经典分析与常微分方程：2025–2026 年度综述

An annual survey of the math.CA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 108 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Spectral Analysis and Asymptotic Methods in Classical Analysis (73.1%) — Investigates spectral properties of limiting operators, fractional derivatives, and asymptotic behaviors of orthogonal polynomials and wave packet systems.
• Sharp Inequalities and Geometric Measure Theory Estimates (10.0%) — Focuses on establishing sharp bounds for Hardy, Poincaré-Sobolev, and Kakeya-type inequalities involving fractal measures and geometric sets.
• Harmonic Analysis on Spaces of Homogeneous Type (7.8%) — Develops weighted norm estimates, maximal function bounds, and singular integral theory for operators on general metric measure spaces.
• Integrability and Symmetry in Differential Equations (3.5%) — Explores exact solvability, symmetry structures, and optimization techniques for ordinary and partial differential equations, including Painlevé systems.
• Asymptotics and Structure of Orthogonal Polynomials (1.9%) — Analyzes asymptotic expansions, zero distributions, and algebraic properties of orthogonal polynomial families under various weight modifications.
• Analytic Number Theory and Approximation Theory (1.6%) — Connects analytic number theory concepts like rational approximations and arithmetic functions with geometric measure theory and function approximation.
• Discrete Harmonic Analysis and Ergodic Theory (1.0%) — Examines convergence properties, Szemerédi-type theorems, and spectral sum rules within discrete settings and ergodic averages.
• Fourier Extension and Riesz Energy Problems (0.8%) — Studies Fourier restriction/extension phenomena and the distribution of critical points for Riesz and Green energy functionals.
• Existence Conditions and Regularity in Dynamical Systems (0.4%) — Identifies necessary and sufficient conditions for existence, resonance, and regularity in eigenvalue problems and competitive dynamical systems.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.