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1. Solvability of the $$\mathrm{L}^{p}$$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

   https://doi.org/10.1007/s00222-024-01300-1  

   Bortz, Simon; Hofmann, Steve; Martell, José María; Nyström, Kaj (January 2025, Inventiones mathematicae)

   Abstract We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is parabolic$$A_{\infty }$$ $A_{\infty }$with respect to surface measure (which property is in turn equivalent to$$\mathrm{L}^{p}$$ $L^p$solvability of the Dirichlet problem for some finite$$p$$ $p$), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the$$A_{\infty }$$ $A_{\infty }$property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has$$\mathrm{L}^{p}$$ $L^p$solvability (for some$$p <\infty $$ $p<\infty$) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type. 

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2. The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

   https://doi.org/10.1007/s00526-024-02660-5  

   Bauer, Martin; Le Brigant, Alice; Lu, Yuxiu; Maor, Cy (February 2024, Calculus of Variations and Partial Differential Equations)

   Abstract We introduce a family of Finsler metrics, called the$$L^p$$ $L^p$-Fisher–Rao metrics$$F_p$$ $F_p$, for$$p\in (1,\infty )$$ $p\in (1,\infty )$, which generalizes the classical Fisher–Rao metric$$F_2$$ $F_2$, both on the space of densities$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$and probability densities$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$. We then study their relations to the Amari–C̆encov$$\alpha $$ $\alpha$-connections$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$from information geometry: on$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$, the geodesic equations of$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$coincide, for$$p = 2/(1-\alpha )$$ $p=2/(1-\alpha )$. Both are pullbacks of canonical constructions on$$L^p(M)$$ $L^p\left(M\right)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$ $\alpha$-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$, the$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$ $L^p\left(M\right)$, but in this case they no longer coincide unless$$p=2$$ $p=2$. Using this transformation, we solve the geodesic equation of the$$\alpha $$ $\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$ $F_p$, and study their relation to$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$. 

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3. Second order $$L_p$$ estimates for subsolutions of fully nonlinear equations

   https://doi.org/10.1007/s00526-025-02929-3  

   Dong, Hongjie; Kitano, Shuhei (February 2025, Calculus of Variations and Partial Differential Equations)

   Abstract We obtain new$$L_p$$ $L_p$estimates for subsolutions to fully nonlinear equations. Based on our$$L_p$$ $L_p$estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of$$L_p$$ $L_p$estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the$$L_p$$ $L_p$spaces or in the space of Radon measures. 

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4. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

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5. Continuum limit of 2D fractional nonlinear Schrödinger equation

   https://doi.org/10.1007/s00028-023-00881-3  

   Choi, Brian; Aceves, Alejandro (March 2023, Journal of Evolution Equations)

   Abstract We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in$$L^2({\mathbb {R}}^2)$$ $L^2\left(R^2\right)$to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter$$\alpha \in (1,2)$$ $\alpha \in (1,2)$approaches the boundaries. 

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