  We develop a geometric invariant Littlewood-Paley theory for arbitrary
tensors on a compact 2 dimensional manifold. We show that all the important
features of the classical LP theory survive with estimates which depend only on
very limited regularity assumptions on the metric. We give invariant
descriptions of Sobolev and Besov spaces and prove some sharp product
inequalities. This theory has being developed in connection to the work of the
authors on the geometry of null hypersurfaces with a finite curvature flux
condition, see \cite{KR1}, \cite{KR3}.
  We are confident however that it can be applied, and extended, to many
different situations.
