Question: Can I invite guests on Shabbos Tisha B’av?
Answer: The Shulchan Aruch (OC 552:10) writes that if Tisha B’av falls either on Sunday, or on Shabbos and so is observed on Sunday, then one may eat meat and drink wine during seuda shelishis. Even though normally the seuda mafsekes, the meal before the fast is eaten in a state of mourning, one isn’t allowed to mourn on Shabbos.
The Magen Avraham (OC 552:14) and Aruch Hashulchan (OC 552:12) write that while it is prohibited to limit oneself from eating properly because of Tisha B’av, one must eat this meal in a solemn mood and mustn’t invite guests.
The Mishna Berura (552:23) writes, however, that if one usually spends seuda shelishis in company of friends then not doing so because of Tisha B’av is a public demonstration of mourning which is prohibited on Shabbos. Likewise, R’ Moshe Feinstein (Igros Moshe OC 4:112:1) writes that one may sing zemiros during seuda shelishis even if one doesn’t regularly sing.In conclusion, one may invite guests throughout Shabbos Tisha B’av, though one shouldn’t do so for seuda shelishis unless they normally do so.