In this paper, we characterize some properties on weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, we obtain sharp conditions for product inequalities, convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. By a unified approach different from others we give a complete answer to the question of finding sharp conditions of certain relations on weighted modulation and Wiener amalgam spaces.

We study the $m$th Gauss map in the sense of F. L. Zak of a projective variety $X\subset {\mathbb{P}}^N$ over an algebraically closed field in any characteristic. For all integers $m$ with $n:=dim(X)⩽m<N$, we show that the contact locus on $X$ of a general tangent $m$-plane is a linear variety if the $m$th Gauss map is separable. We also show that for smooth $X$ with $n<N-2$, the $(n+1)$th Gauss map is birational if it is separable, unless $X$ is the Segre embedding ${\mathbb{P}}^1\times {\mathbb{P}}^n\subset {\mathbb{P}}^{2n-1}$. This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.

In this paper, we study holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is the solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the noncommutative case nonlinearizable semicocycles are shown to exist. We derive simple conditions for linearizability and show that they are sharp.

We study mixed weak-type inequalities for the commutator $[b,T]$, where $b$ is a BMO function, and $T$ is a Calderón–Zygmund operator. More precisely, we prove that, for every $t>0$,

$$uv\left(\right\{x\in {\mathbb{R}}^n:\left|\frac{[b,T](fv)(x)}{v(x)}\right|>t\left\}\right)\le C{\int }_{{\mathbb{R}}^n}\Phi \left(\frac{|f(x)|}{t}\right)u(x)v(x)dx,$$ where $\Phi (t)=t(1+{log}^{+}t)$, $u\in A_1$, and $v\in A_{\infty }(u)$. Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators.

For a given Young function $\varphi$ we also consider singular integral operators $T$ whose kernels satisfy a $L^{\varphi }$-Hörmander property, and we find sufficient conditions on $\varphi$ such that a mixed weak estimate holds for $T$ and also for its higher order commutators $T_b^m$.

We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $LlogL$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.

We give a deviation estimate for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1, and controlled fourth moments. We also give some new properties of Laguerre polynomials.

Let $(SpecR,\mathfrak{m})$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\ge 0$. We evaluate further the dimensions of the local cohomology groups, which were treated by Wahl in 1975 as vanishing theorem C (resp., D) under the assumption that $p$ is a very good prime (resp., good prime) with respect to $(SpecR,\mathfrak{m})$. We use Artin’s classification of rational double points and completely determine the dimensions ${dim}_k{H}_E^1\left(S_X\right)$ and ${dim}_k{H}_E^1\left(S_X\otimes {\mathcal{O}}_X\right(E\left)\right)$, supplementing Wahl’s theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution $X\to SpecR$ and an equisingular family that injects into a versal deformation of the rational double point $(SpecR,\mathfrak{m})$.

In this note, we study the Seifert rational homology spheres with two complementary legs, that is, with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with three exceptional fibers and two complementary legs that bound rational homology balls. The result translates into a statement on the sliceness of some Montesinos knots.

We compute the Picard group of the universal Abelian variety over the moduli stack ${\mathcal{A}}_{g,n}$ of principally polarized Abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for ${\mathcal{A}}_{g,n}$.