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Keith M. Rogers

Publications and source records attributed to Keith M. Rogers.

At least 19 recordsLinked to original sources
arXiv2026

Estimates for $L^p$ variants of Gowers norms

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We also establish two families of degree-lowering inequalities and study their near-extremisers. As a byproduct of this broader theory, we show that the constant in the log-convexity estimate for Gowers norms is strictly less than unity, confirming the aforementioned conjecture of Bennett and Tao. Finally, we characterise the values of the parameters for which these Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

math.CA↗
arXiv2026

Orthonormal Sobolev estimates with fractal measures

We prove a fractal version of Lieb's Hardy-Littlewood-Sobolev inequality for orthonormal functions. On the one hand, this can be viewed as a trace theorem for orthonormal functions. On the other, it allows us to recover the Rozenblum-Tashchiyan bound for the number of negative eigenvalues of $-Δ-μ$, where $μ$ is a shell potential. We also recover Rozenblum's bound for the sum of negative eigenvalues via a Lieb-Thirring kinetic inequality. Our proof is direct, avoiding both Schatten classes and variational arguments. We first reprove Adams' fractal Hardy-Littlewood-Sobolev inequality (for single functions) via Fourier analysis. This yields the required endpoint estimate as well as a bound for the interaction energy of Frostman measures.

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arXiv2024

Reconstruction for the Calderón problem with Lipschitz conductivities

We determine the conductivity of the interior of a body using electrical measurements on its surface. We assume only that the conductivity is bounded below by a positive constant and that the conductivity and surface are Lipschitz continuous. To determine the conductivity we first solve an associated integral equation locally, finding solutions in $H^1(B)$, where $B$ is a ball that properly contains the body. A key ingredient is to equip this Sobolev space with an equivalent norm which depends on two auxiliary parameters that can be chosen to yield a contraction.

math.AP↗
arXiv2019

Improved Fourier restriction estimates in higher dimensions

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.

math.CA↗
arXiv2019

On a higher dimensional version of the Benjamin--Ono equation

We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1Δu+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish sharp space--time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in $L^2$-Sobolev spaces $H^{s}(\mathbb{R}^d)$, with $s>5/3$ if $d=2$ and $s>d/2+1/2$ if $d\ge 3$. We also provide ill-posedness results.

math.AP↗
arXiv2019

Improved bounds for the Kakeya maximal conjecture in higher dimensions

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.

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arXiv2019

Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane

For potentials $V\in L^\infty(\mathbb{R}^2,\mathbb{R})$ and $A\in W^{1,\infty}(\mathbb{R}^2,\mathbb{R}^2)$ with compact support, we consider the Schrödinger equation $-(\nabla +iA)^2 u+Vu=k^2u$ with fixed positive energy $k^2$. Under a mild additional regularity hypothesis, and with fixed magnetic potential $A$, we show that the scattering solutions uniquely determine the electric potential $V$. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.

math.AP↗
arXiv2018

On the polynomial Wolff axioms

We confirm a conjecture of Guth concerning the maximal number of $δ$-tubes, with $δ$-separated directions, contained in the $δ$-neighborhood of a real algebraic variety. Modulo a factor of $δ^{-\varepsilon}$, we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski's projection theorem and Gromov's algebraic lemma.

math.CA↗
arXiv2015

On directional maximal operators in higher dimensions

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sjögren--Sjölin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\text{loc}}^p(\mathbb{R}^n)$, replacing balls by tubes which point in these directions.

math.CA↗
arXiv2015

Rough Potential Recovery in the Plane

We reconstruct compactly supported potentials with only half a derivative in $L^2$ from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial data of solutions to time-dependent Schrödinger equations. We also provide examples of compactly supported potentials, with $s$ derivatives in $L^2$ for any $s<1/2$, which cannot be recovered by these means. Thus the recovery method has a different threshold in terms of regularity than the corresponding uniqueness result.

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arXiv2013

On Conic Fourier Multipliers

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on $L^p(\mathbb{R}^3)$ with $1<p<\infty$.

math.CA↗
arXiv2012

Twisted Hilbert transforms vs Kakeya sets of directions

Given a discrete group $\G$ and an orthogonal action $γ: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_γ\G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_γid_\G$ is $L_p$-bounded iff the orbit $\mathcal{O}_γ(u)$ is finite. This is in sharp contrast with twisted Riesz transforms $R_u \rtimes_γid_\G$, which are always bounded. Our result characterizes Fourier summability in $L_p$ for this class of groups. We also extend de Leeuw's compactification theorem to this setting and obtain stronger estimates for functions with "lacunary" frequency support.

math.OA↗
arXiv2012

Square functions and maximal operators associated with radial Fourier multipliers

We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.

math.CA↗
arXiv2011

Improved bounds for Stein's square functions

We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schrödinger equations.

math.CA↗
arXiv2009

Endpoint maximal and smoothing estimates for Schroedinger equations

For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.

math.CA↗