We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.
We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents $p$. Our method is based on the bilinearization argument due to Bourgain and Bourgain-Demeter. Our result generalizes theirs to all higher dimensions.
We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.
We prove $\ell^{p}L^{p}$ decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell. In our proofs we take a new point of view on the Bourgain--Demeter--Guth induction on scales argument. This point of view substantially simplifies even the proof of $\ell^{2}L^{p}$ decoupling for the moment curve.
We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.
For each $d\geq 0$, we prove decoupling inequalities in $\mathbb R^3$ for the graphs of all bivariate polynomials of degree at most $d$ with bounded coefficients, with the decoupling constant depending uniformly in $d$ but not the coefficients of each individual polynomial. As a consequence, we prove a decoupling inequality for (a compact piece of) every smooth surface in $\mathbb{R}^3$, which in particular solves a conjecture of Bourgain, Demeter and Kemp.
We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>α\}$ where $α>0$ and $f:\mathbb{R}^2\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$-neighborhood of the truncated parabola $\mathbb{P}^1$. These estimates imply the small cap decoupling theorem for $\mathbb{P}^1$ of Demeter, Guth, and Wang, and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter. New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality. Second, in our main result we prove an $L^{2n}$ square function estimate (which implies a corresponding decoupling estimate) for the extension opera…
In this paper, we present the greatest values $α$, $λ$ and $p$, and the least values $β$, $μ$ and $q$ such that the double inequalities $αD(a,b)+(1-α)H(a,b)<T(a,b)<βD(a,b)+(1-β) H(a,b)$, $λD(a,b)+(1-λ)H(a,b)<C(a,b)<μD(a,b)+(1-μ) H(a,b)$ and $p D(a,b)+(1-p)H(a,b)<Q(a,b)<q D(a,b)+(1-q)H(a,b)$ hold for all $a,b>0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respective…
We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. It is shown that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the measures holds, then a family of dyadic shifts satisfies the two weight estimate in an $\mathcal{R}$-bounded sense if and only if it satisfies the direct- and the dual quadratic testing condition. In the case $p=q=2$ this reduces to the result by T. Hytönen, C. Pérez, S. Treil and A. Volberg. The dyadic square function satisfies the two weight estimate if and only if it…
In this paper we state and prove some new inequalities related to the rate of $L^{p}$ approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from $L^{p}$.
In this paper, we prove that the inequalities $α[1/3 Q(a,b)+2/3 A(a,b)]+(1-α)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <β[1/3 Q(a,b)+2/3 A(a,b)]+(1-β)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $λ[1/6 C(a,b)+5/6 A(a,b)]+(1-λ)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<μ[1/6 C(a,b)+5/6 A(a,b)]++(1-μ)C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $α\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777...$, $β\geq 4/5$, $λ\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274...$, and $μ\geq 8/25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ …
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.
For functions $f$ with Fourier transform supported in the truncated cone, we bound superlevel sets $\{x\in\mathbb{R}^3:|f(x)|>α\}$ using an $α$-dependent version of the wave envelope estimate of Guth--Wang--Zhang. Our estimates imply both sharp square function and decoupling inequalities for the cone. We also obtain sharp small cap decoupling for the cone, where small caps $γ$ subdivide canonical $1\times R^{-1/2}\times R^{-1}$ planks into $R^{-β_2}\times R^{-β_1}\times R^{-1}$ sub-planks, for $β_1\in[\frac{1}{2},1]$ and $β_2\in[0,1]$.
