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On certain elementary trilinear operators. (English) Zbl 1014.42015

The author studies generalized multilinear convolution type operators of the form \[ T(f_1, \ldots, f_m)(x) = \int_{|t|\leq 1} \prod_{j=1}^m f_j(S_j(x,t)) dt \] for \(m\) functions \(f\) on \(\mathbb{R}^d\), and linear surjections \(S_j: \mathbb{R}^{d+d} \to \mathbb{R}^d\). One reason for being interested in such operators is that they are a dyadic component of multilinear singular operators such as the bilinear and trilinear Hilbert transform, as well as multilinear fractional singular integrals of Kenig-Stein type. From Minkowski’s inequality and Hölder’s inequality one can see that this multilinear operator obeys the same multilinear \(L^p\) estimates as does the pointwise product operator \((f_1, \ldots, f_m) \to f_1 \ldots f_m\), provided all the exponents are at least equal to 1. The author then investigates what happens for exponents less than one and finds an interesting phenomenon: one has a non-trivial range of estimates below \(L^1\) if and only if the linear transformations \(S_j\) are “rationally commensurate”, meaning that they have a certain type of linear dependence over the rationals. For instance, the operator \[ T(f,g,h)(x) = \int f(x-t) g(x+t) h(x+\theta t) dt \] obeys some estimate below \(L^1\) if and only if \(\theta\) is rational. Even more surprising is the nature of proof in the rational case (the irrational case being easily resolved by a lifting technique). Here they use recent arguments on arithmetic projection used in recent work of Katz and Tao on the Kakeya problem, thus illustrating the remarkable interconnectedness of techniques in harmonic analysis.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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