Type 1 PDCCH Common Search Space is a subset of NR PDCCH Search Space that is dedicated to transmit the PDCCH with RA RNTI, TC-RNTI,C-RNTI on a primary cell. It means UE would monitor this search space during the whole period of RACH process. It means that UE searches this space in order to find DCIs for Msg2(PDSCH), Msg4(PDSCH) in the RACH process

This search space can be explicitely configured by ra-ControlResourceSet in PDCCH-ConfigCommon. The RRC Message or IE that can carry PDCCH-ConfigCommon are SIB1, BWP-DownlinkCommon, BWP-DownlinkDedicated. If the configuration for this search space is not explicitly configured in RRC message(ra-ControlResourceSet and ra-SearchSpace), UE will search the Type 1 PDCCH in Type 0 PDCCH Common Search Space.

Basically Type 1 PDCCH Common Search Space is a type of the spaces defined by RRC message. The RRC message IE that carry the search space information is as follows. It may look simple, but the first two item (ControlResourceSet and SearchSpace) are huge topics on their own. Those are explained in separate pages. Click on ControlResourceSet and SearchSpace below.

Reference

[1] Type0-PDCCH common search space