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1. Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces

   https://doi.org/10.54330/afm.163110  

   Kumagai, Takashi; Shanmugalingam, Nageswari; Shimizu, Ryosuke (January 2025, Annales Fennici Mathematici)

   In the context of a metric measure space \((X,d,\mu)\), we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space \(B^\theta_{p,p}(X)\) is \(k>1\), then \(X\) can be decomposed into \(k\) number of irreducible components (Theorem 1.1). Note that \(\theta\) may be bigger than \(1\), as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is 1. We introduce critical exponents \(\theta_p(X)\) and \(\theta_p^{\ast}(X)\) for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces \(X\) formed by glueing copies of \(n\)-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet. 

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2. Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

   https://doi.org/10.1088/1361-6544/ad140b  

   Bai, Xiang; Miao, Qianyun; Tan, Changhui; Xue, Liutang (January 2024, Nonlinearity)

   Abstract In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as $t\to \mathrm{\infty }$. 

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3. Critical local well‐posedness for the fully nonlinear Peskin problem

   https://doi.org/10.1002/cpa.22139  

   Cameron, Stephen; Strain, Robert M (February 2024, Communications on Pure and Applied Mathematics)

   Abstract We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space . We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure. 

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4. Solving a Dirichlet problem on unbounded domains via a conformal transformation

   https://doi.org/10.1007/s00208-023-02705-8  

   Gibara, Ryan; Korte, Riikka; Shanmugalingam, Nageswari (July 2024, Mathematische Annalen)

   Abstract In this paper, we solve thep-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameterp. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class. 

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5. Potential theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary

   Lehrback, Juha; Shanmugalingam, Nageswari (May 2024, Pute and Applied Functional Analysis)

   We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space.This approach is applicable even if the space does not support any Poincar´e inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric mea- sure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map. 

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