It is shown that one can define a Hilbert space structure over a kaehlerian manifold with global potential in a natural way.

In this article we prove for $1<p<\infty$ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega$. More precisely, $\Omega$ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega$ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as...

In this article we prove the Monotone Convergence Theorem [16].MML identifier: MESFUNC9, version: 7.8.10 4.100.1011

An exact expression for the down norm is given in terms of the level function on all rearrangement invariant spaces and a useful approximate expression is given for the down norm on all rearrangement invariant spaces whose upper Boyd index is not one.

In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space ${[-1,1]}^D$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of ${[-1,1]}^D$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is the case then...

A complete description of the real interpolation space $L={(L_{p₀}\left(\omega ₀\right),...,L_{pₙ}\left(\omega ₙ\right))}_{\theta ⃗,q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces ${\Omega }_i$ (i ∈ I) such that L is an $l_q$ sum of the restrictions of L to ${\Omega }_i$, and L on each ${\Omega }_i$ is a result of interpolation of just two weighted $L_p$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_p$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....

We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt $A_p$ class.