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1. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian; Gusakova, Anna; Reitzner, Matthias; Schütt, Carsten; Thäle, Christoph; Werner, Elisabeth_M (August 2023, Mathematische Annalen)

   Abstract Consider two half-spaces$$H_1^+$$ $H_1^{+}$and$$H_2^+$$ $H_2^{+}$in$${\mathbb {R}}^{d+1}$$ $R^{d+1}$whose bounding hyperplanes$$H_1$$ $H_1$and$$H_2$$ $H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ $S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$ $S^d$, which contains a great subsphere of dimension$$d-2$$ $d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$ $logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. 

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2. Uniform stabilization in Besov spaces with arbitrary decay rates of the magnetohydrodynamic system by finite-dimensional interior localized static feedback controllers

   https://doi.org/10.1007/s40687-024-00490-7  

   Lasiecka, Irena; Priyasad, Buddhika; Triggiani, Roberto (December 2024, Research in the Mathematical Sciences)

   Abstract We consider thed-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain,$$d = 2,3$$ $d=2,3$with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In additional, they will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space$$\displaystyle \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega )\times \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega ), \, 1< p < \frac{2q}{2q-1}, \, q > d,$$ ${\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right)\times {\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right),1<p<\frac{2q}{2q-1},q>d,$of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to$$\textbf{L}^3(\Omega )$$ $L^3\left(\Omega \right)$for$$d = 3$$ $d=3$). Showing maximal$$L^p$$ $L^p$-regularity up to$$T = \infty $$ $T=\infty$for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. 

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3. Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

   https://doi.org/10.1007/s00220-022-04522-7  

   Laurens, Thierry (November 2022, Communications in Mathematical Physics)

   Abstract Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ $u(0,x)\in V(0,x)+H^{-1}\left(R\right)$. Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ $V(0,x)\in H^5(R/Z)$satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$ $V\equiv 0$. In that setting, it is known that$$H^{-1}(\mathbb {R})$$ $H^{-1}\left(R\right)$is sharp in the class of$$H^s(\mathbb {R})$$ $H^s\left(R\right)$spaces. 

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4. 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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5. Sobolev extension in a simple case

   https://doi.org/10.1515/ans-2023-0132  

   Drake, Marjorie; Fefferman, Charles; Ren, Kevin; Skorobogatova, Anna (April 2025, Advanced Nonlinear Studies)

   Abstract In this paper, we establish the existence of a bounded, linear extension operator $T:L^{2,p}\left(E\right)\to L^{2,p}\left({\mathbb{R}}^2\right)$$$T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$$when 1 <p< 2 andEis a finite subset of ${\mathbb{R}}^2$$${\mathbb{R}}^{2}$$contained in a line. 

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