Most people don't build it on him since most people don't play AP Ezreal in the first place so I can't imagine this is a common occurrence. I was playing AP Ezreal mid earlier today for fun and I decided to build Liandry's Torment + Rylai's on him for kiting but I noticed that while my W, E, and R would apply the burn, my Q would not and I can't figure out why.

Now, from reading the League Wiki article regarding this item (http://leagueoflegends.wikia.com/wiki/Liandry's_Torment)
it appears that it applies on all spell damage, and since Ezreal's Q is a spell, I can't imagine why it doesn't apply.

Ezreal's "Q" (Mystic Shot) deals Physical Damage, not Magic. While Liandry's says it slows on Spell Damage, I can only assume they imply Magic Damage.

That can't be it, either, because I also had Lich Bane on him when I tested this which applies on his Q and does magic damage so it should've worked either way.

Considering that Spell Vamp doesn't come into play on item abilities like Lich Bane's magic damage, perhaps Liandry's slow ALSO doesn't proc on Lich Bane?

Lich Bane activates on any ability use, not just abilities that apply magic damage. Mystic shot does physical damage, not magic damage that is why Ryail's doesn't work with his Q.

mystic shot applies on hit effects. spells that apply on hit don't apply "spell damage" procs. the damage type doesn't matter.

id say it for magic damage too

if its physical then na i dont play ez

Mystic Shot deals magic damage. It's like a 0.2 ratio but it's still there. There's no reason why Liandry's shouldn't proc.

Umm jut because something has an ap ratio doesn't mean it does magic damage

Ezreals q applies on hit affects which means lifesteal, and sheen work, and spellvamp, rylais, and liandrys dont