  We present high resolution millimeter continuum ALMA observations of the
disks around the T Tauri stars LkCa 15 and J1610. These disks host
dust-depleted inner regions, possibly carved by massive planets, and are of
prime interest to study the imprints of planet-disk interactions. While at
moderate angular resolution they appear as a broad ring surrounding a cavity,
the continuum emission resolves into multiple rings at a resolution of
~60$\times$40 mas (~7.5 au for LkCa 15, ~6 au for J1610) and ~$7\,\mu$Jy
beam$^{-1}$ rms at 1.3 mm. In addition to a broad extended component, LkCa 15
and J1610 host 3 and 2 narrow rings, respectively, with two bright rings in
LkCa 15 being radially resolved. The rings look marginally optically thick,
with peak optical depths of ~0.5 (neglecting scattering), in agreement with
high angular resolution observations of full disks. We perform hydrodynamical
simulations with an embedded, sub-Jovian-mass planet and show that the observed
multi-ringed substructure can be qualitatively explained as the outcome of the
planet-disk interaction. We note however that the choice of the disk cooling
timescale alone can significantly impact the resulting gas and dust
distributions around the planet, leading to different numbers of rings and gaps
and different spacings between them. We propose that the massive outer disk
regions of transition disks are favorable places for planetesimals and possibly
second generation planet formation of objects with a lower mass than the
planets carving the inner cavity (typically few $M_{\rm Jup}$), and that the
annular substructures observed in LkCa 15 and J1610 may be indicative of
planetary core formation within dust-rich pressure traps. Current observations
are compatible with other mechanisms being at the origin of the observed
substructures, in particular with narrow rings generated at the edge of the CO
and N$_2$ snowlines.
