Question

Which region did the Ottomans control entirely or partially from the 1400s through the end of World War I?

Answer 1

Given the time span you've noted, only territory on the Anatolian peninsula (modern-day Turkey) and a portion of Greek territory.

If you had your starting date in the 1500s rather than the 1400s, then you could name territories in the Middle East as well.  Suleiman the Magnificent, sultan from 1520 to 1566, was responsible for much of the territorial expansion of the Ottoman Empire in the 16th century.   

By the beginning of the 20th century, the Ottoman Empire was known as "the sick man of Europe," and other European powers were waiting to carve up its territory after it fell.  (The Ottoman Empire came to an end after they were on the losing side in World War I.)