In 1977, Coifman and Weiss gave a proof of the $\mathrm{VMO}$-$H^1$ duality. We consider generalized Campanato spaces and atomic Hardy spaces with variable growth condition and give an extension of the duality to these spaces. We also apply this duality to the Riesz transforms.

In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is replaced by $\boldsymbol{e}$. First of all, there is no written proof in the available literature that the group over $\boldsymbol{e}$ is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other hand, we extend Newelski's theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context.

We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces $H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) and the scaling critical space $H^{2-\alpha}(\mathbb{R}^2)$. More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data $\theta_{0} \in H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset $K \subset H^{2-\alpha}(\mathbb{R}^{2})$, which is precompact in some homogeneous Sobolev spaces.

To each link $L$ in $S^{3}$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure of the width trees can bound the values of these invariants from below. We also show that each width tree is associated with a knot in $S^{3}$ and that if it also meets a high enough “distance threshold” it is, up to a certain equivalence, the unique width tree realizing the invariants.

In this paper, we prove that a free abelian group cannot occur as the group of self-homotopy equivalences of a rational CW-complex of finite type. Thus, we generalize a result due to Sullivan–Wilkerson showing that if $X$ is a rational CW-complex of finite type such that $\mathrm{dim}\,H^{*}(X, \mathbb{Z}) < \infty$ or $\mathrm{dim}\,\pi_{*}(X) < \infty$, then the group of self-homotopy equivalences of $X$ is isomorphic to a linear algebraic group defined over $\mathbb{Q}$.

In this paper, we obtain the classification theorem for 3-dimensional complete space-like $\lambda$-translators $x:M^{3} \rightarrow \mathbb{R}^{4}_{1}$ with constant norm of the second fundamental form and constant $f_{4} = \sum_{i,j,k,l} h_{ij} h_{jk} h_{kl} h_{li}$ with $h_{ij}$ being the components of the second fundamental form in the Minkowski space $\mathbb{R}^{4}_{1}$.

We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N \in ] -\infty, 1] \cup [n, +\infty]$ with $\varepsilon$-range introduced by Lu–Minguzzi–Ohta in 2022. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $N \in [n, +\infty[$ and $\varepsilon = 1$, and for $N \in ] -\infty, 1]$ and $\varepsilon = 0$ by the second named author.

We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties of general type in positive characteristic on which Bogomolov's inequality holds for semistable sheaves of any rank. The key ingredient in the proof is a high rank generalization of the slope inequality established by Xiao and Cornalba–Harris. This Bogomolov's inequality is applied to study the positivity of linear systems and semistable sheaves and construct Bridgeland stability conditions on product type surfaces in positive characteristic. We also give some new counterexamples to Bogomolov's inequality and pose some open questions.

We classify 3-braids arising from collision-free choreographic motions of 3 bodies on Lissajous plane curves, and present a parametrization in terms of levels and (Christoffel) slopes. Each of these Lissajous 3-braids represents a pseudo-Anosov mapping class whose dilatation increases when the level ascends in the natural numbers or when the slope descends in the Stern–Brocot tree. We also discuss 4-symbol frieze patterns that encode cutting sequences of geodesics along the Farey tessellation in relation to odd continued fractions of quadratic surds for the Lissajous 3-braids.

The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of Grothendieck and Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

We construct higher-dimensional analogues of the $\mathcal{I}^{\prime}$-curvature of Case and Gover in all CR dimensions $n \geq 2$. Our $\mathcal{I}^{\prime}$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n \geq 2$.

We give a complete classification of complex $\mathbb{Q}$-homology projective planes with numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all moduli curves are rational and defined over $\mathbb{Q}$, and we determine all families explicitly using extremal rational elliptic surfaces and Enriques involutions of base change type.