Fourier restriction for affine arclength measures in the plane (Q2731924)

The main result of the paper is the following NEWLINENEWLINENEWLINETheorem. If \(1\leq p<4/3\) and \(p^{-1}+(3q)^{-1}=1,\) there is a constant \(C=C(p)\) such that the estimate NEWLINE\[NEWLINE\left(\int_a^b|\hat f(t,\phi(t))|^q d\lambda(t)\right)^{1/q}\leq C\|f\|_{L^p(\mathbb R^2)}NEWLINE\]NEWLINE holds whenever \(d\lambda(t)=\phi''(t)^{1/3} dt\) with \(\phi\) being a real-valued function on \((a,b)\) satisfying \(\phi''>0\) and \(\phi'''\geq 0\) on \((a,b).\) The measure \(d\lambda\) is called the affine arclength measure on the curve \(\gamma(t)=(t,\phi(t)).\) NEWLINENEWLINENEWLINES. W. Drury was the first to point out its relevance to certain problems in harmonic analysis. The theorem is an analog, uniform over the indicated class of curves \(\gamma\) in \(\mathbb R^2,\) of the Fefferman-Zygmund restriction theorem for the circle, and its proof is similar, in general, to the Fefferman-Zygmund proof.