In this article we shall investigate the Hardy inequalities and improve them by finding out missing terms. Although the missing terms for the higher order Hardy inequality can not be determined in a unique way, we shall give a canonical form of the remainder. As a direct application we shall study blow-up solutions of a semilinear elliptic boundary value problem and give some lower estimate of the first eigenvalue of the linearized operator. We also improve the weighted Hardy inequalities, which will be fundamental to study singular solutions of quasilinear elliptic equations.

In this paper we study space-time regularity of solutions of the following linear stochastic evolution equation in $\mathcal{S'}(\mathbb{R}^{d})$, the space of tempered distributions on $\mathbb{R}^{d}$: \[ \begin{array}{cc} (*) & \begin{array}{ll}du(t)=Au(t)dt+dW(t),& t \geqslant 0,\\ u(0)=0.&\end{array} \end{array} \] Here A is a pseudodifferential operator on $\mathcal{S'} (\mathbb{R}^{d})$ whose symbol $q : \mathbb{R}^{d} \to \mathbb{C}$ is symmetric and bounded above, and $\{ W(t)\}_{t\geqslant 0}$ is a spatially homogeneous Wiener process with spectral measure $\mu$. We prove that for any $p \in [1,\infty )$ and any nonnegative weight function $\rho \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$, the following assertions are equivalent:

(1) The problem (*) admits a unique $L^{p}(\rho )$-valued solution;

(2) The weight $\rho$ is integrable and \[ \int_{\mathbb{R}^{d}}\frac{1}{C-\mathrm{Re}q(\xi )}d\mu (\xi )<\infty \] for sufficiently large $C$.

Under stronger integrability assumptions we prove that the $L^{p}(\rho )$-valued solution has a continuous, resp. Hölder continuous version.

This paper deals with some additive functionals of the fractional Brownian motion that arise as limits in law of some occupation times of this process. In concrete, this functionals are obtained via the Cauchy principal value and the Hadamard finite part. We derive some regularity properties of theses functionals in Sobolev-Watanabe sense.

The aim of this paper is the description of the integral cohomology rings of the real Stiefel manifolds $V_{n,k}$ in terms of generators and relations. The computation is carried out by using the Gysin exact sequence for the sphere bundle $S^{n-k-1} \to V_{n,k+1} \to V_{n,k}$.

We show that a combination between a remark of I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [2] and the idea, systematically investigated by D. Eisenbud, G. Fløystad and F.-O. Schreyer [3], of taking Tate resolutions over exterior algebras leads to quick proofs of the main results of [2] and [3] (Theorems 7 and 10 below). This combination is expressed by Lemma 6 from the text, a result which can be seen as a formula for computing hyperext groups on projective spaces in terms of linear algebra. We prove it directly, using only the cohomology of invertible sheaves on a projective space and a few basic facts about derived categories.

The category of totally ordered graded module braids and that of the exact interlocking sequences are shown to be equivalent. As an application of this equivalence, we show the existence of a connection matrix for a totally ordered graded module braid without assuming the existence of chain complex braid that induces the given graded module braid.