality is due to our assumption that the cube Qj1;k1 has the smallest side-length. It is now a

12

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

simple consequence of (23) that for x 2= [mj=1 [k (Qj;k ) we have jT (h1; : : : ; hm)(x)j

Z Y  jbji;ki (yji )j K (x; ~y ) bj1;k1 (yj1 ) dyj1 dyi CA n m?1 jgi(yi)j Qj1 ;k1 (R ) i=2fj1 ;:::;jl g i=2 ki i6=j1 Z " Y Yl Yl ? X  l(Qji;ki ) l Z

CA

Y

kgikL1

i=2fj1 ;:::;jl g

Yl ? X

n+ " i=1 (l(Qi;ki ) + jx ? ci;ki j) l (Rn )l?1 i=2

Y Y  X kbji;ki kL1 l(Qji;ki ) "l  CA kgikL1 n+ " i=2 ki (l(Qi;ki ) + jx ? ci;ki j) l i=2fj1 ;:::;jlg  l  m?l Y X ( )1=m l(Q )n+ "l C 0 A( ) m

where

l

Y = C 00A M

ji ;ki ( l ( Q ) + j x ? ci;ki j)n+ "l i;ki ki

i=1

ki

jbji;ki (yji )j dyi2 : : : dyil

l

i=1

i;"=l (x);

X

n+ " l ( Q ji ;ki ) l Mi;"=l (x) = n+ "l ki (l(Qi;ki ) + jx ? ci;ki j) is the Marcinkiewicz function associated with the union of the cubes fQi;ki gk . It is a known fact [22] that

Z

Now, since

Rn

Mi;"=l (x) dx  C j [ki Qi;ki j  C 0 ( )?1=m:

j [mj=1 [k (Qj;k) j  C ( )?1=m ;

inequality (22) will be a consequence of the estimate (24) jfx 2= [mj=1 [k (Qj;k ) : jT (h1; : : : ; hm)(x)j > =2mgj  C ( )?1=m(A )1=l : We prove (24) using an L1=l estimate outside [mj=1 [k (Qj;k ) ; recall here that we are considering the situation where l is not zero. Using the size estimate derived above for jT (h1; : : : ; hm )(x)j outside the exceptional set, we obtain jfx 2= [mj=1 [k (Qj;k ) : jT (h1; : : : ; hm)(x)j > =2mgj

C ?1=l

Z

Rn ?[mj=1 [k (Qj;k )

C ( A)1=l

Z ?

Rn

? A M

M1;"=l (x)dx

1;"=l (x) : : : Ml;"=l (x)

1=l  Z :::

C 0( A)1=l ( )?1=m : : : ( )?1=m

Rn

1=l dx

Ml;"=l (x)dx

1=l

1=l = C 0 ?1=m(A )1=l ?1=m;

which proves (24) and thus (22). We have now proved (22) for any > 0. Selecting = (A + B )?1 in both (21) and (22) we obtain that all the sets Es satisfy (20). Summing over all 1  s  2m we obtain the conclusion of the theorem.

MULTILINEAR CALDERO N-ZYGMUND THEORY

13

For purposes of interpolation that will become apparent in the next section, we will need the following strengthening of Theorem 1. Let us denote by Lp;c 1 the space of all compactly supported functions in Lp;1 for ;1 1 1 0 < p < 1. Also set L1 c = Lc , the set of all compactly supported functions in L . 1 ; 1 (Recall that the de nition in (1) gives L = f0g but, to simplify the statement in ;1 1 the next theorem, we set L1 c = Lc .) Theorem 2. Let T be a multilinear operator with kernel K in m-CZK (A; "). Assume that for some 1  q1 ; q2 ; : : : ; qm  1 and some 0 < q < 1 with 1 + 1 ++ 1 = 1 q q q q

1 2 m q ; 1 q ; 1 q; 1 m 1 T maps Lc      Lc into L . Then T is a bounded operator from the m-fold product L1      L1 into L1=m;1 . Moreover, for some constant Cn;m (that depends

only on the parameters indicated) we have that ?  kT kL1L1!L1=m;1  Cn;m A + kT kLq1 Lqm !Lq;1 : (25) Proof. The proof of this theorem only requires some minor modi cations in the proof of the previous result. Let fj 2 L1 . Without loss of generality, we may assume that they have compact support and norm one. It follows that the gj 's obtained from the fj 's using the Calderon-Zygmund decomposition, fj = gj + bj , must also have compact support. Moreover, it is easy to see that kgj kLqj ;1  C ( )1=mqj0 when qj < 1, while kgj kL1  C ( )1=m as before. These estimates are sucient to deduce (21) while the rest of the arguments remain unchanged.

4. Multilinear interpolation In this section we show how to obtain strong type Lp1   Lpm ! Lp boundedness results for multilinear Calderon-Zygmund operators starting from a single estimate. To avoid unnecessary technical complications (see the remark at the end of Lemma 3) 1 we will be working with L1 c instead of L . Theorem 3. Let T be a multilinear operator with kernel K in m-CZK (A; "). Let 1  q1 ; q2; : : : ; qm; q < 1 be given numbers with 1 = 1 + 1 ++ 1 : q q1 q2 qm Suppose that either (i) or (ii) below hold: (i) T maps Lq1 ;1      Lqm ;1 into Lq;1 if q > 1, (ii) T maps Lq1 ;1      Lqm ;1 into L1 if q = 1. Let p, pj be numbers satisfying 1=m  p < 1, 1  pj  1, and 1 = 1 + 1 ++ 1 : p p1 p2 pm Then all the statements below are valid: (iii) when all pj > 1, then T can be extended to a bounded operator from Lp1   Lpm

14

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

into Lp , where Lpk should be replaced by L1 c if some pk = 1; (iv) when some pj = 1, then T can be extended to a bounded map from Lp1    Lpm into Lp;1, where again Lpk should be replaced by L1 c if some pk = 1. (v) when all pj = 1, then T can be extended to a bounded map from the m-fold 1 product L1 c      Lc into BMO. Moreover, there exists a constant Cn;m;pj ;qi such that under either assumption (i) or (ii), we have the estimate ?  kT kLp1 Lpm !Lp  Cn;m;pj ;qi A + B ; (26) where B = kT kLq1 Lqm !Lq;1 if q > 1, and B = kT kLq1 Lqm !L1 if q = 1. Furthermore, conclusions (iii), (iv), and (v) as well estimate (26) are also valid for all the transposes T j , 1  j  m. Remark. Hypothesis (i) is not strong enough to imply (iii) nor (iv) when q = 1. The reason is that L1;1 does not have a predual and its dual is not useful in interpolation. Before we prove the theorem we set up some notation. We will identify exponents p1 ; : : : ; pm; p for which T maps Lp1   Lpm into Lp with points (1=p1; : : : ; 1=pm; 1=p) in Rm+1. We need to show that T is bounded for (1=p1; : : : ; 1=pm; 1=p) in the convex hull of the m + 2 points E = (1; 1; : : : ; 1; m), O = (0; 0; : : : ; 0; 0), C1 = (1; 0; : : : ; 0; 1), C2 = (0; 1; : : : ; 0; 1), : : : , and Cm = (0; 0; : : : ; 1; 1) We will denote this set by EC1 : : : Cm O. Observe that the simplex C1C2 : : : Cm is contained in the (m ? 1)-dimensional plane P = f(1=p1; : : : ; 1=pm; 1=p) : 1=p1 +    + 1=pm = 1=p = 1g and splits EOC1 : : : Cm into two simplices EC1 : : : Cm and OC1 : : : Cm based on the equilateral polygon C1 : : : Cm in P . See Figure 1. Let qj ; q be as in the statement of the theorem and let Q = (1=q1 ; : : : ; 1=qm; 1=q). In a geometric language, assumption (i) is saying that Q lies in the interior of OC1 : : : Cm while assumption (ii) is saying that Q lies in the interior of C1 : : : Cm. Geometrically speaking, conclusion (iii) is saying that T satis es a strong type bound in the closure of the simplex OC1C2 : : : Cm minus its vertices union the interior of the simplex EC1C2 : : : Cm. Conclusion (iv) is saying that T satis es a weak type bound on the vertices C1 ; : : : ; Cm and on the exterior faces of the simplex EC1 : : : Cm. Let Mj be the midpoints of the line segments OCj . Then the (m ? 1)-dimensional simplices Pj = C1 : : : Cj?1Mj Cj+1 : : : Cm determine the planes of symmetry with respect to the transposes T j . See Figure 1. This means that the re ection of the point P = (1=p1; : : : ; 1=pm; 1=p) in OC1 : : : Cm with the respect to Pj is the point (1=p1; : : : ; 1=pj?1; 1=p0; 1=pj+1; : : : ; 1=pm; 1=p0j ): Observe that the boundedness of the transpose T j at the latter point is equivalent to the boundedness of T at P . The main idea of the proof of the theorem is to obtain appropriate bounds in each of the faces of the polyhedron EC1 : : : CmO by reducing matters to (m ? 1)linear operators. Induction on m will then be used to obtain the required bounds

MULTILINEAR CALDERO N-ZYGMUND THEORY

15

on the faces of this polyhedron which will imply strong type bounds on its interior, by interpolation. In the case m = 1 (required by the induction) all the statements of the theorem are known classical results about linear Calderon-Zygmund operators which we therefore omit.

Figure 1. A geometric description of the proof of Lemma 4 for trilin-

ear operators.

We will need to use the following version of the multilinear Marcinkiewicz interpolation theorem, Theorem A below, obtained by Grafakos and Kalton [11]. Other versions of the multilinear Marcinkiewicz interpolation theorem can be found in Janson [14] and Strichartz [24]. We say that a nite subset  of [0; 1)m is anely independent if the conditions

X 2

  = (0; : : : ; 0)

and

X 2

 = 0

imply  = 0, for all  2 : Theorem A. ([11] Theorem 4.6) Let 0 < pj;k; pj  1 for 1  j  m + 1 and 1  k  m; suppose that for j = 1; : : : ; m + 1 we have m X

1 = 1; pj k=1 pj;k

16

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

and that the set  = f(1=pj;1; : : : ; 1=pj;m) : j = 1; : : : ; m + 1g is anely independent, that is 1 0 1=p 1=p1;2 : : : 1=p1;m 1 1;1 B 1=p2;1 1=p2;2 : : : 1=p2;m 1 CC 6= 0: det B @ ::: ::: ::: ::: ::: A 1=pm+1;1 1=pm+1;2 : : : 1=pm+1;m 1 Assume that an m-linear map T satis es (27) kT (E1 ; : : : ; Em )kLpj ;1  M jE1 j1=pj;1 : : : jEm j1=pj;m for all sets Ej of nite measure and all 1  j  m + 1. Suppose that the point (1=q1 ; : : : ; 1=qm; 1=q) lies in the open convex hull of the points (1=pj;1; : : : ; 1=pj;m; 1=pj ) in Rm+1 : Then T extends to a bounded m-linear map from Lq1      Lqk into Lq with constant a multiple of M . Remark. If (27) is assumed to hold on characteristic functions of compact sets, then it follows that T maps Lqc1      Lqcm into Lq and a simple density argument gives the conclusion of Theorem A. Naturally, the same conclusion holds if (27) is valid for general compactly supported functions fj (instead of Ej ). The proof of Theorem 3 will be a consequence of Theorem A and of the following lemma. Lemma 4. Under either hypothesis (i) or (ii) there exists a point V in the interior of the (m ? 1)-dimensional simplex C1 C2 : : : Cm at which T satis es a strong type bound with constant a multiple of (A + B ). Similarly for every 1  j  m there exists a point V j in the interior of C1 C2 : : : Cm at which T j satis es a strong type bound with constant also a multiple of (A + B ). Let us now prove Theorem 3 assuming Lemma 4. Proof. Using Theorem 2, we obtain a weak type estimate for T at the point E . Duality and Lemma 4 imply that T satis es a strong type bound at certain points Vj which lie in the interior of each of the m faces Sj = OC1 : : : Cj?1Cj+1 : : : Cm of the simplex OC1 : : : Cm. (Vj is the re ection of V j with respect to Pj .) We will use this information and induction on m to obtain strong type bounds for T in the closure of each face Sj minus its vertices. At the vertices C1 ; : : : ; Cj?1; : : : ; Cj+1; : : : ; Cm of Sj we will prove weak type bounds. We achieve this by using the inductive hypothesis that the theorem is true for (m ? 1)-linear operators. For simplicity let us only work with the face Sj for j = m. Fix a function fm in n L1 c (R ) and de ne the (m ? 1)-linear operator Tfm (f1; : : : ; fm?1 ) = T (f1; f2; : : : ; fm): By Lemma 3, it follows that Tfm has a kernel in (m ? 1)-CZK (cn;mkfmkL1 A; "). Moreover, Tfm satis es a strong type estimate at a point Vm in the interior of the face Sm with bound a constant multiple of (A + B )kfmkL1 since T satis es a strong

MULTILINEAR CALDERO N-ZYGMUND THEORY

17

type estimate at the point Vm with bound a constant multiple (A + B ). The induction hypothesis now gives that Tfm is bounded on the closure of the (m ? 1)-dimensional simplex OC1 : : : Cm?1 minus its vertices. It also gives that Tfm satis es weak type estimates at the vertices C1; : : : ; Cm?1. Moreover, all the bounds in the estimates are constant multiples of AkfmkL1 + (A + B )kfm kL1 . It follows that T satis es a strong type estimate on the interior of the face Sm with bound a multiple of (A + B ) with the restriction that its last argument lies in L1 c . Similarly T satis es a weak type estimate at the vertices C1; : : : ; Cm?1 with bound a multiple of (A + B ) with the same restriction on its last argument. Next we observe that if Dj are points in the interior of the faces Sj then the points D1 ; : : : ; Dm and E are anely independent in the sense of Theorem A. Once strong type estimates have been obtained on the faces Sj , Theorem A implies strong type estimates in the closure of the simplex OC1 : : : Cm minus its m + 1 vertices, union the interior of the simplex EC1 : : : Cm. Note that we are using here the remark after Theorem A since in one of the arguments of T only compactly supported functions appear. The weak type bounds on the sides of the simplex EC1 : : : Cm follow by interpolation between E and the points Cj at which we already know that a weak type estimate holds. The weak type estimates on each of the edges ECj are obtained by complex interpolation. This concludes the proof of (iii) and (iv) for T . To obtain conclusion (v) observe that the induction hypothesis gives that Tfm 1 maps the (m ? 1)-fold product L1 c      Lc into BMO with bound a multiple of (A + B )kfmkL1 . Since fm is an arbitrary element of L1 c , assertion (v) follows for T . Since Lemma 4 gives the same conclusion for all the transposes T j of T , it follows that the same result is also valid for all the of T j 's as claimed in the statement of the Theorem 3. We now prove Lemma 4. Proof. Let F be the point of intersection of the line segment QE with the simplex C1 C2 : : : Cm. Under hypothesis (i) pick Fe on the line QE , close to F and in the interior of the simplex OC1 : : : Cm. Under assumption (ii) just let Fe = F = Q. Multilinear complex interpolation between the points E and Q implies that T satis es a weak type estimate at the point Fe with constant bounded by a multiple of (A + B ). The estimate at the point Fe implies in particular that (28)

kT (f1; : : : ; fm)kL ;1  C (A + B )

m Y

k=1

kfk kL k ;1

for some 1; : : : m ; > 1. Under assumption (ii) this step is vacuous since estimate (28) already holds. Now let Gj and Gej be the re ections of F and Fe about Pj . Duality gives the following estimates at the points Gej Y (29) kT j (f1 ; : : : ; fm)kL 0;1  C (A + B ) kfk kL k ;1 kfj kL 0j ;1 : k6=j

This is because for s > 1, Ls;1 is the dual space of Ls0 ;1 and these two spaces are `norming duals' of each other (i.e. each of the two norms can be realized as a

18

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

supremum of integrals against functions in the unit ball of the other space). Now let Hj be the intersection of the line Gej E with the (m ? 1)-dimensional simplex C1 C2 : : : Cm. Pick a point Hej on the line Gej E but inside OC1 : : : Cm and near Hj . See Figure 1. Theorem 2 implies that T j satis es a weak type estimate at the point E . Multilinear complex interpolation between the points E and Ge gives the following Lorentz space estimate at the point Hej Y kT j (f1; : : : ; fm)kL j ;1  C (A + B )kfj kL jj;1 kfk kL kj ;1 ; k6=j

for some 1 < j ; kj < 1. Now re ect the points Hj and Hej about Pj to obtain points Rj and Rej at which T satis es the Lorentz space estimates Y kT (f1; : : : ; fm)kL jj0;1  C (A + B )kfj kL j0 ;1 kfk kL kj ;1 : k6=j

We now have m +1 points Re1 ; : : : ; Rem; and E at which T satis es restricted weak type estimates of the form Ls1;1   Lsm;1 ! Ls;1 with constant bounded by a multiple of (A + B ). We observed earlier that if Dj are points in the interior of the faces Sj , then the points D1; : : : ; Dm ; E are anely independent in the sense of Theorem A. Since the notion of ane independence is stable under small perturbations, we conclude that the m + 1 points Re1; : : : ; Rem ; and E are anely independent when the Rej 's are very close to the Rj 's. Theorem A implies that T satis es a strong type estimate at every point P = (1=p1; : : : ; 1=pm; 1=p) in the interior of the polyhedron Re1 : : : Rem E with bound a multiple of (A + B ). In this way we obtain a point V in the interior of the (m ? 1)-dimensional simplex C1 C2 : : : Cm at which T satis es a strong type bound with constant a multiple of (A + B ). To obtain the points V j as in the statement of the lemma we argue as follows. Let Wj be the re ections of Q with respect to Pj . Then T j satis es a strong type estimate at the point Wj . Repeat the argument above with T j playing the role of T and Wj playing the role of the starting point Q. We nd points V j in the interior of the (m ? 1)-dimensional simplex C1C2 : : : Cm at which T j satis es a strong type bound with constant a multiple of (A + B ). This concludes the proof of the Lemma. Remark. As mentioned earlier, to avoid technical complications we have obtained estimates only on L1 c . In many instances, such estimates and certain ad hoc procedures allow extensions of linear or multilinear operators to all of L1. One way to achieve this is described in the book of Meyer and Coifman [18], Section 13.3, in their treatment of multilinear multiplier operators. Remark. Extensions to L1 can also be obtained in certain cases by duality. Notice that Theorem 3 gives that T j maps the m-fold product Lm    Lm into L1 : Using duality, we can then extend T as a bounded operator Lm      Lm  L1  Lm      Lm ! Lm0 : We obtain similar extensions for all points in the boundary of OC1 : : : Cm , except at the vertices O; C1; : : : ; Cm.

MULTILINEAR CALDERO N-ZYGMUND THEORY

19

We now discuss how to achieve this extension at the vertex O. This will allow us to obtain a multilinear version of the theorem of Peetre, Spanne, and Stein on the boundedness of a linear Calderon-Zygmund operators from L1 to BMO. Proposition 1. Under either hypothesis (i) or (ii) of Theorem 3, T has an extension that maps L1      L1 ! BMO with bound a constant multiple of (A + B ). By duality, T also maps L1      H 1      L1 ! L1: (where H 1 is the Hardy space predual of BMO). Proof. Fix a C 1 function supported in the ball of radius two in Rn and satisfying 0  (x)  1 and (x) = 1 when 0  jxj  1 and let k (x) = (2?k x) as in Lemma 1. Theorem 3 gives that T maps 1 2 2 L1 c      Lc  L ! L : Since T is well de ned on this product of spaces the expression T ( k f1 ; : : : k fm) is a well de ned L2 function whenever fj 2 L1 . For f1; : : : ; fm 2 L1 let

G( k f1 ; : : : ; k fm ) = ?

Z

jy j>1 1min j m j

K (0; ~y)( k f1)(y1) : : : ( k fm )(ym) d~y:

Lemma 1 implicitly contains a proof that the limit ?T ( f ; : : : ; f ) + G( f ; : : : ; f ) = T (f ; : : : ; f ); lim k 1 k m k 1 k m 1 m k!1

exists pointwise almost everywhere and de nes a locally integrable function. (The smoothness of the functions fj in the proof this lemma was needed to make sense of the expression T ( k f1 ; : : : ; k fm ) when T was only de ned on S (Rn)   S (Rn ).) To check that this extension of T maps L1      L1 into BMO, let us observe that

kT (f1; : : : ; fm )kBMO  lim sup T ( k f1; : : : ; k fm ) + G( k f1; : : : ; k fm ) BMO k!1 = lim sup kT ( k f1 ; : : : ; k fm)kBMO k!1 cn;m(A + B ) lim sup k k f1 kL1 : : : k k fm kL1 k!1 =cn;m(A + B )kf1kL1 : : : kfmkL1 ; where the last inequality follows by assertion (v) of Theorem 3.

5. The multilinear T 1 theorem As we saw in Section 4, if a multilinear operator with kernel in m-CZK (A; ") maps Lq1    Lqm into Lq for a single point (1=q1; : : : ; 1=qm; 1=q) with q > 1, then T maps Lp1     Lpm into Lp in the full range of possible exponents. It is therefore natural to ask under what conditions T maps Lq1      Lqm into Lq for one (m + 1)-tuple

20

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

(1=q1 ; : : : ; 1=qm; 1=q). A necessary and sucient condition for this to happen is given by the multilinear T 1 theorem discussed in this section. The linear T 1 theorem was obtained by David and Journe [10]. Its original formulation involves three conditions equivalent to L2 boundedness. These conditions are that T 1 2 BMO, T 1 2 BMO, and that a certain weak boundedness property, which we do not need to state here, holds. An equivalent formulation of the T 1 theorem, also found in [10] and better suited for our purposes, is the following: A linear operator T with kernel in 1-CZK (A; ") maps L2 (Rn) ! L2 (Rn) if and only if ?  supn kT (e2i(  ))kBMO + kT (e2i(  ) )kBMO < 1: 2R

In this section we will state and prove a multilinear version of the T 1 theorem using the characterization stated above. We will base some of our arguments on yet another formulation of the T 1 theorem given by Stein [23]. Let us consider the set of all C 1 functions supported in the unit ball of Rn satisfying k@ kL1  1; for all multiindices j j  [n=2]+1. Such functions are called normalized bumps. For a normalized bump , x0 2 Rn, and R > 0, de ne the function ?  R;x0 (x) =  x?Rx0 : The formulation in [23], Theorem 3, page 294, says that a necessary and sucient condition for an operator T with kernel in 1-CZK (A; ") to be L2 -bounded is that for some constant B > 0 we have kT (R;x0 )kL2 + kT (R;x0 )kL2  B Rn=2 for all normalized bumps , all R > 0 and all x0 2 Rn. Moreover, the norm of the operator T on L2 (and therefore on Lp) is bounded by a constant multiple of (A + B ). We are now in a position to state the multilinear T 1 theorem. Recall that in view of Lemma 1, T (e2i1 (  ) ; : : : e2im (  )) is a well de ned element of D0(Rn). Theorem 4. Fix 1 < q1; : : : ; qm ; q < 1 with 1 +    + 1 = 1: (30) q q q 1

m

Let T be a continuous multilinear operator from S (Rn )   S (Rn ) ! S 0 (Rn) with kernel K in m-CZK (A; "). Then T has a bounded extension from Lq1      Lqm into Lq if and only if (31) supn : : : sup n kT (e2i1 (  ); : : : ; e2im (  ) )kBMO  B and also (32)

1 2R

m 2R

sup : : : sup n kT j (e2i1 (  ); : : : ; e2im (  ) )kBMO  B

1 2Rn

m 2R

for all j = 1; : : : ; m. Moreover, if (31) and (32) hold then we have that kT kLq1 Lqm !Lq  cn;m;qj (A + B ); for some constant cn;m;qj depending only on the parameters indicated.

MULTILINEAR CALDERO N-ZYGMUND THEORY

21

Proof. We begin the proof by observing that the necessity of conditions (31) and (32) follows from Proposition 1. The thrust of this theorem is provided by their suciency, i.e. the fact that if (31) and (32) hold, then T is extends to a bounded operator from Lq1      Lqm into Lq . Let us say that T is BMO-restrictedly bounded with bound C if kT (R1 1;x1 ; : : : ; Rmm;xm )kBMO  C < 1; and kT j (1R1 ;x1 ; : : : ; Rmm;xm )kBMO  C < 1 for all 1  j  m, all j normalized bumps, all Rj > 0, and all xj 2 Rn. We will need the following lemma whose proof we postpone until the end of this section. Lemma 5. If (31) and (32) are satis ed, then T is BMO-restrictedly bounded with bound a multiple of B > 0. We will now show by induction on m that if T is BMO-restrictedly bounded with bound B > 0, then it must map Lq1      Lqm ! Lq for some 1 < q; qj < 1 satisfying 1 +    + 1 = 1;

q1 qm q with norm controlled by a multiple of (A + B ). To start the induction, we explain why this fact is true when m = 1. Note that for a point y outside the ball B (x; 2R), the size estimate on the kernel of T gives (33) jT (R;x)(y)j  CARnjx ? yj?n: By Lemma 5, the BMO norm of the function T (R;x) is bounded by a multiple of B . Pick z at distance 5R from x. As usual, let gE denote the average of the function g on a ball E . Then, kT (R;x)kL2(B(x;2R))  kT (R;x) ? T (R;x)B(x;2R) kL2(B(x;2R)) + kT (R;x)B(x;2R) ? T (R;x)B(z;R) kL2(B(x;2R)) + kT (R;x)B(z;R) kL2(B(x;2R))  cBRn=2 + cBRn=2 + cARn=2 ; where we have used (33) and basic properties of BMO functions. The same computations apply to T . It follows that (34) kT (R;x)kL2 + kT (R;x)kL2  c(A + B )Rn=2 : As mentioned before, see Stein [23], this last condition implies that T maps L2 into L2 with bound a multiple of (A + B ). This completes the case m = 1 of the induction. Suppose now that the required conclusion of the BMO-restrictedly boundedness condition is valid for (m ? 1)-linear operators. Let T be an m-linear operator which is BMO-restrictedly bounded with bound B > 0. Consider the (m ? 1)-linear operator TmRm ;xm (f1; : : : ; fm?1) = T (f1 ; : : : ; fm?1 ; Rmm;xm )

22

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

obtained from T by freezing an arbitrary normalized bump in the last entry. It is easy to see that TmRm ;xm satis es the (m ? 1)-linear BMO-restrictedly boundedness condition with bound B , because of identities (16). The induction hypothesis implies that TmRm;xm is bounded from Lq1      Lqm?1 into Lq for some 1 < qj ; q < 1 satisfying 1=q1 +    + 1=qm?1 = 1=q. Since  is compactly supported, Lemma 3 gives that TmRm ;xm has a kernel in (m ? 1)-CZK (A; "). Theorem 3 (v) now gives that 1 TmRm ;xm maps the (m ? 1)-fold product L1 c    Lc into BMO with norm at most a multiple of (A + B ). Thus the estimate kT (g; R2 2;x2 ; : : : ; Rmm;xm )kBMO  c(A + B )kgkL1 (35) holds for all g 2 L1 c . Similar estimates hold when the function g above appears in any other entry 2  j  m. Now for 1  j  m consider the operators Tgj de ned by Tgj (f1; : : : ; fm?1 ) = T (f1; : : : ; fj?1; gj ; fj+1; : : : ; fm?1 ); for functions gj 2 L1 c . Inequality (35) is saying that Tg1 satis es the (m ? 1)-linear BMO-restrictedly boundedness condition with constant a multiple of (A + B )kg1kL1 . Similar conclusions are valid for Tgj . The inductive hypothesis implies that Tgj maps Lq1      Lqj?1  Lqj+1      Lqm ! Lq for some 1 < qk = qk (j ); q = q(j ) < 1 satisfying X 1 1 = ; q q k 1km k6=j

with bound a multiple of Akgj kL1 + (A + B )kgj kL1 . It follows that qj+1     Lqm ! Lq ; T : Lq1      Lqj?1  L1 c L with norm controlled by a multiple of (A + B ). Therefore, for 1  j  m there exist points Qj in the interior of the faces Sj , as de ned in Section 4, at which T satis es strong type estimates with bound a multiple of (A + B ). Furthermore, Theorem 2 gives that T maps L1      L1 into L1=m;1. We have observed that the m + 1 points Q1 ; : : : ; Qm , and E are anely independent. We can then use Theorem A to interpolate between these points and obtain a point Q in the interior of OC1C2 : : : Cm at which T satis es a strong type bound with the required constant. This concludes the proof of the theorem modulo the proof of Lemma 5. We now prove Lemma 5. Proof. At a formal level the proof of this lemma is clear since we can write each bump as the inverse Fourier transform of its Fourier transform and interchange the integrations with the action of T to obtain T (R1 1;x1 ; : : : ; Rmm;xm ) Z Z (36) R ;x 1 1 = 1 (1) : : : mR1 ;xm (m)T (e2i1 (  ); : : : ; e2im (  ) ) d~: Rn

\

Rn

\

To justify this identity we provide the following argument.

MULTILINEAR CALDERO N-ZYGMUND THEORY

23

Let us set Rj j ;xj = j . Pick a smooth and compactly supported function g with mean value zero and let k be as in Lemma 1. Observe that ( k 1; : : : ; k m) converges to (1; : : : ; m) in S (Rn)      S (Rn) and therefore lim hT ( k 1; : : : ; k m ); gi = hT (1; : : : ; m); gi: k!1 The continuity and multilinearity of T also allow us to write hT (1Z; : : : ; mZ ); gi ?  (37) : : : b1(1) : : : cm (m) hT k e2i1 (  ); : : : ; k e2im (  ) ; gi d~: = lim k!1 Rn

Rn

Pick k0 so that the support of g is contained in ball of radius 2k0 centered at the origin. Set Fk = T ( k e2i1 (  ) ; : : : ; k e2im (  )) + G( k e2i1 (  ) ; : : : ; k e2im (  )) where G is de ned in Lemma 1. Using that g has mean-value zero we obtain (38) hT ( k e2i1 (  ); : : : ; k e2im (  ) ); gi = hFk ; gi = hFk0 ; gi + hFk ? Fk0 ; gi: The proof of Lemma 1 gives that jhFk ? Fk0 ; gij is bounded uniformly on 1; : : : ; m by a constant that depends on g. On the other hand hT ( k0 e2i1 (  ) ; : : : ; k0 e2im (  ) ); gi (39) =hK; g k0 e2i1 (  )    k0 e2im (  ) i; where K is the Schwartz kernel of T . It follows that the expression in (39) is controlled by a nite sum of the L1 norms of derivatives of g k0 e2i1 (  ) : : : k0 e2im (  ) on a compact set (that depends on g). This is in turn bounded by Cg (1 + j1j)N : : : (1 + jmj)N for some N > 0 and some constant Cg depending on g. The Lebesgue dominated convergence theorem allows us to pass the limit inside the integrals in (37) to obtain Z Z ?  hT (1; : : : ; m); gi = : : : b1 (1) : : : cm (m) hT e2i1 (  ); : : : ; e2im (  ) ; gid~: Rn

Rn

Using the H 1-BMO duality we obtain that the distribution T (1; : : : ; m) can be identi ed with a BMO function satisfying kT (1; : : : ; m)kBMO  B kb1kL1 : : : kcmkL1  cB: In the last inequality we used the fact that all the derivatives of the normalized bumps up to order [n=2] + 1 are bounded and also the fact that kbj kL1 = kRj j ;xj kL1 is independent of Rj > 0 and of xj 2 Rn. We now mention another characterization of boundedness of multilinear operators with Calderon-Zygmund kernels. This formulation can be used in speci c applications to justify formal computations involving the action of an operator on an m-tuple of characters.

\

24

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

Proposition 2. Let (40)

k

be as in Lemma 1. Fix 1 < q1 ; : : : ; qm ; q < 1 with 1 +    + 1 = 1:

q1

q Let T be a continuous multilinear operator from S (Rn )   S (Rn ) ! S 0 (Rn) with kernel K in m-CZK (A; "). Then T has a bounded extension from Lq1      Lqm into Lq if and only if (41) sup supn : : : sup n kT ( k e2i1 (  ); : : : ; k e2im (  ) )kBMO  B k>0 1 2R

qm

m 2R

and also (42) sup supn : : : sup n kT j ( k e2i1 (  ) ; : : : ; k e2im (  ) )kBMO  B k>0 1 2R

m 2R

for all j = 1; : : : ; m. Moreover, if (41) and (42) hold then we have that kT kLq1 Lqm !Lq  cn;m;qj (A + B ); for some constant cn;m;qj depending only on the parameters indicated. Proof. Using our de nition for T (e2i1 (  ) ; : : : ; e2im (  ) ), it follows that (41) and (42) imply (31) and (32) respectively. It is possible to obtain a version of Theorem 4 involving a certain multilinear weak boundedness property and the action of T and its transposes on the m-tuple (1; : : : ; 1). In fact, using other methods, Christ and Journe established in [5] a multilinear T 1 theorem for forms. Consider the (m + 1)-linear form de ned on functions in D(Rn) via U (f1 ; : : : ; fm; fm+1 ) = hT (f1; : : : ; fm); fm+1 i: It is proved in [5] that the estimates ? Y kf k 1 kf k 2 kf k 2 jU (f1 ; : : : ; fm+1)j  C j L k L l L j 6=k;l

are equivalent to imposing an appropriate multilinear weak boundedness condition on U , together with the hypotheses Uj (1) 2 BMO. The distributions Uj (1) are de ned by hUj (1); gi = U (1; : : : ; 1; g; 1; : : :; 1), with g, a test function with mean zero, in the j -position. The version of the the T 1 theorem we gave is more suitable for some applications. We end this section with an example. Other applications of our multilinear T 1 theorem are given in the next section. Example. Consider the class of multilinear pseudodi erential operators Z Z T (f1 ; : : : ; fm )(x) = : : : (x; ~)fb1(1) : : : fcm(m) e2ix(1 ++m) d1 : : : dm Rn

Rn

with symbols  satisfying j@x @ 11 : : : @ mm (x; 1; : : : ; m)j  C ; (1 + j1j +    + jmj)j j?(j 1j++j mj);

MULTILINEAR CALDERO N-ZYGMUND THEORY

25

for all ; 1; : : : ; m n-tuples of nonnegative integers. We will denote the class of all such symbols by m-S10;1. It is easy to see that such operators have kernels in m-CZK . For these operators we have that T (e2i1 (  ) ; : : : ; e2im (  ) ) = (x; 1 ; : : : ; m )e2ix(1 ++m ); which is uniformly bounded in j 2 Rn. It follows from Theorem 4 that a necessary and sucient condition for T to map a product of Lp spaces into another Lebesgue space with the usual relation on the indices, is that T j (e2i1 (  ); : : : ; e2im (  )) are in BMO uniformly in k 2 Rn. In particular this is the case if all the transposes of T have symbols in m-S10;1. Therefore we have obtained the following multilinear extension of a result of Bourdaud [2]. Corollary 1. Let T be a multilinear pseudodi erential operator with symbol in the class m-S10;1 . Suppose that all of the transposes T j also have symbols in m-S10;1 . Then T extends as bounded operator from Lp1      Lpm into Lp, when 1 < pj < 1 and 1 +    + 1 = 1: (43) p1 pm p Moreover, if one pj = 1, then T maps Lp1      Lpm into Lp;1 and in particular it maps L1      L1 ! L1=m;1 . In general the symbols of the transposes of an operator with symbol in m-S10;1 are hard to compute. Nevertheless, this can be explicitly achieved for the class of operators studied in the next section. 6. Translation invariant multilinear operators Let h(f )(x) = f (x ? h) be the translation of a function f on Rn by h 2 Rn. We say that a multilinear operator T from S (Rn )      S (Rn) ! S 0 (Rn) commutes with translations, or that it is translation invariant, if for all f1 ; : : : ; fm 2 S (Rn) and all h 2 Rn we have (44) h(T (f1; : : : ; fm)) = T (hf1; : : : ; hfm ): When m = 1, an operator that satis es (44) and maps Lp ! Lq for some 1  p; q  1 must be given by convolution with a tempered distribution K0 on Rn, i.e. it has the form Tf (x) = (K0  f )(x): An analogous result is true for multilinear operators. Proposition 3. Let T be a continuous multilinear operator originally de ned from S (Rn )      S (Rn ) into S 0 (Rn). Assume that T commutes with translations and that it extends to a bounded operator from Lp1      Lpm into Lp for some indices 1  p1; : : : ; pm; p  1. Then there exists a tempered distribution K0 on (Rn)m such that for all f1 ; : : : ; fm in S (Rn) we have ?  (45) T (f1; : : : ; fm )(x) = K0  (f1    fm) (x; : : : ; x); where  denotes convolution on (Rn )m , and (f1    fm)(y1; : : : ; ym) = f1 (y1) : : : fm (ym):

26

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

Formally speaking, this proposition is saying that the Schwartz kernel K of T has the special form K (x; y1 ; : : : ; ym) = K0 (x ? y1; : : : ; x ? ym). Proof. We indicate the main ideas. Fix f1 ; : : : ; fm 2 S (Rn ). Using identity (10) and the property that T commutes with translations we obtain that m @ T (f ; : : : ; f ) = X T (f1; : : : ; fj?1; @ fj ; fj+1; : : : ; fm); 1 m @xk @xk j =1

and hence any distributional partial derivative of T (f1; : : : ; fm ) is an Lp function. Then T (f1; : : : ; fm) agrees almost everywhere with a continuous function whose value at zero is controlled by a nite sum of Lp norms of derivatives of T (f1 ; : : : ; fm ). De ne a continuous multilinear functional L on S (Rn )      S (Rn ) by setting L(f1 ; : : : ; fm) = T (f1; : : : ; fm )(0): Since (S (Rn)      S (Rn))0 can be identi ed with S 0 ((Rn)m ), there exists a distribution u 2 S 0 ((Rn)m ) such that L(f1 ; : : : ; fm ) = hu; f1    fm i: Let ue be the re ection of u, i.e. ue(F ) = u(Fe), where Fe(z) = F (?z). Then K0 = ue is the required distribution. Using the Fourier transform we can write (at least in the distributional sense) the multilinear operator

T (f1; : : : ; fm)(x) = as (46)

Z

T (f1; : : : ; fm )(x) =

(Rn )m

Z

K0(x ? y1; : : : ; x ? ym)f1 (y1) : : : fm (ym) d~y; (1; : : : ; m)fb1 (1) : : : fcm (m)e2i(1 ++m )xd~;

(Rn )m

where  is the Fourier transform of K0 in (Rn)m. Under this general setting,  may be a distribution but we are only interested here in the case where  is a function. We want to consider translation invariant operators which are given by (46), with  a function, and which extend to bounded operators from some product of Lp spaces into another Lebesgue space. Observe that (46) is a priori well de ned for f1 ; : : : ; fm in S (Rn) when the function  is locally integrable and has some tempered growth at in nity, i.e. it satis es an estimate of the form (47) j(1; : : : ; m)j  C (j1j +    + jmj)N when j1j +    + jmj > R for some C; N; R > 0. In the sequel, whenever we write (46), we will assume that  is locally integrable and satis es (47). De nition. A locally integrable function  de ned on (Rn)m and satisfying (47) is called a (p1 ; : : : ; pm; p) multilinear multiplier if the corresponding operator T given by (46) extends to a bounded operator from Lp1 (Rn)      Lpm (Rn) into Lp(Rn). We denote by Mp1;:::;pm;p(Rn) the space of all (p1; : : : ; pm; p) multilinear multipliers on

MULTILINEAR CALDERO N-ZYGMUND THEORY

27

Rn. We de ne the norm of  in Mp1;:::;pm;p(Rn) to be the norm of the corresponding operator T from Lp1      Lpm into Lp, i.e. kmkMp1 ;:::;pm;p = kTmkLp1 Lpm !Lp :

In view of the correspondence between kernels K0 and multipliers , multilinear operators which commute with translations will also be called multilinear multiplier operators. It is natural to ask whether the symbols of multilinear multiplier operators which are bounded from Lp1      Lpm into Lp, where the indices satisfy 1 +    + 1 = 1; (48) p1 pm p are themselves bounded functions. This is of course the case when m = 1, since such operators are always L2 bounded. The following theorem gives some basic properties of multilinear multipliers and in particular answers this question. Proposition 4. The following are true: (i) If  2 C, ; 1 and 2 are in Mp1 ;:::;pm;p , then so are  and 1 + 2 , and kkMp1;:::;pm ;p = jjkkMp1;:::;pm ;p ;

?



k1 + 2 kMp1 ;:::;pm ;p  Cp k1 kMp1;:::;pm;p + k2kMp1 ;:::;pm ;p : (ii) If (1 ; : : : ; m) 2 Mp1;:::;pm ;p and 1 ; : : : ; m 2 Rn, then (1 + 1 ; : : : ; m + m ) is in Mp1;:::;pm ;p with the same norm. 1 1 1 (iii) If (1 ; : : : ; m) 2 Mp1;:::;pm;p and  > 0, then  n( p1 ++ pm ? p )(1 ; : : : ; m) is in Mp1 ;:::;pm ;p with the same norm. (iv) If (1 ; : : : ; m) 2 Mp1;:::;pm ;p and A is an orthogonal matrix in Rn, then (A1 ; : : : ; Am) is in Mp1;:::;pm;p with the same norm. (v) Let j be a sequence of functions in Mp1;:::;pm;p such that kj kMp1 ;:::;pm ;p  C for

all j = 1; 2; : : : . If j are uniformly bounded by a locally integrable function on Rn, they satisfy (47) uniformly in j , and they converge pointwise to  a.e. as j ! 1, then  is in Mp1;:::;pm;p with norm bounded by C . (vi) Assume that for all j and some N > 0 we have j(1; : : : ; m)j  C (1 + j1j +    + jmj)N and that  is in Mp1 ;:::;pm ;p, for some 1  p1 ; : : : ; pm  1 and 0 < p < 1 satisfying (48). Then  is a bounded function with L1 norm less than its Mp1;:::;pm ;p norm and thus Mp1;:::;pm ;p can be naturally embedded in L1 . (vii) Let 1  p1 ; : : : ; pm  1 and 0 < p < 1 satisfying (48). Then the spaces Mp1;:::;pm;p(Rn) are complete, and thus they are Banach spaces when p  1 and quasi-Banach spaces when p < 1. Proof. (i)-(iv) are straightforward. (v) easily follows from the Lebesgue dominated convergence theorem and Fatou's lemma, while (vii) is a consequence of (v). We prove (vi). Let B be the norm of T : Lp1      Lpm ! Lp. Let us rst assume that

28

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

 is a C 1 function. This assumption can be disposed using suitable regularization. For xed ~a = (a1 ; : : : ; am) 2 (Rn)m and f1 ; : : : ; fm 2 S (Rn) we have that Z fb1 (1) : : : fcm (m)(~a + "~)e2ix(1 ++m) d~ (Rn )m

converges to (~a)f1(x) : : : fm (x) as " ! 0. Moreover the functions (~a + "~) are in Mp1;:::;pm;p uniformly in ~a and " > 0. Fatou's lemma (recall p < 1) and the fact that  is in Mp1;:::;pm;p give that j(~a)j kf1 : : : fm kLp  B kf1 kLp1 : : : kfm kLpm : Picking f1 =    = fm we obtain the required conclusion. We also have the following result, whose linear version was obtained by Hormander [12]. Proposition 5. Suppose that a multilinear multiplier operator T has a compactly supported kernel and maps the m-fold product Lp1      Lpm into Lp , where 1 < pj < 1 and 0 < p < 1. Then ?  p  p11 +    + p1m ?1: (49) Proof. Fix f1 ; : : : ; fm 2 D(Rn). Then, T (f1 + h f1; : : : ; f1 + hfm ) = T (f1; : : : ; fm ) + T (hf1; : : : ; hfm) = T (f1; : : : ; fm ) + h(T (f1; : : : ; fm)) for h suciently large. Taking Lp norms and letting h goes to in nity we obtain 1 1 1 2 p kT (f1; : : : ; fm)kLp  2 p1 ++ pm kT k kf1kLp1 : : : kfmkLpm ; which implies (49). As examples of operators that map Lp1  Lp2 ! Lp when p > ( p11 + p12 )?1, we mention the bilinear fractional integrals

I (f1; f2 )(x) =

Z

jtj1

f (x + t)g(x ? t) jtj ?n dt:

These operators map Lp1 (Rn)  Lp2 (Rn) ! Lp(Rn) when 0 < < n, 1 < p1; p2 < 1, and 1 + 1 = + 1: p1 p2 n p See the articles by Grafakos and Kalton [11] and also by Kenig and Stein [15] for details. It is very natural to ask for sucient conditions on bounded functions  on (Rn)m so that the corresponding operators are continuous from Lp1    Lpm into Lp, when the indices satisfy (48). When m = 1, the classical Hormander-Mihlin multiplier theorem says that if a function  on Rn satis es j@ ( )j  C j j?j j

MULTILINEAR CALDERO N-ZYGMUND THEORY

29

for j j  [n=2] + 1, then  is an Lp multiplier for 1 < p < 1. The multilinear analogue of the Hormander-Mihlin multiplier theorem was obtained by Coifman and Meyer when p > 1. The point of the next proposition is the extension of this result to the range p > 1=m. Proposition 6. Suppose that a(1 ; : : : ; m) is a C 1 function on (Rn)m ? f0g which satis es j@ 11 : : : @ mm a(1; : : : ; m)j  C 1;:::; m (j1j +    + jmj)?(j 1j++j mj) (50) for all multiindices 1 ; : : : ; m . Let T be as in (46). Then T is a bounded operator from Lp1      Lpm into Lp , when 1 < pj < 1 and 1 +    + 1 = 1: (51) p1 pm p p Moreover, if one pj = 1, then T maps L 1      Lpm into Lp;1 and in particular it maps L1      L1 ! L1=m;1 . Proof. First we observe that conditions (50) easily imply that the inverse Fourier transform of a, satis es (52) j@ 11 : : : @ mm a_ (x1 ; : : : ; xm)j  C 1;:::; m (jx1 j +    + jxm j)?(mn+j 1j++j mj) for all multiindices 1 ; : : : ; m . It follows that the kernel K (x; y1; : : : ; ym) = a_(x ? y1; : : : ; x ? ym) of the operator T satis es the required size and smoothness conditions (4), (5), and (6). The Lp1   Lpm ! Lp boundedness of T for a xed point (1=p1; : : : ; 1=pm; 1=p) satisfying (51) will follow from the multilinear T 1 theorem (Theorem 4) once we have veri ed the required BMO conditions. As in the example in the previous section, we have that T (e2i1 (  ); : : : ; e2im (  ))(x) = a(1; : : : ; m )e2ix(1 ++m ) which is in L1 and thus in BMO uniformly in 1 ; : : : m . The same calculation is valid for the m transposes of T since their corresponding multipliers also satisfy (50). The weak type results follow from Theorem 1. We now turn our attention to sucient conditions on a singular kernel K0 so that the corresponding translation invariant operator (53)

T (f1; : : : ; fm)(x) =

Z

Rn

:::

Z

Rn

K0(x ? y1; : : : ; x ? ym)f1 (y1) : : : fm (ym)d~y

maps Lp1      Lpm into Lp when the indices satisfy (51). The next theorem gives a satisfactory sucient condition. In what follows j(u1; : : : ; um)j will denote the euclidean norm of ~u = (u1; : : : ; um) thought as an element in Rnm. Theorem 5. Let K0(u1; : : : ; um) be a locally integrable function on (Rn)m ? f0g which satis es the size estimate (54) jK0(u1; : : : ; um)j  Aj(u1; : : : ; um)j?nm;

30

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

the cancellation condition Z (55)

K0(u1; : : : ; um) d~u  A < 1; R1 0.

MULTILINEAR CALDERO N-ZYGMUND THEORY

31

Example. Let R1 be the bilinear Riesz transform in the rst variable Z Z x ? y1 f1(y1)f2 (y2) dy1dy2: R1 (f1; f2)(x) = p.v. R R j(x ? y1 ; x ? y2 )j3 By Corollary 2, this operator maps Lp1 (R)  Lp2 (R) into Lp(R) for 1=p1 +1=p2 = 1=p, 1 < p1; p1 < 1, 1=2 < p < 1. It also maps L1  L1 into L1=2;1 . However, it does not map L1  L1 into L1=2 . In fact, letting f1 = f2 = [0;1], an easy computation shows that R1 (f1; f2)(x) behaves at in nity like jxj?2.

It is also natural to ask whether the corollary above is true under less stringent conditions on the function . For instance, is the conclusion of Corollary 2 true when

is an odd function in L1 (Snm?1 )? It is a classical result obtained by Calderon and Zygmund [4] using the method of rotations, that homogeneous linear singular integrals with odd kernels are always Lp bounded for 1 < p < 1. We now indicate what happens if the method of rotation is used in the multilinear setting. Let be an odd integrable function on Snm?1 . Using polar coordinates in Rnm we can write Z +1  Z dt T (f1; : : : ; fm)(x) =

(1 ; : : : ; m ) f1 (x ? t1 ) : : : fm(x ? tm ) t d~: Smn?1 0 Replacing  by ?, changing variables, and using that is odd we obtain Z +1  Z dt T (f1 ; : : : ; fm )(x) =

(1 ; : : : ; m ) f1(x + t1 ) : : : fm(x + tm ) t d~: mn ? 1 S 0 Averaging these two identities we conclude that Z +1  Z 1 dt T (f1; : : : ; fm)(x) = 2

(1 ; : : : ; m ) f1(x ? t1 ) : : : fm (x ? tm ) t d~: Smn?1 ?1 To be able to complete the method of rotations we need to know whether the operator inside the curly brackets above is uniformly bounded in ~ 2 Smn?1 . We call the operator Z +1 H~(f1 ; : : : ; fm )(x) = f1 (x ? t1 ) : : : fm(x ? tm ) dtt ?1 the directional m-linear Hilbert transform (in the direction ~). The observations above involving the method of rotations motivate the following Question. Is the operator H~ bounded from Lp1 (Rn)      Lpm (Rn) into Lp(Rn) uniformly in ~ when 1 < p1; : : : ; pm; p < 1 satisfy (51)? Some progress has been recently achieved on this question by Thiele and independently by Grafakos and Li, for m = 2 and n = 1. The boundedness of the directional bilinear Hilbert transforms in dimension one was recently obtained by Lacey and Thiele [16], [17] with constants depending on the direction.

32

LOUKAS GRAFAKOS AND RODOLFO H. TORRES

References [1] A. Benedek, A. Calderon. and R. Panzone, Convolution operators on Banach-space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356{365. [2] G. Bourdaud, Une algebre maximale d'operateurs pseudodi erentiels, Comm. Partial Di . Eq. 13 (1988), 1059{1083. [3] A. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85{139. [4] A. Calderon and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289{309. [5] M. Christ and J-L. Journe, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), 51{80. [6] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315{331. [7] R. R. Coifman and Y. Meyer, Commutateurs d' integrales singulieres et operateurs multilineaires, Ann. Inst. Fourier, Grenoble 28 (1978), 177{202. [8] R. R. Coifman and Y. Meyer, Au-dela des operateurs pseudo-di erentiels, Asterisk 57, 1978. [9] R. R. Coifman and Y. Meyer, Non-linear harmonic analysis, operator theory, and P.D.E., in Beijing Lectures in Harmonic Analysis, E. M. Stein, ed., Annals of Math. Studies 112, Princeton Univ. Press, Princeton, New Jersey, 1986. [10] G. David and J-L. Journe, A boundedness criterion for generalized Calderon{Zygmund operators, Ann. of Math. 120 (1984), 371{397. [11] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, preprint. [12] L. Hormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93{140. [13] L. Hormander, The Analysis of Linear Partial Di erential Operators I, 2nd edition, SpringerVerlag, Berlin Heidelberg New York 1990. [14] S. Janson, On interpolation of multilinear operators, in Function spaces and applications (Lund, 1986), Lecture Notes in Math. 1302, Springer, Berlin-New York, 1988. [15] C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, preprint. [16] M. T. Lacey and C. M. Thiele, Lp bounds for the bilinear Hilbert transform, 2 < p < 1, Ann. Math. 146 (1997), 693{724. [17] M. T. Lacey and C. M. Thiele, On Calderon's conjecture, Ann. Math. (1999), to appear. [18] Y. Meyer and R. R. Coifman Wavelets: Calderon-Zygmund and multilinear operators, Cambridge Univ. Press, Cambridge, United Kingdom, 1997. [19] J. Peetre, On convolution operators leaving Lp; spaces invariant, Ann. Mat. Pura Appl. 72 (1966), 295{304. [20] S. Spanne, Sur l'interpolation entre les espaces Lk p; , Ann. Scuola Norm. Sup. Pisa 20 (1966), 625{648. [21] E. M. Stein, Singular integrals, harmonic functions, and di erentiability properties of functions of several variables, in Singular integrals, Proc. Sympos. Pure Math. 10, 1966. [22] E. M. Stein, Singular integrals and di erentiability properties of functions, Princeton University Press, Princeton NJ, 1970. [23] E. M. Stein, Harmonic analysis: Real variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton NJ, 1993. [24] R. Strichartz, A multilinear version of the Marcinkiewicz interpolation theorem, Proc. Amer. Math. Soc. 21 (1969), 441{444.

MULTILINEAR CALDERO N-ZYGMUND THEORY

33

Loukas Grafakos, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address :

[email protected]

Rodolfo H. Torres, Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

E-mail address :

[email protected]