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1. Quantitative Sobolev regularity of quasiregular maps

   https://doi.org/10.54330/afm.155498  

   Di_Plinio, Francesco; Green, A Walton; Wick, Brett D (January 2025, Annales Fennici Mathematici)

   We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities. 

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2. P -Partitions and Quasisymmetric Power Sums

   https://doi.org/10.1093/imrn/rnz375  

   Liu, Ricky Ini; Weselcouch, Michael (February 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract The $$(P, \omega )$$-partition generating function of a labeled poset $$(P, \omega )$$ is a quasisymmetric function enumerating certain order-preserving maps from $$P$$ to $${\mathbb{Z}}^+$$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $$\{\psi _{\alpha }\}$$. Using this expansion, we show that connected, naturally labeled posets have irreducible $$P$$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $$\psi _{\alpha }$$-expansion of the $$(P, \omega )$$-partition generating function akin to the Murnaghan–Nakayama rule. 

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3. Sobolev spaces and Poincaré inequalities on the Vicsek fractal

   https://doi.org/10.54330/afm.122168  

   Baudoin, Fabrice; Chen, Li (October 2022, Annales Fennici Mathematici)

   In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete \(p\)-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for \(p>1\). As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain \(L^p\)-Poincaré inequalities for all values of \(p \ge 1\). 

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4. Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

   https://doi.org/10.3390/math10030522  

   Behzadan, Ali; Holst, Michael (February 2022, Mathematics)

   In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<1, and 1<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here. 

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5. On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

   https://doi.org/10.1007/s00229-025-01656-5  

   Gibara, Ryan; Kangasniemi, Ilmari; Shanmugalingam, Nageswari (July 2025, Manuscripta mathematica)

   We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure μ with μ(X) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞). Our objective is to understand the relationship between the Dirichlet space D^(1,p)(X), defined using upper gradients, and the Newton-Sobolev space N^(1,p)(X)+ℝ, for 1 ≤ p < ∞. We show that when X is of uniformly locally p-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space ℍⁿ with n ≥ 2, these two spaces coincide precisely when 1 ≤ p ≤ n-1. We also provide additional characterizations of when a function in D^(1,p)(X) is in N^(1,p)(X)+ℝ in the case that the two spaces do not coincide. 

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