In this paper, we obtain some weighted integral inequalities for differential forms, which can be considered as generalizations of the Poincaré inequality, the Caccioppoli-type estimate, and the weak reverse Hölder inequality, respectively. These results can be used to study the integrability of differential forms and to estimate the integrals of differential forms. We also give some applications of the above results.

Following the ideas of M.Geck and R.Rouquier in [19], we show that there exists a “canonical basic set” of Specht modules in bijection with the set of simple modules of Ariki-Koike algebras at roots of unity. Moreover, we determine the parametrization of this set and we give the consequences of these results on the representation theory of Ariki-Koike algebras.

We study the characterizing problem of the compactness of radial operators on the harmonic Bergman space. We show that under an oscillation condition, the compactness is equivalent to the boundary vanishing conditions of the certain Berezin transforms. As an application, we characterize compact Toeplitz operators with radial symbol on the harmonic Bergman space.

We consider a discontinuous Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. By modifying some techniques of [2], [11] and [14] we extend and generalize some approach and results of classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation such a way that the considered problem can be interpreted as an eigenvalue problem of suitable self-adjoint operator, then we construct the Green function and resolvent operator and derive an asymptotic formulas for eigenvalues and normalized eigenfunctions.

We show that if $f$ is a positive definite distribution which is analytic on some neighborhood of the origin, then $f$ is analytic everywhere in $\mathbb{R}^{n}$.

Let $X$ be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on $X$ with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of zeta functions associated with groups of cycles. We also consider an analogue in Arakelov geometry.

In this paper, we generalize the main result of [Mzk2] (to the effect that very general noetherian log schemes may be reconstructed from naturally associated categories) to the case of log schemes locally of finite type over Zariski localizations of the ring of rational integers which are, moreover, equipped with certain “archimedean structures”.