arXiv subjects
Publications and source records attributed to René Quilodrán.
We prove the existence of functions that extremize the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space $\mathbb{R}^4$ and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.
We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{iξ\cdotω}$, for some $ξ$, thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq 2$ and general even exponents.
Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $σ_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(fσ_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.
For an appropriate class of convex functions $ϕ$, we study the Fourier extension operator on the surface $\{(y, |y|^2+ϕ(y)):y\in\mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations, and compute the corresponding operator norms exactly where only lower bounds were previously known.
We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $1 4$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.
We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper.
We study the problem of existence of extremizers for the $L^2$ to $L^p$ adjoint Fourier restriction inequalities on the hyperboloid in dimensions 3 and 4, in which cases $p$ is an even integer. We will use the method developed by Foschi to show that extremizers do not exist.
It is known that extremizers for the $L^2$ to $L^6$ adjoint Fourier restriction inequality on the cone in $\mathbb{R}^3$ exist. Here we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the cone. If we use the knowledge of the exact form of the extremizers, as found by Carneiro, then we can show that nonnegative extremizing sequences converge, after the application of symmetries.
It was proved independently by Foschi and Hundertmark, Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for L^2 functions on the paraboloid in the two lowest-dimesional cases. Here we prove that Gaussians are critical points for the L^p to L^q adjoint Fourier restriction inequalities if and only if p=2. Also, Gaussians are critial points for the L^2 to L^r_t L^q_x Strichartz inequalities for all admissible pairs (r,q) in (1,infinity)^2.
We show that given a collection of A lines in \R^n, n\geq 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(A^{n/(n-1)}). An analogous result for smooth algebraic curves is also proven.