Restriction theorems for the Fourier transform to some manifolds in 𝐑ⁿ

“…the conditions on p and q are also necessary. Also in [1], [5], [4] and [6] restriction theorems for curves of finite type are obtained. Concerning the homogeneous case, the type set E is studied in [2] for ϕ(x) = ( n j=1 |x j | r ) α .…”

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confidence: 99%

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in R³

2004

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Abstract. Let ϕ : R 2 → R be a homogeneous polynomial function of degree m ≥ 2, let Σ = {(x, ϕ(x)) : |x| ≤ 1} and let σ be the Borel measure on Σ defined by σ(A) = B χ A (x, ϕ(x)) dx where B is the unit open ball in R 2 and dx denotes the Lebesgue measure on R 2 . We show that the composition of the Fourier transform in R 3 followed by restriction to Σ defines a bounded operator fromFor m ≥ 6 the results are sharp except for some border points.1. Introduction. Let ϕ : R n → R be a smooth enough function, let B be the open unit ball in R n and letwhere f denotes the usual Fourier transform of f defined by f (ξ) = f (u)e −i u,ξ du. Let σ be the Borel measure on Σ defined by σ(A) = B χ A (x, ϕ(x)) dx and let E be the type set for the operator R, i.e. the set of pairs (1/p, 1/q)for some c > 0 and all f ∈ S(R n+1 ), where the spaces L p (R n+1 ) and L q (Σ) are taken with respect to the Lebesgue measure in R n+1 and the measure σ respectively. The L p (R n+1 )-L q (Σ) boundedness properties of the restriction operator R have been widely studied. It is well known that for Σ as above, ifIn [10], it is proved, for the case where ϕ is a nondegenerate quadratic form in R n+1 , that (1/p, 1/2) ∈ E if (n + 4)/(2n + 4) ≤ 1/p ≤ 1, and the method given there provides a general tool to obtain, from suitable estimates for σ, L p (R n+1 )-L 2 (Σ) estimates for R. Moreover, a general theorem, due to Stein,2000 Mathematics Subject Classification: Primary 42B10, 26D10.

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“…Initial results in higher dimensions for the smaller range 1 ≤ p < (d 2 + 2d)/(d 2 + 2d − 2) are due to Prestini [28], with strict inequality p ′ > d(d + 1)q/2 for the local result. For the same range of p, Christ [13] showed boundedness on the edge p ′ = d(d + 1)q/2.…”

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confidence: 99%