Decouplings for surfaces in R^4
We prove a sharp decoupling for non degenerate surfaces in $\R^4$. This puts the recent progress on the Lindelöf hypothesis into a more general perspective.
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arXiv subjects
Ciprian Demeter
Publications and source records attributed to Ciprian Demeter.
$l^p$ decouplings for hypersurfaces with nonzero Gaussian curvature
We prove an $l^p$ decoupling inequality for hypersurfaces with nonzero Gaussian curvature and use it to derive a corresponding $l^p$ decoupling for curves not contained in a hyperplane. This extends our earlier work from [2]
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Incidence theory and restriction estimates
We investigate the interplay between the discrete restriction phenomenon and incidence theory. Two angles are explored. One is a refinement of the machinery developed by Thomas Wolff, which when combined with a recent subcritical estimate of Bourgain leads to new Strichartz estimates for irrational tori. The other one connects various additive energies with the Szemerédi-Trotter Theorem. The combination of the two approaches recovers the best known Strichartz estimates for the classical torus in dimensions $n\ge 3$, without any appeal to number theory.
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On the number of non-congruent lattice tetrahedra
We prove that there are "many" non-congruent tetrahedra in the truncated lattice $[0,q]^3\cap \mathbb{Z}^3$.
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Some new light on a few classical results
The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Cordoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, that does not rely on product theory. Our arguments are inspired by the vector valued restricted type interpolation used in [1].
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New bounds for the discrete Fourier restriction to the sphere in four and five dimensions
We improve the range for the discrete Fourier restriction to the four and five dimensional spheres. We rely on two new ingredients, incidence theory and Siegel's mass formula.
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Improved estimates for the discrete Fourier restriction to the higher dimensional sphere
We improve the exponent for the discrete Fourier restriction to the $n$ dimensional sphere, from $p=\frac{2(n+1)}{n-3}$ to $p=\frac{2n}{n-3}$, when $n\ge 4$.
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Endpoint bounds for the quartile operator
It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p_3=2/3. We show that the quartile operator maps L^{p_1}(R) \times L^{p_2}(R) into L^{2/3,\infty}(R) when p_1,p_2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^{p_1}(R) \times L^{p_2,2/3}(R) into L^{2/3,\infty}(R). We also provide restricted weak-type estimates and boundedness on Orlicz-Lorentz spaces near p_1=1,p_2=2 which improve, in the Walsh case, on results of Bilyk and Grafakos, and Carro-Grafakos-Martell-Soria. Our main tool is the multi-frequency Calderón-Zygmund decomposition first used by Nazarov, Oberlin and Thiele.
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Logarithmic L^p bounds for maximal directional singular integrals in the plane
We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L^p almost orthogonality principle for Fourier restrictions to cones.
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Bilinear Fourier restriction theorems
We provide a general scheme for proving $L^p$ estimates for certain bilinear Fourier restrictions outside the locally $L^2$ setting. As an application, we show how such estimates follow for the lacunary polygon. In contrast with prior approaches, our argument avoids any use of the Rubio de Francia Littlewood--Paley inequality.
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On the finite linear independence of lattice Gabor systems
In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell's theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the spectral theory of random Schrödinger operators; in the one-dimensional setting, we recover the full strength of Linnell's result for general lattices.
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$L^2$ bounds for a Kakeya type maximal operator in $\R^3$
We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to $N^{1/4}\sqrt{\log N}$. Apart from the logarithmic terms these bounds are optimal.
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Proof of the HRT conjecture for (2,2) configurations
We prove that for any 4 points in a (2-2) configuration, there is no linear dependence between the associated time-frequency translates of any $L^2(\R)$ function
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Singular Integrals Along N Directions in R^2
We prove optimal bounds in L^2(R^2) for the maximal oper- ator obtained by taking a singular integral along N arbitrary directions in the plane. We also give a new proof for the optimal L^2 bound for the single scale Kakeya maximal function.
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Multilinear singular operators with fractional rank
We prove bounds for multilinear operators on $\R^d$ given by multipliers which are singular along a $k$ dimensional subspace. The new case of interest is when the rank $k/d$ is not an integer. Connections with the concept of {\em true complexity} from Additive Combinatorics are also investigated.
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Improved range in the Return Times Theorem
We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<{3/2}$.
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On some maximal multipliers in $L^p$
We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1<p<\infty$.
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{Spaces of Infinite Measure and Pointwise Convergence of the Bilinear Hilbert and Ergodic Averages Defined by $L^{p}$-Isometries
We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sigma-finite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue space isometries and act on $L^{p_{1}}\times L^{p_{2}}$ ($ 1<p_{1},p_{2}<\infty $, $p_{1}^{-1}+p_{2}^{-1}<3/2$). In the setting of an arbitrary measure space, this yields the a.e. convergence of these discrete bilinear averages when they act on $L^{p_{1}}\times L^{p_{2}}$ and are defined by an invertible measure-preserving point transformation.
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