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Multilinear Calderón-Zygmund theory. (English) Zbl 1032.42020

In this paper the authors consider a systematic treatment of multilinear Calderón-Zygmund operators introduced earlier in the papers of Coifman and Meyer and of M. Lacey and C. Thiele [Ann. Math. (2) 146, 693-724 (1997; Zbl 0914.46034); ibid. 149, 475-496 (1999; Zbl 0934.42012)]. The first main result reads as follows: Let m-linear operators be T:[𝒮( n )] m 𝒮 ' ( n ) for which there is a function K defined away from the diagonal x=y 1 ==y m in ( n ) m+1 satisfying

|K(y 0 ,y 1 ,,y m )|c n,m A ( k,l=0 m |y k -y l |) nm


|K(y 0 ,,y j ,,y m )-K(y 0 ,,y j ' ,,y m )|c n,m A|y j -y j ' | ε ( k,l=0 m |y k -y l |) nm+ε ,

whenever 0jm and |y j -y j ' |1 2max 0km |y j -y k |. Let q j [1,) be given numbers with 1/q= j=1 m 1/q j . Suppose that T maps L q 1 ,1 ××L q m ,1 into L q, if q>1 or L 1 if q=1. Then for any p j [1,] such that 1/mp<, T extends to a bounded map from L p 1 ××L p m into L p if all p j >1 and into L p, if some p j =1. If some p k =, L p k should be replaced by L c . Moreover, T extends to a bounded map from L ××L to BMO. Next, the authors obtain the version of the multilinear T1 theorem by G. David and J.-L. Journé [Ann. Math. (2) 120, 371-397 (1984; Zbl 0567.47025)]. It is proved that if T(e ξ 1 ,,e ξ m ) and T *j (e ξ 1 ,,e ξ m ) (ξ 1 ,,ξ m n , 1jm) are bounded subsets of BMO, then T has a bounded extension from L q 1 ××L q m into L q if 1<q,q j <. Here jth transpose T *j of T is defined via

T *j (f 1 ,,f m ),h=T(f 1 ,,f j-1 ,h,f j+1 ,,f m ),f j

for all f 1 ,,f m , g in 𝒮( n ). This multilinear Calderón-Zygmund theory is applied to obtain some new continuity results for multilinear translation invariant operators, multlinear pseudodifferential operators, and multilinear multipliers.

42B20Singular and oscillatory integrals, several variables