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1. On Weissler’s Conjecture on the Hamming Cube I

   https://doi.org/10.1093/imrn/rnaa363  

   Ivanisvili, P; Nazarov, F (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract Let $$1\leq p \leq q <\infty $$ and let $$w \in \mathbb{C}$$. Weissler conjectured that the Hermite operator $$e^{w\Delta }$$ is bounded as an operator from $$L^{p}$$ to $$L^{q}$$ on the Hamming cube $$\{-1,1\}^{n}$$ with the norm bound independent of $$n$$ if and only if $$\begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*}$$It was proved in [ 1], [ 2], and [ 17] in all cases except $$2<p\leq q <3$$ and $$3/2<p\leq q <2$$, which stood open until now. The goal of this paper is to give a full proof of Weissler’s conjecture in the case $p=q$. Several applications will be presented. 

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2. Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

   https://doi.org/http://dx.doi.org/10.1090/tran/7536  

   Cavero, J.; Hofmann, S.; Martell, J.M. (January 2019, Transactions of the American Mathematical Society)

   Let $$\Omega\subset\re^{n+1}$$, $$n\ge 2$$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $$\partial\Omega$$ is $$n$$-dimensional Ahlfors regular. Consider $$L_0$$ and $$L$$ two real symmetric divergence form elliptic operators and let $$\omega_{L_0}$$, $$\omega_L$$ be the associated elliptic measures. We show that if $$\omega_{L_0}\in A_\infty(\sigma)$$, where $$\sigma=H^n\lfloor_{\partial\Omega}$$, and $$L$$ is a perturbation of $$L_0$$ (in the sense that the discrepancy between $$L_0$$ and $$L$$ satisfies certain Carleson measure condition), then $$\omega_L\in A_\infty(\sigma)$$. Moreover, if $$L$$ is a sufficiently small perturbation of $$L_0$$, then one can preserve the reverse H√∂lder classes, that is, if for some $$1<\infty$$, one has $$\omega_{L_0}\in RH_p(\sigma)$$ then $$\omega_{L}\in RH_p(\sigma)$$. Equivalently, if the Dirichlet problem with data in $$L^{p'}(\sigma)$$ is solvable for $$L_0$$ then so it is for $$L$$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $$A_\infty(\sigma)$$ then necessarily $$\Omega$$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable. 

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3. An inequality for the normal derivative of the Lane–Emden ground state

   https://doi.org/10.1515/acv-2022-0005  

   Frank, Rupert L.; Larson, Simon (May 2022, Advances in Calculus of Variations)

   Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions.Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality.Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity. 

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4. Robust Recovery of Robinson Property in $L^p$-Graphons: A Cut-Norm Approach

   https://doi.org/10.37236/12435  

   Ghandehari, Mahya; Mishura, Teddy (October 2024, The Electronic Journal of Combinatorics)

   This paper investigates the Robinson graphon completion/recovery problem within the class of $L^p$-graphons, focusing on the range $$5 5$, any $L^p$-graphon $$w$$ can be approximated by a Robinson graphon, with error of the approximation bounded in terms of $$\Lambda(w)$$. When viewing $$w$$ as a noisy version of a Robinson graphon, our method provides a concrete recipe for recovering a cut-norm approximation of a noiseless $$w$$. Given that any symmetric matrix is a special type of graphon, our results can be applicable to symmetric matrices of any size. Our work extends and improves previous results, where a similar question for the special case of $$L^\infty$$-graphons was answered. 

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5. Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

   https://doi.org/10.3934/mine.2023034  

   Silvestre, Luis; Snelson, Stanley (January 2022, Mathematics in Engineering)

   The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $$ L^\infty $$ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time. 

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