Let $A$ be a right coherent ring and $\mathcal{X}$ be a contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct a recollement of abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is a full subcategory of ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting of all functors vanishing on projective modules. As a result, a relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Some examples and applications will be provided.

We consider the multilinear pseudo-differential operators with symbols in a generalized $S_{0,0}$-type class and prove the boundedness of the operators from $(L^2, \ell^{q_1}) \times \cdots \times (L^2, \ell^{q_{N}})$ to $(L^2, \ell^{r})$, where $(L^2, \ell^{q})$ denotes the $L^2$-based amalgam space. This extends the previous result by the same authors, which treated the bilinear pseudo-differential operators and gave the $L^2 \times L^2$ to $(L^2, \ell^{1})$ boundedness.

We obtain a residue formula for an obstruction to the existence of coupled Kähler–Einstein metrics described by Futaki–Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler–Einstein metric but still admits coupled Kähler–Einstein metrics.

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, that admit a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund–Hübsch–Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.

We analyze the structure of simply-connected Enriques surfaces in characteristic two whose K3-like coverings are normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct such surfaces and the resulting twistor lines in the moduli stack of Enriques surfaces, including the case that the K3-like covering is a normal rational surface rather then a normal K3 surface. Among other things, we show that elliptic double points indeed do occur. In this case, there is only one singularity. The main idea is to apply flops to Frobenius pullbacks of rational elliptic surfaces, to get the desired K3-like covering. Our results hinge on Lang's classification of rational elliptic surfaces, the determination of their Mordell–Weil lattices by Shioda and Oguiso, and the behavior of unstable fibers under Frobenius pullback via Ogg's formula. Along the way, we develop a general theory of Zariski singularities in arbitrary dimension, which is tightly interwoven with the theory of height-one group schemes actions and restricted Lie algebras. Furthermore, we determine under what conditions tangent sheaves are locally free, and introduce a theory of canonical coverings for arbitrary proper algebraic schemes.

In this paper, we present exact missing terms for the Poincaré inequalities on homogeneous groups. We also discuss some consequences in the Euclidean cases. Thus, we have succeeded in finding a simple proof of the sharp Poincaré inequality for the Dirichlet Laplacian.

For a relatively minimal fibration $f : X \to \mathbb{P}^1$ of non-hyperelliptic curves of genus $g$, we know the Picard number $\rho(X) \leq 3g + 8$. We study the case where $\rho(X) = 3g + 8$ and the Mordell–Weil group of $f$ is trivial. Such an $f$ occurs only if $g \equiv 0$ or $1 \pmod{3}$, and we describe such $f : X \to \mathbb{P}^1$ explicitly.

Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps.

We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models.

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.

The aim of this paper is to give a mass transportation proof for a full family of sharp Gagliardo–Nirenberg inequalities in dimension one. In fact, we shall establish a duality principle which derives this family of inequalities as a consequence. We also characterize all optimizers for these inequalities via the mass transportation method.

We consider a mixed function of type $H(\boldsymbol{z}, \bar{\boldsymbol{z}}) = f(\boldsymbol{z}) \bar{g} (\boldsymbol{z})$ where $f$ and $g$ are holomorphic functions which are non-degenerate with respect to the Newton boundaries. We assume also that the variety $f = g = 0$ is a non-degenerate complete intersection variety. In our previous paper, we considered the case that $f, g$ are convenient so that they have isolated singularities. In this paper we do not assume the convenience of $f$ and $g$. In non-convenient case, two hypersurfaces may have non-isolated singularities at the origin. We will show that $H$ still has both a tubular and a spherical Milnor fibrations under the local tame non-degeneracy and the toric multiplicity condition. We also prove the equivalence of two fibrations.