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1. 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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2. The Sphere of Semiadditive Height 1

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

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

   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. An Unconditional Explicit Bound on the Error Term in the Sato–tate Conjecture

   https://doi.org/10.1093/qmath/haac004  

   Hoey, Alexandra ; Iskander, Jonas ; Jin, Steven ; Trejos Suárez, Fernando ( April 2022 , The Quarterly Journal of Mathematics)

   Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states that the angles θp are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric power L-functions due to Newton and Thorne, we explicitly bound the error term in the Sato–Tate conjecture when f corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when f has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid N, \theta_p\in I\}$ and $\pi(x) := \# \{p \leq x \}$, we prove the following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.

    

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4. 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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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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