A restricted 2-plane transform related to Fourier Restriction in codimension 2

Abstract: The $2-$plane transform is the operator that maps a function to its averages along affine $2-$planes. We consider the operator obtained by restricting the allowed directions of the $2-$planes to those normal to a fixed surface $S$ (quadratic, for simplicity) of codimension $2$. By duality and discretisation, $L^p\to L^q$ estimates for such an operator imply Kakeya-type estimates for the supports of Fourier-transformed wave-packets adapted to the surface $S$ (wave-packet decompositions being a powerful tool in proving Fourier Restriction results). We connect this operator to Gressman's theory of affine invariant measures by showing that if the surface is well-curved à la Gressman (meaning, the affine invariant surface measure on S is non-vanishing) then the restricted $2-$plane transform is $L^p\to L^q$ bounded in the maximal range of $(p,q)$ exponents allowed. The proof relies on a characterisation of well-curvedness in Geometric Invariant Theory terms, which will be discussed. Joint work with S. Dendrinos and A. Mustata.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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