Hypoelliptic differential operators and nilpotent groups

𝐿^{𝑝} estimates for nonvariational hypoelliptic operators with 𝑉𝑀𝑂 coefficients

“…The above result generalizes the L p -estimates proved by Rothschild-Stein [35], since our operators do not belong to the class considered there, whenever the a ij are not C ∞ (in fact, they can be discontinuous). Moreover, we prove the following regularity result: Theorem 0.2.…”

“…A first result of this kind was achieved in '75 by Folland [18], assuming the existence of an underlying structure of homogeneous group for which the operator L is left-invariant and homogeneous of degree two. Later, in '76, Rothschild and Stein [35] proved a similar result for any system of vector fields satisfying Hörmander's condition. This deep generalization was made possible by a technique consisting in lifting the vector fields X i to a higher dimensional space, obtaining new vector fieldsX i which are free up to some step s, and can be locally approximated by other vector fields Y i which fit the assumptions of Folland's quoted paper.…”

“…The L p -estimates proved both in [18] and in [35] are local, so that an L p theory for the Dirichlet problem for general Hörmander's operators of kind (0.1) is still lacking. Also, note that the operators considered in the above papers have always C ∞ coefficients, in contrast with the fact that L p -estimates for linear PDEs do not require intrinsically a high regularity of the coefficients.…”

“…1. We apply Rothschild-Stein's technique of "litfing and approximation" ( [35]), as well as Folland's results for the homogeneous situation ( [18]): the operator L is "lifted" to an operatorL depending also on extra variables; the coefficients a ij are then frozen; the resulting operator can be locally approximated by a left invariant homogeneous operator, for which a homogeneous fundamental solution exists. These facts allow us to write an "explicit" representation formula for X i X j u in terms of singular intergals and commutators of singular integrals.…”

“…To have a first idea of the proof, the reader who is already familiar with the paper of Rothschild-Stein [35] can give a glance to § §2.1-2.2 and then pass to §3.1, to see how the theory of singular integrals on homogeneous spaces is involved.…”