I know that there are complexity classes that are "below" P like L or NL and stuff like that. But I was curious to know where sub linear time algorithms fitted in this picture and maybe how they were related to the P vs NP problem or if they had their own version of something similar. I am aware that this question might lead to a list of open problem in sublinear time algorithms or something similar, which is a fine answer to my question. I am also curious to know what we know about this theory and what we don't know or what are open problems to tackle.
[I tried adding the tag sublinear time algorithms but I am not sure if it exists. Feel free to add a tag that might help]