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Stefan Buschenhenke

Publications and source records attributed to Stefan Buschenhenke.

13 recordsLinked to original sources
arXiv2026

$L^p$-Estimates for maximal averages along mixed homogeneous hypersurfaces in $\mathbb{R}^{3}$

In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $\Phi$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $\Phi$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $\Phi,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $\Phi.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an ``exceptional'' class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.

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arXiv2025

$L^p$-estimates for FIO-cone multipliers

The classical cone multipliers are Fourier multiplier operators which localize to narrow $1/R$-neighborhoods of the truncated light cone in frequency space. By composing such convolution operators with suitable translation invariant Fourier integral operators (FIOs), we obtain what we call FIO-cone multipliers. We introduce and study classes of such FIO-cone multipliers on $\Bbb R^3$, in which the phase functions of the corresponding FIOs are adapted in a natural way to the geometry of the cone and may even admit singularities at the light cone. By building on methods developed by Guth, Wang and Zhang in their proof of the cone multiplier conjecture in $\Bbb R^3,$ we obtain $L^p$-estimates for FIO-cone multipliers in the range $4/3\le p\le 4$ which are stronger by the factor $R^{-|1/p-1/2|}$ than what a direct application of the method of Seeger, Sogge and Stein for estimating FIOs would give. An important application of our theory is to maximal averages along smooth analytic surfaces in $\Bbb R^3.$ It allows to confirm a conjecture on the the critical Lebesgue exponent for a prototypical surface from a small class of ``exceptional'' surfaces, for which this conjecture had remained open.

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arXiv2022

Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part II -- A geometric conjecture and its proof for generic 2-surfaces

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small surface-patches near a given point $x^0$ exhibiting singularities of type $\mathcal A$ in the sense of Arnol'd at this point; this is the situation which had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponent such that $\mathcal M_S$ is $L^p$-bounded for $p>p_c,$ we are able to identify $p_c$ for all analytic surfaces of type $\mathcal A$ (with the exception of a small subclass), by means of quantities which can be determined from associated Newton polyhedra. Besides the well-known notion of height at $x^0,$ a new quantity, which we call the effective multiplicity, turns out to play a crucial role here. We also state a conjecture on how the critical exponent $p_c$ might be determined by means of a geometric measure theoretic condition, which measures in some way the order of contact of arbitrary ellipsoids with $S,$ even for hypersurfaces in arbitrary dimension, and show that this conjecture holds indeed true for all classes of 2-hypersurfaces $S$ for which we have gained an essentially complete understanding of $\mathcal M_S$ so far. Our results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger for arbitrary analytic 2-surfaces.

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arXiv2021

Factorisation in Restriction theory and near extremisers

We give an alternative argument to the application of the so-called Maurey- Nikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple method applies to any compact quadratic surface, in particular compact parts of the paraboloid and the hyperbolic paraboloid. This is achieved by constructing near extremisers with big "mass", which itself might be of interest.

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arXiv2020

Fourier restriction for smooth hyperbolic 2-surfaces

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

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arXiv2020

On the nonlinear Brascamp-Lieb inequality

We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic analysis and partial differential equations. Our proof consists of running an efficient, or "tight", induction on scales argument, which uses the existence of gaussian near-extremisers to the underlying linear Brascamp-Lieb inequality (Lieb's theorem) in a fundamental way. A key ingredient is an effective version of Lieb's theorem, which we establish via a careful analysis of near-minimisers of weighted sums of exponential functions.

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arXiv2020

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

We consider a surface with negative curvature in $\Bbb R^3$ which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

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arXiv2020

Partitions of flat one-variate functions and a Fourier restriction theorem for related perturbations of the hyperbolic paraboloid

We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which $|h'''|$ is of a fixed size $λ,$ we can apply methods devised in preceding papers, but since we loose control on higher order derivatives of $h$ we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of $h'''.$

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arXiv2019

On Fourier restriction for finite-type perturbations of the hyperboloid

In this note, we continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy,$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function of finite type. Our results build on previous joint work in which we have studied the case $h(y)=y^3/3$ by means of the bilinear method. As it turns out, the understanding of that special case becomes also crucial for the treatment of arbitrary finite type perturbation terms $h(y).$

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arXiv2019

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear method gave here sharp results for p > 10/3 (Lee 05, Vargas 05, Stovall 17), and this result was recently improved to p > 3.25 (Cho-Lee 17, Kim 17). This paper aims to be a first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the end-point, for p > 10/3. In the application of the bilinear method, we show that the behavior at small scale in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces.

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arXiv2018

The nonlinear Brascamp-Lieb inequality for simple data

We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant as the size of the neighbourhood approaches zero, answering a question of Cowling, Martini, Müller and Parcet. Our proof consists of running an efficient, or "tight", induction on scales argument which uses the existence of gaussian extremisers to the underlying linear Brascamp-Lieb inequality in a fundamental way.

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arXiv2015

A Fourier Restriction Theorem For A Twodimensional Surface Of Finite Type

The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general $L^p(R^3)\to L^q(S)$ Fourier restriction estimates, by studying a prototypical class of two-dimensional surfaces with strongly varying curvature conditions. Our approach is based on an adaptation of the so-called bilinear method. We discuss several new features arising in the study of this problem.

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