I am not using a HUD so I don't have stats on villain.

PokerStars, Hold'em No Limit - $0.01/$0.02 - 5 players
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UTG (Hero): $3.03 (152 bb)
CO: $1.91 (96 bb)
BU: $2.14 (107 bb)
SB: $2.31 (116 bb)
BB: $1.78 (89 bb)

Pre-Flop: ($0.03) Hero is UTG with J♠ J♥
Hero raises to $0.08, 2 players fold, SB 3-bets to $0.28, 1 fold, Hero 4-bets to $3.03 (all-in), SB calls $2.03 (all-in)

4BB is a very large open size, and unless you are doing this with your entire range you better be sure, that the opponents are really bad, because otherwise you are announcing your hand and allowing them to play very well against you. When you face a 3-bet, the standard play is to call with JJ. Getting it in for 116BB with JJ HJ vs. SB requires some special read, which you did not tell us, you had. A relatively normal player will probably only call you with exactly QQ+ and sometimes AK, so you are kind of value owning yourself.

I've played enough micro zoom. I don't know if you've played, but 3betting at that limit usually shows that he really has a worthy hand.
The bad thing about zoom is that you have no information on how the villain plays and without a hud, it got worse.

It's a tough spot, and very player dependent.

If SB is positionally aware, then 3 betting a 4X UTG open whilst being out of position shows real strength.

If he is just a 2 NL drooler, then it could mean almost anything. I usually use QQ as the inflection point at which I'm prepared to get all in preflop, though that can be changed either way with reads on a certain player.

I am not a fan of the play here as you will be called by hands that crush you while giving hands like 88-1010 a reason to fold. I am not saying all villains will always fold 1010 there but many of them will facing that big or a raise/jam playing .01/.02. Therefore you have made a negative expectation bet here for the most part where better hands snap you off and you lose 80% of the time and worse hands fold and you lose value. JJ will be tough to play post flop as they always are but calling there gives your opponents a chance to stay in with hands you beat. Most villains will C bet when they 3 bet so calling there at least gives you a chance they are on something worse to pickup another street of value and gives you a chance to see further action and if you should release your hand.

Against unknown villain 4bet is wrong decision, only better hands will call (sometimes AK). Just call 3bet.