Journal of the London Mathematical Society-Second Series最新文献

This paper concerns the bubbling phenomena for the $�L^2$-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single-bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non-linear Schrödinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non-local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders $�{�\left|�t�\right|�}^{�2�+�}$ and $�{�\left|�t�\right|�}^{�3�-�}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.

Let $��\alpha �\in �(�0�,�1�)�$. We show that any $�\alpha$-Hölder homeomorphism from the unit circle in the plane to the plane can be extended to an $�\alpha$-Hölder homeomorphism from the whole unit disc.

Let $�F$ be a diagonal cubic form over $�Z$ in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of $�F$ in regions of diameter $��X�\to �\infty �$. Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general $�F$.

We consider the class $��S_{\perp }^m��\left(�\Omega �\right)��$ of $�m$-dimensional surfaces in $��\overline{\Omega }�\subset �R^n�$ that intersect $��S�=�\partial �\Omega �$ orthogonally along the boundary. A piece of an affine $�m$-plane in $��S_{\perp }^m��\left(�\Omega �\right)��$ is called an orthogonal slice. We prove estimates for the area by the $�L^p$ integral of the second fundamental form in three cases: first, when $�\Omega$ admits no orthogonal slices, second for $��$

Let $�{\text{Fl}}_{�n�,�q�}$ be the simplicial complex whose vertices are the nontrivial subspaces of $�F_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let $��{\Delta }_k^{+}��\left(�{\text{Fl}}_{�n�,�q�}�\right)��$ be the $�k$-dimensional weighted upper Laplacian on $�{\text{Fl}}_{�n�,�q�}$. The spectrum of $��{\Delta }_k^{+}��\left(�{\text{Fl}}_{�n�,�q�}�\right)��$

We investigate long time asymptotics of the modified Camassa–Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé-type asymptotic formulae in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $�\overline{\partial }$ nonlinear steepest descent analysis to the associated Riemann–Hilbert problem.

We introduce several new notions of (sectional) curvature bounds for Lorentzian pre-length spaces: On the one hand, we provide convexity/concavity conditions for the (modified) time separation function, and, on the other hand, we study four-point conditions, which are suitable also for the non-intrinsic setting. Via these concepts, we are able to establish (under mild assumptions) the equivalence of all previously known formulations of curvature bounds. In particular, we obtain the equivalence of causal and timelike curvature bounds as introduced by Kunzinger and Sämann.

We study when blowup algebras are $�F$-split or strongly $�F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of $�F$-split filtrations and symbolic $�F$-split ideals.

We introduce a class of uniformly 2-nondegenerate CR hypersurfaces in $�C^N$, for $��N�>�3�$, having a rank 1 Levi kernel. The class is first of all remarkable by the fact that for every $��N�>�3�$ it forms an explicit infinite-dimensional family of everywhere 2-nondegenerate hypersurfaces. To the best of our knowledge, this is the first such construction. Besides, the class contains infinite-dimensional families of nonequivalent structures having a given constant nilpotent CR symbol for every such symbol. Using methods that are able to handle all cases with $��N�>�5�$ simultaneously, we solve the equivalence problem for the considered structures whose symbol is represented by a single Jordan block, classify their algebras of infinitesimal symmetries, and classify the locally homogeneous structures among them. We show that the remaining considered structures, which have symbols represented by a direct sum of Jordan blocks, can be constructed from the single block structures through simple linking and extension processes.

A sublinear bilipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function, the associated sublinearly Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that sublinearly-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi-geodesic rays. As an application, we distinguish a pair of right-angled Coxeter groups brought up by Behrstock up to SBE. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.