I have Hostas planted under oaks and I am surprised they survive at all due to the invasive roots. There are just mats of very fine roots all under the oaks, sometimes I don't know how anything manages to emerge though them.
Here are some of the Hostas growing under the oaks, really near them. All my Hostas are shaded by the canopy of oak trees.
'Bressingham Blue': Not thriving, but it is a division of the plant shown in the next photo, it is very slow growing here.
'Zounds': This plant has been successful in some plantings, and it has died in at least three plantings, I currently have these two.
'Undulata Albomarginata': Without a doubt this plant seems to love it here, it grows anywhere I put it. I have it spread all over.
'August Moon' seems to live here, but it is not that impressive.
Two large growing hostas that seem to like it here under the oaks are:
'Sum and Substance':
I have quite a few others also, but if I were to recommend any for this area (these are growing under the shade of the oaks, but not so near to the trees themselves) it would be.
Of course 'Francee', and one other that has done very well here is 'So Sweet':
Just wanted to add that you might be interested in SunHosta, I have four of them bought off the discount rack at Lowe's and they are all doing fine, even the one that gets a lot of direct sun. But they have had plenty of rain, and that might make all the difference!