Yes, you can use your PS4 DualShock 4 controller with the PS3 wirelessly
A recent PS3 update added several small changes, bug fixes and features that mainly overlooked, although one is of interest to people who want to use the superior DualShock 4 controller that comes with the PS4 with their PlayStation 3s.
The update added wireless support for the new controller, albeit with caveats, so it is now possible to pair the next-generation device to a current generation machine and use it to play games.
It did work previously but only when plugged into a PS3 USB port, which meant that gamers had to sit close to their machines. The wireless support is therefore welcome.
Some issues still remain, however. As the PS3 sees the DualShock 4 as a generic controller, it doesn't work with some of the functionality. Obviously the large touch panel and share buttons are out of the question, but Sony is still yet to allow the PlayStation 3 to work with the rumble or sixaxis technologies.
The former might not bother you, but the latter does mean that certain PS3 games that require sixaxis control are currently not compatible.
In addition, while the DualShock 4 PS button will switch on a PS3 once paired (through Bluetooth), it won't perform any other function. You cannot quite back to the system menu, for example.
If you're willing to forgive the foibles though, Reddit user shafiggy says that all you need to do to pair your PS4 controller to the PS3 is register is as a new device under the "manage Bluetooth devices" menu. While the PS3 searches for devices, press the PS and share buttons on the controller at the same time until it starts to blink. It will show up as "wireless controller" in the list that appears.
One final caveat is that once paired to a PS3, it will only power on that console, not the PS4 as well. You will have to pair it to the next-gen machine again to re-enable that feature.