Yes. However, technically he is going All In for any amount UP TO $44. How much he ACTUALLY goes in for depends on whether player 2 has at least $37 still in his stack AND wants to make the call.
If player 2 folds, no problem. Player one can go All In but it will be only for $11.
If player 2 cannot cover the $44 bet but still wants to play, he could still go All In and player 1's All In bet amount will become whichever is greater between player 2's stack and player 3's stack (of $11). This would technically become a "side pot" and what happens then would depend on whose hand won. If player 3 wins outright, player 2 (who didn't fold in this case) pays him $11 if he has that much and player 1 pays him $11 - after which player 1 is either out of the game or has to buy in again.
Note also that since this is a cash game and the stakes sound like "friendly" poker, the "house rules" might not allow side pots.
If player 2 can & does cover the $44 bet, then player 1 DID go All In for $44. In that case, player 1 and player 3 have a side pot (if they are allowed).
The rule is that saying "All In" implies but doesn't actually state an amount. If you win, you get whatever was bet. If you lose, you owe only the amount of the highest bet against you. Or stated another way, you can never lose more than the amount that somebody else bet against you.