The study of star formation activity (SF), active-galactic-nuclei (AGN) and their relation with feedback and accretion mechanisms plays a fundamental role to understand how galaxies evolves. Local Luminous and Ultra-Luminous Infrared Galaxies (U/LIRGs) serves as a key point to facilitate such studies. Radio observations can be made with high-angular resolution, are unaffected by dust, allowing to resolve their nuclear regions with great detail. This is essential for the inference of physical properties of distant U/LIRGs in later studies, where angular resolution is limited. As part of the e-MERLIN Legacy Project & Collaboration (LIRGI), this work has as objectives to stablish resolved calibrated star formation (SF) rates for local (z < 0.1) U/LIRGs. We focus on the relevance of disentangling the SF emission from AGN emission in a multi-frequency and multiscale approach, sampling regions from a few parsecs to several kilo-parsecs.

With recent published results, our strategy is to use multiscale imaging and image processing techniques to decompose the radio emission into multi-components, at various spatial scales. With e-MERLIN observations at 6 GHz and VLA-A configuration observations above 22 GHz, we reach angular resolutions from 0.05'' to 0.1'', respectively. We can resolve the nuclear regions at ~ 20 - 100 pc scales in order to probe compact radio components and disentangle them from the nuclear diffuse emission at scales ~ 100 - 200 pc. With the VLA-A configuration below 10 GHz and VLA-B/C configuration above 15 GHz, we characterise the large scale diffuse emission (~ 1 kpc), mostly associated with SF activity, at > 0.3" angular scales. With this, we can build up a multiscale spectral energy distribution to separate thermal and non-thermal emission mechanisms.

The next step is to use VLBI observations at angular resolutions below 10 mas to completely disentangle core-compact components (unresolved by e-MERLIN) from any nuclear diffuse emission. Combining multiple visibilities allows us to characterise the structure of these sources at all scales. Hence, we can apply our decomposition method simultaneously to the three instruments as a proxy to remove the contribution of core-compact components (AGNs) and map the diffuse emission. We will be able to provide constrained and calibrated estimates to our multiscale tracer for star formation. In an attempt to tackle the problem of underestimation of the radio emission coming from SF activity, we discuss the relevance of a simultaneous multiscale characterisation of radio sources to constrain their total star formation rates and associated physical properties.