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The luminosity distance is defined operationally by the bolometric relation $F=L/(4\pi d_L^2)$ for an isotropic source. It incorporates geometric spreading, reduced photon energy and dilated photon arrival times. If $D_M$ is the transverse comoving distance with the present scale factor included, then $d_L=(1+z)D_M$ in a spacetime with an FLRW metric with photon number conserved.

Let ${\nu }_e=(1+z){\nu }_o$ be the emitted frequency. A narrow emitted band carries energy $P({\nu }_e)d{\nu }_{e}d{t}_{e}d\Omega$. At the observer the energy is smaller by $1+z$, the time interval is $d{t}_o=(1+z)d{t}_e$, the frequency interval is $d{\nu }_o=d{\nu }_e/(1+z)$, and the illuminated area is $D_M^{2}d\Omega$. Hence the spectral spectral flux density per unit observed frequency isFor the specified power law, $P((1+z){\nu }_o)=(1+z{)}^{-\alpha }P({\nu }_o)$, so the power-law spectral K correction isThere is no extra $4\pi$ because $P$ is power per steradian. For total spectral luminosity $L_{\nu }=4\pi P_{\nu }$, that factor would instead appear in the denominator. The frequency-band factor is essential: bolometric flux has two redshift-dimming factors, but spectral flux has one of them compensated by the bandwidth change.