According to the polynomial- reductiontime-reduction definition
"If problem Y$Y$ can be reduced to problem X$X$ in polynomial time, we denote this by Y ≤(P)X $Y \leq_p X$."
If X$X$ is one of already known NP-complete problem then we can say that Y$Y$ is NP-complete.
From my understanding, P$p$ should be always polynomial and this cannot be an exponential function such as 2^n$2^n$ or 3^n$3^n$.
However, my question is if Y$Y$ or X$X$ already has a lower bound that is not polynomial, which is 2^n$2^n$ or 3^n$3^n$ then can I still say that Y$Y$ is NP-complete?
In other words, does lower bounds for X$X$ and Y$Y$ matter?
