Nuclear starbursts and AGN activity are the main heating processes in luminous infrared galaxies (LIRGs) and their relationship is fundamental to understand galaxy evolution. In this paper, we study the star-formation and AGN activity of a sample of 11 local LIRGs imaged with subarcsecond angular resolution at radio (8.4GHz) and near-infrared ($2.2mu$m) wavelengths. This allows us to characterize the central kpc of these galaxies with a spatial resolution of $simeq100$pc. In general, we find a good spatial correlation between the radio and the near-IR emission, although radio emission tends to be more concentrated in the nuclear regions. Additionally, we use an MCMC code to model their multi-wavelength spectral energy distribution (SED) using template libraries of starburst, AGN and spheroidal/cirrus models, determining the luminosity contribution of each component, and finding that all sources in our sample are starburst-dominated, except for NGC6926 with an AGN contribution of $simeq64$%. Our sources show high star formation rates ($40$ to $167M_odotmathrm{yr}^{-1}$), supernova rates (0.4 to $2.0mathrm{SN}mathrm{yr}^{-1}$), and similar starburst ages (13 to $29mathrm{Myr}$), except for the young starburst (9Myr) in NGC6926. A comparison of our derived star-forming parameters with estimates obtained from different IR and radio tracers shows an overall consistency among the different star formation tracers. AGN tracers based on mid-IR, high-ionization line ratios also show an overall agreement with our SED model fit estimates for the AGN. Finally, we use our wide-band VLA observations to determine pixel-by-pixel radio spectral indices for all galaxies in our sample, finding a typical median value ($alphasimeq-0.8$) for synchrotron-powered LIRGs.
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