A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM

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1 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. We give a short proof of the well known Coifman-Meyer theorem on multilinear operators.. Introduction The main task of the present paper is to present a new proof of the classical Coifman- Meyer theorem on multilinear singular integrals, see [2], [3], [5], [6]. Let m L (IR n ) be a bounded function which is smooth away from the origin and satisfies the following Marcinkiewicz-Mihlin-Hörmander type condition α m(ξ) ξ α, () for sufficiently many multiindices α. For f,..., f n S(IR) Schwartz functions on the real line, we define the n-linear operator T m by the formula T m (f,..., f n )(x) = IR m(ξ) f (ξ )... f n (ξ n )e 2πix(ξ +...+ξ n) dξ...dξ n. (2) n The following theorem holds, see [2], [3], [5], [6]. Theorem.. As defined, the multilinear operator T m maps L p... L pn as < p i, i n, /p /p n = /p and 0 < p <. L p as long When such an n + -tuple (p,..., p n, p), has the property that 0 < p < and p j = for some j n then, for some technical reasons (see [5], [7]), by L one actually means L c the space of bounded measurable functions with compact support. The case p has been proven in [3] while the general case p > /n has been independently settled in [6] and [5]. The interesting fact that p can be a number smaller than goes back to [2]. The usual argument to prove the theorem (see [2], [3], [5], [6]) uses the celebrated T theorem of David and Journe [4] and relies on BMO theory, Carleson measures and C.Fefferman s duality theorem between the Hardy space H and BMO. As the reader will see, our proof is conceptually simpler and does not use any of the aforementioned ingredients. It is based on a careful stopping time argument involving the Hardy-Littlewood maximal function and the Littlewood-Paley square function. Throughout the paper we will write A B iff there is a universal constant C > 0 so that A CB.

2 2 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE 2. Model operators For simplicity we treat the n = 2 case only. It is a standard fact by now (see for instance the papers [7], [8]) that the study of our bilinear operators T m can be reduced to the study of finitely many discrete model operators Π j P, j = 0,, 2, 3 of the form Π j P (f, f 2 ) = ɛ P I P /2 f, φ P f 2, φ P2 φ P3. (3) Here P is a collection of lacunary dyadic tiles corresponding to lattice points (k, n) in Z 2 and (ɛ P ) P is an arbitrary sequence of uniformly bounded constants. More precisely, P i i =, 2, 3 are defined by P i = I P ω Pi where I P = [n2 k, (n + )2 k+ ], ω Pi = [0, 2 k ] if i = j and ω Pi = [2 k, 2 k+ ] if i j, while φ Pi i =, 2, 3 are L 2 normalized wave packets corresponding to the Heisenberg boxes P i i =, 2, 3. This means that the function φ Pi is a smooth L 2 normalized bump function adapted to the interval I P whose Fourier transform φ Pi is supported inside the interval ω Pi for i =, 2, 3. We should emphasize here that the tile P is uniquely determined by the interval I P. To explain this reduction in a few words, let Q := I J be a dyadic rectangle in the plane, having the property that diam(q) dist(q, {0}). Let also φ I, φ J be two L normalized smooth functions such that supp( φ I ) I and supp( φ J ) J. If we replace the symbol m(ξ, ξ 2 ) by φ I (ξ ) φ J (ξ 2 ) we observe that the right hand side of (2) becomes (f φ I )(x)(f 2 φ J )(x). On the other hand, inequality () implies that one can think of m as being essentially constant on each such Q and so the integral in (2) when smoothly restricted to this cube, becomes roughly ɛ Q (f φ I )(x)(f 2 φ J )(x). Then, one covers the plane by a collection of carefully selected such Whitney cubes and discretize again in the x-variable. In the end, one obtains a formula in which the general T m is written as an average of operators of the type of the model operators above. The details can be found in [7], [8]. Now our analysis of the operators Π j P will be independent on j = 0,, 2, 3 and so we can assume without loss of generality that j = and write from now on, for simplicity, Π P instead of Π P (the reader will observe that for j, the only difference is that the roles of the Hardy Littlewood maximal function M and the Littlewood Paley square function S, get permuted). It is therefore enough to prove the theorem for the bilinear operator Π P. 3. The proof First, let us observe that it is very easy to obtain the necessary L p estimates in the particular case when all the indices are strictly between and. To see this, let f L p, g L q, h L r for < p, q, r < with /p + /q + /r =. Then, IR Π P(f, g)(x)h(x) dx I P f, φ /2 P g, φ P2 h, φ P3 =

3 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 3 IR f, φ P I P /2 g, φ P2 I P /2 h, φ P3 I P /2 χ I P (x) dx (4) ( ) ( ) /2 ( /2 f, φ P sup IR I P χ g, φ P2 2 h, φ P3 2 /2 I P (x) χ IP (x) χ IP (x)) dx I P I P IR Mf(x)Sg(x)Sh(x) dx Mf p Sg q Sh r f p g q h r, where M is the maximal function of Hardy and Littlewood and S is the discrete square function of Littlewood and Paley, see [9] and [7]. This means that theorem. is nontrivial only when one index is, or less or equal than. To prove the general case we just need to show that the bilinear operator Π P maps L L L /2, because then, by interpolation and symmetry the theorem follows as in [7]. Let f, g L be such that f = g =. We now recall Lemma 5.4 in []. Lemma 3.. Let 0 < p < and A > 0. Then the following statements are equivalent up to constants: (i) f p, A. (ii) For every set E with 0 < E <, there exists a subset E E with E E and f, χ E A E /p. Here p is defined by /p + /p = (note that p can be a negative number!). Proof To see that (i) implies (ii), set E := E \ {x : f(x) CA E /p }. If C is a sufficiently large constant, then (i) implies E E and the claim follows. To see that (ii) implies (i), let λ > 0 be arbitrary E := {x : Re(f(x)) > λ}. Then by (ii) we have λ E λ E A E /p, and (i) easily follows (replacing Re by Re, Im, Im as necessary). Using this Lemma 3. in the particular case p = /2 and the scale invariance, it is enough to show that given E IR E =, one can find a subset E E with E such that I P /2 f, Φ P g, Φ P2 h, Φ P3 (5) where h := χ E. Fix such a set E with E =. To construct the subset E, we first consider Ω 0 = {x IR : M(f)(x) > C} {x IR : S(g)(x) > C} {x IR : M(g)(x) > C}. Also, define

4 4 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Ω = {x IR : M( Ω0 )(x) > 00 }. Clearly, we have Ω < /2, if C is a big enough constant which we fix from now on. Then, we define E := E \ Ω = E Ω c and observe that indeed E. After this, we split our sum in (5) into two parts = I P Ω c + I P Ω c = := I + II. We also assume that the set P is finite, since our estimates do not depend on its cardinality. First, we estimate term I. Since I P Ω c, it follows that I P Ω 0 I P < or equivalently, 00 I P Ω c 0 > 99 I 00 P. We are now going to describe three decomposition procedures, one for each function f, g, h. Later on, we will combine them, in order to handle our sum. First, define then define Ω = {x IR : M(f)(x) > C 2 } T = {P P : I P Ω > 00 I P }, Ω 2 = {x IR : M(f)(x) > C 2 2 } T 2 = {P P \ T : I P Ω 2 > 00 I P }, and so on. (The constant C > 0 is the one which we fixed before). Since there are finitely many tiles, this algorithm ends after a while, producing the sets {Ω n } and {T n } such that P = n T n. Independently, define then define Ω = {x IR : S(g)(x) > C 2 } T = {P P : I P Ω > 00 I P }, Ω 2 = {x IR : S(g)(x) > C 2 2 } T 2 = {P P \ T : I P Ω 2 > 00 I P },

5 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 5 and so on, producing the sets {Ω n } and {T n } such that P = nt n. We would like to have such a decomposition available for the function h also. To do this, we first need to construct the analogue of the set Ω 0, for it. We will therefore pick N > 0 a big enough integer such that for every P P we have I P Ω c N > 99 I 00 P where we defined Ω N = {x IR : S(h)(x) > C2N }. Then, similarly to the previous algorithms, we define then define Ω N+ = {x IR : S(h)(x) > C2N 2 } T N+ = {P P : I P Ω N+ > 00 I P }, Ω N+2 = {x IR : S(h)(x) > C2N 2 2 } T N+2 = {P P \ T N+ : I P Ω N+2 > 00 I P }, and so on, constructing the sets {Ω n } and {T n } such that P = nt n. Then we write the term I as n,n 2 >0,n 3 > N P T n,n 2,n 3 I P 3/2 f, Φ P g, Φ P2 h, Φ P3 I P, (6) where T n,n 2,n 3 := T n T n 2 T n 3. Now, if P belongs to T n,n 2,n 3 this means in particular that P has not been selected at the previous n, n 2 and n 3 steps respectively, which means that I P Ω n < I 00 P, I P Ω n 2 < I 00 P and I P Ω n 3 < I 00 P or equivalently, I P Ω c n > 99 I 00 P, I P Ω c n2 > 99 I 00 P and I P Ω c n 3 > 99 I 00 P. But this implies that I P Ω c n Ω c n2 Ω c n 3 > I P. (7) In particular, using (7), the term in (6) is smaller than n,n 2 >0,n 3 > N n,n 2 >0,n 3 > N I P f, Φ 3/2 P g, Φ P2 h, Φ P3 I P Ω c n c Ω n2 c Ω n 3 = P T n,n 2,n 3 Ω c n Ω c n2 Ω c n 3 n,n 2 >0,n 3 > N P T n,n 2,n 3 I P 3/2 f, Φ P g, Φ P2 h, Φ P3 χ IP (x) dx Ω c n Ω c n2 Ω c n 3 Ω Tn,n 2,n 3 M(f)(x)S(g)(x)S(h)(x) dx

6 6 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE where On the other hand we can write n,n 2 >0,n 3 > N Ω Tn,n 2,n 3 := 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3, (8) P T n,n 2,n 3 I P. Ω Tn,n 2,n 3 Ω Tn {x IR : M(χ Ωn )(x) > 00 } Ω n = {x IR : M(f)(x) > C 2 } n 2n. Similarly, we have Ω Tn,n 2,n 3 2 n 2 and also Ω Tn,n 2,n 3 2 n2α, for every α, since E. In particular, it follows that Ω Tn,n 2,n 3 2 n θ 2 n 2θ 2 2 n 3αθ 3 (9) for any 0 θ, θ 2, θ 3 <, such that θ + θ 2 + θ 3 =. Now we split the sum in (8) into n,n 2 >0,n 3 >0 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3 + n,n 2 >0,0>n 3 > N 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3. To estimate the first term in (0) we use the inequality (9) in the particular case θ = θ 2 = /2, θ 3 = 0, while to estimate the second term we use (9) for θ j, j =, 2, 3 such that θ <, θ 2 < and αθ 3 > 0. With these choices, the geometric sums in (0) are finite. This ends the discussion on I. Now term II is much simpler, being just an error term. We split (0) where and easily observe that Then, term II is smaller than d>0 d ;I P Ω P := d>0 P d P d := {P P : dist(i P, Ω c ) I P d ;I P Ω I P f, Φ P I P /2 g, Φ P 2 I P /2 h, Φ P 3 I P /2 d>0 for any big number K > 0, and this ends the proof. 2 d } I P Ω. () d ;I P Ω I P 2 d 2 d 2 Kd,

7 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 7 References [] Auscher P., Hofmann S., Muscalu C., Tao T. and Thiele C. Carleson measures, trees, extrapolation and T b theorems, Publ. Mat., vol. 46, , [2002]. [2] Coifman R. and Meyer Y., On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. vol. 22, 35-33, [975]. [3] Coifman R. and Meyer Y., Ondelettes et operateurs III. Operateurs multilineaires, Hermann, Paris, [99]. [4] David G. and Journe J-L., A boundedness criterion for generalized Calderon-Zygmund operators, Ann. of Math., vol. 20, , [984]. [5] Grafakos L. and Torres R., Multilinear Calderon-Zygmund theory, Adv. Math., vol. 65, 24-64, [2002]. [6] Kenig C. and Stein E., Multilinear estimates and fractional integration, Math. Res. Lett., vol. 6, -5, [999]. [7] Muscalu C., Tao T. and Thiele C., Multilinear operators given by singular multipliers, J. Amer. Math. Soc. 5, [2002], [8] Muscalu C., Pipher J., Tao T. and Thiele, C., Bi-parameter paraproducts, Preprint, [2003]. [9] Stein, E., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, [993]. Department of Mathematics, Cornell University, Ithaca, NY address: camil@math.cornell.edu Department of Mathematics, Brown University, Providence, RI address: jpipher@math.brown.edu Department of Mathematics, UCLA, Los Angeles, CA address: tao@math.ucla.edu Department of Mathematics, UCLA, Los Angeles, CA address: thiele@math.ucla.edu

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