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Restriction estimates for the paraboloid over finite fields

Posted in Fourier Analysis, math.CA, Paper by Mark Lewko on September 19, 2010

Allison and I recently completed a paper titled Restriction estimates for the paraboloid over finite fields. In this note we obtain some endpoint restriction estimates for the paraboloid over finite fields.

Let S denote a hypersurface in \mathbb{R}^{n} with surface measure d\sigma. The restriction problem for S is to determine for which pairs of (p,q) does there exist an inequality of the form

\displaystyle ||\hat{f}||_{L^{p'}(S,d\sigma)} \leq C ||f||_{L^{q'}(\mathbb{R}^n)}.

We note that the left-hand side is not necessarily well-defined since we have restricted the function \hat{f} to the hypersurface S, a set of measure zero in  \mathbb{R}^{n}. However, if we can establish this inequality for all Schwartz functions f, then the operator that restricts \hat{f} to S (denoted by \hat{f}|_{S}), can be defined whenever f \in L^{q}. In the Euclidean setting, the restriction problem has been extensively studied when S is a sphere, paraboloid, and cone. In particular, it has been observed that restriction estimates are intimately connected to questions about certain partial differential equations as well as problems in geometric measure theory such as the Kakeya conjecture. The restriction conjecture states sufficient conditions on (p,q) for the above inequality to hold. In the case of the sphere and paraboloid, the question is open in dimensions three and higher.

In 2002 Mockenhaupt and Tao initiated the study of the restriction phenomena in the finite field setting. Let us introduce some notation to formally define the problem in this setting. We let F denote a finite field of characteristic p >2. We let S^{1} denote the unit circle in \mathbb{C} and define e: F \rightarrow S^1 to be a non-principal character of F. For example, when F = \mathbb{Z}/p \mathbb{Z}, we can set e(x) := e^{2\pi i x/p}. We will be considering the vector space F^n and its dual space F_*^n. We can think of 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, so the total mass is equal to 1 here).

For a complex-valued function f on F^n, we define its Fourier transform \hat{f} on F_*^n by:

\displaystyle \hat{f}(\xi) := \sum_{x \in F^n} f(x) e(-x \cdot \xi).

For a complex-valued function g on F_*^n, we define its inverse Fourier transform g^{\vee} on F^n by:

\displaystyle g^{\vee}(x) := \frac{1}{|F|^n} \sum_{\xi \in F_*^n} g(\xi) e(x\cdot \xi).

It is easy to verify that (\hat{f})^\vee = f and \widehat{(g^{\vee})} = g.

We define the paraboloid \mathcal{P} \subset F_*^n as: \mathcal{P} := \{(\gamma, \gamma \cdot \gamma): \gamma \in F_*^{n-1}\}. This is endowed with the normalized “surface measure” d\sigma which assigns mass |\mathcal{P}|^{-1} to each point in \mathcal{P}. We note that |\mathcal{P}| = |F|^{n-1}.
For a function f: \mathcal{P} \rightarrow \mathbb{C}, we define the function (f d\sigma)^\vee: F^n \rightarrow \mathbb{C} as follows:

\displaystyle (f d\sigma)^\vee (x) := \frac{1}{|\mathcal{P}|} \sum_{\xi \in \mathcal{P}} f(\xi) e(x \cdot \xi).

For a complex-valued function f on F^n and q \in [1, \infty), we define

\displaystyle ||f||_{L^q(F^n, dx)} := \left( \sum_{x \in F^n} |f(x)|^q \right)^{\frac{1}{q}}.

For a complex-valued function f on \mathcal{P}, we similarly define

\displaystyle ||f||_{L^q(\mathcal{P},d\sigma)} := \left( \frac{1}{|\mathcal{P}|} \sum_{\xi \in \mathcal{P}} |f(\xi)|^q \right)^{\frac{1}{q}}.

Now we define a restriction inequality to be an inequality of the form

\displaystyle ||\hat{f}||_{L^{p'}(S,d\sigma)} \leq \mathcal{R}(p\rightarrow q) ||f||_{L^{q'}(\mathbb{R}^n)},

where \mathcal{R}(p\rightarrow q) denotes the best constant such that the above inequality holds. By duality, this is equivalent to the following extension estimate:

||(f d\sigma)^\vee||_{L^q(F^n, dx)} \leq \mathcal{R}(p\rightarrow q) ||f||_{L^p(\mathcal{P},d\sigma)}.

We will use the notation X \ll Y to denote that quantity X is at most a constant times quantity Y, where this constant may depend on the dimension n but not on the field size, |F|. For a finite field F, the constant \mathcal{R}(p\rightarrow q) will always be finite. The restriction problem in this setting is to determine for which (p,q) can we upper bound \mathcal{R}(p\rightarrow q) independently of |F| (i.e. for which (p,q) does \mathcal{R}(p \rightarrow q) \ll 1 hold).

Mockenhaupt and Tao solved this problem for the paraboloid in two dimensions.  In three dimensions, we require -1 not be a square in F (without this restriction the parabaloid will contain non-trivial subspaces which lead to trivial counterexamples, but we will not elaborate on this here). For such F, they showed that \mathcal{R}(8/5+\epsilon \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{18}{5}+\epsilon) \ll 1 for every \epsilon>0. When \epsilon=0, their bounds were polylogarithmic in |F|. Mockenhaupt and Tao’s argument for the \mathcal{R}(8/5 \rightarrow 4) estimate proceeded by first establishing the estimate for characteristic functions. Here one can expand the L^4 norm and reduce the problem to combinatorial estimates. A well-known dyadic pigeonhole argument then allows one to pass back to general functions at the expense of a logarithmic power of |F|. Following a similar approach (but requiring much more delicate Gauss sum estimates), Iosevich and Koh proved that \mathcal{R}(\frac{4n}{3n-2}+ \epsilon \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{2n^2}{n^2-2n+2} + \epsilon) \ll 1 in higher dimensions (in odd dimensions some additional restrictions on F are required). Again, however, this argument incurred a logarithmic loss at the endpoints from the dyadic pigeonhole argument.

In this note we remove the logarithmic losses mentioned above. Our argument begins by rewriting the L^4 norm as ||(fd\sigma)^{\vee}||_{L^4}=||(fd\sigma)^{\vee}(fd\sigma)^{\vee}||_{L^2}^{1/2}. We then adapt the arguments of the prior papers to the bilinear variant ||(fd\sigma)^{\vee}(gd\sigma)^{\vee}||_{L^2}^{1/2} in the case that f and g are characteristic functions.

To obtain estimates for arbitrary functions f, we can assume that f is non-negative real-valued and decompose f as a linear combination of characteristic functions, where the coefficients are negative powers of two (we can do this without loss of generality by adjusting only the constant of our bound). We can then employ the triangle inequality to upper bound ||(fd\sigma)^{\vee}||_{L^4} by a double sum of terms like ||(\chi_j d\sigma)^{\vee}(\chi_k d\sigma)^{\vee}||_{L^2}^{1/2}, where \chi_i and \chi_j are characteristic functions, weighted by negative powers of two. We then apply our bilinear estimate for characteristic functions to these inner terms and use standard bounds on sums to obtain the final estimates.

Our method yields the following theorems:

Theorem For the paraboloid in 3 dimensions with -1 not a square, we have \mathcal{R}(8/5 \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{18}{5}) \ll 1.
Theorem For the paraboloid in n dimensions when n \geq 4 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, we have \mathcal{R}(\frac{4n}{3n-2} \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{2n^2}{n^2-2n+2}) \ll 1.

We recently learned that in unpublished work Bennett, Carbery, Garrigos, and Wright have also obtained the results in the 3-dimensional case. Their argument proceeds rather differently than ours and it is unclear (at least to me) if their argument can be extended to the higher dimensional settings.

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