Weighted Sobolev spaces of radially symmetric functions

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.Comment: 38 page

Significant discharge of CO2 from hydrothermalism associated with the submarine volcano of El Hierro Island

The residual hydrothermalism associated with submarine volcanoes, following an eruption event, plays an important role in the supply of CO2 to the ocean. The emitted CO2 increases the acidity of seawater. The submarine volcano of El Hierro, in its degasification stage, provided an excellent opportunity to study the effect of volcanic CO2 on the seawater carbonate system, the global carbon flux, and local ocean acidification. A detailed survey of the volcanic edifice was carried out using seven CTD-pH-ORP tow-yo studies, localizing the redox and acidic changes, which were used to obtain surface maps of anomalies. In order to investigate the temporal variability of the system, two CTD-pH-ORP yo-yo studies were conducted that included discrete sampling for carbonate system parameters. Meridional tow-yos were used to calculate the amount of volcanic CO2 added to the water column for each surveyed section. The inputs of CO2 along multiple sections combined with measurements of oceanic currents produced an estimated volcanic CO2 flux = 6.0 105 ± 1.1 105 kg d−1 which is ~0.1% of global volcanic CO2 flux. Finally, the CO2 emitted by El Hierro increases the acidity above the volcano by ~20%.En prens

Geology and petrology of the Sasso di Castro ophiolites and associated plagiogranites.

The Sasso di Castro ophiolite is a huge olistolith within the chaotic complex (late Cretaceous) of the external Liguride unit at Futa Pass, Tuscan Apennines. The ophiolite consists of massive basalts cut by few dykes and overlain by pillow basalts which grade up into pillow basalt breccia covered by red cherts. The whole sequence was affected by ocean-floor metamorphism under static conditions from lower amphibolite to zeolite facies. As opposed to other occurrences in the Northern Apennines, the Sasso di Castro plagiogranites (trondhjemites and very scarce diorites) occur as two small stock-like bodies intruded in the lower part (massive basalt) of the effusive complex. The largest intrusion produced a small thermometamorphic aureole whose hornfelses reached lower amphibolite facies. The thermal effect was associated with fragmentation of the wall-rock and locally also with plastic deformation. Chemical exchange between the host massive basalts and the intruded plagiogranite body has been recognized. The plagiogranites, particularly the trondhjemites, have extremely low K2O (50) ratio. However the presence of a small amount of modal biotite indicates crystallization from a relatively K2O-rich 'granitic' liquid. Low K2O and CaO contents and absence of K-feldspar in the Sasso di Castro plagiogranite probably result from K/Na and Ca/Na exchange in feldspars related to hydrothermal ocean-ridge metamorphism. All the analysed basalts were sampled closely to one another in a quarry. The ratios between refractory elements such Ti, Zr, Nb, Y allow the subdivision of the Sasso di Castro basalts into two groups with high and low Zr/Y; they can be interpreted as two series fractionated from two different parental magmas, which are more enriched in incompatible element than normal MORB. Melting models of the Sasso di Castro basalts as well as of basaltic rocks from Northern Apennines, Western Alps and Corsica ophiolites point to at least two significantly different types of mantle sources: one LREE enriched (Maggiorasca, external Ligurides, Montgenevre, Balagne), the other LREE depleted (Inzecca, Eastern Liguria). It is shown that the Western Mediterranean ophiolite basalts (which are more enriched in incompatible elements than normal MORB) are clearly distinguishable from both transitional and enriched MORB

Solutions of logarithmic type for elliptic and hypoelliptic equations

LetP be an elliptic or hypoelliptic linear partial operator. We study the followingL1-BMO regularity problem: doesPu=f 08L1 implyu 08BMO? The related problem of the existence forP of fundamental solutions of logarithmic type is also discussed

Late Quaternary deformation history of the volcanic edifice of Panarea (Aeolian Arc)

A series of raised paleoshorelines is documented along the emergent coastal slopes of Panarea and surrounding islets at elevations of 115 (paleoshoreline Ia) and 100 metres asl (Ib), 62.5 m (II), 35 m (III), 12 m (IV), 10-12 (Va) and 5 m (Vb). According to stratigraphic constraints and cross-cutting relations, these paleoshorelines are correlated with discrete high sea-level stillstands during marine oxygen-isotope stages (MIS) 5e, 5c, 5a and 3. Coastal elevation changes suggest the occurrence of a long-term, sustained uplift trend of the volcanic edifice since the last interglacial (last 124 ka). The uplift rates are not constant but display a progressive deceleration from maximum values of 1.5-1.58 m/ka, in the period between 124 and 100 ka, down to the lowest values of 0.66-0.69 m/ka, which tend to be constant starting from 81 ka BP. The long-term deformation pattern of Panarea suggests that a transitory, volcano-related component of uplift interplayed with the regional tectonic component affecting the sub-volcanic basement which has undergone a persistent and widespread uplift since the mid-Pleistocene. The volcano-related component of uplift, prevailing between 124 and 100-81 ka, is interpreted as the result of visco-elastic deformation mechanisms which characterize the progressive re-equilibration of the shallow magmatic system following the incoming quiescence of the volcanic edifice. The long-term uplift values at Panarea are higher than in the main portion of the western-central Aeolian Arc, where a mean uplift rate of 0.34 m/ka was estimated since the last interglacial (last 124 ka). Such a pattern of deformation on a regional scale may be a response to active deformation processes connected with the southeastward rollback of the subducting Ionian slab which is still active only in correspondence of the eastern sector of the Aeolian Arc (including Panarea). In the short-term, a localized submergence trend is documented at the nearby islet of Basiluzzo for the last 2000 years, likely connected to neo-tectonic movements along main NE-SW trending faults