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1. The L _p chord Minkowski problem

   https://doi.org/10.1515/ans-2022-0041  

   Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming (January 2023, Advanced Nonlinear Studies)

   Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 . 

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2. Desingularization and p-Curvature of Recurrence Operators

   https://doi.org/10.1145/3476446.3535478  

   Zhou, Yi; van Hoeij, Mark (July 2022, ISSAC'2022)

   Linear recurrence operators in characteristic p are classified by their p-curvature. For a recurrence operator L, denote by χ(L) the characteristic polynomial of its p-curvature. We can obtain information about the factorization of L by factoring χ(L). The main theorem of this paper gives an unexpected relation between χ(L) and the true singularities of L. An application is to speed up a fast algorithm for computing χ(L) by desingularizing L first. Another contribution of this paper is faster desingularization. 

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3. On the Korányi spherical maximal function on Heisenberg groups

   https://doi.org/10.1007/s00208-022-02533-2  

   Srivastava, Rajula (December 2022, Mathematische Annalen)

   Abstract We prove$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in (Ganguly and Thangavelu in J Funct Anal 280(3):108832, 2021) for the associated lacunary maximal operator. 

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4. Projections onto L^p Bergman spaces of Reinhardt domains

   https://doi.org/10.1016/j.aim.2024.109790  

   Chakrabarti, Debraj; Edholm, Luke D (August 2024, Advances in Mathematics)

   For $$1<\infty$$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be an absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $$p \neq 2$$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces. 

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5. Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

   https://doi.org/10.1090/bproc/164  

   Chirvasitu, Alexandru (July 2024, Proceedings of the American Mathematical Society, Series B)

   An extended derivation (endomorphism) of a (restricted) Lie algebra $L$is an assignment of a derivation (respectively) of $L^{\prime }$for any (restricted) Lie morphism $f:L\to <\#comment/>L^{\prime }$, functorial in $f$in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L^{\prime }$to every $f$; and (b) if $L$is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. 

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