Restriction of Fourier transforms to curves: an endpoint estimate with affine arclength measure (Q2856659)

The Fourier restriction operator associated to a curve \(\gamma:I\rightarrow \mathbb{R}^d\) is defined as follows NEWLINE\[NEWLINE \mathcal{R}f(t)=\widehat{f}(\gamma(t)), NEWLINE\]NEWLINE where \(f\) is a Schwartz function on \(\mathbb{R}^d\) and \(\widehat{f}\) denotes its Fourier transform. Metrical properties of the operator \(\mathcal{R}\) are studied with respect to the affine arclength measure \(d\lambda=\omega(t)dt\) with weight NEWLINE\[NEWLINE \omega(t)=|\tau(t)|^{\frac{2}{d^2+d}}, NEWLINE\]NEWLINE where \(\tau(t)=\det(\gamma'(t),\dots, \gamma^{(d)}(t)).\)NEWLINENEWLINENEWLINENEWLINE In dimensions \(d\geq 3\) for ''monomial'' curves NEWLINE\[NEWLINE \gamma(t)=(t^{a_1},t^{a_2},\dots,t^{a_d}), \;\;\;\; 0<t<\infty, NEWLINE\]NEWLINE and for ''simple'' polynomial curves NEWLINE\[NEWLINE \gamma(t)=(t,t^2/2!,\dots,t^{d-1}/(d-1)!,P(t)),\;\;\;\; t\in\mathbb{R}, NEWLINE\]NEWLINE where \(P\) is an arbitrary polynomial, the validity of the endpoint inequality NEWLINE\[NEWLINE \Bigg(\int_I |\widehat{f}\circ \gamma|^{p_d}d\lambda \Bigg)^{1/p_d}\lesssim \|f\|_{L^{p_d,1}(\mathbb{R}^d)}, \;\;\;\;p_d=\frac{d^2+d+2}{d^2+d}NEWLINE\]NEWLINE is established. The inequality was previously known only for the model case \(\gamma(t)=(t,t^2,\dots,t^d)\).