NL200: AJ in BB against known degenerate Ax player
 

Yes, I suck and I have a lot or work to do on my NL game. That's why you guys see me posting around here in the last few days. I want to get better and I need your help.

In this hand, villan on button had not been at my table too long. I didn't notice I had a note on him until the flop. When he called the $8 bet on the flop I opened my notes on him and saw this:

"Ax any pos; will call off $$$ with TPWK. Waits to river to raise big if he hit 2 pr"

PokerStars No-Limit Hold'em, $2 BB (8 handed) FTR converter on zerodivide.cx

MP2 ($15.65)
CO ($119.45)
Button ($321.90)
Hero ($506.65)
BB ($354.40)
UTG ($104.35)
UTG+1 ($198)
MP1 ($227.35)

Preflop: Hero is SB with A[img]/images/graemlins/spade.gif[/img], J[img]/images/graemlins/diamond.gif[/img].
<font color="#666666">2 folds</font>, MP1 calls $2, <font color="#666666">1 fold</font>, CO calls $2, Button calls $2, Hero completes, BB checks.

Flop: ($10) A[img]/images/graemlins/club.gif[/img], K[img]/images/graemlins/spade.gif[/img], 4[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(5 players)</font>
Hero checks, BB checks, <font color="#CC3333">MP1 bets $8</font>, CO folds, Button calls $8, Hero calls $8, BB folds.

On the flop I'm not really worried about MP1 as I think that he usually raises any hand that beats me here pf (except 44). After seeing my note, I'm think there is a very good chance button is on Ax with a weaker kicker. Should I be raising here or waiting until the turn?

Turn: ($34) 9[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(3 players)</font>
Hero checks, MP1 checks, <font color="#CC3333">Button bets $12</font>, Hero calls $12, MP1 folds.

I think I went a little chicken here. I'm not fully confident enough in my kicker to put on the big check-raise and get lots of $$ in the pot, yet I obviously can't fold to this bet given the note on him. I decide to check call down.

River: ($58) Q[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, Button checks.

Final Pot: $58

Given the information provided, how would you play this differently on all streets? Thx.