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1. 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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2. 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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3. 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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4. COVERING INTEGERS BY $x^2 + dy^2$

   https://doi.org/10.1017/S1474748024000513  

   Green, Ben; Soundararajan, Kannan (May 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$. 

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5. Theta functions, fourth moments of eigenforms and the sup-norm problem II

   https://doi.org/10.1017/fmp.2024.9  

   Khayutin, Ilya; Nelson, Paul D; Steiner, Raphael S (January 2024, Forum of Mathematics, Pi)

   Abstract Letfbe an$$L^2$$-normalized holomorphic newform of weightkon$$\Gamma _0(N) \backslash \mathbb {H}$$withNsquarefree or, more generally, on any hyperbolic surface$$\Gamma \backslash \mathbb {H}$$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$$\mathbb {Q}$$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform$$\varphi $$of eigenvalue$$\lambda $$on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras. 

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