发表机构
Center for Applied Mathematics and KL-AAGDM, Tianjin University; Tianjin University（天津大学应用数学中心与KL-AAGDM; 天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立分数阶趋于零时负Sobolev范数的渐近公式，并推广到一般测度空间，应用于测度绝对连续、随机分布正则性、分数布朗运动局部时及截断极大算子和鞅的极限。
AI 中文摘要
本文建立了当分数阶趋于零时负Sobolev范数的渐近公式。在欧几里得空间中，在温和的有界性条件下，阶数与范数的$p$次幂之积收敛到相应$L^p$范数的维度无关常数倍。证明依赖于热核正则化、弱紧性和Abel-Tauber论证。该结果进一步推广到配备一族有界连续算子的一般测度空间，涵盖非线性和非半群情形。我们还给出了分析和概率中的若干应用，包括测度的绝对连续准则、随机分布的 regularity 结果、分数布朗运动的平方可积局部时的构造，以及截断极大算子和鞅的极限公式。
英文摘要
This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.
Comments18 pp
