  We study charge transport properties in a domain-wall geometry, whose near
horizon IR geometry is a Lifshitz black hole and whose UV geometry is AdS. The
action for the gauge field contains the standard Maxwell term plus the Weyl
tensor coupled to Maxwell field strengths. In four dimensions we calculate the
conductivity via both the membrane paradigm and Kubo's formula. Precise
agreements between both methods are obtained. Moreover, we perform an analysis
of the four-dimensional electro-magnetic duality in our domain-wall background
and find that the relation between the longitudinal and transverse components
of the current-current correlation functions and those of the `dual'
counterparts holds, irrespective of the near horizon IR geometry. Conductivity
at extremality is also investigated. Generalizations to higher dimensions are
performed.
