Estimates of BMO-type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs

Let us consider the class of nonvariational uniformly hypoelliptic operators: \[ Lu\equiv\sum_{i,j=1}^{q}a_{ij}\left( x\right) X_{i}X_{j}u \] where: $X_{1},X_{2},\ldots,X_{q}$ is a system of H\"{o}rmander vector fields in $\mathbb{R}^{n}$ ($n>q$), $\left\{ a_{ij}\right\} $ is a $q\times q$ uniformly elliptic matrix, and the functions $a_{ij}\left( x\right) $ are continuous, with a suitable control on the modulus of continuity. We prove that: \[ \left\Vert X_{i}X_{j}u\right\Vert _{BMO\left( \Omega^{\prime}\right) }\leq c\left\{ \left\Vert Lu\right\Vert _{BMO\left( \Omega\right) }+\left\Vert u\right\Vert _{BMO\left( \Omega\right) }\right\} \] for domains $\Omega^{\prime}\Subset\Omega$ that are regular in a suitable sense. Moreover, the space $BMO$ in the above estimate can be replaced with a scale of spaces of the kind studied by Spanne. To get this estimate, several results are proved, regarding singular and fractional integrals on general spaces of homogeneous type, in relation with function spaces of $BMO$ type.