In the years leading up to the war, Hitler was surprisingly non-belligerent about the parts of Italy with German speaking populations - by these I am referring to South Tyrol and Trent. This could have been a potential sticking point in Italo-German relations but Hitler was willing to let the issue go and in fact reached an agreement with Mussolini called the 'South Tyrol Option Agreement'.

Basically, the German population of South Tyrol was permitted to either immigrate to Germany, or remain behind while losing minority rights and eventually assimilating.

However, late on in the war when the Italian government declared a separate armistice, the Germans provisionally annexed these regions to the greater German Reich where it was administered as part of the Reichsgau Tyrol-Voralberg. Whether the Germans intended to keep ahold of these territories indefinitely or would eventually cede them back to Italy isn't really clear to me, and at that point the war was basically lost in any case.

Hitler wrote about the question at length for the first time in his "Second Book".

The "Second Book" written in 1928 is a manuscript for a sequel to Mein Kampf but unlike Mein Kampf it is not a mix between biography and political treatise. The Second Book was pretty much a direct response to the 2,6% for the NSDAP in the 1928 elections. Hitler thought the bad outcome was related to the public not understanding/fully appreciating his political positions so he retreaded to Munich to write the Manuscript which mostly deals with the most pertinent questions of the 1928 elections, among them the question of Southern Tyrol.

Around that time Mussolini had instituted his policies of "Italinization" in Southern Tyrol and pretty much the entire German political establishment with foreign minister Gustav Streseman leading the effort, took a stance against the Italian policies. The only exception was the NSDAP. Hitler's position was that Italy was too important an ally to alienate them because of this. His main argument revolves around Italy being an important ally in the struggle against the "real enemy" meaning Judeo-Bolshevism and therefore, the issue can be sidelined as of now. From the introduction to the Zweite Buch:

Against this powerful coalition, which from the most varied points of view is seeking to make the question of the Southern Tyrol the pivot of German foreign policy, the National Socialist Movement fights by unswervingly advocating an alliance with Italy against the ruling Francophile tendency. Thereby the Movement, in contradistinction to the whole of public opinion in Germany, emphatically points out that the Southern Tyrol neither can nor should be an obstacle to this policy. This view is the cause of our present isolation in the sphere of foreign policy and of the attacks against us. Later, to be sure, it will ultimately be the cause of the resurgence of the German nation.

Unlike the change of stances with so many other political issues laid down in his writing in the 1920s, with this one he and the Nazi movement followed that one. After annexing Austria (with the blessing of Italy) in 1938, a German-Italian treaty was signed on the matter in 1939. It specified that German-speakers from Southern Tyrol were given the option of migrating to Germany or staying behind and losing their minority rights. A lot of the people who opted for Germany were prepared as settlers for soon-to-be-conquered territories such as the Crimea but in essence, the position that Souther Tyrol was to remain with Italy was not reversed.