Lorentz spaces and Lie groups (Q1912238)

Let \(G\) be a connected Lie group and \(X_1, \dots, X_k\) a Hörmander system of left invariant vector fields on \(G\), i.e. vector fields which, together with their successive brackets, generate the tangent space at every point. Let \(\Delta = \sum X^2_j\) be the corresponding Laplace operator on \(G\). The fractional powers of \(\Delta\), defined by means of the heat kernel, are bounded operators between certain Lorentz spaces (generalized \(L^p\) spaces) of \(G\). This is the main result of the paper; it is proved by means of two interpolation theorems and a convolution theorem involving these spaces.