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1. 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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2. The Sphere of Semiadditive Height 1

   https://doi.org/10.1093/imrn/rnad044  

   Yuan, Allen (May 2023, International Mathematics Research Notices)

   Abstract We construct a lift of the $$p$$-complete sphere to the universal height $$1$$ higher semiadditive stable $$\infty $$-category of Carmeli–Schlank–Yanovski, providing a counterexample, at height $$1$$, to their conjecture that the natural functor $$ _n \to \operatorname{\textrm{Sp}}_{T(n)}$$ is an equivalence. We then record some consequences of the construction, including an observation of Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg–MacLane spaces. 

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3. On ℓ p -Gaussian–Grothendieck Problem

   https://doi.org/10.1093/imrn/rnab311  

   Chen, Wei-Kuo; Sen, Arnab (November 2021, International Mathematics Research Notices)

   Abstract For $$p\geq 1$$ and $$(g_{ij})_{1\leq i,j\leq n}$$ being a matrix of i.i.d. standard Gaussian entries, we study the $$n$$-limit of the $$\ell _p$$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $$p=\infty $$, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $$1\leq p<2$$ and $$2<p<\infty .$$ For the former, we compute the limit of the $$\ell _p$$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $$n^{-1}$$. 

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4. The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

   https://doi.org/10.1007/s00208-021-02219-1  

   Hofmann, Steve; Li, Linhan; Mayboroda, Svitlana; Pipher, Jill (July 2021, Mathematische Annalen)

   Grafakos, Loukas (Ed.)

   We establish the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $$p\in (1,\infty)$$, the Dirichlet problem for the elliptic equation $$Lu= \dv A\nabla u=0$$ in the upper half-space $$\mathbb{R}^{n+1},\, n\geq 2,$$ is uniquely solvable when the boundary data is in $$L^p(\mathbb{R}^n,dx)$$, provided that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $$L$$ belongs to the $$A_\infty$$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity. 

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5. The space is primary for 1 < p < ∞

   https://doi.org/10.1017/fms.2022.25  

   Lechner, Richard; Motakis, Pavlos; Müller, Paul F.X.; Schlumprecht, Thomas (January 2022, Forum of Mathematics, Sigma)

   Abstract The classical Banach space $$L_1(L_p)$$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $$L_1(L_p) (1 < p < \infty )$$ is primary, meaning that whenever $$L_1(L_p) = E\oplus F$$ , where E and F are closed subspaces of $$L_1(L_p)$$ , then either E or F is isomorphic to $$L_1(L_p)$$ . More generally, we show that $$L_1(X)$$ is primary for a large class of rearrangement-invariant Banach function spaces. 

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