Milliarcsecond-scale spectrum of the persistent radio source associated with FRB 20190417A and constraints for FRB 20181030A
We aim to confirm the compact nature and constrain the radio spectra of candidate persistent radio sources (PRSs) associated with repeating fast radio bursts (FRBs). We performed European VLBI Network (EVN) observations at 5 and 8 GHz targeting two candidates identified in a recent VLA survey. We measured flux densities and upper limits at milliarcsecond resolution and combined them with published VLBI data at lower frequencies to derive spectral constraints. We detect a compact source associated with FRB 20190417A at 5 GHz with a flux density of $150\pm45$ uJy, while no detection is obtained at 8 GHz. The source is unresolved and has a brightness temperature $T_{\rm b}>10^{5}$ K, confirming its non-thermal nature. Combining our measurement with VLBI data at 1.4 GHz, we derive a spectral index $\alpha = -0.19 \pm 0.29$, 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 $L_{\nu}$-|RM| relation. Including this source yields a scatter of $\sigma_\Delta = 0.65$, corresponding to $\hat{\alpha}|\epsilon| = 1.5 \pm 0.7$, consistent with forward shocks in the free-expansion phase or young pulsar wind nebulae. For the candidate PRS associated with FRB 20181030A, we report upper limits of 80 uJy at 5 GHz and 150 uJy at 8 GHz, corresponding to $L_{5\,\mathrm{GHz}} \lesssim 3.8 \times 10^{25}\ {\rm erg\ s^{-1}\ Hz^{-1}}$, and implying a steep spectral index ($\alpha \lesssim -1.2$) if the VLA emission arises from a compact component. Our results highlight the importance of VLBI in isolating compact emission from FRB engines and provide one of the few spectral constraints for PRSs at milliarcsecond resolution. The consistency of FRB 20190417A with the $L_{\nu}$-|RM| relation supports a nebular origin for the persistent emission.