We prove that if $\Lambda \in M_p\left({ℝ}^N\right)$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_p\left({ℝ}^N\right)$ is the space of multipliers of $L^p\left({ℝ}^N\right)$.

We consider quasilinear operators T of joint weak type (a, b; p, q) (in the sense of [2]) and study their properties on spaces Lφ,E with the norm||φ(t) f*(t)||Ê, where Ê is arbitrary rearrangement-invariant space with respect to the measure dt/t. A space Lφ,E is said to be "close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to 1/a or to 1/p. For all possible kinds of such "closeness", we give sharp estimates for the function ψ(t) so as to obtain...

Let $E^{\varphi }\left(\mu \right)$ be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator $T:E^{\varphi }\left(\mu \right)\to C\left(\Omega \right)$ is extreme if and only if $T^{*}\omega \in ExtB\left({\left(E^{\varphi }\left(\mu \right)\right)}^{*}\right)$ on a dense subset of $\Omega$, where $\Omega$ is a compact Hausdorff topological space and $\langle T^{*}\omega ,x\rangle =\left(Tx\right)\left(\omega \right)$. This is done via the description of the extreme points of the space of continuous functions $C(\Omega ,L^{\varphi }\left(\mu \right))$, $L^{\varphi }\left(\mu \right)$ being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme...

We investigate which points in the unit sphere of the Besicovitch--Orlicz space of almost periodic functions, equipped with the Luxemburg norm, are extreme points. Sufficient conditions for the strict convexity of this space are also given.