PDS 70b is the most robustly detected young planet imaged in the gap of a transition disk so far, found at a separation of ~195 mas (~22 au) from its host star and at a position angle of about 155 deg. This system is therefore a unique laboratory to characterize the properties of young planetary systems at the stage of their formation. We aim to trace direct and indirect imprints of PDS 70b on the gas and dust emission of the circumstellar disk in order to study the properties of this ~5 Myr young planetary system. We obtained ALMA band 7 observations of PDS 70 in dust continuum and $^{12}CO$ (3-2) and combined them with archival data resulting in an unprecedented angular resolution of about 70 mas (~8 au). We derive an upper limit on circumplanetary material at the location of PDS 70b of ~0.01 $M_{oplus}$ and find a highly structured circumstellar disk in both dust and gas. The outer dust ring peaks at ~0.65 (74 au) and reveals a possible second unresolved peak at ~0.53 (60 au). The integrated intensity of CO also shows evidence of a depletion of emission at ~0.2 (23 au) with a width of ~0.1 (11 au). The gas kinematics show evidence of a deviation from Keplerian rotation inside $lesssim$0.8 (91 au). This implies a pressure gradient that can account for the location of the dust ring well beyond the location of PDS 70b. Further in, we detect an inner disk which appears to be connected to the outer disk by a possible bridge feature in the North-West region in both gas and dust. We compare the observations to hydrodynamical simulations including a planet with different masses covering the estimated mass range previously derived from NIR photometry (~5-9 M$_{Jup}$). We find that even a planet with a mass of 10 M$_{Jup}$ may not be sufficient to explain the extent of the wide gap and an additional low-mass companion may be needed to account for the observed disk morphology.
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