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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. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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3. Mixed Norm l 2 Decoupling for Paraboloids

   https://doi.org/10.1093/imrn/rnad226  

   Dasu, Shival; Jung, Hongki; Li, Zane_Kun; Madrid, José (September 2023, International Mathematics Research Notices)

   Abstract We prove the sharp mixed norm $$(l^{2}, L^{q}_{t}L^{r}_{x})$$ decoupling estimate for the paraboloid in $d + 1$ dimensions. 

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4. 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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5. On Bertrand's and Rodriguez Villegas' higher-dimensional Lehmer conjecture, with an appendix by Gernando Rodriguez Villegas

   Ted Chinburg, Eduardo Friedman (January 2023, Pacific journal of mathematics)

   Let L be a number field and let E ⊂ OL∗ be any subgroup of the units of L. If rankZ(E) = 1, Lehmer’s conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rankZ(E) = rankZ(OL∗), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L : Q]. By sharpening a 1997 conjecture of Daniel Bertrand’s, Fernando Rodriguez Villegas “interpolated” between these two extremes of rank with a new higher-dimensional version of Lehmer’s conjecture. Here we prove a high-rank case of the Bertrand-Rodriguez Villegas conjecture. Namely, it holds if L contains a subfield K for which [L : K] ≫ [K : Q] and E contains the kernel of the norm map from OL∗ to OK∗ . 

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