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1. Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

   https://doi.org/10.1515/crelle-2024-0016  

   Dong, Hongjie; Peng, Fa; Zhang, Yi Ru-Ya; Zhou, Yuan (April 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We introduce a distributional Jacobian determinant $detD{V}_{\beta }\left(Dv\right)$\det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient $V_{\beta }\left(Dv\right)={\left|Dv\right|}^{\beta }(v_{x_1},-v_{x_2})$V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any $\beta >-1$\beta>-1, whenever $v\in W_{\mathrm{loc}}^{1,2}$v\in W^{1\smash{,}2}_{\mathrm{loc}}and $\beta {\left|Dv\right|}^{1+\beta }\in W_{\mathrm{loc}}^{1,2}$\beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when $\beta \ne 0$\beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant $detD{V}_{\beta }\left(Du\right)$\det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with $\beta =0$\beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole ${\mathbb{R}}^2$\mathbb{R}^{2}with $\underset{R\to \mathrm{\infty }}{lim\; inf}\underset{c\in \mathbb{R}}{inf}\frac{1}{R}{⨍}_{B(0,R)}\left|u\left(x\right)-c\right|dx<\mathrm{\infty },$\liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then $u=b+a\cdot x$u=b+a\cdot xfor some $b\in \mathbb{R}$b\in\mathbb{R}and $a\in {\mathbb{R}}^2$a\in\mathbb{R}^{2}.Denoting by $u_p$u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that $detD{V}_{\beta }\left(D{u}_p\right)\to detD{V}_{\beta }\left(Du\right)$\det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as $p\to \mathrm{\infty }$p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that $V_{\beta }\left(D{u}_p\right)$V_{\beta}(Du_{p})is always quasiregular mappings, but $V_{\beta }\left(Du\right)$V_{\beta}(Du)is not in general. 

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2. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa; Teräväinen, Joni (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

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3. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}${\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m${A_{m}}with $m\le n${m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}${\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$\square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4${D_{4}}which do not arise from an embedding of $A_r${A_{r}}with $r\le 3${r\leq 3}in $D_4${D_{4}}. 

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4. Sobolev extension in a simple case

   https://doi.org/10.1515/ans-2023-0132  

   Drake, Marjorie; Fefferman, Charles; Ren, Kevin; Skorobogatova, Anna (April 2025, Advanced Nonlinear Studies)

   Abstract In this paper, we establish the existence of a bounded, linear extension operator $T:L^{2,p}\left(E\right)\to L^{2,p}\left({\mathbb{R}}^2\right)$$$T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$$when 1 <p< 2 andEis a finite subset of ${\mathbb{R}}^2$$${\mathbb{R}}^{2}$$contained in a line. 

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5. Ordinary modules for vertex algebras of 𝔬𝔰𝔭 _1|2𝑛

   https://doi.org/10.1515/crelle-2024-0060  

   Creutzig, Thomas; Genra, Naoki; Linshaw, Andrew (August 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that the affine vertex superalgebra $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$V^{k}(\mathfrak{osp}_{1|2n})at generic level 𝑘 embeds in the equivariant 𝒲-algebra of $\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}times $4n$4nfree fermions.This has two corollaries:(1) it provides a new proof that, for generic 𝑘, the coset $\mathrm{Com}(V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right),V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right))$\operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n}))is isomorphic to ${\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})for $\mathrm{\ell }=-\left(n+1\right)+\left(k+n+1\right)/\left(2k+2n+1\right)$\ell=-(n+1)+(k+n+1)/(2k+2n+1), and(2) we obtain the decomposition of ordinary $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$V^{k}(\mathfrak{osp}_{1|2n})-modules into $V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})-modules.Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for $\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}, we show that the simple algebra $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$L_{k}(\mathfrak{osp}_{1|2n})is an extension of the simple subalgebra $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$L_{k}(\mathfrak{sp}_{2n})\otimes{\mathcal{W}}_{\ell}(\mathfrak{sp}_{2n}).Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$L_{k}(\mathfrak{osp}_{1|2n})-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects.It is equivalent to a certain subcategory of ${\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$\mathcal{W}_{\ell}(\mathfrak{sp}_{2n})-modules.A similar result also holds for the category of Ramond twisted modules.Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$L_{k}(\mathfrak{sp}_{2n})-modules are rigid. 

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