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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)., v2: 10 pages, minor corrections

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\u{\i} 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., Comment: 45 pages

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 $L^p(\mathbb{R}^n)$., Comment: 21 pages

In this work we extend consideration of the well-known polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in Rn+1 for n≥2. Inspired by work of Stein and Wainger on the original polynomial Carleson operator, we develop a method to treat polynomial Carleson operators along the paraboloid via van der Corput estimates. A key new step in the approach of this paper is to approximate a related maximal oscillatory integral operator along the paraboloid by a smoother operator, which we accomplish via a Littlewood–Paley decomposition and the use of a square function. The most technical aspect then arises in the derivation of bounds for oscillatory integrals involving integration over lower-dimensional sets. The final theorem applies to polynomial Carleson operators with phase belonging to a certain restricted class of polynomials with no linear terms and whose homogeneous quadratic part is not a constant multiple of the defining function |y|2 of the paraboloid in Rn+1.

Using ideas from Guth-Maldague-Wang and working over $\mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_{\varepsilon}((\log M)^{2 + \varepsilon})$., 21 pages; revised version incorporating referee comments

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., Comment: Fixed up some typos. arXiv admin note: text overlap with arXiv:2003.05216

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., Comment: changes in presentation, new references

We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $\nu_{\gamma} (E_{\lambda,\gamma/p}[u])$ where \[ E_{\lambda,\gamma/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+\gamma/p}}>\lambda\Big\} \] and the measure $\nu_{\gamma}$ is given by $\mathrm{d} \nu_\gamma(x,y)=|x-y|^{\gamma-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{\lambda>0} \lambda \, \nu_{\gamma} (E_{\lambda,\gamma/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda\to\infty$ when $\gamma>0$ and the limit for $\lambda\to 0^+$ when $\gamma1$ the characterizations hold for all $\gamma \neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $\gamma\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $\gamma+1$. For $\gamma=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $\nu_0(E_{\lambda,0}[u])$., Comment: 40 pages

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., Comment: To appear in Ann. Fenn. Math

We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As corollaries we derive alternative estimates in some exceptional cases (involving $W^{1,1}$) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail., Comment: 14 pages

Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the W ˙ 1 , p norm of a function in C c ∞ ( R N ) . The formula was obtained by replacing the L p ( R 2 N ) norm in the Gagliardo semi-norm for W ˙ s , p ( R N ) with a weak- L p ( R 2 N ) quasi-norm and setting s = 1 . This provides a characterization of such W ˙ 1 , p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1] . 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 ( R N ) , which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on L p ( R N ) , which complements a formula by Maz′ya and Shaposhnikova [12] . As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F 2 s , p ( R N ) (i.e. the Bessel potential space ( I − Δ ) − s / 2 L p ( R N ) ), as well as its homogeneous counterpart F ˙ 2 s , p ( R N ) , for s ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) .

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L 1 vector fields, with vector fields in a critical Sobolev space on M : | ∫ M 〈 f , ϕ 〉 d V | ≤ C ‖ f ‖ L 1 ( d V ) ‖ ∇ ϕ ‖ L m ( d V ) . This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.

We offer a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on $\mathbb{R}^n$. We use this idea to extend the result to real hyperbolic spaces $\mathbb{H}^n$., 21 pages; to appear in a Special Issue of Chinese Annals of Mathematics in honour of Haim Brezis

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 operator associated to a non-degenerate curve in $\mathbb{R}^n$. The proof is via a combinatorial argument that builds on the idea that if $\gamma$ is a non-degenerate curve in $\mathbb{R}^n$, then as long as $x_1,\ldots, x_{2n}$ are chosen from a sufficiently well-separated set, then $ \gamma(x_1)+\cdots+\gamma(x_n) = \gamma(x_{n+1}) + \cdots + \gamma(x_{2n}) $ essentially only admits solutions in which $x_1,\ldots,x_n$ is a permutation of $x_{n+1},\ldots, x_{2n}$., Comment: 16 pages

Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along the parabola $(t, ut^2) $. For $U\subset(0,\infty)$ we consider $L^p$ estimates for the maximal functions $\sup_{u\in U}|M^{(u)} f|$ and $\sup_{u\in U}|H^{(u)} f|$, when $1, Comment: 14 pages, final version to appear in Proceedings of the Edinburgh Mathematical Society

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve $\square_b$ works whenever $\overline{\partial}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng, Malchiodi and Yang, which they then applied to study the CR Yamabe problem in 3 dimensions., Comment: to appear in Calc. Var. Partial Differential Equations

Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ where $u\in \mathbb R$. For a set $U$ of positive numbers consider the maximal function $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. We obtain an (essentially) optimal result for the $L^p$ operator norm of $\mathcal{H}^U$ when $2, Comment: 44 pages, fixed typos. A shortened version of this paper will appear in Mathematische Annalen

Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$., Comment: 28 pages; revised version incorporating referee comments

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., Comment: v2: 19 pages, small corrections following referee report

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schr\"odinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter $\ell^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n \ge 4$., Comment: 46 pages, 1 figure

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hat{\theta}$ so that $\hat{\theta}$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hat{\theta})$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat{X} \setminus \{p\}, T^{1,0} \hat{X}, G^2_p \hat{\theta})$ and let $\Box_{b}$ denote the corresponding Kohn Laplacian on $\hat{X} \setminus \{p\}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p \Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p \hat{\theta} \wedge d\hat{\theta}$, and that the associated partial inverse and the Szeg\"{o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{b}$ that grow at a specific rate at $p$. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng-Malchiodi-Yang., Comment: 72 pages

We study the incompressible two dimensional Navier–Stokes equation with initial vorticity in the homogeneous Sobolev space \(\dot{W}^{1,1}(\mathbb{R}^{2})\). This complements our earlier work for the case when the initial vorticity is in the inhomogeneous Sobolev space \(W^{1,1}(\mathbb{R}^{2})\).

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space $\dot{NL}^{1,Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain-Brezis \cite{MR2293957}. We then use this to obtain a Gagliardo-Nirenberg inequality for $\bar{\partial}_b$ on the Heisenberg group $\mathbb{H}^n$.

We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest., Comment: typos fixed

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem., 27 pages

We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calder\'on commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form., Comment: 12 pages

Let $M^{2n+1}$ ($n \geq 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\dbarb$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\dbarb$ complex for $(0,q)$ forms when $q \ne 1$ nor $n-1$. We also prove an analogous $L^1$ inequality when $M$ satisfies condition $Y(q)$. The main technical ingredient is a new kind of $L^1$ duality inequality for vector fields that satisfy Hormander's condition., Comment: to appear in Math. Res. Lett.

Let {F i } N i=1 be a system of similitudes in R n . We study necessary and sufficient conditions for their associated self-similar measures to be doubling on its support. An equivalent condition is obtained when {F i } satisfies the open set condition. The condition allows us to construct many examples of interest. In the case where the open set condition is not satisfied, we study an infinitely convoluted Bernoulli measure (associated with the golden ratio p = (√5 - 1) /2) and give a necessary and sufficient condition for it to be doubling on its support [0,1].

The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalities for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal amount of smoothing, as well as a variant of that which describes quantitatively an improvement of the inequality as one gets away from certain characteristic varieties.

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities., Comment: References added to the work of M. Ben-Artzi and also Haim Brezis in ARMA. Introduction revised

In this note, we consider the constant-mean-curvature wave equation in ( 1 + 2 ) -dimensions. We show that it does not admit any self-similar blow-up. We also remark that the equation is locally well-posed for initial data in H ˙ 3 / 2 .

Using the div-curl inequalities of Bourgain-Brezis [?MR2057026] and van Schaftingen [?MR2078071], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow $L^1_x$ norms of the inhomogeneity in the right hand side of the estimate., Comment: 9 pages

We study the wave analog of the Liouville equation for constant Gauss curvature on S 2 . We show the global existence in time for the sub-critical equation with data in H 1 ( S 2 ) × L 2 ( S 2 ) . For the super-critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on S 2 . Use is made of the Moser–Trudinger inequalities for proving this result. For general data without symmetry assumptions and for the critical equation, we establish that if there is a blow-up, then it will occur at a single point of S 2 . We also consider the wave analog of the constant mean curvature equation in the Plateau formalism. This is a system. We establish finite time blow up if the energy of the initial data is larger than 8 π and with an additional subsidiary assumption. The threshold energy of 8 π arises because it is the energy of the basic bubble discovered by Brezis and Coron.

We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x n / n !. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.