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

E. Ferreyra, T. Godoy, M. Urciuolo (2004)

Studia Mathematica

Let φ:ℝ² → 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 ) ) d x where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from L p ( ³ ) to L q ( Σ , d σ ) for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Restrictions of Fourier transforms to curves

S. W. Drury (1985)

Annales de l'institut Fourier

Let x ( t ) = ( t , 1 2 t 2 , 1 6 t 3 ) a certain curve in R 3 . We investigate inequalities of the type { | f ^ ( x ( t ) ) | b d t } 1 / b C f a for f 𝒮 ( R 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

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