In this paper, we give a natural, and generalized reverse Hölder inequality, which says that if $\omega_{i}\in A_\infty$, then for every cube $Q$, \begin{equation} \int_Q\prod_{i =1}^m{\omega_i}^{\theta_i} \geq \prod_{i = 1}^m \left( \frac{\int_Q{\omega_i}}{[\omega_i]_{A_\infty}} \right)^{\theta_i}, \end{equation} where $\sum_{i=1}^m\theta_{i}=1$, $0\leq\theta_i\leq1$.

As a consequence, we get a more general inequality, which can be viewed as an extension of the reverse Jensen inequality in the theory of weighted inequalities. Based on this inequality (0.1), we then give some results concerning multilinear Calderón-Zygmund operators and maximal operators on weighted Hardy spaces, which improve some known results significantly.

Let $p$ be an odd prime, and let B$_\infty$ denote the Z$_p$-extension over the rational field. Let $l$ be an odd prime different from $p$. The question whether the $l$-class group of B$_\infty$ is trivial has been considered in our previous papers mainly for the case where $l$ varies with $p$ fixed. We give a criterion, for checking the triviality of the $l$-class group of B$_\infty$, which enables us to discuss the triviality when $p$ varies with $l$ fixed. As a consequence, we find that, if $l$ does not exceed 13 and $p$ does not exceed 101, then the $l$-class group of B$_\infty$ is trivial.

We prove the equivalence of parabolic Harnack inequalities and sub-Gaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form.

For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. This functor-valued invariant of a knot is an extension of the knot quandle. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariants, and study their properties.

Let $\iota$: C$^2 \hookrightarrow S$ be a compactification of the two dimensional complex space C$^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 ([2]) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f$: C$^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $\rho_f$ denote the order of $f$. We will prove that if $\rho_f$ < 2, then every global symmetric holomorphic tensor must vanish; in particular, if $\dim X=2$ and $X$ is kähler, then $X$ is a rational surface. Without the kähler condition there is no such conclusion, as we will show by a counter-example using a Hopf surface. This may be the first instance that the kähler or non-kähler condition makes a difference in the value distribution theory.

We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.

We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.

We show that the intersections of two real forms, certain totally geodesic Lagrangian submanifolds, in Hermitian symmetric spaces of compact type are antipodal sets. The intersection number of two real forms is invariant under the replacement of the two real forms by congruent ones. If two real forms are congruent, then their intersection is a great antipodal set of them. It implies that any real form in Hermitian symmetric spaces of compact type is a globally tight Lagrangian submanifold. Moreover we describe the intersection of two real forms in the irreducible Hermitian symmetric spaces of compact type.

In the 1980s, Harada introduced a class of algebras now called Harada algebras. The aim of this paper is to study Harada algebras in representation theoretical point of view. The paper concludes the following two results. The first is the classification of modules over left Harada algebras whose projective dimension is at most one. The second is the classification of tilting modules over left Harada algebras, which is done by giving a bijection between tilting modules over Harada algebras and tilting modules over direct products of upper triangular matrix algebras over a field.

Let $f$ and $g$, of weights $k' > k \geq 2$, be normalised newforms for $\Gamma_0(N)$, for square-free $N > 1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $\lambda\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,(k+k'-2)/2)$. Under certain hypotheses, we prove that $\lambda$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we also give a precise pullback formula for a genus-four Eisenstein series, and a general formula for the Petersson norm of a Yoshida lift.

Given such a congruence, using the 4-dimensional $\lambda$-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order $\lambda$, as required by the Bloch-Kato conjecture on values of $L$-functions.