arXiv subjects
Publications and source records attributed to Betsy Stovall.
We prove that maximizers exist and that maximizing sequences possess subsequences that converge modulo symmetries to maximizers for the $L^p \to L^q$ Fourier extension inequalities associated to the hyperbolic paraboloid in three ambient dimensions.
We prove new $L^p$ boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group $\mathbb H_n$. Our results hold for a range $1\le p\le p_n$ where $p_n\to 2$ as $n\to\infty$. As shown by the first named author in 1990 a Stein-Tomas type Fourier restriction theorem fails to hold on $\mathbb H_n$ and thus previous results based on the approach by Fefferman and Stein from the Euclidean setting only allowed to cover the cases $p=1$ and $p=\infty$. Our results on Bochner-Riesz means follow from a more general $p$-sensitive spectral multiplier theorem which is the main result of this article. This is obtained as a consequence of $L^p$ estimates for square functions associated with the Heisenberg wave operator.
We consider two incidence problems for integral curves of vector fields. The first is an analogue of the Euclidean joints problem, in which lines are replaced by integral curves of smooth vector fields taken from some finite-dimensional set. The second is a bilinear and more rigid version of the first, in which we have two vector fields and one family of integral curves tangent to each and wish to know how many intersecting pairs are possible. In both cases, a curvature condition of Hörmander makes possible nontrivial bounds on the number of incidences in terms of the number of curves.
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.
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 establish lower bounds for the operator norms of the Fourier restriction/extension operators associated to monomial curves with affine arclength measure. Furthermore, we prove that the set of all extremizing sequences of such an operator is precompact modulo the operator's symmetry group if and only if the operator norm is strictly larger than this threshold. For the proof, we introduce a number of new ingredients, some of which may be applicable to analogous questions on more general manifolds.
We prove both necessary and sufficient conditions for $L^p$-bound\-ed\-ness of certain multilinear generalized Radon transforms that arise as Heisenberg group analogues of the Brascamp--Lieb inequalities on Euclidean space. The necessary and sufficient conditions coincide in some important special cases, but differ in others.
We show that the restriction and extension operators associated to the moment curve possess extremizers and that $L^p$-normalized extremizing sequences of these operators are precompact modulo symmetries.
In this article, we develop a linear profile decomposition for the $L^p \to L^q$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p \max\{p,\tfrac{d+2}d p'\}$, or if $q=\tfrac{d+2}d p'$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator.
We consider $r$-variation operators for the family of spherical means, with special emphasis on $L^p\to L^q$ estimates.
In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These inequalities are unconditional (and optimal) in the bilinear range $q > \frac{10}{3}$.
In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.
In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case $q=2$. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.
We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying a natural nilpotency hypothesis.
Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. For example, when is there a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order? We address this question in a quantitative way, which strengthens and generalizes previous works on the quantitative theory of sub-Riemannian (aka Carnot-Carathéodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, the second author, and others. Furthermore, we provide a diffeomorphism invariant version of these theories. This is the first part in a three part series of papers. In this paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results from the backbone of the series. The methods of this paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs.
We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.
If $T$ is the operator given by convolution with surface measure on the sphere, $(E,F)$ is a quasi-extremal pair of sets for $T$ if $\langle Tχ_E, χ_F \rangle \gtrsim |E|^{d/(d+1)}|F|^{d/(d+1)}$. In this article, we explicitly define a family $\mathcal{F}$ of quasi-extremal pairs of sets for $T$. We prove that $\mathcal{F}$ is fundamental in the sense that every quasi-extremal pair $(E,F)$ is comparable (in a rather strong sense) to a pair from $\mathcal{F}$. This extends work carried out by M. Christ for convolution with surface measure on the paraboloid.
We establish strong-type endpoint $L^p(\mathbb R^d) \to L^q(\mathbb R^d)$ bounds for the operator given by convolution with affine arclength measure on polynomial curves for $d \geq 4$. The bounds established depend only on the dimension $d$ and the degree of the polynomial.