  In this paper we will establish different weighted Poincar\'{e} inequalities
with variable exponents on Carnot-Carath\'{e}odory spaces or Carnot groups. We
will use different techniques to obtain these inequalities. For vector fields
satisfying H\"{o}rmander's condition in variable non-isotropic Sobolev spaces,
we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast
}(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses,
and in this case we obtain the first order weighted Poincar\'{e} inequalities
with variable exponents. In the case of Carnot groups we also set up the higher
order weighted Poincar\'{e} inequalities with variable exponents. For these
results the crucial part is proving the boundedness of the fractional integral
operator on Lebesgue spaces with weighted and variable exponents on spaces of
homogeneous type. Moreover, using other techniques, we extend some of these
results when the exponent satisfies a jump condition and the weight is in a
smaller Muckenhoupt class.
  Finally, we will use these weighted Poincar\'{e} inequalities to establish
the existence and uniqueness of a minimizer to the Dirichlet energy integral
for a problem involving a degenerate $p(\cdot )$-Laplacian with zero boundary
values in Carnot groups.
