I find that often researchers in math do not define every piece of notation that they use, however, I am not sure what criteria they use to deem that a definition should be left out, or should be included.
For example, in Terence Tao's paper, he did not provide a definition of the gamma function. But the gamma function is standard, so there shouldn't be any problems.
In these other papers on optimization, for example, the researchers uniformly do not define what they mean by $\Omega, \omega, O$, or $o$ and assume that they are well known (even though I can think of conflicting definitions in CS versus math, see chapter 1 of CLRS versus Luenberger and Ye's optimization text)
I am sure there are many other examples. Now I am beginning to learn how to communicate in math, but I am not sure what I should define, what I should leave out. And if I define everything, it soon becomes a rabbit hole, but if I don't define, I get yelled at for being fast and loose and imprecise.
How do you determine what to define?
You can look at similar papers (to the one you're writing) in your particular field, and you can emulate what others have done. I am assuming you'd be referencing other papers for example, so you can take a clue from these, regarding what is assumed well known or standard in that field. You can also think of what you believe is well known or could be easily looked up by a reader and decide whether it is worth leaving something out or if it would be of benefit to the reader or clarity of exposition to include it. Referees will help here too as they may suggest exclusions or additions.