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1. 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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2. 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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3. Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

   https://doi.org/10.1515/acv-2024-0005  

   Mengesha, Tadele; Schikorra, Armin; Seesanea, Adisak; Yeepo, Sasikarn (November 2024, Advances in Calculus of Variations)

   Abstract We extend the Calderón–Zygmund theory for nonlocal equations tostrongly coupled system of linear nonlocal equations ${\mathcal{ℒ}}_A^{s}u=f${\mathcal{L}^{s}_{A}u=f}, where the operator ${\mathcal{ℒ}}_A^s${\mathcal{L}^{s}_{A}}is formally given by ${\mathcal{ℒ}}_A^{s}u={\int }_{{ℝ}^n}\frac{A(x,y)}{{\left|x-y\right|}^{n+2s}}\frac{\left(x-y\right)\otimes \left(x-y\right)}{{\left|x-y\right|}^2}\left(u\left(x\right)-u\left(y\right)\right)𝑑y.$\mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-%y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For $0<s<1${0<1}and $A:{ℝ}^n\times {ℝ}^n\to ℝ${A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}}taken to be symmetric and serving asa variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot ,y)${A(\,\cdot\,,y)}is uniformly Holder continuous and ${inf}_{x\in {ℝ}^n}A(x,x)>0${\inf_{x\in\mathbb{R}^{n}}A(x,x)>0}, then for $f\in L_{\mathrm{loc}}^p${f\in L^{p}_{\rm loc}}, for $p\ge 2${p\geq 2}, the solution vector $u\in H_{\mathrm{loc}}^{2s-\delta ,p}${u\in H^{2s-\delta,p}_{\rm loc}}for some $\delta \in (0,s)${\delta\in(0,s)}. 

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4. A new zero-free region for Rankin–Selberg 𝐿-functions

   https://doi.org/10.1515/crelle-2025-0009  

   Harcos, Gergely; Thorner, Jesse (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝜋 and ${\pi }^{\prime }$\pi^{\prime}be cuspidal automorphic representations of $\mathrm{GL}\left(n\right)$\mathrm{GL}(n)and $\mathrm{GL}\left(n^{\prime }\right)$\mathrm{GL}(n^{\prime})with unitary central characters.We establish a new zero-free region for all $\mathrm{GL}\left(1\right)$\mathrm{GL}(1)-twists of the Rankin–Selberg 𝐿-function $L(s,\pi \times {\pi }^{\prime })$L(s,\pi\times\pi^{\prime}), generalizing Siegel’s celebrated work on Dirichlet 𝐿-functions.As an application, we prove the first unconditional Siegel–Walfisz theorem for the Dirichlet coefficients of $-L^{\prime }(s,\pi \times {\pi }^{\prime })/L(s,\pi \times {\pi }^{\prime })$-L^{\prime}(s,\pi\times\pi^{\prime})/L(s,\pi\times\pi^{\prime}).Also, for $n\le 8$n\leq 8, we extend the region of holomorphy and nonvanishing for the twisted symmetric power 𝐿-functions $L(s,\pi ,{\mathrm{Sym}}^n\otimes \chi )$L(s,\pi,\mathrm{Sym}^{n}\otimes\chi)of any cuspidal automorphic representation of $\mathrm{GL}\left(2\right)$\mathrm{GL}(2). 

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5. Negative moments of the Riemann zeta-function

   https://doi.org/10.1515/crelle-2023-0091  

   Bui, Hung M; Florea, Alexandra (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta \left(s\right)${\zeta(s)}. For example, integrating ${\left|\zeta \left(\frac{1}{2}+\alpha +it\right)\right|}^{-2k}${|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto $2T${2T}, we obtain an asymptotic formula when the shift α is roughly bigger than $\frac{1}{\log T}${\frac{1}{\log T}}and $k<\frac{1}{2}${k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log \frac{1}{\alpha }\ll \log \log T${\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function. 

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