Sobolev-Morrey spaces related to an ultraparabolic equation

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

“…Concerning the regularity of the weak solutions to (1), we recall the papers [5,24,31], where the coefficients a i j satisfy a suitable vanishing mean oscillation condition. In [28] we proved some pointwise estimate for the weak solutions to (1) by adapting a classical iterative method introduced by Moser [26,27] to the non-Euclidean framework of the homogeneous Lie groups.…”

Linear and Nonlinear Ultraparabolic Equations of Kolmogorov Type Arising in Diffusion Theory and in Finance

“…A priori L p estimates and Hölder regularity properties of solutions to Kolmogorov equations with weakly continuous coefficients have been proved in [8], [34], [43] and [44]. Let L be the following operator…”

“…In [8] and [43], interior regularity properties of strong solutions to the equation Lu = f in Ω ⊆ R N +1 , with L as in (5.1), were studied. In order to recall those results, we need to introduce some notation.…”

“…The technique used in [8] and [43] is inspired to the one first introduced by Chiarenza, Frasca and Longo [11] for classical elliptic operators in non divergence form. The starting point is a representation formula which, in the present context, takes the following form.…”

“…The key tools used in [8] and [43] are some deep continuity results for singular integrals, and their commutators with VMO G functions, modelled on the Lie group and metric structures described in Section 2.…”