arXiv subjects
Publications and source records attributed to Mauro Sanchiz.
We investigate the weak Banach--Saks property in the setting of Hölder spaces over metric spaces. We show that, for every infinite metric space $(M,d)$ and every $α\in (0,1]$, the Hölder space $C^α(M)$ fails to have the weak Banach--Saks property.
We prove two weak compactness criteria in Musielak-Orlicz spaces for $N$-functions satisfying the $Δ_2$-condition. They extend criteria from Andô for Orlicz spaces to this setting of non-symmetrical Banach function spaces. As consequences, we prove criteria for a sequence in a Musielak-Orlicz space to be weakly convergent, and show that Musielak-Orlicz spaces with the subsequence splitting property are weakly Banach-Saks. The study includes the case of Musielak-Orlicz sequence spaces.
We characterize the strictly singular inclusions $\ell_{p_n}\hookrightarrow\ell_{q_n}$ between Nakano sequence spaces providing a useful criterion, namely $\varliminf_{n\rightarrow\infty}\vert p_n-q_n\vert>0$ (also recently obtained by Lang and Nekvinda in [12] with a different proof). It is also noted that no inclusion operator between Nakano sequence spaces is compact, neither $L$-weakly compact nor $M$-weakly compact. An easy criterion is given for the weak compactness of the inclusion.
Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot)}(μ)$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$-weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot)}(μ) \hookrightarrow L^{q(\cdot)}(μ)$ is not disjointly strictly singular. No restrictions on the exponent are imposed.
The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. In the second part strictly singular and disjointly strictly singular operators between spaces $\lpv$ are studied. New results on the disjoint strict singularity of the inclusions $ L^{p(\cdot)}(Ω) \hookrightarrow L^{q(\cdot)}(Ω)$ are given.