In this paper, we explore the mechanisms that regulate the formation and evolution of stellar black hole binaries (BHBs) around supermassive black holes (SMBHs). We show that dynamical interactions can efficiently drive in-situ BHB formation if the SMBH is surrounded by a massive nuclear cluster (NC), while orbitally segregated star clusters can replenish the BHB reservoir in SMBH-dominated nuclei. We discuss how the combined action of stellar hardening and mass segregation sculpts the BHB orbital properties. We use direct N-body simulations including post-Newtonian corrections up to 2.5 order to study the BHB-SMBH interplay, showing that the Kozai-Lidov mechanism plays a crucial role in shortening binaries lifetime. We find that the merging probability weakly depends on the SMBH mass in the $10^6-10^9{rm ~M}_odot$ mass range, leading to a merger rate $Gamma simeq 3-8$ yr$^{-1}$ Gpc$^{-3}$ at redshift zero. Nearly $40%$ of the mergers have masses in the BH mass gap, $50-140{rm ~M}_odot$, thus indicating that galactic nuclei are ideal places to form BHs in this mass range. We argue that gravitational wave (GW) sources with components mass $m_1>40{rm ~M}_odot$ and $m_2<30{rm ~M}_odot$ would represent a strong indicator of a galactic nuclei origin. The majority of these mergers could be multiband GW sources in the local Universe: nearly $40%$ might be seen by LISA as eccentric sources and, a few years later, as circular sources by LIGO and the Einstein Telescope, making decihertz observatories like DECIGO unique instruments to bridge the observations during the binary inspiral.
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