  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 ($3\sigma$) on any multi-wavelength bursts as
$5.03 \times 10^{47}$ erg in the hard X-ray band (15-150 keV), $7.98 \times
10^{45}$ erg in the soft X-ray band (0.3-10 keV), and $4.51 \times 10^{44}$ 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 $<6\times10^{7}$, and the
ratio between optical (U band) and radio as $<1.19\times10^{6}$. The 3$\sigma$
luminosity upper limits at the position of PRS are 1.04$\times10^{47}$ (15-150
keV), 8.81$\times10^{42}$ (0.3-10 keV), 9.26$\times10^{42}$ (UVW1), and
2.54$\times10^{42}$ 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.
