Burchfield in Fowler's Modern English Usage  writes about the hopefully dispute under the entry of:  sentence adverb. The adverb [hopefully] was regarded as acceptable when it meant in a hopeful manner, as in to set to work hopeufully, i.e. its traditionlal use since the 17c.; but not acceptable when used to mean  it is hoped [that] , let us hope. as in We asked her when she expected to move into her new apartment, and she answered, 'Hopfully on Tuesday'. (Burchfield 702)

"Certain adverbs in -ly have acquired the ability to qualify a predication or assertion as a whole.  Such adverbs are all elliptical uses of somewhat longer phrases.  It the last third of the 20c., this little-used and scarecely observed mechanism of the language has broken loose.  Any number of adverys in -ly have com into common used as sentence adverbs." (Burchfield 703)

Biber, et al. Longman Grammar Of Spoken And Written English, (1999) refer to these words as attitude stance adverbs. (Biber 10.3)

Stance adverbs "have the primary function of commenting on the content or style of a clause…"  They fall into three categories:

(1) epistemic — It was, definitely, a waste of time. (personal belief, "truth or value of the proposition, commenting on: certainty, reality, sources, limitations and precision of the proposition.");

(2) attitude —  Fortunately, it was completed on time. (expresses the speaker's attitude or evaluation of the situation);

(3) style —  Frankly, it was a waste of time. (commenting on the style or form of the utterance, clarifying  how the speaker is speaking, how the utterance should be understood).

hopefully. "First, it was widely condemned from the 1960s to the 1980s.  Briefly, the objections are that (1) hopefully properly means "in a hopeful manner" and shouldn't be used in the radically different sense 'I hope' or 'it is to be hoped.' (2) if the extended sense is accepted, the original sense will be forever lost; and in constructions such as 'Hopefully, it won't rain this afternoon,' the writer illogically ascribes an emotion (hopefulness) to a nonperson… Second, whatever the merits of those arguments, the battle is now over. Hopefully is now a part of AmE and it has all but lost its traditional meaning… " — (Garner 427)

Garner adds that it is a "skunked term".  Avoid it in all senses if you are concerned with your credibility.  Stage 4 on the Language Change Index:commonly found but still shunned by a few stalwarts "die-hard snoots" .
 

Huddleston et al., The Cambridge Grammar of the English Language, (2002)  refer to these structures as clause adjuncts.  (771)

Evaluative adjuncts:  "With adjuncts of this kind the residual proposition is presented as a fact, and the adjunct expresses the speaker's evaluation of it." i. Amazingly, he escaped with only a scratch;  ii. He escaped with only a scratch, which was amazing. (reworded as a relative clause); iii. It was amazing that he escaped with only a scratch. (reworded with an it-clause)

The term adjunct covers modifiers to the verb phrase or clause together with related supplements. (dependents).  (8 §17)

Merriam-Webster's Dictionary of English Usage, 1994. The sentence adverb is an adverb or adverbial phrase that is connected with a whole sentence rather than with with a single word or phrase in the sentence.  Sentence adverbs are  a common feature of present-day English, and they go by many names… The chief virtue of a sentence adverb is its compactness: it permits the writer or speaker to express in a single or short phrase what would otherwise take a much longer form. — Merriam-Webster  (Merriam-Webster 512)

Some handbooks point out that conjunctive adverbs like therefore, nevertheless, and however can also be considered sentence adverbs because to the extent they are adverbial they modify clauses rather than any particular part of the clause. 

To sum up: hopefully had been in sporadic American use as a sentence modifier for some thirty years before it suddenly caught fire in the early 1960s.  What is newly popular will often be disparaged, and criticism followed rapidly, starting in 1962 and reaching a high point around 1975.  ... You can use it if you need it, or avoid it if you do not like it.  There never was anything really wrong with it; it was censured, as Bolinger 1980 notes, because it was new, and it is not very new any more. (Merriam-Webster 837)
 

Swan, Michael. Practical English Usage, (2009) refers to these words as comment adverbs, which give the speaker's opinion of an action. (22.2)

Fortunately, she has decided to help us; Stupidly, I forgot my keys; Surprisingly, they failed; I stupidly forgot my keys. (mid-position) (24.5 Comment Adverbs); Hopefully, inflation will soon be under control. "It is to be hope that" or "I hope" (251)

Quirk, Randolph and Sidney Greenbaum. A Comprehensive Grammar of the English Language (1989) refer to these words as style disjuncts: "Style disjuncts convey the speaker's comment on the style and form of what he is saying, defining in some way under what conditions he is speaking as the 'authority' for the utterance. (8.123-33)

Style Disjunct  (a)  modality and manner (e.g., truthfully, bluntly, if I may say so); (b)  respect (e.g., in broad terms, personally)

Content Disjunct  (a) degree of conditions for truth of content (e.g., really, certainly, if he'd listened); (b) value judgment of content (e.g., understandably, wisely, to everyone's surprise)