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1. A Weighted Formulation of Refined Decoupling and Inequalities of Mizohata–Takeuchi-Type for the Moment Curve

   https://doi.org/10.1007/s12220-025-02262-3  

   Carbery, Anthony; Li, Zane Kun; Pang, Yixuan; Yung, Po-Lam (February 2026, The Journal of Geometric Analysis)

   Abstract Let$$\Gamma $$ $\Gamma$ be a compact patch of a well-curved$$C^{n+1}$$ $C^{n+1}$ curve in$$\mathbb {R}^n$$ $R^n$ with induced Lebesgue measure$$\textrm{d} \lambda $$ $\text{d}\lambda$ , and let$$g \mapsto \widehat{g \,\textrm{d}\lambda }$$ $g\mapsto \widehat{g\text{d}\lambda }$ be the Fourier extension operator for$$\Gamma $$ $\Gamma$ . Then we have, for arbitrary non-negative weightsw,$$\begin{aligned} \int _{B_R} |\widehat{g \,\textrm{d}\lambda }|^2w \le C_{n,a} R^{a} \sup _S \left( \int _S w\right) \int _\Gamma |g|^2 \, \textrm{d} \lambda \end{aligned}$$ $\begin{array}{c}{\int }_{B_R}|\widehat{g\text{d}\lambda }{|}^{2}w\le C_{n,a}{R}^a\underset{S}{sup}\left({\int }_S,w\right){\int }_{\Gamma }{\left|g\right|}^2\text{d}\lambda \end{array}$ for any$$a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$$ $a>\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)}$ , where the$$\sup $$ $sup$ is over all 1-neighbourhoodsSof hyperplanes whose normals are parallel to the tangent at some point of$$\Gamma $$ $\Gamma$ . This represents partial progress on the Mizohata–Takeuchi conjecture for curves in dimensions$$n \ge 3$$ $n\ge 3$ , improving upon the exponent$$a=n-1$$ $a=n-1$ 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. 

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2. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$ conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$ , and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$ ,$$r>0$$ $r>0$ , we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$ where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$ such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$ and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$ . Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$ where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$ is$$n$$ $n$ -rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$ ,$$\Omega ^{-}$$ ${\Omega }^{-}$ are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$ satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$ and$$r>0$$ $r>0$ , we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$ the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$ . We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$ measure zero,$$x$$ $x$ is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$ and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$ if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$ The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$ conjecture of Carleson. 

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3. Instantaneous everywhere-blowup of parabolic SPDEs

   https://doi.org/10.1007/s00440-024-01263-7  

   Foondun, Mohammud; Khoshnevisan, Davar; Nualart, Eulalia (October 2024, Probability Theory and Related Fields)

   Abstract We consider the following stochastic heat equation$$\begin{aligned} \partial _t u(t,x) = \tfrac{1}{2} \partial ^2_x u(t,x) + b(u(t,x)) + \sigma (u(t,x)) {\dot{W}}(t,x), \end{aligned}$$ $\begin{array}{c}{\partial }_{t}u(t,x)=\frac{1}{2}{\partial }_x^{2}u(t,x)+b\left(u(t,x)\right)+\sigma \left(u(t,x)\right)\dot{W}(t,x),\end{array}$ defined for$$(t,x)\in (0,\infty )\times {\mathbb {R}}$$ $(t,x)\in (0,\infty )\times R$ , where$${\dot{W}}$$ $\dot{W}$ denotes space-time white noise. The function$$\sigma $$ $\sigma$ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the functionbis assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition$$\begin{aligned} \int _1^\infty \frac{\textrm{d}y}{b(y)}<\infty \end{aligned}$$ $\begin{array}{c}{\int }_1^{\infty }\frac{\text{d}y}{b\left(y\right)}<\infty \end{array}$ implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that$$\textrm{P}\{ u(t,x)=\infty \quad \hbox { for all } t>0 \hbox { and } x\in {\mathbb {R}}\}=1.$$ $\text{P}\left\{u\right(t,x)=\infty \text{for all}t>0\text{and}x\in R\}=1.$ The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022). 

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4. An explicit economical additive basis

   https://doi.org/10.1017/S096354832510014X  

   Jain, Vishesh; Pham, Huy Tuan; Sawhney, Mehtaab; Zakharov, Dmitrii (November 2025, Combinatorics, Probability and Computing)

   Abstract We present an explicit subset$$A\subseteq \mathbb{N} = \{0,1,\ldots \}$$such that$$A + A = \mathbb{N}$$and for all$$\varepsilon \gt 0$$,\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*} This answers a question of Erdős. 

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5. Phase mixing for solutions to 1D transport equation in a confining potential

   https://doi.org/10.3934/krm.2022002  

   Chaturvedi, Sanchit; Luk, Jonathan (January 2022, Kinetic and Related Models)

   Consider the linear transport equation in 1D under an external confining potential \begin{document}$$ \Phi $$\end{document}: \begin{document}$$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $$\end{document} For \begin{document}$$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $$\end{document} (with \begin{document}$$ \varepsilon >0 $$\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $$\end{document}, with an inverse polynomial decay rate \begin{document}$$ O({\langle} t{\rangle}^{-2}) $$\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $$\end{document}D under the external potential \begin{document}$$ \Phi $$\end{document}$. 

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