PokerStars - $0.02 PL Hi FAST (6 max) - Omaha Hi - 6 players
Hand converted by PokerTracker 4

Hero (BTN): $5.46
SB: $4.51 (VPIP: 26.32, PFR: 10.53, 3Bet Preflop: 11.11, Hands: 19)
BB: $2.08 (VPIP: 25.00, PFR: 25.00, 3Bet Preflop: 0.00, Hands: 8)
UTG: $1.80 (VPIP: 50.00, PFR: 50.00, 3Bet Preflop: 0.00, Hands: 2)
MP: $6.09 (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 3)
CO: $1.33 (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 3)

SB posts SB $0.01, BB posts BB $0.02

Pre Flop: (pot: $0.03) Hero has Q Q A J

fold, fold, fold, Hero raises to $0.07, fold, BB calls $0.05

Flop: ($0.15, 2 players) 7 2 7
BB checks, Hero bets $0.11, BB raises to $0.43, fold

BB wins $0.36

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The only thing I can think of is that this is either a really stupid check-raise bluff OR 7xxx, in which case I'm way behind. I don't think villain would do this with JJxx I also do think hands like 987x are big in a calling range. I bet because I thought I had the best hand but didn't want to give a free card if villian had two hearts, when villain check-raises, I don't think this is a fancy bluff.

I play your limit and beat it solidly, but I'm not one of the really strong players that are around here.

Now that this is out of the way: I think, you should check up on some of JNandez free content considering smaller stakes to get a better idea of how to beat them. There is no reason to be obsessed with giving free cards.

Instead what to consider, when deciding wether to
a) cbet or check and
b) which size to cbet
is how your complete range of hands look like in a given situation and how you want to play that range.

Questions that can help you get an idea:
1) Who has the range advantage?
2) What hands do I have in this spot, given the preflop action? (Usually varies at least by position).
3) Compared to all my hands, is this a strong/medium or weak hand?
4) What do I want to do with my strong/medium/weak hands? For example, a strong hand usually wants to build a pot, but if the board doesn't allow for many really strong hands, you might want to bet smaller even with the very best ones, in order to "protect" the hands that are not as strong.
5) Which hands do I want to bluff/give up with? Even in PLO everybody flops much more weak/trash hands then strong ones. If you bluff with all of them, you are going to bluff too much. People will discuss optimal frequencies of bluffing depending on sizing and whatnot. For now, I think, you either want to have some outs or good blockers, preferably both.
6) If I bet this size for value, how does a typical calling range look like?
7) If I bet this size for protection, which hands will be typically folded?

This is what I think, about your actual hand, but it's just my opinion:

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The board is pretty dry, there isn't much except for the flushdraw.
=> There are only few hands, I can get value from and no real need for protection. I will bet a small size with my entire betting range, somewhere between 25% to 50% pot. Usually 33%.
Getting raised with QQ is a snapfold, nobody at this limit (except the occasional maniac) raises as a bluff often enough to make this a profitable call.

bet less on the flop, clear fold to the c/r