Space regularity for evolution operators modeled on H\xf6rmander vector fields with time dependent measurable coefficients

Marco Bramanti

doi:10.1007/s00028-020-00629-3

Abstract

We consider a heat-type operator mathcal {L} structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, with a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if mathcal {L}u is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives of every order satisfy a 1/2-Hölder continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.

Highlights

Let G = RN, ◦, Dλ be a Carnot group and let X1, . . . , Xq be the generators of its Lie algebra, so that the canonical sublaplacian q
Is a weak solution to Lu = F, u (0, ·) = 0 and F is smooth, with respect to the space variables, in some domain (0, T ) ×, the same is true for u, with quantitative regularity estimates on u in terms of Lu
Required to span g, we will say that g is a stratified Lie algebra of step s, G is a Carnot group and its generators X1, X2, . . . , Xq satisfy Hörmander’s condition at step s in G

Summary

Preliminaries about Carnot groups

Let us recall some standard definitions and results that will be useful in the following. Given any differential operator P with smooth coefficients on G, we say that P is left invariant if for every x, y ∈ G and every smooth function f. Xq satisfy Hörmander’s condition at step s in G. required to span g, we will say that g is a stratified Lie algebra of step s, G is a Carnot group (or a stratified homogeneous group) and its generators X1, X2, . Let us point out a relation between left and right invariant operators which will be very useful in the following. 2.2]) Let L, R be any two differential operators on G with smooth coefficients, left and right invariant, respectively. For every couple of measurable functions f, ψ defined on G such that the following convolutions are well defined, we have i) if P is a left invariant differential operator .

Subelliptic estimates for heat-type operators with t-measurable coefficients

Regularization of distributional solutions