  In this paper, we solve the $p$-Dirichlet problem for Besov boundary data on
unbounded uniform domains with bounded boundaries when the domain is equipped
with a doubling measure satisfying a Poincar\'{e} inequality. This is
accomplished by studying a class of transformations that have been recently
shown to render the domain bounded while maintaining uniformity. These
transformations conformally deform the metric and measure in a way that depends
on the distance to the boundary of the domain and, for the measure, a parameter
$p$. We show that the transformed measure is doubling and the transformed
domain supports a Poincar\'{e} inequality. This allows us to transfer known
results for bounded uniform domains to unbounded ones, including trace results
and Adams-type inequalities, culminating in a solution to the Dirichlet problem
for boundary data in a Besov class.
