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Uniform estimates on multi-linear operators with modulation symmetry. (English) Zbl 1041.42013
In earlier work [J. Am. Math. Soc. 15, 469–496 (2002; Zbl 0994.42015)] the authors investigated boundedness properties of multilinear singular integrals associated to an \(n\)-linear form \[ \Lambda_m (f_1,\dots, f_n)=\int \delta (\xi_1+\cdots +\xi_n) m(\xi) \widehat f_1(\xi)\cdots \widehat f_n(\xi) \, d\xi. \] They proved that if \(m\) and its derivatives decay at appropriate rates away from a suitably nondegenerate singular subspace \(\Gamma'\) of \(\Gamma = \{\xi_1+\cdots +\xi_n=0\}\), then \(\Lambda\) is bounded from \(\prod_{k=1}^n L^{p_k}\) to \(\mathbb R\) in the Hölder range \(\sum 1/p_k =1\).
In the present paper, the authors restrict to the case in which \(\Gamma'\) is the line in direction \(V=(v_1,\dots,v_n)\) with \(\sum v_k=0\), all \(v_k\neq 0\), and \(n\geq 3\). Bounds on \(\Lambda\) are then obtained with weaker decay conditions on \(m\) described as follows: for \(x,y\in \Gamma\) one sets \(d_v (x,y) = \max_{1\leq k\leq n} | x_k-y_k| /| v_k| \) and \(d_v (x,\Gamma')=\inf_{y\in \Gamma'} d_v(x,y)\). One then assumes that, for all partial derivatives up to some suitable finite order, one has \(| \partial_\xi^\alpha m(\xi)| \leq C\prod_{k=1}^n | v_k d_v(\xi,\Gamma')| ^{-\alpha_k}\). Then \(| \Lambda_m (f_1,\dots, f_n)| \leq C\prod_{k=1}^n \| f_k\| _{L^{p_k}}\) whenever each \(p_k>2\) and \(\sum_k 1/p_k=1\).
The proof relies on the same intricate phase space machinery used in the authors’ earlier work. The sharper bounds attained here are only known to apply to one-dimensional singularities because of difficulty in finding a suitable analogue of \(d_v\) for higher dimensional singularities.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B38 Linear operators on function spaces (general)
47G10 Integral operators
Citations:
Zbl 0994.42015
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