In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential.

Let $G$ be a finite-dimensional Lie group, and let $E$ be a locally convex topological $G$-module. If $E$ is sequentially complete, then $E$ and the space $E^{\infty }$ of smooth vectors are $C_c^{\infty }\left(G\right)$-modules, but the module multiplication need not be continuous. The pathology can be ruled out if $E$ is (or embeds into) a projective limit of Banach $G$-modules. Moreover, in this case $E^{\omega }$ (the space of analytic vectors) is a module for the algebra $\mathcal{A}\left(G\right)$ of superdecaying analytic functions introduced by Gimperlein, Krötz, and Schlichtkrull. We prove that $E^{\omega }$ is a topological $\mathcal{A}\left(G\right)$-module if $E$ is a Banach space or, more generally, if every countable set of continuous seminorms on $E$ has an upper bound. The same conclusion is obtained if $G$ has a compact Lie algebra. The question of whether $C_c^{\infty }\left(G\right)$ and $\mathcal{A}\left(G\right)$ are topological algebras is also addressed.

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion-free sheaves on a K3 surface with Picard number one is the moduli space of $\mu$-stable locally free sheaves if the rank of the sheaves is not a square number.

We consider $L^2$-supercritical and $H^1$-subcritical focusing nonlinear[4] Schrödinger equations. We introduce a subset $\mathrm{PW}$ of $H^1\left({\mathbb{R}}^d\right)$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. To this end, we divide $\mathrm{PW}$ into two disjoint components ${\mathrm{PW}}_{+}$ and ${\mathrm{PW}}_{-}$. Then, it turns out that any solution starting from a datum in ${\mathrm{PW}}_{+}$ behaves asymptotically free, and solution starting from a datum in ${\mathrm{PW}}_{-}$ blows up or grows up, from which we find that the ground state has two unstable directions. Our result is an extension of the one by Duyckaerts, Holmer, and Roudenko to the general powers and dimensions, and our argument mostly follows the idea of Kenig and Merle.

The $KO$-theory of the flag manifold $G/T$ is determined by calculating the Atiyah–Hirzebruch spectral sequence when $G$ is one of the exceptional Lie groups $G_2$, $F_4$, $E_6$, where $T$ is a maximal torus of $G$.

This work studies boundedness properties of the fractional maximal operator on metric measure spaces under standard assumptions on the measure. The main motivation is to show that the fractional maximal operator has similar smoothing and mapping properties as the Riesz potential. Instead of the usual fractional maximal operator, we also consider a so-called discrete maximal operator which has better regularity. We study the boundedness of the discrete fractional maximal operator in Sobolev, Hölder, Morrey, and Campanato spaces. We also prove a version of the Coifman– Rochberg lemma for the fractional maximal function.