Air carrier Captain reported receiving an ECAM advisory for cabin pressure on initial climb; resulting in the flight crew returning to the departure airport where they landed safely.

Narrative

After takeoff from ZZZ climbing through 6000 feet my First Office noticed a flashing ECAM advisory for cabin pressure; with cabin vertical speed; cab alt v/s" flashing. My First Officer was the flying pilot "FP" and I was the pilot monitoring "PM". After looking at the pressurization system page I noticed our cab alt v/s was maxed at 2000'/min and the cabin altitude was quickly catching up to our flight altitude. I asked ZZZ departure for an immediate level off of 8000' to troubleshoot the problem. After level off the cab alt v/s was still climbing so I asked for 6000' to stabilize the v/s. At 6000' flight altitude the cabin altitude finally stopped climbing and the cab alt v/s slowly started to decrease. With no memory item or QRC needed I looked up ECAM advisory conditions in the QRH under the miscellaneous section. I found the Cab Press with the Cab Vertical Speed condition. I followed the recommended action of mode select manual; after 10 seconds mode select auto with no change in cabin altitude. We asked for flight vectors to continue to troubleshoot. I decided to call dispatch to connect with maintenance control. Maintenance control advised us that this exact condition happened the day before and pressurization was not restored in flight. Maintenance control advised us to return to ZZZ for landing. I called our Crew to discuss our issue and informed them that we needed to return to ZZZ for an overweight landing and maintenance. Additionally I informed our customers of the same information. Since we were overweight for landing I ran the overweight landing QRH procedure and advised ZZZ departure that we would require an overweight air return to ZZZ. I asked for the longest available runway being used at ZZZ. The First Officer and I discussed the overweight landing procedure that we would follow for landing and the First Officer continued as the flying pilot. Touching down at ZZZ our v/s was 200'/min; well within the 360'/min max for overweight landing. We transitioned to manual breaks and used a long rollout to protect break temperatures. The break temperatures never got above 125c. The landing and taxi to the gate were uneventful. A logbook write up for the overweight landing and ECAM advisory was completed."