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1. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo ; Wang, Jinhuan ( June 2024 , Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following$L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality$\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}$where parameters$q$and$p$satisfy the conditions$p>d\geq 1$,$q\geq 0$. The best constant$C_{q,\infty ,p}$is given by$\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}$where$u_{c,\infty }$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (x-x_0))$for any real numbers$A$,$\lambda >0$and$x_{0}\in \mathbb {R}^d$. In fact, the generalized Lane-Emden equation in$\mathbb {R}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For$q=0$or$d=1$,$u_{c,\infty }$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that$u_{c,m}\to u_{c,\infty }$and$C_{q,m,p}\to C_{q,\infty ,p}$as$m\to +\infty$for$d=1$, where$u_{c,m}$and$C_{q,m,p}$are the function achieving equality and the best constant of$L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively.

    

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2. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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3. The two-phase problem for harmonic measure in VMO and the chord-arc condition

   https://doi.org/10.1090/btran/197  

   Tolsa, Xavier ; Toro, Tatiana ( November 2024 , Transactions of the American Mathematical Society, Series B)

   Let$\Omega ^+\subset \mathbb {R}^{n+1}$be a bounded$\delta$-Reifenberg flat domain, with$\delta >0$small enough, possibly with locally infinite surface measure. Assume also that$\Omega ^-= \mathbb {R}^{n+1}\setminus \overline {\Omega ^+}$is an NTA (non-tangentially accessible) domain as well and denote by$\omega ^+$and$\omega ^-$the respective harmonic measures of$\Omega ^+$and$\Omega ^-$with poles$p^\pm \in \Omega ^\pm$. In this paper we show that the condition that$\log \dfrac {d\omega ^-}{d\omega ^+} \in \operatorname {VMO}(\omega ^+)$is equivalent to$\Omega ^+$being a chord-arc domain with inner unit normal belonging to$\operatorname {VMO}(\mathcal {H}^n|_{\partial \Omega ^+})$.

    

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4. Type-I permanence

   https://doi.org/10.1090/ert/648  

   Chirvasitu, Alexandru ( July 2023 , Representation Theory of the American Mathematical Society)

   We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding$\mathbb {N}\trianglelefteq \mathbb {E}$of locally compact groups and a twisted action$(\alpha ,\tau )$thereof on a (post)liminal$C^*$-algebra$A$the twisted crossed product$A\rtimes _{\alpha ,\tau }\mathbb {E}$is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup$\mathbb {N}\trianglelefteq \mathbb {E}$is type-I as soon as$\mathbb {E}$is. This happens for instance if$\mathbb {N}$is discrete and$\mathbb {E}$is Lie, or if$\mathbb {N}$is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions.

   In the same spirit, call a locally compact group$\mathbb {G}$type-I-preserving if all semidirect products$\mathbb {N}\rtimes \mathbb {G}$are type-I as soon as$\mathbb {N}$is, andlinearlytype-I-preserving if the same conclusion holds for semidirect products$V\rtimes \mathbb {G}$arising from finite-dimensional$\mathbb {G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie.

    

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5. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn ; Wake, Preston ( October 2022 , Proceedings of the American Mathematical Society, Series B)

   We show that for primes$N, p \geq 5$with$N \equiv -1 \bmod p$, the class number of$\mathbb {Q}(N^{1/p})$is divisible by$p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv -1 \bmod p$, there is always a cusp form of weight$2$and level$\Gamma _0(N^2)$whose$\ell$th Fourier coefficient is congruent to$\ell + 1$modulo a prime above$p$, for all primes$\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$extension of$\mathbb {Q}(N^{1/p})$.

    

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