Hardy Space Estimates for Multilinear Operators, Ii

We continue the study of multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces Lp(Rn) into the Hardy spaces Hr(Rn). At the endpoint case r = n/n+m+ 1, where m is the highest vanishing moment of the multilinear operator, we prove a weak type result. 0. Introduction A well known by now theorem of P.L. Lions says that the determinant of the Jacobian of a function from R → R maps the product of Sobolev spaces L1 × · · · × L1 into the Hardy space H. Coifman, Lions, Meyer and Semmes, [CLMS], went below H by showing that for p, q > 1, the Jacobian-determinant maps Lp1(R) × L q 1(R) into H(R), where r−1 = p−1 + q−1, as long as r > 2/3. Their result can be generalized to give the n-dimensional version that the determinant of the Jacobian maps L1(R)×· · ·×Ln(R) into H(R), as long as the harmonic mean r of the pj ’s is strictly greater than n/n+ 1. In this work we prove a positive result in the endpoint case r = n/n + 1. We treat more general multilinear operators with vanishing integral since our methods show that this is the only assumption needed. We also study the case of multilinear operators with higher moments vanishing. The number of vanishing moments is related to the lowest r for which these operators map products of Lebesgue spaces into H. If such an operator has all moments of order ≤ m vanishing, then it maps products of Lebesgue spaces into H for r > n/n+m+ 1. Also, a weak type estimate holds in the endpoint case r = n/n+m+ 1 and no boundedness result holds for r < n/n+m+ 1. 1. Statements of results Throughout this article, N and K will denote fixed integers ≥ 2. We are given a matrix Typeset by AMS-TEX 1 of convolution Calderón-Zygmund kernels {K i } K i=1,j=1 on R. We call T j i the associated Calderón-Zygmund operator. We denote by L(f1, . . . , fK) the K-linear operator (1.1) L(f1, . . . , fK) = N ∑ i=1 (T 1 i f1) . . . (T K i fK). originally defined for smooth compactly supported functions f1, . . . , fK . For p ≤ 1, we denote by H the usual real variable Hardy space as defined in [S] or [FST], i.e. the set of all distributions f on R for which the maximal function supt>0 |φt ∗ f(x)| is in L, where φt(x) = 1 tnφ( x tn ) and φ is smooth, nonzero and compactly supported. We also denote by Hp,∞ the weak H as defined in [FRS] (or [FSO] in the case p = 1), i.e. the set of all f in R for which the maximal function supt>0 |φt ∗ f(x)| is in weak L. The weak L (quasi)norm of this maximal function is by definition the ‖ ‖Hp,∞ (quasi)norm of f . Our first result concerns the general multilinear operators L of the type above and it presents very clearly the method that will be used in this article. Note however, that there is an unpleasant restriction about the exponents that will be lifted later. Theorem I . Assume that for all (f1, . . . , fK) ∈ (C∞ 0 ) , the K-linear operator L satisfies: ∫ L(f1, . . . , fK) dx = 0. Suppose that p1, . . . , pK > 1 are given and let r = (p−1 1 + · · · + p −1 K ) −1 be their harmonic mean. Assume that the harmonic mean of any proper subset of the pj’s is greater than 1. Then 1) If r > 1, L maps L1 × · · · × LK → L. 2) If 1 ≥ r > n/n+ 1, L maps L1 × · · · × LK → H. 3) If r = n/n+ 1, L maps L1 × · · · × LK → Hr,∞. Next, we treat the case of multilinear operators with vanishing higher moments. The significance of the number of vanishing moments is that it gives the lowest r for which such operators map into H. We also get rid of the assumption that the harmonic mean of any subset of the pj ’s is always greater than 1. We are assuming however, that the K-linear operators L that have a special form. When K = 2, we consider operators L of the general form (1.1), i.e. inner products of two vectors of Calderón-Zygmund operators. For K ≥ 3, we consider operators built inductively as follows: We are assuming that for any j there exist Λji = Λ j i (f1, . . . , fj−1, fj+1, . . . , fK) (K−1)-linear operators already defined by the induction hypothesis with the same number 2 of vanishing moments, such that (1.2) L(f1, . . . , fK) = M ∑ i=1 T j i (fj) Λ j i (f1, . . . , fj−1, fj+1, . . . , fK) Condition (1.2) essentially says that the multilinear operators L look like determinants of matrices. They are built by induction starting from arbitrary bilinear operators as the ones in theorem I (when K = 2) and at each stage they look like sums of products of multilinear operators of one smaller degree multiplied by a Calderón-Zygmund operator. These sums have a certain degree of symmetry because it follows from a repeated application of (1.2) that for each j1, . . . , jl, there exist (K − l)-linear operators Λ1l i with the same number of vanishing moments such that L(f1, . . . , fK) = ∑ i (T j1 i fj1) . . . (T jl i fjl) Λ j1,...,jl i (remaining fj ’s). In most applications we have in mind, the multilinear operators have this form, for example determinants of matrices. In the case of bilinear operators, K = 2, there are no additional assumptions about the operators L and this is why we state and prove this case separately. Also, this case is going to serve as the first step of an inductive argument that will be used later. Theorem IIa. Assume that for some m, 0 ≤ m ≤ n− 1 and for all f, g ∈ C∞ 0 (R) the bilinear operator B(f, g) = ∑N i=1(T 1 i f)(T 2 i g) satisfies: ∫ xB(f, g) dx = 0 for all multiindices α with |α| ≤ m. Suppose that p, q > 1 are arbitrary and let r = (p−1 + q−1)−1 be their harmonic mean. Then 1) If r > 1, B maps L × L → L. 2) If 1 ≥ r > n/n+m+ 1, B maps L × L → H. 3) If r = n/n+m+ 1, B maps L × L → Hr,∞. Next, we generalize theorem IIa for K-linear operators of the form (1.2) and for these type of operators we don’t have any additional assumption about the pj ’s Theorem IIb. Assume that for some m, 0 ≤ m ≤ n(K − 1) − 1 and for all fj ∈ C∞ 0 (R) the K-linear operator L(f1, . . . , fK) has the form (1.1), where each Λ j i satisfy ∫ xΛji dx = 0 for all multiindices α with |α| ≤ m. 3 Suppose that p1, . . . , pK > 1 are arbitrary and let r = ( ∑ k p −1 k ) −1 be their harmonic mean. Then 1) If r > 1, L maps L1 × · · · × LK → L. 2) If 1 ≥ r > n/n+m+ 1, L maps L1 × · · · × LK → H. 3) If r = n/n+m+ 1, L maps L1 × · · · × LK → Hr,∞. Remarks: a. The assumption m ≤ n(K − 1) − 1 is necessary in theorem II, since otherwise r = n/n+m+ 1 < 1/K which would contradict that pj > 1. b. The hypothesis that the harmonic mean of any subset of the pj ’s is greater than 1 seems to be necessary in conclusions 2) and 3) of theorem I. It is obviously not needed in conclusion 1) of theorem I and it is always automatically satisfied when r = 1 or when K = 2. This condition imposes an upper bound on the degree K of multilinearity of the K-linear operator L. For, let pj = p > 1 and let r < 1 be the harmonic mean of the pj ’s. Then Kr = p. The assumption on the harmonic mean of any subset of the pj ’s gives p/(K − 1) > 1. We conclude that K < 1/(1 − r) which is a restriction on the size of K. Note, however, that when r = 1 there is no upper bound on K nor any restriction about the exponents and our theorem implies for example, that any K-linear operator as above with mean value zero maps L1 × · · · × LK → H when ∑ p−1 j = 1. c. The vanishing integral hypothesis for L in theorem I can be relaxed to the milder condition that for all f1 smooth with compact support and for some f2, . . . , fK in the corresponding Lebesgue spaces the integrals ∫ L(f1, f2, . . . , fK) dx vanish. Then conclusion 2) of theorem I will be that the operator g → L(g, f2, . . . , fK) maps L1 to H with norm no larger than a constant times the product of the Lj norms of the fj ’s, j = 2, . . . ,K. Conclusion 3) of theorem I will be similar. 2. Proof of theorem I We fix p1, . . . , pK > 1 and we let r be their harmonic mean. Clearly only the case r ≤ 1 is interesting because the case r > 1 is just Hölder’s inequality together with the L boundedness of Calderón Zygmund operators. Fix a smooth compactly supported function φ in R, an x0 ∈ R and define φt,x0(x) = 1 tnφ( x−x0 t ).Without loss of generality we may assume that φ is supported in |x| ≤ 1. We need to show that supt>0 | ∫ φt,x0L(f1, . . . , fK) dx| is in L when r > n/n+1 and in Lr,∞ when r = n/n+1. We also fix a smooth cutoff η(x) such that η ≡ 1 on |x| < 2 and supported in |x| < 4. We call for simplicity η0(x) = η(x0−x t ) and η1(x) = 1 − η0(x). The reader should remember the dependence of η0, η1 on t. We 4 now decompose L(f1, . . . , fK) = L0 + L1 + · · ·+ LK+1, where L0 = L(η0f1, η0f2, . . . , η0fK) L1 = K ∑ j=1 L(f1, . . . , η1fj , . . . , fK) L2 = − ∑ 1≤j,l≤K j0 |T k i (η1f)(x)− T k i (η1)(x0)| ≤ sup t>0 ∣∣∣∣ ∫ (Kk i (x− y)−K i (x0 − y)) η1(y)f(y) dy∣∣∣∣ ≤C sup t>0 ∫ |y−x0|≥t |x− x0| |y − x0||f(y)| dy ≤ C|f |(x0) where by g(x0) we denote the Hardy-Littlewood maximal function of g at the point x0. We also use the notation (T j i )∗ for the maximal truncated operator of T j i . The term L0 is the main term term of the decomposition and is treated last. We begin with term L1. We write it as K ∑ j=1 L(f1, . . . , (η1fj)(x)− (η1fj)(x0), . . . , fK) + K ∑ j=1 L (f1, . . . , (η1fj)(x0), . . . , fK) . We then have: sup t>0 ∣∣∣∣ ∫ φt,x0L1 dx∣∣∣∣ ≤ K ∑