Two weight estimates for difference quotients
We prove local and global two weight estimates in which we bound difference quotients of a function in terms of certain weighted $L^p$ norms of its gradient.
math.CA↗
arXiv subjects
Po-Lam Yung
Publications and source records attributed to Po-Lam Yung.
Function spaces for decoupling
We introduce new function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that yield a natural reformulation of the $\ell^{q}L^{p}$ decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless $p=q$, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.
math.AP↗
A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve
Let $Γ$ be a compact patch of a well-curved $C^{n+1}$ curve in $\mathbb{R}^n$ with induced Lebesgue measure ${\rm d} λ$, and let $g \mapsto \widehat{g \,{\rm d}λ}$ be the Fourier extension operator for $Γ$. Then we have, for arbitrary non-negative weights $w$, \begin{equation*} \int_{B_R} |\widehat{g \,{\rm d}λ}|^2w \leq C_{n,a} R^{a} \sup_S \left(\int_S w\right)\int_Γ|g|^2 \, {\rm d} λ \end{equation*} for any $a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$, where the $\sup$ is over all $1$-neighbourhoods $S$ of hyperplanes whose normals are parallel to the tangent at some point of $Γ$. This represents partial progress on the Mizohata-Takeuchi conjecture for curves in dimensions $n \geq 3$, improving upon the exponent $a=n-1$ which can be obtained as a consequence of the Agmon-Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.
math.CA↗
On Polynomial Carleson operators along quadratic hypersurfaces
We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by $(y,Q(y))\subseteq \mathbb{R}^{n+1}$, for an arbitrary non-degenerate quadratic form $Q$, admits an a priori bound on $L^p$ for all $1<p<\infty$, for each $n \geq 2$. This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of $\{p_2,\ldots,p_d\}$ for any set of fixed real-valued polynomials $p_j$ such that $p_j$ is homogeneous of degree $j$, and $p_2$ is not a multiple of $Q(y)$. The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case $Q(y)=|y|^2$.
math.CA↗
Strichartz inequalities: some recent developments
Strichartz inequalities, originating from Fourier restriction theory, play a central role in the analysis of dispersive partial differential equations. They serve as a cornerstone for many subsequent developments. We survey some of them in memory of Strichartz, highlighting connections to recent developments in Fourier decoupling.
math.CA↗
A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem
We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely does solution counting in older partial progress on Vinogradov's mean value theorem correspond to in Fourier decoupling theory.
math.CA↗
A short proof of $\ell^2$ decoupling for the moment curve
We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\mathbb{R}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley (arXiv:1708.01220).
math.NT↗
The Neumann problem on the Clifford torus in $\mathbb{S}^3$
We discuss the solution of the Neumann problem associated with the CR Yamabe operator on a subset $Ω$ of the CR manifold $\mathbb{S}^3$ bounded by the Clifford torus $Σ$. We also discuss the Yamabe-type problem of finding a contact form on $Ω$ which has zero Tanaka--Webster scalar curvature and for which $Σ$ has constant $p$-mean curvature.
math.AP↗
A new type of superorthogonality
We provide a simple criterion on a family of functions that implies a square function estimate on $L^p$ for every even integer $p \geq 2$. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known.
math.CA↗
Spaces of Besov-Sobolev type and a problem on nonlinear approximation
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.
math.FA↗
On the strict majorant property in arbitrary dimensions
In this work we study $d$-dimensional majorant properties. We prove that a set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least $d+2$ frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $p \not\in 2 {\mathbb N}$ of length 2. Any infinite set of frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $p \not\in 2 {\mathbb N}$ of length $2$. Finally, given any $p>0$ with $p \not\in 2{\mathbb N}$, we exhibit a set of $d+2$ frequencies on the moment curve in ${\mathbb R}^d$ that violate the strict majorant property on $L^p([0,1]^d).$
math.CA↗
Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
math.CA↗
Limiting Sobolev and Hardy inequalities on stratified homogeneous groups
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
math.CA↗
Sobolev spaces revisited
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.
math.CA↗
Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail
In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms.
math.CA↗
Decoupling for two quadratic forms in three variables: a complete characterization
We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $\mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables.
math.CA↗
A new formula for the $L^p$ norm
Recently, Brezis, Van Schaftingen and the second author established a new formula for the $\dot{W}^{1,p}$ norm of a function in $C^{\infty}_c(\mathbb{R}^N)$. The formula was obtained by replacing the $L^p(\mathbb{R}^{2N})$ norm in the Gagliardo semi-norm for $\dot{W}^{s,p}(\mathbb{R}^N)$ with a weak-$L^p(\mathbb{R}^{2N})$ quasi-norm and setting $s = 1$. This provides a characterization of such $\dot{W}^{1,p}$ norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula. In this paper, we obtain an analog for the case $s = 0$. In particular, we present a new formula for the $L^p$ norm of any function in $L^p(\mathbb{R}^N)$, which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on $L^p(\mathbb{R}^N)$, which complements a formula by Maz'ya and Shaposhnikova. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space $F^s_{p,2}(\mathbb{R}^N)$ (i.e. the Bessel potential space $(I-Δ)^{-s/2} L^p(\mathbb{R}^N)$), as well as its homogeneous counterpart $\dot{F}^s_{p,2}(\mathbb{R}^N)$, for $s \in (0,1)$, $p \in (1,\infty)$.
math.CA↗
Improved discrete restriction for the parabola
Using ideas from Guth-Maldague-Wang and working over $\mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_{\varepsilon}((\log M)^{2 + \varepsilon})$.
math.NT↗