Fast radio bursts (FRBs) are bright, millisecond-duration radio bursts of cosmic origin. There have been several dozen FRBs found to repeat. Among them, those precisely localized provide the best opportunity to probe their multi-wavelength counterparts, local environment, and host galaxy that would reveal their origins. Here we report our X-ray, ultraviolet (UV) and optical observations with the $Swift$ satellite that were performed simultaneously in the radio band with the Five-hundred-meter Aperture Spherical radio Telescope (FAST) observations of the repeating FRB 20190520B, aiming at detection of possible multi-wavelength bursts in association with radio bursts and multi-wavelength counterpart of the persistent radio source (PRS). While a total of 10 radio bursts were detected by FAST at the same time of $Swift$ observations, we detected neither X-ray, UV or optical bursts in accompany of the radio bursts, nor persistent multi-wavelength counterpart of the PRS. We obtained the energy upper limits (3σ) on any multi-wavelength bursts as 5.03 × 10⁴⁷ erg in the hard X-ray band (15-150 keV), 7.98 × 10⁴⁵ erg in the soft X-ray band (0.3-10 keV), and 4.51 × 10⁴⁴ erg in the U band, respectively. The energy ratio between soft X-ray (0.3-10 keV) and radio emission of the bursts is constrained as <6×10⁷, and the ratio between optical (U band) and radio as <1.19×10⁶. The 3σ luminosity upper limits at the position of PRS are 1.04×10⁴⁷ (15-150 keV), 8.81×10⁴² (0.3-10 keV), 9.26×10⁴² (UVW1), and 2.54×10⁴² erg s⁻¹ (U), respectively. We show that the PRS is much more radio loud than representative pulsar wind nebulae, supernova remnants, extended jet of Galactic X-ray binaries and ultraluminous X-ray sources, suggestive of boosted radio emission of the PRS.
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