arXiv subjects
Publications and source records attributed to James Tautges.
We investigate the sharp endpoint extension inequality for the moment curve in finite fields. We determine the optimal constant and characterize the maximizers in two complementary regimes: (i) low dimensions $d\leq 20$; (ii) large field cardinality $q\geq \frac{d(d-1)}{2 \log 6} + \frac{(2d-1)}{3}$. Our proof strategy relies on an intriguing interplay between analysis, algebra and combinatorics.
In this article, we establish various facts about extremizers for $L^p$-improving convolution operators $T\colon L^p \rightarrow L^q$ associated with compactly-supported probability measures on either $\mathbb{R}^d$ or $\mathbb{T}^d$ . If $σ$ has positive Fourier decay, we prove that extremizers exist and extremizing sequences are precompact modulo translation for all "non-endpoint" $(p,q)$. These extremizers also satisfy an interesting positivity property and belong to $C_{loc}^\infty \cap L^\infty$.
We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb R^{1+d}$. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. Global maximizers for the $L^2$ Fourier extension inequality on the cone in $\mathbb R^{1+d}$ have been characterized in the lowest-dimensional cases $d\in\{2,3\}$. We further prove that these functions are critical points for the $L^p$ to $L^q$ Fourier extension inequality if and only if $p = 2$.
We consider the adjoint restriction inequality associated to the hypersurface $\{(τ, ξ) : τ= \pm|ξ|^2, \;ξ\in \mathbb{R}^d\}$ at the Stein-Tomas exponent. Extremizers exist in all dimensions and extremizing sequences are precompact modulo symmetries conditional on a certain inequality, which we verify in the case $d \in \{1,2\}$.
Consider the adjoint restriction inequality associated with the hypersurface $\{ (τ, ξ) \in \mathbb{R}^{d+1} : τ= |ξ|^2 \} \cup \{(τ, ξ) \in \mathbb{R}^{d+1} : τ- τ_0 = |ξ- ξ_0|^2\}$ for any $(τ_0, ξ_0) \neq 0$. We prove that extremizers do not exist for this inequality and fully characterize extremizing sequences in terms of extremizers for the adjoint restriction inequality for the paraboloid.