  Fast radio bursts (FRBs) are short duration ($\sim$millisecond) radio
transients with cosmological origin. The simple sharp features of the FRB
signal have been utilized to probe two fundamental laws of physics, namey,
testing Einstein's weak equivalence principle and constraining the rest mass of
the photon. Recently, \cite{hessels18} found that after correcting for
dispersive delay, some of the bursts in FRB 121102 have complex time-frequency
structures that include sub-pulses with a time-frequency downward drifting
property. Using the delay time between sub-pulses in FRB 121102, here we show
that the parameterized post-Newtonian parameter $\gamma$ is the same for
photons with different energies to the level of
$\left|\gamma_1-\gamma_2\right|<2.5\times 10^{-16}$, which is 1000 times better
than previous constraints from FRBs using similar methods. We also obtain a
stringent constraint on the photon mass, $m_{\gamma} < 5.1\times10^{-48}$ g,
which is 10 times smaller than previous best limits on the photon mass derived
through the velocity dispersion method.
