Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.Comment: 60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references adde

Harnack inequality and regularity for degenerate quasilinear elliptic equations

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,\alpha}$ local estimates for solutions of a degenerate equation in non divergence form

A new CO carbonaceous chondrite from Acfer, Algeria

Many small fragments, totally weighing 118 g were found in the Acfer area by an Italian dealer. Chondrules size and types (predominance of granular olivine type), occurrence of twinned clinoenstatite and absence of plagioclase suggested a classification as CO carbonaceous chondrite

A new CH carbonaceous chondrite from Acfer, Algeria

A single stone weighing 1456 g was found in November 2002 in the Acfer area, Algeria. Oxygen isotope, chondrules-matrix ratio as well as other petrographic features point to a classification as CH carbonaceous chondrite

A new CK carbonaceous chondrite from Hammada Al Hamra, Libya

A single stone weighing 198 g was found in 2001 in the Hammada al Hamra region of Libya. Petrographic features (mean chondrules dimensions, coarse grained matrix and presence of AOIs and CAIs) point to a classification as CK carbonaceous chondrite

Harnack inequality for fractional sub-Laplacians in Carnot groups

In this paper we prove an invariant Harnack inequality on Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups