This paper is devoted to studying the singular integral with rough kernel associated to surfaces, which contain many classical surfaces as model examples. Also, the kernel of our operator lacks smoothness on the unit sphere as well as in the radial direction. We obtain the $L^p$ boundedness of the singular integral under a sharp size condition on its kernels in an extrapolation argument. In addition, the corresponding results for maximal truncated singular integral operators are also established.

Let $X$ be a compact toric surface. Then there exists a sequence of torus equivariant blow-ups of $X$ such that the blown-up toric surface admits a cscK metric.

We obtain a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds.

Bilinear Fourier multiplier operators corresponding to multipliers that are singular at the origin are considered. New criterions on such multipliers to assure the boundedness of the corresponding operators from $L^p \times L^q$ to $L^r$, $1/p+1/q=1/r$, are given in the range $1<p,q \le \infty$, $2/3<r<\infty$.

For a variety $X$ which admits a Cox ring, we introduce a functor from the category of quasi-coherent sheaves on $X$ to the category of graded modules over the homogeneous coordinate ring of $X$. We show that this functor is right adjoint to the sheafification functor and therefore left-exact. Moreover, we show that this functor preserves torsion-freeness and reflexivity. For the case of toric sheaves, we give a combinatorial characterization of its right derived functors in terms of certain right derived limit functors.

The CR equivalence problem between CR manifolds with slice structure is studied. Let $N$ be a connected holomorphically nondegenerate real analytic hypersurface and $M(p)$ a finitely nondegenerate real analytic hypersurface parametrized by $p \in N$. Let $M$ be a totality of $N$ and $M(p)$ with moving $p$ in $N$. Assume that $M$ and $\widetilde{M}$ (with a same structure as $M$) are CR equivalent and that $N$ and $\widetilde{N}$ are also CR equivalent. Then we prove that, for any $p \in N$, there exists $\tilde{p}\in \widetilde{N}$ such that $M(p)$ is CR equivalent to $\widetilde{M}(\tilde{p})$.

Furuichi and Yanagi showed a Schrödinger uncertainty relation for the Wigner-Yanase-Dyson skew information, which is a special monotone pair skew information. In this paper, we give a Schrödinger uncertainty relation based on a monotone pair skew information, and extend the result of Furuichi and Yanagi. Moreover, we show that some monotone pair skew information becomes a metric adjusted skew information and therefore the convexity of it follows from known results.

H. Beirão da Veiga proved that, for a straight channel in $\boldsymbol{R}^n$ ($n$ arbitarily large) and for a given flux with the time periodicity, there exists a unique time periodic Poiseuille flow in a straight channel in $\boldsymbol{R}^n$. Furthermore, the existence of a time periodic solution in a perturbed channel (Leray's problem) is shown for the Stokes problem (arbitary dimension) and for the Navier-Stokes problem ($n\le4$). Concerning the Navier-Stokes case, a quatitative condition requaired to show the existence of a time periodic solution depends not just on the flux of the time periodic Poiseuille flow but also on the domain it self. In this paper, by applying the result of H. Beirão da Veiga and C. J. Amick, we succeed in proving the independence of such a condition on the particular domain.

Using variational methods based on the critical point theory and suitable truncation and comparison techniques, we study existence, multiplicity and nonexistence of positive solutions for a parametric nonlinear Neumann problem driven by the $p$-Laplacian. Our hypotheses cover the case of nonlinearities of concave-convex type whose exponents depend on the parameter.