A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$
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| Formato: | Preprint |
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2022
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| author | Dendrinos, Spyridon Mustata, Andrei Vitturi, Marco |
| author_facet | Dendrinos, Spyridon Mustata, Andrei Vitturi, Marco |
| contents | We draw a connection between the affine invariant surface measures constructed by P. Gressman and the boundedness of a certain geometric averaging operator associated to surfaces of codimension $2$ and related to the Fourier Restriction Problem for such surfaces. For a surface given by $(ξ, Q_1(ξ), Q_2(ξ))$, with $Q_1,Q_2$ quadratic forms on $\mathbb{R}^d$, the particular operator in question is the $2$-plane transform restricted to directions normal to the surface, that is \[ \mathcal{T}f(x,ξ) := \iint_{|s|,|t| \leq 1} f(x - s \nabla Q_1(ξ) - t \nabla Q_2(ξ), s, t)\,ds\,dt, \] where $x,ξ\in \mathbb{R}^d$. We show that when the surface is well-curved in the sense of Gressman (that is, the associated affine invariant surface measure does not vanish) the operator satisfies sharp $L^p \to L^q$ inequalities for $p,q$ up to the critical point. We also show that the well-curvedness assumption is necessary to obtain the full range of estimates. The proof relies on two main ingredients: a characterisation of well-curvedness in terms of properties of the polynomial $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$, obtained with Geometric Invariant Theory techniques, and Christ's Method of Refinements. With the latter, matters are reduced to a sublevel set estimate, which is proven by a linear programming argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_15530 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$ Dendrinos, Spyridon Mustata, Andrei Vitturi, Marco Classical Analysis and ODEs 44A12 (Primary) 42B10, 14L24 (Secondary) We draw a connection between the affine invariant surface measures constructed by P. Gressman and the boundedness of a certain geometric averaging operator associated to surfaces of codimension $2$ and related to the Fourier Restriction Problem for such surfaces. For a surface given by $(ξ, Q_1(ξ), Q_2(ξ))$, with $Q_1,Q_2$ quadratic forms on $\mathbb{R}^d$, the particular operator in question is the $2$-plane transform restricted to directions normal to the surface, that is \[ \mathcal{T}f(x,ξ) := \iint_{|s|,|t| \leq 1} f(x - s \nabla Q_1(ξ) - t \nabla Q_2(ξ), s, t)\,ds\,dt, \] where $x,ξ\in \mathbb{R}^d$. We show that when the surface is well-curved in the sense of Gressman (that is, the associated affine invariant surface measure does not vanish) the operator satisfies sharp $L^p \to L^q$ inequalities for $p,q$ up to the critical point. We also show that the well-curvedness assumption is necessary to obtain the full range of estimates. The proof relies on two main ingredients: a characterisation of well-curvedness in terms of properties of the polynomial $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$, obtained with Geometric Invariant Theory techniques, and Christ's Method of Refinements. With the latter, matters are reduced to a sublevel set estimate, which is proven by a linear programming argument. |
| title | A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$ |
| topic | Classical Analysis and ODEs 44A12 (Primary) 42B10, 14L24 (Secondary) |
| url | https://arxiv.org/abs/2209.15530 |