In this paper, some boundedness for commutators of fractional integrals is obtained on Herz–Morrey spaces with variable exponent applying some properties of variable exponent and bounded mean oscillation (BMO) functions.

In this paper, we show that the depth of an isolated log canonical center is determined by the cohomology of the $-1$ discrepancy divisors over it. A similar result also holds for normal isolated Du Bois singularities.

In this paper we study the pseudoprocess driven by ${\partial }_t=-A{\partial }_x^3$. Our method is an approximation by the pseudo–random walk. We obtain their joint distribution of the first hitting time and the first hitting place. In addition, this result is provided by the alternate method of Shimoyama.

In part I of this project we examined low-regularity local well-posedness for generic quasilinear Schrödinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an $l^1$-summability over cubes in order to account for Mizohata’s integrability condition, which is a necessary condition for the $L^2$ well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent $L^2$-nature of the Schrödinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in $H^s$-spaces.

A compact Levi flat hypersurface in a complex manifold is said to be of $q$-concave type if it admits a neighborhood system consisting of $q$-concave manifolds in the sense of Andreotti and Grauert. The real analytic Levi flat hypersurfaces of $1$-concave type in Hopf surfaces are classified.

We provide conditions that classify sequences of random graphs into two types in terms of cover times. One type (type $1$) is the class of random graphs on which the cover times are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (type $2$) is the class of random graphs on which the cover times are of the order of the maximal hitting times. The conditions are described by some parameters determined by random graphs: the volumes, the diameters with respect to the resistance metric, and the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton–Watson trees, the incipient infinite cluster of a critical Galton–Watson tree, and the Sierpinski gasket graph.

Starting from the $1$-dimensional complex-valued Ornstein–Uhlenbeck process, we present two natural ways to obtain the associated eigenfunctions of the $2$-dimensional normal Ornstein–Uhlenbeck operator in the complex Hilbert space $L_{\mathbb{C}}^2\left(\mu \right)$. We call the eigenfunctions Hermite–Laguerre–Itô polynomials. In addition, the Mehler summation formula for the complex process is shown.

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like $SU(\infty )$, in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable $\sigma$-$C^{\ast }$-algebras that generalizes both the twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern–Connes-type character taking values in a bivariant local cyclic homology. We analyze the structure of the dual Chern–Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate the twisted periodic cyclic homology via locally convex algebras and the local cyclic homology via $C^{\ast }$-algebras (in the compact case).

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh–Marcinkiewicz means in terms of the modulus of continuity on the Hardy space $H_{2/3}$.

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph ${\mathcal{G}}_M$ with labeling in $H^2\left(BT\right)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the “graph cohomology” ring $H_T^{\ast }\left({\mathcal{G}}_M\right)$ of ${\mathcal{G}}_M$ is defined to be a subring of ${\oplus }_{v\in V\left({\mathcal{G}}_M\right)}{H}^{\ast }\left(BT\right)$, where $V\left({\mathcal{G}}_M\right)$ is the set of vertices of ${\mathcal{G}}_M$, and is known to be often isomorphic to the equivariant cohomology $H_T^{\ast }\left(M\right)$ of $M$. In this paper, we determine the ring structure of $H_T^{\ast }\left({\mathcal{G}}_M\right)$ with $\mathbb{Z}$ (resp., $\mathbb{Z}\left[\frac{1}{2}\right]$) coefficients when $M$ is a flag manifold of type A, B, or D (resp., C) in an elementary way.

Let $G$ be a simply connected, compact Lie group, let $P\to S^4$ be a principal $G$-bundle, and let $\mathcal{G}\left(P\right)$ be the gauge group of this bundle. When $G$ is a matrix group and $p$ is an odd prime, we use new methods to improve on the $p$-local homotopy decompositions of $\mathcal{G}\left(P\right)$ appearing in separate work of the first two authors and the third author.

We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that if $\mathcal{X}$ is a sufficiently nice subcategory of an abelian category, then the derived dimension of $\mathcal{A}$ is at most $\mathcal{X}$-dim$\mathcal{A}$, provided that $\mathcal{X}$-dim$\mathcal{A}$ is greater than one. We provide some applications.