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1. On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

   https://doi.org/10.1090/tran/9315  

   Gong, Xianghong (March 2025, Transactions of the American Mathematical Society)

   We study regularity of solutions $u$ to $\stackrel{¯<\#comment/>}{\mathrm{\partial <\#comment/>}}u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$ , where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-<\#comment/>q)$ positive Levi eigenvalues at each point of boundary $\mathrm{\partial <\#comment/>}D$ . Under the necessary condition that a locally $L^2$ solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the Hölder–Zygmund space ${\mathrm{\Lambda <\#comment/>}}^r\left(D\right)$ with $r>1$ . For $q>1$ , the same regularity for the solutions is achieved when $\mathrm{\partial <\#comment/>}D$ is either sufficiently smooth or of $(n-<\#comment/>q)$ positive Levi eigenvalues everywhere on $\mathrm{\partial <\#comment/>}D$ . 

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2. On a conjectural symmetric version of Ehrhard’s inequality

   https://doi.org/10.1090/tran/9177  

   Livshyts, Galyna (May 2024, Transactions of the American Mathematical Society)

   We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-<\#comment/>1}\left(s\right)={\int <\#comment/>}_0^s{t}^{k-<\#comment/>1}{e}^{-<\#comment/>\frac{t^2}{2}}dt$ and $c_{k-<\#comment/>1}=J_{k-<\#comment/>1}(+\mathrm{\infty <\#comment/>})$ , we conjecture that the function $F:[0,1]\to <\#comment/>\mathbb{R}$ , given by $F\left(a\right)=\underset{k=1}{\overset{n}{\sum <\#comment/>}}{1}_{a\in <\#comment/>E_k}\cdot <\#comment/>\left({\mathrm{\beta <\#comment/>}}_k{J}_{k-<\#comment/>1}^{-<\#comment/>1}\right(c_{k-<\#comment/>1}a)+{\mathrm{\alpha <\#comment/>}}_k)$ (with an appropriate choice of a decomposition $[0,1]={\cup <\#comment/>}_i{E}_i$ and coefficients ${\mathrm{\alpha <\#comment/>}}_i,{\mathrm{\beta <\#comment/>}}_i$ ) satisfies, for all symmetric convex sets $K$ and $L$ , and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$ , $F\left(\mathrm{\gamma <\#comment/>}\right(\mathrm{\lambda <\#comment/>}K+(1-<\#comment/>\mathrm{\lambda <\#comment/>})L\left)\right)\ge <\#comment/>\mathrm{\lambda <\#comment/>}F\left(\mathrm{\gamma <\#comment/>}\right(K\left)\right)+(1-<\#comment/>\mathrm{\lambda <\#comment/>})F\left(\mathrm{\gamma <\#comment/>}\right(L\left)\right).$ We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set $K$ , itsGaussian concavity power $p_s(K,\mathrm{\gamma <\#comment/>})$ is greater than or equal to $p_s(R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k},\mathrm{\gamma <\#comment/>})$ , for some $k\in <\#comment/>\{1,\dots <\#comment/>,n\}$ . We call the sets $R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$ round $k$ -cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set $K$ in ${\mathbb{R}}^n$ , $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>\underset{F\in <\#comment/>L^2(K,\mathrm{\gamma <\#comment/>})\cap <\#comment/>Lip\left(K\right):\int <\#comment/>F=1}{sup}\left(2T_{\mathrm{\gamma <\#comment/>}}^F\right(K)-<\#comment/>Var(F\left)\right)+\frac{1}{n-<\#comment/>\mathbb{E}{X}^2},$ where $T_{\mathrm{\gamma <\#comment/>}}^F\left(K\right)$ is the $F-<\#comment/>$ torsional rigidity of $K$ with respect to the Gaussian measure.Moreover, the equality holds if and only if $K=R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$ for some $R>0$ and $k=1,\dots <\#comment/>,n$ .As a consequence, we get $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>Q(\mathbb{E}|X{|}^2,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^4,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^2,r(K\left)\right),$ where $Q$ is a certain rational function of degree $2$ , the expectation is taken with respect to the restriction of the Gaussian measure onto $K$ , $\Vert <\#comment/>\cdot <\#comment/>{\Vert <\#comment/>}_K$ is the Minkowski functional of $K$ , and $r\left(K\right)$ is the in-radius of $K$ . The result follows via a combination of some novel estimates, the $L2$ method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185;Geometric aspects of functional analysis, Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function $F$ whose composition with $\mathrm{\gamma <\#comment/>}\left(K\right)$ is concave in the Minkowski sense for all symmetric convex sets. 

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3. A single-point Reshetnyak’s theorem

   https://doi.org/10.1090/tran/9415  

   Kangasniemi, Ilmari; Onninen, Jani (May 2025, Transactions of the American Mathematical Society)

   We prove a single-value version of Reshetnyak’s theorem. Namely, if a non-constant map $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ from a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfies the estimate $\left|Df\right(x){|}^n\le K{J}_f(x)+\mathrm{\Sigma }(x\left)\right|f\left(x\right)-y_0{|}^n$ at almost every $x\in \mathrm{\Omega }$ for some $K\ge 1$ , $y_0\in {\mathbb{R}}^n$ and $\mathrm{\Sigma }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }\left(\mathrm{\Omega }\right)$ , then $f^{-1}\left\{y_0\right\}$ is discrete, the local index $i(x,f)$ is positive in $f^{-1}\left\{y_0\right\}$ , and every neighborhood of a point of $f^{-1}\left\{y_0\right\}$ is mapped to a neighborhood of $y_0$ . Assuming this estimate for a fixed $K$ at every $y_0\in {\mathbb{R}}^n$ is equivalent to assuming that the map $f$ is $K$ -quasiregular, even if the choice of $\mathrm{\Sigma }$ is different for each $y_0$ . Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$ -quasiregularity. As a corollary of our single-value Reshetnyak’s theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem. 

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4. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan; Smirnov, Ilya (February 2023, Proceedings of the American Mathematical Society)

   Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\ge <\#comment/>2$ . We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$ , then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$ -primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$ such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$ for all $\mathfrak{m}$ -primary ideals $I\subseteq <\#comment/>R$ . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$ . 

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5. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$ be analytic on $[0,1]$ with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$ for some constants $A$ and $\mathrm{\alpha <\#comment/>}>2$ and all $k⩾<\#comment/>1$ . We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$ under random linear scrambling with $n=2^m$ points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$ for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$ . We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$ random linearly scrambled estimates, when $k/m$ is bounded away from zero. When $f$ has a $p$ ’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$ -Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$ for any $\mathrm{ϵ<\#comment/>}>0$ , if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$ as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$ . The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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