  We study the inverse energy transfer in forced two-dimensional (2D)
Navier--Stokes turbulence in a doubly periodic domain. It is shown that an
inverse energy cascade that carries a nonzero fraction of the injected energy
to the large scales via a power-law energy spectrum $\propto k^{-\alpha}$
requires that $\alpha\ge5/3$. This result is consistent with the classical
theory of 2D turbulence that predicts a $k^{-5/3}$ inverse-cascading range,
thus providing for the first time a rigorous basis for this important feature
of the theory. We derive bounds for the Kolmogorov constant $C$ in the
classical energy spectrum $E(k)=C\epsilon^{2/3}k^{-5/3}$, where $\epsilon$ is
the energy injection rate. Issues related to Kraichnan's conjecture of energy
condensation and to power-law spectra as the quasi-steady dynamics become
steady are discussed.
