So here is a theoretical spot. BTN oppens with 54dd, SB folds, BB calls. Both players are 100bb deep. Villain is unknown winning reg.

Flop: AhKh2s (7bb);

BB checks, BTN bets 2.5bb, BB calls.

Turn: AhKh2s 9d (12bb);

BB checks, BTN?

How do we approach this spot? If we continue bluffing with all of our zero equity air, then we will have too many bluffs? However, if we check-back all of them then we won't have enough bluffs. How do we decide whether to continue bluffing or give up? Is there any systematic way of working on this spot?

Basically it's a function of how much the turn shifts hand strengths. In this case a 9 is a complete blank, so you retain your uncapped advantage and can continue to bet air at an extremely high frequency.

If the turn was say a 3h then that would dramatically alter the situation. Now the BB is no longer capped and you have to be very careful of what you continue betting.

Completely agree with getmeoffcompletely and well said.

I would add that as ranges get narrower (i.e. as you progress streets) you can start choosing hands that at least don't block villain's folding range and if possible block villain's calling range. Neither of these is always possible but it's something to consider to narrow your bluffing candidates.

For example if you get to the river here unimproved and the board runs out without hearts you would prefer not to have hearts in your hand so there's a greater chance villain was on a heart draw. Same reasoning for having say a J in your hand. It blocks villain's straight draws that he might be continuing with as well as blocked something like KJ that might go 1 or 2 streets but fold to a 3rd barrel.

Also the larger number of better hands in villain's range here the better a bluff it might be. Your example is pretty good in that regard as it maximizes folding a bunch of hands better than it that villain can't really call with (some suited jacks, some suited queens, bigger suited connectors, some pairs, etc).

Contrast that with QJ. It still will want to bluff for a variety of reasons (blocking effects, bluffing candidates for later streets, board coverage for later streets, etc) but there's fewer overall better hands that it folds because it is the highest ranked no pair hand on that board.

Those are just a couple of extra considerations to help you narrow down your bluffing candidates when you need to do so.

Well, I agree with you to some degree, However, on the turn I have a lot of hands that I bet flop for value but don't want to continue barreling. For example, KQ, KJ, A5 and 88, 99 that I bet flop for protection/value. As you see our value bet range is significantly smaller, so we need to have a way to narrow our bluff range.

Let's talk little bit about bet sizing to better set up this spot for discussion. As 9 didn't improve my range it means I would want to continue betting small, probably 1/3 pot. And if we bet 33%, then villain definitely wouldn't fold flush draw. It means that not having h in our hand doesn't have immediate merit, but has merit on the river. Does it mean we would start eliminating h hands on the turn?

You are still uncapped on the turn and in general your range should be getting more polar which means your bets should be getting larger street by street. The only time this probably doesn't hold true is when the equity significantly shifts in your opponent's favor.

Because of that I don't think 1/3 pot is a good sizing here on the turn. I suppose you could support that sizing and a larger sizing but you would have to start mixing with your best hands and it seems like too complex of a strategy for probably very little EV gain.

To answer the second part of your question just consider where your heart draws end up when you get to the turn. Of all of the junk you bet the flop with, they have probably moved on the edge of your bluffing range into your checking range equity wise because they don't loose equity as quickly as say gutters, random overs, etc. In reality your probably mixing hearts some percentage of the time so that you can have nut hands when your draw comes in on the river.

So I constructed turn range for us on equilab and indeed it's more polarized and we would want to use bigger size. Thanks for pointing that out, though, I still think villain won't necessarily fold his flush draws. So, looking at my ranges here, on the turn I have 40% less value bets than on the flop and it's still not very clear how to narrow our pure bluff range. I guess the best way is to just cut out top of my pure air, like you said in your first post. So, on this turn I stop bluffing with Jx and Tx air and continue bluffing with 9 high or lower. 54dd then would become bet turn, bet river huge hand I guess.

Yea I would say you have a good grasp on it.

Keep in mind that bluffing properties can change based on the actual runout as well. You probably already knew that but I wouldn't to explicitly state that here.