The main purpose of this article is to establish the Caffarelli–Kohn– Nirenberg-type (CKN-type) inequalities for all $\alpha \in \mathbf{R}$ and to study the related matters systematically. Roughly speaking, we discuss the characterizations of the CKN-type inequalities for all $\alpha \in \mathbf{R}$ as the variational problems, the existence and nonexistence of the extremal solutions to these variational problems in proper spaces, and the exact values and the asymptotic behaviors of the best constants in both the noncritical case and the critical case.

In the study of the CKN-type inequalities, the presence of weight functions on both sides prevents us from employing effectively the so-called spherically symmetric rearrangement. Further the invariance of ${\mathbf{R}}^n$ by the group of dilatations creates some possible loss of compactness. As a result we see that the existence of extremals, the values of best constants, and their asymptotic behaviors essentially depend upon the relations among parameters in the inequality.

A strictly convex curve is a $C^{\infty }$-regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve $\Gamma$, and take two distinct tangent lines $l_1$ and $l_2$ of $\Gamma$. A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of $\Gamma$, which are tangent to $l_1$ and $l_2$, as a corollary of Ghys’s theorem on diffeomorphisms of $S^1$. In this paper, we prove a refinement of Foreman’s result. We then prove a projectively dual version of our refinement, which is a claim about semiosculating conics passing through two fixed points on $\Gamma$. We also show that the dual version of Foreman’s four-vertex theorem is almost equivalent to the Ghys’s theorem.

Let $\Omega$ be a bounded domain of ${\mathbb{R}}^n$ ($n\ge 1$) containing the origin. In the present paper we establish the weighted Hardy–Sobolev inequalities with sharp remainders. For example, when $\alpha =1-n/p$ and $1<p<+\infty$ hold, we establish the following inequality.

There exist positive numbers ${\Lambda }_{n,p,\alpha },C$, and $R$ such that we have

(0.1) $$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^p{|x|}^{\alpha p}dx& \ge & {\mathrm{\Lambda }}_{n,p,\alpha }{\int }_{\mathrm{\Omega }}\frac{{|u(x)|}^p}{{|x|}^n}{A}_1{(|x|)}^{-p}dx\\ \\ & & +C{\int }_{\mathrm{\Omega }}\frac{{|u(x)|}^p}{{|x|}^n}{A}_1{(|x|)}^{-p}{A}_2{(|x|)}^{-2}dx\end{array}$$

for any $u\in W_{\alpha ,0}^{1,p}(\Omega )$. Here $A_1\left(\right|x\left|\right)=log\frac{R}{\left|x\right|}$, and $A_2\left(\right|x\left|\right)=log{A}_1\left(\right|x\left|\right)$. This is called the critical Hardy–Sobolev inequality with a sharp remainder involving a singular weight $A_1\left(\right|x\left|{)}^{-p}{A}_2\right(\left|x\right|{)}^{-2}$, in the sense that the improved inequality holds for this weight but fails for any weight more singular than this one. Here ${\Lambda }_{n,p,\alpha }$ is a sharp constant independent of each function $u$. Further we establish the Hardy–Sobolev inequalities in the subcritical case $(\alpha >1-n/p)$ and the supercritical case $(\alpha <1-n/p)$.

As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by $-div\left(\right|x{|}^{\alpha p}|\nabla u{|}^{p-2}\nabla u)-\mu /\left|x{|}^n{A}_1\right(\left|x\right|{)}^{-p}|u{|}^{p-2}u$ (the critical case) will tend to zero as $\mu$ increases to ${\Lambda }_{n,p,\alpha }$. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.

In this paper we show that the mean Euler characteristic of equivariant symplectic homology is an effective obstruction against the existence of displaceable exact contact embeddings. As an application we show that certain Brieskorn manifolds do not admit displaceable exact contact embeddings.

In this paper we prove the Miyawaki conjecture related to the spinor $L$- function of a Siegel cusp form of weight $12$ and degree $3$ as a special example of results related to Miyawaki lifts of odd degree. We work with the Fourier–Jacobi model of Ikeda lifts discovered by Yamazaki and Hayashida.

We give an alternative proof of Kovács’s vanishing theorem. Our proof is based on the standard arguments of the minimal model theory. We do not need the notion of Du Bois pairs. We reduce Kovács’s vanishing theorem to the well-known relative Kawamata–Viehweg–Nadel vanishing theorem.

In this paper we show that a metrizable topological vector space over $\mathbf{R}$ is topologically an open cone. This generalizes the partial results obtained by Henderson.

A family of partial differential operators on the Heisenberg group is introduced and studied. These operators may be regarded as analogues of the ultrahyperbolic operator on Euclidean space. Each of them is conformally invariant under the special linear group. The main focus is on the space of smooth solutions that extend to smooth sections of a suitable line bundle over a generalized flag manifold that contains the Heisenberg group as a dense open subset. The space of polynomial solutions is also considered from the point of view of conformal invariance.