Can you activate the loyalty ability of a planeswalker you just gained control of this turn?

Loyalty abilities have a special timing restriction found in CR 606.3: a player may activate a loyalty ability only if no loyalty ability of that permanent has been activated this turn, and the planeswalker has been under that player's control continuously since the beginning of that player's most recent turn.

This rule applies regardless of how you gained control — whether the planeswalker entered the battlefield under your control this turn (e.g., you cast it or it was put into play), or you took control of an opponent's planeswalker mid-turn via a spell or ability like Sway of Stars or a Control Magic variant. In both cases, the continuous-control-since-start-of-turn requirement is not met.

This is sometimes called the planeswalker equivalent of 'summoning sickness.' Just as a creature without haste can't attack or use tap abilities the turn it comes under your control (CR 302.6), a planeswalker's loyalty abilities are similarly locked until you've controlled it since your turn began.

Example: On your opponent's turn, they cast Generous Gift targeting your creature. You respond by playing nothing, but later that same turn you cast a spell that gives you control of their Liliana of the Veil. On your next turn, you may now activate Liliana's loyalty ability, because you have controlled her continuously since the start of your current turn.

If you had stolen the planeswalker during your own turn, you would have to wait until your next turn to use its loyalty abilities, since the continuous control requirement resets at the beginning of each of your turns (CR 606.3).