In the Biur Halachah (Siman 555), a discussion is brought regarding this, where the Eliyah Rabbah in the name of the Knesset HaGedolah is lenient, whereas the Matteh Yehuda holds that the law of Tisha B’Av is like the law of Chol HaMoed, and the Chavot Yair rules in accordance with his view. However, in practice, we are lenient after midday (Chatzot) to write anything. By the way, I saw that the Minchat Elazar used to write letters after midday, and he would write upon them: “The day of the birth of Ben David” (the Moshiach).