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We consider relativistic electrons in magnetospheric flows close to asupermassive black hole and study the mechanism of quasi-linear diffusion (QLD)to investigate the correlation between $\gamma$-ray and radio emission inactive galactic nuclei. Moving in the nonuniform magnetic field the particlesexperience a force that is responsible for the conservation of the adiabaticinvariant. This force, together with the radiation reaction force, tends todecrease the pitch angles. Contrary to this, the QLD attempts to increase thepitch angle, and this maintains the synchrotron emission regime. To examine thebalance between the QLD and the aforementioned dissipative factors weinvestigate the quasi-stationary state by applying the kinetic equation.Considering the magnetospheric plasma close to the supermassive black hole, weexamined the efficiency of the QLD for different parameters. By examining thecyclotron instability, we show that despite the efficient dissipative factors,the cyclotron modes excite transverse and longitudinal-transversal waves whichleads to the QLD. We find that the QLD provides a connection of emission in the$\gamma$-ray and radio domains. We show that under favourable conditions theradio emission from 22 MHz to 9 GHz on the mpc scale is associated with$\gamma$-ray emission from 900 GeV down to 9 GeV on the same scale.

The generation of connected high and very high energy $\gamma$-rays and radio emission in active galactic nuclei