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1. 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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2. Vertex operator superalgebras and the 16-fold way

   https://doi.org/10.1090/tran/8454  

   Dong, Chongying; Ng, Siu-Hung; Ren, Li (July 2021, Transactions of the American Mathematical Society)

   Let V V be a vertex operator superalgebra with the natural order 2 automorphism σ \sigma . Under suitable conditions on V V , the σ \sigma -fixed subspace V 0 ¯ V_{\bar 0} is a vertex operator algebra and the V 0 ¯ V_{\bar 0} -module category C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a modular tensor category. In this paper, we prove that C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a fermionic modular tensor category and the Müger centralizer C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 of the fermion in C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is generated by the irreducible V 0 ¯ V_{\bar 0} -submodules of the V V -modules. In particular, C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 is a super-modular tensor category and C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We provide a construction of a vertex operator superalgebra V l V^l for each positive integer l l such that C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We prove that these modular tensor categories C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} are uniquely determined, up to equivalence, by the congruence class of l l modulo 16. 

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3. On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

   https://doi.org/10.1090/tran/9315  

   Gong, Xianghong (March 2025, Transactions of the American Mathematical Society)

   We study regularity of solutions $u$to $\stackrel{¯<\#comment/>}{\mathrm{\partial <\#comment/>}}u=f$on a relatively compact $C^2$domain $D$in a complex manifold of dimension $n$, where $f$is a $(0,q)$form. Assume that there are either $(q+1)$negative or $(n-<\#comment/>q)$positive Levi eigenvalues at each point of boundary $\mathrm{\partial <\#comment/>}D$. Under the necessary condition that a locally $L^2$solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$derivative when $q=1$and $f$is in the Hölder–Zygmund space ${\mathrm{\Lambda <\#comment/>}}^r\left(D\right)$with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\mathrm{\partial <\#comment/>}D$is either sufficiently smooth or of $(n-<\#comment/>q)$positive Levi eigenvalues everywhere on $\mathrm{\partial <\#comment/>}D$. 

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4. POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES

   https://doi.org/10.1017/fms.2018.11  

   DU, XIUMIN; GUTH, LARRY; LI, XIAOCHUN; ZHANG, RUIXIANG (January 2018, Forum of Mathematics, Sigma)

   We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $$n\geqslant 3$$ , $$\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$$ $=f(x)$ almost everywhere with respect to Lebesgue measure for all $$f\in H^{s}(\mathbb{R}^{n})$$ provided that $s>(n+1)/2(n+2)$ . The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya. 

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5. Random walks on direct products of groups

   https://doi.org/10.4171/JEMS/1550  

   Golsefidy, Alireza Salehi; Srinivas, Srivatsa (November 2024, Journal of the European Mathematical Society)

   Suppose\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})is generated by a symmetric setSof cardinalitynwherepis a prime number. Suppose the Cheeger constants of the Cayley graphs of\operatorname{SL}_{2}(\mathbb{F}_{p})with respect to\pi_{L}(S)and\pi_{R}(S)are at leastc_{0}, where\pi_{L}and\pi_{R}are the projections to the left and the right components of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}), respectively. Then the Cheeger constant of the Cayley graph of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})with respect toSis at leastcwherecis a positive number which only depends onnandc_{0}. This gives an affirmative answer to a question of Lindenstrauss and Varjú. 

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