Let's assume villain is playing a standard 12% flatting range:
JJ:0.75,TT,99,88,77,66,55,44,33,22,AK:0.5,AQs:0.75 ,AQo,AJs,ATs,A9s,A8s,A7s,A6s,A5s,A4s,A3s,A2s,KQs,K Js,KTs,QJs,QTs,JTs,J9s,T9s,98s,87s,76s:0.5
Should we bet the flop? Villain's pure air range is 30% of his hands, and they have 23% equity. Typically when we bet to fold out overcard equity, we need a rough 2:1 ratio of air:made (when air has 3-6 outs), so we aren't going to bet purely to fold out villain's equity.
So lets look at his calling range vs his raising range on this flop.
Calling: AK, AQ, any pair worse than 99 without a straight draw (32% of his range, 18% equity)
Raising: 99+, sets, A7s, pairs with 8 outs straight draws, most flush draws. 38% of his range, has 55% equity vs us, this is 56 combos, with 19.5 of them being sets or overpairs that beat us. Since we are beat equity wise here we don't really want to 3bet the flop.
So when he check/raises us, we have enough equity to continue. However, villain can easily blow us off our hand on later streets by dropping 17 bluff combos on the turn and 10 on the river (making him perfectly balanced and dropping our EV to 0). If he does this, when we call the flop he will bet the turn 70% of the time, and lets assume the remaining 30% of the time we win the pot. So EV of calling the raise is -60*.7+190*.3=15. But wait! He will hit his hand on the turn sometimes, giving him more value. What cards are safe for us? T+, 2, 3 without completing the flush, 21 cards, so over half the time he will gain an additional 12 combos of hands that beat us. He also beats us on some of our safe cards, but they only add 3 combos on average. With 12 additional value hands on the turn he can bet any two that he check/raised and make us fold, so our equition turns into (-60*.7+190*.3)*(26/47)+(-60*21/47)=-18.5. So because his 'bluffs' have so much equity, we can't really call this flop raise.
Basically, we have equity to continue but not if we are playing a talented player who will make us make big mistakes on upcoming streets.
.38*-30+.30*(60*.23)+.32*(30*.82)=.62 So +ev to bet. But doesn't mean it is maximally +ev to bet. Math gets real complicated here but if we run this situation through PIOSolver (giving hero a 15% opening range), we are checking back TT almost always on this flop, which is a good sign that we should probably check back against good opponents.