Milliarcsecond-scale spectrum of the persistent radio source associated with FRB 20190417A and constraints for FRB 20181030A

Aims. We confirm the compact nature and constrain the radio spectra of candidate persistent radio sources (PRSs) associated with repeating fast radio bursts (FRBs), and we increase the number of PRSs with spectral indices measured using very long baseline interferometry (VLBI). Methods. We performed observations with the European VLBI Network (EVN) at 5 and 8 GHz targeting two candidate PRSs that were identified in a recent VLA survey of repeating FRBs. We measured the flux densities and upper limits at milliarcsecond resolution and combined them with published data at lower frequencies to derive spectral constraints. Results. We detect a compact source associated with FRB 20190417A at 5 GHz with a flux density of 150 ± 45 μJy, but no detection is obtained at 8 GHz. The source is unresolved on milliarcsecond scales, implying a projected physical size ≲15 pc and a brightness temperature T b > 3.8 × 10 5 K, confirming its non-thermal nature. The combination of our measurement with the 1.4 GHz VLBI data yields a spectral index α = −0.19 ± 0.29, consistent with a nearly flat spectrum. This makes FRB 20190417A only the second PRS with a spectral index constrained using VLBI data. The inferred luminosity places the source on the proposed L ν –|RM| relation. When this source is included, the scatter is σ Δ = 0.67, corresponding to α̂ | ϵ | ∼ 1.5. This value is roughly consistent with scenarios involving forward shocks in the free-expansion phase or young pulsar wind nebulae. For the candidate PRS associated with FRB 20181030A, we report 5 σ upper limits of 80 μJy at 5 GHz and 150 μJy at 8 GHz, corresponding to L 5 GHz ≲ 3.8 × 10 25 erg s −1 Hz −1 , and implying a steep spectral index ( α ≲ −1.2) if the VLA emission arises from a compact component. Conclusions. Our results highlight the importance of VLBI in isolating compact emission from FRB engines, and they provide one of the few spectral constraints for PRSs at milliarcsecond resolution. The consistency of FRB 20190417A with the L ν –|RM| relation supports a nebular origin for the persistent emission.