In what cases is it used?

Aþk olsun literally means let love exist. 

In what cases is it used?

I have been told it can be used when somebody you know refuses to get what you asked for. (So it is an expression to show them your disappointment, or that you are upset with them.)

Like responding with ´show some love´.

thank you

  Im still waiting for a turkish native because  Henry”s answer doesnt satisfy me ( Im also curious  

aþk olsun!,bravo! well done!; also expresses disappointment, that´s too bad of you!

One time i borrow a little money from a friend of mine ( 3 TL ) and when i returned it she refused and said ”aþk olsun” i understood in this situation something to be ”for God sake im not looking at 3 TL ” ( tone of ”aþk olsun was like dissapointment)

There are a few more opinions about the meaning for aþk olsun

here

 Thanks!

"Ask olsun" is the opening form of adress, in a long ritual of initial greetings between two Mevlevis.

 No English expression can perfectly correspond to our "aþkolsun".It means bravo, well done, and often used ironically to show disappointment.

Mainly because Mevlevis are hard to come by in English speaking countries. Common greeting in English speaking countries is a short, crispy and civilized "Hiya! ",