A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then, sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for the type index $\alpha \in [-1/2, \infty)^{d}$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^{1}$ norms in place of $H^{1}$ norms are obtained in both settings.

We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces $\dot{B}^{n/p}_{p,1} \times \dot{B}^{n/p-1}_{p,1}$ for all $1 \leq p < 2n$. However, if the data is in a larger scaling invariant class such as $p > 2n$, then the system is not well-posed. In this paper, we demonstrate that for the critical case $p = 2n$ the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin and Haspot are indeed sharp in the framework of the homogeneous Besov spaces.

Let $(X, L)$ denote a quasi-polarized manifold of dimension $n \geq 5$ defined over the field of complex numbers such that the canonical line bundle $K_{X}$ of $X$ is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of $K_{X} + mL$ in this case, and we prove that $h^{0}(K_{X} + mL) > 0$ for every positive integer $m$ with $m \geq n - 3$. In particular, a Beltrametti–Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.

In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its Jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.

We prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in $\mathbb{C} \mathbb{P}^1$, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in $\mathbb{C} \mathbb{P}^2$, is equal to zero.

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.

In this paper, we treat moduli spaces of parabolic connections. We take an affine open covering of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation for each affine open subset.

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities.

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C$, which is the germ of submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT $\mathcal{C}^{C}_{x} \subset \mathbb{P} D_{x}$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that $\mathcal{C}_{x}^{C}$ should be Legendrian with respect to the induced contact structure on $\mathbb{P} D_{x}$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT $\mathcal{C}^C_{x} \subset \mathbb{P} D_{x}$ at some point $x \in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 < p \leq q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.

Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiołek (1996) formulated a differential-geometric criterion (we call the M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) satisfies the M-criterion if the background manifold is a sphere, on the other hand, there are zonal flows satisfy the M-criterion if the background manifold is an ellipsoid (even it is sufficiently close to the sphere). The conjugate point is created by the fully nonlinear effect of the inviscid fluid flow with differential geometric mechanism.

We consider the Moore–Nehari equation, $u'' + h(x, \lambda) |u|^{p-1} u = 0$ in $(-1, 1)$ with $u(-1) = u(1) = 0$, where $p > 1$, $h(x, \lambda) = 0$ for $|x| < \lambda$, $h(x, \lambda) = 1$ for $\lambda \leq |x| \leq 1$ and $\lambda \in (0, 1)$ is a parameter. We prove the existence of a solution which has exactly $m$ zeros in the interval $(-1, 0)$ and exactly $n$ zeros in $(0, 1)$ for given nonnegative integers $m$ and $n$.