Similar to ordinary differential equations, rough paths and equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when approximate space-valued weakly geometric $\alpha$-rough by signatures of curves bounded variation, given some tuning the H\"older parameter. We show that these are satisfied on Hilbert spaces. As an application, obtain Wong-Zaka...

Geometric property (T) was defined by Willett and Yu, first for sequences of graphs later more general discrete spaces. Increasing with geometric are expanders, they examples coarse spaces which the maximal Baum-Connes assembly map fails to be surjective. Here, we give a broader definition bounded geometry spaces, includes non-discrete We define generalisation this class show that it is invaria...

Much has been written on variable exponent spaces in recent years. Most of the literature deals with normed space structure such spaces. However, because variability exponent, underlying modular these is radically different from that induced by norm. In this article, we focus our attention progress made toward study sequence Lebesgue exponents. particular, present a survey state art regarding g...