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1. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy; Feneuil, Joseph; Mayboroda, Svitlana (March 2022, Mathematische Annalen)

   Résumé Dans David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions. arXiv:2010.09793 ) il est démontré que pour les domaines à bord uniformément rectifiable, la fonction de Green vérifie des estimations faibles de bonne approximation par des fonctions affines, avec une réciproque vraie dans certains cas encourageants. Ici on part de la rectifiabilité uniforme et on démontre les estimations fortes naturelles d’approximation de la fonction de Green, et aussi des solutions, par des applications affines (ou, de manière équivalente, des multiples de la distance au bord adoucie). L’étude inclut les analogues naturels du Laplacien dans les domaine dont la frontière est de grande co-dimension. On considère les opérateurs elliptiques $$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma -n} \nabla $$ L β , γ = div D d + 1 + γ - n ∇ associés à un domaine $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n dont le bord $$\Gamma $$ Γ est Ahlfors régulier et uniformément rectifiable de dimension $$d < n-1$$ d < n - 1 et à la distance au bord maintenant usuelle $$D = D_\beta $$ D = D β définie par $$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ D β ( X ) - β = ∫ Γ | X - y | - d - β d σ ( y ) pour $$X \in \Omega $$ X ∈ Ω , où $$\beta >0$$ β > 0 et $$\gamma \in (-1,1)$$ γ ∈ ( - 1 , 1 ) sont des paramètres et $$\sigma $$ σ une mesure Ahlfors régulière sur $$\Gamma $$ Γ . Les auteurs ont montré précédemment que la mesure elliptique associée à $$L_{\beta ,\gamma }$$ L β , γ est bien définie et est mutuellement absolument continue par rapport à $$\sigma $$ σ , avec un poids de $$A_\infty $$ A ∞ . Ici on démontre que la fonction de Green G avec pôle à l’infini associée à $$L_{\beta ,\gamma }$$ L β , γ est bien approchée par les multiples de D , au sens où la fonction $$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ | D ∇ ( ln ( G D 1 - γ ) ) | 2 vérifie une condition de Carleson sur $$\Omega $$ Ω . Ces nouvelles estimations sont différentes en nature. Les estimations de David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions. arXiv:2010.09793 ) reposaient sur des arguments de compacité; ici on a besoin d’estimations plus précises, obtenues par intégration par parties et en utilisant les propriétés algébriques de la fonction $$D_\alpha $$ D α dans le cas“magique” de David et al. (Duke Math J, to appear). 

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2. Deep Network With Approximation Error Being Reciprocal of Width to Power of Square Root of Depth

   https://doi.org/10.1162/neco_a_01364  

   Shen, Zuowei; Yang, Haizhao; Zhang, Shijun (January 2021, Neural Computation)

   null (Ed.)

   A new network with super-approximation power is introduced. This network is built with Floor (⌊x⌋) or ReLU (max{0,x}) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters N∈N+ and L∈N+, we show that Floor-ReLU networks with width max{d,5N+13} and depth 64dL+3 can uniformly approximate a Hölder function f on [0,1]d with an approximation error 3λdα/2N-αL, where α∈(0,1] and λ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [0,1]d with a modulus of continuity ωf(·), the constructive approximation rate is ωf(dN-L)+2ωf(d)N-L. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ωf(r) as r→0 is moderate (e.g., ωf(r)≲rα for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially d times a function of N and L independent of d within the modulus of continuity. 

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3. Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

   https://doi.org/10.3233/ASY-231872  

   Shapiro, Jacob (February 2024, Asymptotic Analysis)

   We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity. 

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4. Breakdown of Regularity of Scattering for Mass-Subcritical NLS

   https://doi.org/10.1093/imrn/rnaa072  

   Lee, Gyu Eun (April 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the scattering problem for the nonlinear Schrödinger equation $$i\partial _t u + \Delta u = |u|^p u$$ on $$\mathbb{R}^d$$, $$d\geq 1$$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $$\Sigma$$ holds and the wave operator is well defined on $$\Sigma$$. We show that there exists $$0<\beta <p$$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $$L^2\to L^2$$ of class $$C^{1+\beta }$$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology. 

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5. 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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