Abstract. Given an n-normed space with n ≥ 2, we offer a simple way to derive an (n−1)-norm from the n-norm and realize that any n-normed space is an (n − 1)-normed space. We also show that, in certain cases, the (n − 1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n − 1)-norm. Using this fact, we prove a fixed point theorem for some n-Banach spaces.

We study the space Z p , 1 ^ p ^ oo, and its natural n-norm, which can be viewed as a generalisation of its usual norm. Using a derived norm equivalent to its usual norm, we show that V is complete with respect to its natural n-norm. In addition, we also prove a fixed point theorem for W as an n-normed space.

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A large fraction of volcanic eruptions do not expel magma at the Earth’s surface. Although less known than magmatic eruptions, gas-driven eruptions expel fragments of preexisting rocks, volcanic gases, and steam, causing substantial casualties. The destructive potential of these eruptions lies in the difficulty in identifying clear warning signals. Some gas-driven eruptions have been preceded by some physicochemical changes, but these were extremely short-term (from minutes to hours), and no long-term trends have been clearly evidenced so far. Here, we show that unheralded gas-driven eruptions can be forecast in the long term using seismic signals recorded at nearby active craters. In particular, we have found that the most recent gas-driven eruptions at Kawah Ijen (Indonesia) and Ruapehu and Tongariro (New Zealand) volcanoes were all preceded by a systematic relative increase in lower-frequency (4.5–8 Hz) seismic amplitude compared to higher frequencies (8–16 Hz) over time scales of months to years. We show that this precursory activity reflects significant increases in seismic attenuation affecting preferentially high-frequency travelling waves; this probably results from the accumulation of volatiles in the shallow crust, which increases pore pressure in small-scale rock heterogeneities and eventually leads to gas-driven eruptions. Our results highlight the feasibility of better constraining the onset and the end of an unrest episode, which is of paramount importance for agencies in charge of volcano monitoring.

Let Tρ be the generalized fractional integral operator associated to a function ρ : (0, ∞) → (0, ∞), as defined in [16]. For a function W on R n , we shall be interested in the boundedness of the multiplication operator f → W • Tρf on generalized Morrey spaces. Under some assumptions on ρ, we obtain an inequality for W • Tρ, which can be viewed as an extension of Olsen's and Kurata-Nishigaki-Sugano's results.