Maria FANCIULLO

The first research consists in the study of regularity in Campanato spaces for systems of partial differential equations. The Holder continuity regularity for the solutions and their gradients of second order nonlinear non variational systems satisfying the generalized Cordes condition has been proved, obtaining global results if the dimension n of the space is small. There are also partial regularity results for every n.

In 2004 with prof. J.J. Manfredi and A. Domokos she studies linear subelliptic operators satisfying Cordes condition and defined by Hormander vector fields. As an application, C^{1,\alpha} regularity for p-sublaplacian operator in the Grushin plane has been obtained. Then, with A. Domokos the best constant for the Friedrichs-Knapp-Stein inequality in some particular Lie groups of step two has been obtained.

Moreover in recent years nonlinear non variational operators satisfying Cordes condition are studied in Heisenberg group and, more general in Carnot group. For such operators, with G. Di Fazio, W^{2,p}_{loc} regularity has been otained.

Another research topic is the regularity and the a priori estimates for degenerate equations and systems with VMO coefficients. The obtained results consist in Morrey and BMO regularity for the gradient of solutions of systems defined by Hormander vector fields. Useful is the result, in collaboartion with A. Caruso, about the density of C^\infty in BMO with Carnot-Caratheodory metric.

In collaboration with M. Bramanti a priori BMO estimates have been studied for the second derivatives of solutions of linear non variational equations defined on Carnot groups, with VLMO (Vanishing logarithmic mean oscillation) coefficients. Moreover C^{k+2,\alpha} regularity has been obtained for solutions of linear and quasilinear equations defined by Hormander vector fields.

In collaboration with G. Di Fazio and P. Zamboni she studies L^p_w the Holder continuous regularity for the gradient of solutions of linear and variational degenerate equations and systems. The degeneracy is due to a Muckenhoupt weight w.

Another research topic concerns Harnack inequality. Harnack inequality has been obtained for solutions of linear equations in divergence form with natural growth and degenerate with a Muckenhoupt weight. Moreover Harnack inequality has been proved for quasilinear variational equation with a strong A_\infty degeneracy.

Harnack inequalities have been obtained also for strong degenerate linear and quasilinear equations, that are equations defined by Hormander vector fields and also degenerate with a Muckenhoupt weight.
Similar results for quasilinear equations defined by Grushin vector fields and degenerate because of a strong A_\infty weight. Fundamental tool to prove Harnack inequality is a Fefferman-Phong type inequality for functions in Stummel-Kato classes to which the coefficients of the terms of lower order belong.

The last result on Harnack inequality regard linear equations with coefficients of the terms of lower order belonging to a suitable Stummel-Kato type classes, defined by a new formula of representation introduced by Franchi, Perez and Wheeden in 2003.

Recently with P.D. Lamberti the problem of the extension operators on Sobolev- Morrey spaces has been studied.
In collaboration with M. Bramanti the Fefferman-Stein for sharp maximal functions and John-Nirenberg inequality have been obtained for BMO functions in locally homogeneous spaces.