Abstract
In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on {{mathbb{R}}}^{n} or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.
Highlights
In the case of G = Rn and R = −, Eq (1.1) is the usual wave equation with the timedependent propagation speed and its well-posedness results for Hölder regular functions a have been obtained by Colombini, De Giorgi and Spagnolo in their seminal paper [12]
Already in the cases of G being the Euclidean space Rn, the Heisenberg group Hn, or any stratified Lie group, the case (iii) above allows one to consider R to be an operator of any order, as long as it is a positive left- invariant homogeneous hypoelliptic differential operator
For G being a compact Lie group and −R any Hörmander’s sum of squares on G the problem (1.1) was studied in [27], and so the results of the present paper provide a nilpotent counterpart of the results there
Summary
In this paper we are interested in the well-posedness of the following Cauchy problem:. (iii) G is a graded Lie group in the sense of Folland and Stein [24] and R is any positive Rockland operator on G, i.e. any positive left-invariant homogeneous hypoelliptic differential operator on G. Already in the cases of G being the Euclidean space Rn, the Heisenberg group Hn, or any stratified Lie group, the case (iii) above allows one to consider R to be an operator of any order, as long as it is a positive left- (or right-) invariant homogeneous hypoelliptic differential operator. Since Eq (1.1) in local coordinates is the space-dependent variable multiplicities problem, very little is known about its well-posedness In this direction, only very special results for some second order operators are available, see e.g. Nishitani [38] or Melrose [34]. An example of a positive Rockland operator is the positive sub-Laplacian on a stratified Lie group: if G is a stratified Lie group and
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