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1. The weak-type Carleson theorem via wave packet estimates

   https://doi.org/10.1090/tran/9303  

   Di_Plinio, Francesco; Fragkos, Anastasios (March 2025, Transactions of the American Mathematical Society)

   We prove that the weak- $L^p$norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim <\#comment/>(p-<\#comment/>1{)}^{-<\#comment/>1}$as $p\to <\#comment/>1^{+}$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak- $L^p$type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space ${\mathrm{Q}\mathrm{A}}_{\mathrm{\infty <\#comment/>}}\left(w\right)$, which contains the weighted Antonov space $L\log <\#comment/>L\log <\#comment/>\log <\#comment/>\log <\#comment/>L(\mathbb{T};w)$, converge almost everywhere whenever $w\in <\#comment/>A_1$. This is an extension of the results of Antonov [Proceedings of the XXWorkshop on Function Theory (Moscow, 1995), 1996, pp. 187–196] and Arias De Reyna, where $w$must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near- $L^1$Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint $p=1$, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele [Math. Res. Lett. 17 (2010), pp. 529–545]. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author. 

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2. Characterization of positive definite, radial functions on free groups

   https://doi.org/10.1090/bproc/158  

   Chuah, Chian Yeong; Liu, Zhenchuan; Mei, Tao (April 2023, Proceedings of the American Mathematical Society, Series B)

   This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby [Proc. Amer. Math. Soc. 143 (2015), pp. 1477–1489]. We obtain characterizations of radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^2$length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^p$length on the free real line with infinite generators for $0>p\le <\#comment/>2$. We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well. 

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3. On the maximal spectral type of nilsystems

   https://doi.org/10.1090/bproc/229  

   Ackelsberg, Ethan; Richter, Florian; Shalom, Or (September 2024, Proceedings of the American Mathematical Society, Series B)

   Let $(G/\mathrm{\Gamma <\#comment/>},R_a)$be an ergodic $k$-step nilsystem for $k\ge <\#comment/>2$. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that $L^2\left(G/\mathrm{\Gamma <\#comment/>}\right)$decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124(2014), pp. 261–295] for $2$-step nilsystems and a result by Stepin [Uspehi Mat. Nauk24(1969), pp. 241–242] for nilsystems $G/\mathrm{\Gamma <\#comment/>}$with connected, simply connected $G$. 

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4. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

   https://doi.org/10.1090/tran/8976  

   Livshyts, Galyna (June 2023, Transactions of the American Mathematical Society)

   We show that for any even log-concave probability measure $\mathrm{\mu <\#comment/>}$on ${\mathbb{R}}^n$, any pair of symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $\mathrm{\mu <\#comment/>}\left(\right(1-<\#comment/>\mathrm{\lambda <\#comment/>})K+\mathrm{\lambda <\#comment/>}L{)}^{c_n}\ge <\#comment/>(1-<\#comment/>\mathrm{\lambda <\#comment/>}\left)\mathrm{\mu <\#comment/>}\right(K{)}^{c_n}+\mathrm{\lambda <\#comment/>}\mathrm{\mu <\#comment/>}(L{)}^{c_n},$where $c_n\ge <\#comment/>n^{-<\#comment/>4-<\#comment/>o\left(1\right)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. 

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5. Stability and chaos in dynamical last passage percolation

   https://doi.org/10.1090/cams/35  

   Ganguly, Shirshendu; Hammond, Alan (June 2024, Communications of the American Mathematical Society)

   Many complex disordered systems in statistical mechanics are characterized by intricate energy landscapes. The ground state, the configuration with lowest energy, lies at the base of the deepest valley. In important examples, such as Gaussian polymers and spin glass models, the landscape has many valleys and the abundance of near-ground states (at the base of valleys) indicates the phenomenon ofchaos, under which the ground state alters profoundly when the disorder of the model is slightly perturbed. In this article, we compute the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semidiscrete polymers advance through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is ageodesic, of extremal energy given its endpoints. We perturb Brownian LPP by evolving the disorder under an Ornstein-Uhlenbeck flow. We prove that, for polymers of length $n$, a sharp phase transition marking the onset of chaos is witnessed at the critical time $n^{-<\#comment/>1/3}$. Indeed, the overlap between the geodesics at times zero and $t>0$that travel a given distance of order $n$will be shown to be of order $n$when $t\ll <\#comment/>n^{-<\#comment/>1/3}$; and to be of smaller order when $t\gg <\#comment/>n^{-<\#comment/>1/3}$. We expect this exponent to be universal across a wide range of interface models. The present work thus sheds light on the dynamical aspect of the KPZ class; it builds on several recent advances. These include Chatterjee’s harmonic analytic theory [Superconcentration and related topics, Springer, Cham, 2014] of equivalence ofsuperconcentrationandchaosin Gaussian spaces; a refined understanding of the static landscape geometry of Brownian LPP developed in the companion paper (see S. Ganguly and A. Hammond [Electron. J. Probab. 28 (2023), 80 pp.]); and, underlying the latter, strong comparison estimates of the geodesic energy profile to Brownian motion (see J. Calvert, A. Hammond, and M. Hegde [Astérisque 441 (2023), pp. v+119]). 

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