My question is the following:


Can a (probabilistic, deterministic, ndtm) oracle turing machine $A$ calling an oracle residing in a superior (more difficult) complexity class $B$, have less power then the oracle called ($A^B < B$).   
 My believe is that this new class should be at least as powerful as $B$?


For example, assuming that i've $P^X$, where $X$ resides in a superior complexity class.


Could $P^X$ be weaker than $X$ in some cases and, if yes, is a meaningful complexity class?



weaker oracle machine ?
If you think of Turing machines that have an additional oracle tape and the complexity 
class $C$ under consideration is below linear time, then for any nontrivial oracle $X$ we have $X\not\in C^X$, just because you don't have enough time to actually ask the oracle.  

In all other cases (i.e., if you allow for linear time computations), you will have $X\in C^X$, witnessed by the machine that takes a natural number $n$ as an input and then goes to the $n$-th cell in the oracle tape and checks whether this contain 0 or 1 (or "yes" or "no" or whatever).


