On the lifting and approximation theorem for nonsmooth vector fields

We prove a version of Rothschild-Stein's theorem of lifting and approximation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one.Comment: 46 pages, LaTeX. Minor changes in Section

Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system

La tecnica della funzione massimale sharp nelle stime a priori W2,p per operatori non variazionali

We consider a nonvariational degenerate elliptic operator, structured on a system of left invariant, 1-homogeneous, Hörmander vector fields on a Carnot group, where the coefficient matrix is symmetric, uniformly positive on a bounded domain and the coefficients are locally VMO. We discuss a new proof (given in [BT] and also based on results in [BF]) of the interior estimates in Sobolev spaces, first proved in [BB-To]. The present proof extends to this context Krylov' technique, introduced in [K1], consisting in estimating the sharp maximal function of second order derivatives. Si considera un operatore nonvariazionale ellittico degenere, strutturato su un sistema di campi vettoriali di Hörmander, invarianti a sinistra e 1-omogenei su un gruppo di Carnot, dove la matrice dei coefficienti è simmetrica, uniformemente positiva su un dominio limitato e i coefficienti sono localmente VMO. Si discute una nuova dimostrazione (contenuta in [BT] e basata anche su risultati in [BF]) delle stime all'interno in spazi di Sobolev, provate in [BB-To]. La presente dimostrazione estende a questo contesto la tecnica di Krylov, introdotta in [K1], che consiste nello stimare la funzione massimale sharp delle derivate del second'ordine

On the proof of Hörmander's hypoellipticity theorem

This is a survey paper about the proof of the hypoellipticity theorem by Hörmander (Acta Math. 1967). We will compare three different proofs of this result: the original one by Hörmander, the proof given by Kohn (Proc. Sympos. Pure Math., 1973) and independently by Oleĭnik and Radkevič in their 1973 monograph, and the more recent proof of a special case of this result, concerning sublaplacians on Carnot groups, given by Bramanti and Brandolini (Nonlinear Analysis, 2015)

Local real analysis in locally homogeneous spaces

We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations