Purely non-atomic weak L^p spaces

Let \msp be a purely non-atomic measure space, and let $1 < p < \infty$. If \weakLp\msp is isomorphic, as a Banach space, to \weakLp\mspp for some purely atomic measure space \mspp, then there is a measurable partition $\Omega = \Omega_1\cup\Omega_2$ such that $(\Omega_1,\Sigma\cap\Omega_1,\mu_{|\Sigma\cap\Omega_1})$ is countably generated and $\sigma$-finite, and that $\mu(\sigma) = 0$ or $\infty$ for every measurable $\sigma \subseteq \Omega_2$. In particular, \weakLp\msp is isomorphic to $\ell^{p,\infty}$

On the complemented subspaces of the Schreier spaces

It is shown that the Schreier space X admits a set of continuum cardinality whose elements are mutually incomparable complemented subspaces spanned by subsequences of the natural Schauder basis of X.Comment: 26 pages, AMS-LaTe

In situ photogalvanic acceleration of optofluidic kinetics: a new paradigm for advanced photocatalytic technologies

A multiscale-designed optofluidic reactor is demonstrated in this work, featuring an overall reaction rate constant of 1.32 s¯¹ for photocatalytic decolourization of methylene blue, which is an order of magnitude higher as compared to literature records. A novel performance-enhancement mechanism of microscale in situ photogalvanic acceleration was found to be the main reason for the superior optofluidic performance in the photocatalytic degradation of dyes as a model reaction