  We propose and theoretically substantiate a new method to study the non-local
conductivity of two-dimensional electron systems (2DES) using the tools of
near-field microscopy. We show that the height dependence of induced dipole
moment of illuminated near-field probe is substantially different for various
transport regimes of charge carriers in 2DES. For hydrodynamic transport
regime, the induced dipole moment scales as $z_0^{-2}$, where $z_0$ is the
elevation of probe above the 2DES. Both for Drude and classical ballistic
regimes of conduction, the dipole moment scales as $z_0^{-3}$. In the former
case, the dipole moment is carrier density-independent, while in the latter it
largely depends on carrier density. More generally, we find that the induced
dipole moment of the probe is proportional to the Laplace transform of
wave-vector dependent conductivity and inverse dielectric function of 2DES over
the wave vectors $q$. Our results should provide a simple tool for studies of
non-local conductivity in solids that was challenging to address with other
techniques.
