Failure of weak-type endpoint restriction estimates for quadratic manifolds.

Abstract It is well-known that the Fourier extension operator for the paraboloid in $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> cannot be weak-type bounded at the restriction endpoint $$q = 2d/(d-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , since such an estimate would imply bounds for the Kakeya maximal function which contradict the existence of Besicovitch sets. We generalize this approach to prove that the Fourier extension operator for an n -dimensional quadratic manifold $$\mathcal {M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> cannot be weak-type bounded at the restriction endpoint. The key step in this proof is constructing a set $$K \subset \mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> containing a translate of every plane normal to $$\mathcal {M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> which can be covered by $$\lesssim \delta ^{-d}\left( \frac{\log \log (1/\delta )}{\log (1/\delta )}\right) ^{n/(d-n)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≲</mml:mo> <mml:msup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mfenced> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> many $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -balls. Such a set rules out endpoint bounds for the associated Kakeya maximal function and hence weak-type endpoint estimates for the restriction operator.