  We consider the classical XY model (O(2) nonlinear sigma-model) on a class of
lattices with the (fractal) dimensions 1<D<2. The Berezinskii's harmonic
approximation suggests that the model undergoes a phase transition in which the
low temperature phase is characterized by stretched exponential decay of
correlations. We prove an exponentially decaying upper bound for the two-point
correlation functions at non-zero temperatures, thus excluding the possibility
of such a phase transition.
