The emission of microwave radiation by extended air showers produced by high energy cosmic rays has been investigated for more than half a century. We discuss the expected emitted power as a function of the cosmic ray energy and of the microwave frequency, for both coherent and incoherent emission mechanisms. We show that the available experimental data are not sufficient to clearly identify the emission mechanisms and quantify the emission yield. We infer that the bremsstralhung radiation emission could be exploited for the detection of astronomical $\gamma$-rays with energy above 10 GeV in the 1-10 GHz frequency range, and propose an experimental scheme to verify such idea.
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