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arXiv · 2201.12201

Testing conditions for multilinear Radon-Brascamp-Lieb inequalities

Philip T Gressman
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Abstract

This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in one direction and on a careful analysis of Knapp-type examples in the other. Various applications are discussed which demonstrate new and subtle interplay between curvature and transversality and establish nontrivial mixed-norm $L^p$-improving inequalities in the model case of convolution with affine hypersurface measure on the paraboloid.

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Philip T Gressman. 2022-01-28. Testing conditions for multilinear Radon-Brascamp-Lieb inequalities. https://arxiv.org/abs/2201.12201

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