In this paper we investigate the behavior of the Hardy-Littlewood Maximal Operator. It is well known that for absolutely integrable functions the Hardy-Littlewood Maximal Operator is finite almost everywhere. In this paper it is shown that for each set $E \subset [-\pi,\pi)$ with Lebesgue measure zero there exists a function of vanishing mean oscillation (VMO) such that the Hardy-Littlewood Maximal Operator of this function is infinite for all points of the set $E$. So for VMO-functions the Hardy-Littlewood Maximal Operator has divergence behavior similar to that of absolutely integrable functions. Some applications of these results for the behavior of the Poisson-Integral of VMO-functions are also given.

In this paper we obtain weighted inequalities for Hankel convolution operators. Also, a weighted version of Mihlin-Hörmander theorem for Hankel multipliers is given. Some inequalities for maximal functions play an important role.

Let $G$ be a compact Lie group acting on a closed manifold $M$. Witten's method is used to prove Morse inequalities for the cohomology of the orbit space $M/G$.

In this paper, we look at strongly minimal sets definable in a differentially closed field of characteristic 0. In [3], Hrushovski and Sokolović show that such sets are essentially Zariski geometries. Thus either thre is a definable strongly minimal field nonorthogonal to $D$, or $D$ is locally modular and nontrivial, or $D$ is trivial. We show that the strongly minimal sets defined by a certain family of differential equations are trivial. We also prove a theorem wich provides a test for the orthogonality of types over an ordinary differential field.

In this paper we prove that, with at most $O(N^{13/15+\varepsilon})$ exceptions, all positive even integers $n \leq N$ with $n \equiv 4 (\mathrm{mod}\,24)$ can be written as sums of four squares of primes.

Let $A$ be a graded $C^{\ast}$-algebra. We characterize Kasparov's $K$-theory group $\hat{K}_{0}(A)$ in terms of graded $\ast$-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

A theory of capacities has been extentively studied for Besov spaces [1]. However not much seems to have been done regarding non-linear potentials. We develop some of this here as consequences of the form of certain metric projections.

The non-linear potential theory is used to derive the form of tangent cones for a class of convex sets in Besov spaces. Tangent cones for obstacle problem arise when studying differentiability of metric projection. Characterising the tangent cones is the first step in these considerations. This has been done in some of the Sobolev spaces using Hilbert space methods. In this article we describe tangent cones for obstacle problems precisely, using non-linear potential theoretic ideas, for all Besov spaces $B^{p,q}_{\alpha}$, $1 \lt p \lt \infty$, $1 \lt q \lt \infty$, $\alpha \gt 0$.

This paper is concerned with trees constructed with geodesic arcs in hyperbolic space. We shall be interested in determining how close to geodesics (in hyperbolic space) are the geodesics paths (within the tree), and also in bounding the Hausdorff dimension of the so-called rim of the tree. These estimates and bounds are very useful in studying the Hausdorff dimension of some limit sets appearing in the theory of Fuchsian groups and related to the asymptotic behaviour of geodesics of Riemann surfaces.

In this note, the relationship between a non-constant entire function $f$ and its linear differential polynomial $L(f)$ has been obtained when they share two finite values, ignoring multiplicities, by applying value distribution theory. This confirms Frank's conjecture as a special case. Entire solutions of certain types of non-linear differential equations are also discussed.

We characterize $G_{2}$-manifolds that are locally conformally equivalent to a calibrated one as those $G_{2}$-manifolds $M$ for which the space of differential forms annihilated by the fundamental $3$-form of $M$ becomes a differential subcomplex of de Rham's complex. Special properties of the cohomology of this subcomplex are exhibited when the holonomy group of $M$ can be reduced to a subgroup of $G_{2}$. We also prove a theorem of Nomizu type for this cohomology which permits its computation for compact calibrated $G_{2}$-nilmanifolds.

Convex polytopes arising from complete multipartite graphs and their toric ideals will be studied. First, it is proved that such toric ideals possess squarefree quadratic initial ideals. Second, we show that these convex polytopes are compressed and compute their $f$-vectors, Ehrhart polynomials and normalized volumes explicitly. Finally, all complete multipartite graphs which yield initial ideals coming from finite partially ordered sets will be classified.

We develop a theory of $L^{2}$-von Neumann modules, which encompasses a reformulation of Connes' Spatial Derivative, and the Relative Tensor Product of Sauvageot. We demonstrate the naturality of the relative tensor product construction in the category of $L^{2}$-von Neumann bimodules. Finally, we give evidence for the claim that the relative tensor product is essentially the only tensor product which should be used when considering this tensor category.

It is shown that on a Kähler surface the non-negativity (resp. non-positivity) of the isotropic curvature is implied by the non-negativity (resp. non-positivity) of the holomorphic bisectional curvature. The compact Hermitian surfaces of non-negative isotropic curvature are described. The full list of compact half conformally flat Hermitian surfaces of non-positive isotropic curvature is also given.