Endpoint restriction estimates for the paraboloid over finite fields
Let \(F\) denote a finite field of characteristic \(p>2\). Let \(e:F\to S^1\) (the unit circle in \(\mathbb C\)) be a non-principal character of \(F\). One considers the vector space \(F^n\) as endowed with the counting measure \(dx\) which assigns mass \(1\) to each point and \(F_*^n\) as endowed with the normalized counting measure \(d\xi\) which assigns mass \(|F|^{-n}\) to each point (where \(|F|\) denotes the size of \(F\)). Let \(\mathcal P=\{(\gamma,\gamma\cdot\gamma)\in F_*^n;\,\gamma\in F_*^{n-1}\}\) be the paraboloid in \(F_*^n\), endowed with the normalized surface measure \(d\sigma\) which assigns \(|\mathcal P|^{-1}\) to each point in \(\mathcal P\). For a function \(f:\mathcal P\to \mathbb C\), we define the inverse Fourier transform \((fd\sigma){\check{\phantom{t}}}\) of \(fd\sigma\) by NEWLINE\[NEWLINE(fd\sigma) {\check{\phantom{t}}}(x)=|\mathcal P|^{-1}\sum_{\xi\in\mathcal P}f(\xi)e(x\cdot \xi).NEWLINE\]NEWLINE For \(q\in[1,\infty)\), we denote NEWLINE\[NEWLINE\|f\|_{L^q(F^n,dx)} =(\sum_{x\in F^n}|f(x)|^q)^{1/q}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|f\|_{L^q(\mathcal P,d\sigma)} =(|\mathcal P|^{-1}\sum_{\xi\in \mathcal P}|f(\xi)|^q)^{1/q}.NEWLINE\]NEWLINE A restriction inequality is NEWLINE\[NEWLINE\|\hat f\|_{L^{p'}(\mathcal P,d\sigma)}\leq \mathcal R(p\to q)\|f\|_{L^{q'}(F^n,dx)},NEWLINE\]NEWLINE where \(\mathcal R(p\to q)\) is the best constant in the above. By duality, this is equivalent to NEWLINE\[NEWLINE\|(fd\sigma){\check{\phantom{t}}}\|_{L^{q}(F^n,dx)} \leq \mathcal R(p\to q)\|f\|_{L^{p}(\mathcal P,d\sigma)}.NEWLINE\]NEWLINE The authors' main results are the following: Theorem 1. For the paraboloid in \(3\) dimensions with \(-1\) not a square, \(\mathcal R(8/5\to 4)\) and \(\mathcal R(2\to 18/5)\) are uniformly bounded for \(|F|\).NEWLINENEWLINETheorem 2. For the paraboloid in \(n\) dimensions when \(n\geq4\) is even or when \(n\) is odd and \(|F|=q^m\) for a prime \(q\) congruent to \(3\) modulo 4 such that \(m(n-1)\) is not a multiple of \(4\), \(\mathcal R(4n/(3n-2)\to 4)\) and \(\mathcal R(2\to 2n^2/(n^2-2n+2))\) are uniformly bounded for \(|F|\).NEWLINENEWLINETheorem 1 improves upon Proposition 5.2 in [\textit{G. Mockenhaupt} and \textit{T. Tao}, Duke Math. J. 121, No. 1, 35--74 (2004; Zbl 1072.42007)], and Theorem 2 improves upon Theorems 1, 2 and 3 in [\textit{A. Iosevich} and \textit{D. Koh}, Math. Z. 266, No. 2, 471--487 (2010; Zbl 1203.42013)].
scientific article; zbMATH DE number 6038834
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