Well, “lower” is a verb. “Higher” is not. The confusion is that “higher” and “lower” are both adjectives, with opposite meaning, and at first glance one might think that if “lower” is also a verb, with a meaning related to the adjective “low,” perhaps “higher” would likewise be a verb related to “high.” But it isn’t. The verb normally used as the opposite of “lower” is “raise.”
You can grate cheese, and you can greet guests, but you cannot ‘great’ or ‘greater’ anything, no matter how big or voluminous it is.
I don’t have any Tommy Hilfigerish stuff.
I don’t want any Tommy Hilfigerine stuff.
I don’t need any Tommy Hilfigeral stuff.
I don’t like any Tommy Hilfigerous stuff.
I don’t buy any Tommy Hilfigeresque stuff.
I don’t accept any Tommy Hilfigerful stuff.
(that makes six… so far)
I missed the part of the OP’s question where “higher” was used as a verb. So, “raise the volume” is appropriate, as has been said.
And none of these (neologisms) serve the same function.
They all indicate stuff which is of a type analogous or identical to TH. The original context refers to only stuff that is of or from the company Tommy Hilfiger.
That’s the old partitive genitive raising its cantankerous and grammar-maven-adored head again.
In some cases, the English possessive constructions, corresponding to the genitive in more synthetic languages, refer not to relation or ownership but to a selection from among a group. This is particularly common where an ambiguity would otherwise arise.
“Churchill’s portrait” would probably be understood as a painting of the bulldog-faced wartime Prime Minister, but given that he was an accomplished amateur painter as well, might be a portrait of the wife of one of his Ministers painted by him. We clarify this as: “a portrait of Churchill” means one painted by someone else for which he sat; “a portrait of Churchill’s” means one portrait of somebody from among the group of portraits painted by Churchill. (Actually, he tended more to landscapes, but the example is valid – someone renowned enough to have his own portrait painted and who was also a painter.)
Russian is particularly partial to the partitive genitive – a negative transitive verb in Russian takes not the accusative but the genitive: “U menya yest’ khleb” means “I have bread” but “U menya n’yet khlebu” means “I have no bread” and is literally “To me there is nothing of bread” – they conceive of “all possible objects or quantities of a substance” as a category from which one has none of it.
While there are numerous cases in which possession is properly shown by the attachment of -'s to the last word of a noun phrase, caution must be exercised. For example, it would be awkward to talk of the prerogatives accorded to Colin Powell as “the Secretary of State’s rights” – the phrase could be taken as a sort of small-Federal-government ombudsman for the states!