The full form of YW is you’re welcome
“YW” is shorthand for “you’re welcome”.

This texting abbreviation is most often said in response to someone saying thank you or “ty” to you for something you said or did. It is your way of showing gratitude for the person’s gratitude if that makes sense. Similar ways of saying “yw” are “uw”, “urw” and “urwelcome.” There is also “yvw”, which is “you’re very welcome”; In certain scenarios, such as forums or gaming, you might see “ywa”, “you’re welcome anyway” or “ywaa”, “you’re welcome as always”. Despite these variations, “yw” is the most commonly used or seen.

“Yw” can also have other meanings, such as “yeah, whatever”.

“Yw”, lowercase could also stand for “young woman”,

The origin of “yw” is through the popularity of texting and instant messaging. It started appearing in the late 1990s and early 2000s in text messages, chat rooms, and instant messaging and been around ever since.

Example 1:
Texter 1: Ty for lunch. My treat next time.
Texter 2: Yw. That sounds great. Tomorrow, 11?
Texter 1: Sounds great. See you then.