arXiv subjects

David Beltran

Publications and source records attributed to David Beltran.

28 records · Page 2Linked to original sources

arXiv2019

Bilinear identities involving the $k$-plane transform and Fourier extension operators

We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$-plane transform. As the estimates are $L^2$-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical $L^2(\mathbb{R}^2)$-bilinear identity for Fourier extension operators associated to curves in $\mathbb{R}^2$.

math.CA↗

arXiv2019

Regularity of the centered fractional maximal function on radial functions

We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.

math.CA↗

arXiv2019

Endpoint Sobolev continuity of the fractional maximal function in higher dimensions

We establish continuity mapping properties of the non-centered fractional maximal operator $M_β$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-β)$ the map $f \mapsto |\nabla M_βf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < β< 1$ if $f$ is radial and for $1 \leq β< d$ for general $f$. The results for $1\leq β< d$ extend to the centered counterpart $M_β^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_β^c$ implies its continuity.

math.CA↗

arXiv2019

Sharp local smoothing estimates for Fourier integral operators

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

math.AP↗

arXiv2019

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

math.AP↗

arXiv2018

Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.

math.CA↗

arXiv2018

Sparse bounds for pseudodifferential operators

We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.

math.CA↗

arXiv2017

A Fefferman-Stein inequality for the Carleson operator

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy \textit{a priori} weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding Fefferman-Stein inequality for the Carleson operator $\mathcal{C}$, that is $\mathcal{C}: L^p(M^{\lfloor p \rfloor +1}w) \to L^p(w)$ for any $1<p<\infty$ and any weight function $w$, with bound independent of $w$. We also provide a maximal-multiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by Pérez.

math.CA↗

arXiv2017

Control of pseudodifferential operators by maximal functions via weighted inequalities

We establish general weighted $L^2$ inequalities for pseudodifferential operators associated to the Hörmander symbol classes $S^m_{ρ,δ}$. Such inequalities allow to control these operators by fractional "non-tangential" maximal functions, and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar-Stein almost orthogonality principle and a quantitative version of the symbolic calculus.

math.CA↗

arXiv2016

Subdyadic square functions and applications to weighted harmonic analysis

Through the study of novel variants of the classical Littlewood-Paley-Stein $g$-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on $\mathbb{R}^d$ satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted $L^2$ inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón-Zygmund theory.

math.CA↗