Affiliation: | Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy ; Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende, 87036 Rende (CS), Italy |
Abstract: | Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain ( ). We consider the differential operator ![begin{equation*}mathcal{L}=sum _{i=1}^qa_{ij}(x)X_iX_j, end{equation*}](http://www.ams.org/tran/2000-352-02/S0002-9947-99-02318-1/gif-abstract/img9.gif)
where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: ![begin{equation*}mu|xi|^2leqsum _{i,j=1}^qa_{ij}(x)xi _ixi _jleqmu^{-1}|xi|^2 end{equation*}](http://www.ams.org/tran/2000-352-02/S0002-9947-99-02318-1/gif-abstract/img11.gif)
for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate: ![begin{equation*}left|X_iX_jfright|_{mathcal{L}^p(Omega')}leq cleft{left|mathcal{L}fright|_{mathcal{L}^p(Omega)}+left|fright |_{mathcal{L}^p(Omega)}right} end{equation*}](http://www.ams.org/tran/2000-352-02/S0002-9947-99-02318-1/gif-abstract/img18.gif)
for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular. |