Uniform estimates for the local restriction of the Fourier transform to curves (Q2846972)

The restriction of the Fourier transform to lower dimensional manifolds and \(L^p\)-estimates for this operator have a long history, starting from the original observation of S. M. Stein and the Stein-Tomas restriction theorem. Since then, many authors have studied this phenomenon. The paper under review is such one where the Fourier restriction to certain curves in \(\mathbb R^d\) is studied.NEWLINENEWLINELet \(\gamma : I \to \mathbb R^d\) be a curve where \(I \subset \mathbb R\) is an interval. Define the affine arclength measure \(d\sigma\) by NEWLINE\[NEWLINEd\sigma(\phi) = \int_I\phi(\gamma(t)) |L_\gamma(t)|^{2/(d(d+1))}\,dt,NEWLINE\]NEWLINE where NEWLINE\[NEWLINEL_\gamma(t) = \det (\gamma'(t), \cdots , \gamma^{(d)}(t) ).NEWLINE\]NEWLINENEWLINENEWLINEThe main result is the following:NEWLINENEWLINELet \(\Gamma(t) = (t^{a_1} \theta_1(t), \cdots, t^{a_d}\theta_d(t)), t \geq 0,\) where for each \(1 \leq i \leq d,\) \(a_i \in \mathbb R \setminus \{0\},\) all \(a_i\) are distinct, \(\theta_i\) is real-valued and in \(C^d(\mathbb R),\) the limit \(\lim_{t \to 0} \theta_i(t)\) exists and is not equal to zero, and for all \(1 \leq m \leq d,\) NEWLINE\[NEWLINE \lim_{t \to 0} t^m \theta_i^{(m)}(t) = 0.NEWLINE\]NEWLINE Then there exists \(C,\) only depending on \(d\) and \(p,\) and \(\delta,\) depending on the \(a_i,\) the \(\theta_i\) and \(d,\) such that NEWLINE\[NEWLINE \int_0^\delta|\widehat{f} (\Gamma(t))|^q |L_\Gamma(t)|^{2/(d(d+1))}\,dt \leq C \|f\|_p^qNEWLINE\]NEWLINE where NEWLINE\[NEWLINEp' = \frac{d(d+1)}{2}q,\quad \text{and} \quad 1 \leq p \leq \frac{d^2+d+2}{d^2+d}.NEWLINE\]NEWLINE In the particular case where all \(\theta_i\) are constant, \(\delta\) can be taken to be equal to \(\infty.\)