A non-linear generalisation of the Loomis-Whitney inequality and applications

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Convolutions of singular measures and applications to the Zakharov system (2010, methodology): “striction to Σ3 of the convolution of two measures supported on Σ1,Σ2. Our main results are global L2 estimates. We build on the result on nonlinear Brascamp-Lieb inequalities proved in [3], see also [4]. More precisely, we utilize the m = 3 case of [3, Theorem 1.3] in order to extend the trilinear ”
Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, background): “y replacing pj ∈ [1,∞] with 1/pj ∈ [0,1] and then fj with f pj j . A particularlypalatableapplicationofTheorem5.6inthe setting ofsuchmultilinear Radon-like transforms is the following. Corollary 5.7 ([14],[8]). If G: (Rd−1)d−1 → Ris a smooth function such that |det(∇y1G(0),...,∇y d−1 G(0))| ≥ ν, the”
On the multilinear restriction and Kakeya conjectures (2005, background): “It has been shown recently (in [5], by very different techniques), that if Φj(x, ξ) = x · Γj(ξ), where Γj : R → Rd−1 are smooth submersions, then one may indeed set ε = 0 in the conclusion of Theorem 6.”