The capitalized cap cost reduction is also known as a cap cost reduction. The cap cost reduction is simply the down payment. Do not let this term confuse you. Cap cost reduction=down payment. This term refers to the amount paid in cash due at signing that reduces the amount financed in the lease and ultimately reduces your lease monthly payment.
The cap cost reduction amount includes any additional money you are putting down (as if you were normally financing a new vehicle) as well as any dealer rebates and trade-in vehicle.
I would not recommend putting any additional down payment only because if your leased car is totaled, you do not get any of the down payment back. Of course this is a risk you need to consider. If you need to put money down to reduce the lease monthly payment, then chances are the vehicle is too expensive for you to afford.