Multilinear Hankel Operator

We extend to multilinear Hankel operators a result on the regularity of truncations of Hankel operators. We prove and use a continuity property on the bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces. In this note, we want to extend to multilinear Hankel operators a result obtained by [BB] on the boundedness properties of truncation acting on bounded Hankel infinite matrices. Let us first recall this result. A matrix B = (bmn)m,n∈N is called of Hankel type if bmn = bm+n for some sequence b ∈ l(N). We can consider B as an operator acting on l(N). In this case, we write B = Hb. If we identify l (N) with the complex Hardy space H(D) of the unit disc then Hb can be realized as the integral operator acting on f ∈ H(D) by Hbf(z) = 1 2π ∫ T b(ζ)f(ζ) 1− ζz dσ(ζ). In other words, Hbf = C(bg) where C denotes the Cauchy integral, g(ζ) = f(ζ) and b is the Symbol of the Hankel operator b(ζ) = ∑∞ k=0 bkζ . If f(z) = ∑ n∈N anz , one has Hbf(z) = ∑ m∈N( ∑ n∈N anbm+n)z . This well known fact justify the terminology of Hankel matrices. One can consider truncations of matrices as follows. For β, γ ∈ R, the truncated matrix Πβ,γ(B) is the matrix whose entry at position m,n is bmn if m ≥ βn + γ and zero otherwise. In [BB], they prove, among others, that such truncations, for β 6= −1, preserve the boundedness of Hankel operators. Their method is to show that truncations may be viewed as some bilinear periodic Hilbert transforms and to use the result of Lacey-Thiele (see [LT]) in the periodic setting. We are here interested in the same problem but for multilinear Hankel operators. For n ∈ N, we define the multilinear Hankel operator H (n) b as follows. Let f1, . . . , fn ∈ H (D), H (n) b (f1, . . . , fn)(z) = 1 2π ∫ T b(ζ)f1(ζ) . . . fn(ζ) 1− ζz dσ(ζ) = Hb(f1 × · · · × fn)(z). 1991 Mathematics Subject Classification. 47B35(42A50 47A63 47B10 47B49).