His authorship is in doubt? Find me a serious scholar who thinks that’s possible. The people who doubt authorship are the upscale equivalent of 9/11 truthers. And basically they are snobs.
The theory runs along the lines that it is entirely improbable that someone could have been such a genius as to write what Shakespeare wrote, so, it could not have been Shakespeare, but some other genius. Or, in the alternative, they pose the even more improbable hypothesis, that the only way to to explain the conundrum of one such improbable genius having lived is to posit that there were two such geniuses.
Some genius wrote Shakespeare, and for the sake of convenience we might as well call that author what he called himself, i.e. Shakespeare.