Can you start a sentence with "Hopefully,..."? [duplicate]

Solution 1:

The case of hopefully is a bit different here.

It is a sentence adverb, a disjunct. It is modifying the whole sentence. Rather than in a hopeful manner, it means it is hoped in this case.

_Wikipedia

Oxford's dictionary entry states this meaning and usage as valid, though it warns that some people think it is incorrect:

Although this is the most common use of hopefully, it is a fairly new use and some people think it is not correct.

It is worth noting that this usage has been accepted by oxford. See this article.

...the acceptance of hopefully when used as a sentence adverb. That is to say, the AP Stylebook no longer considers the use of hopefully to mean ‘it is hoped’ (as opposed to the less common meaning of ‘in a hopeful manner’) to be a sin worthy of stoning.

I think all but the most pedantic of speakers would accept this usage as correct.