Integer weight case: Modular forms in $S_{2k}(N) := S_{2k}(\Gamma_{0}(N))$ can come from lower levels, and we'd like to know when that happens. This is where we call a modular form $f$ "old" if in fact it belongs to $S_{2k}(M)$ for some $M \mid N$ and $M < N$. How do we characterize such forms? We notice that if $g \in S_{2k}(M)$ then $g \mid V(d) \in S_{2k}(dM)$. One can then define the space of "oldforms" by
$$S_{2k}^{old}(N) := \bigoplus_{\substack{dM \mid N \\ M \neq N}} S_{2k}(M) \mid V(d).$$
The space $S_{2k}^{new}(N)$ of "newforms" will then be the orthogonal complement in $S_{2k}(N)$ of the oldform space with respect to the Petersson inner product. A "newform" is then a normalized form in $S_{2k}^{new}(N)$ that is an eigenform for all Hecke operators and for all of the Atkin-Lehner involutions $W_{p}$ for $p \mid N$ and $W(N)$.
Half-weight case: Winfried Kohnen has built a theory of newforms for the half-weight subspaces $S_{k+1/2}^{+}(4N) := \{f \in S_{k+1/2}(4N) : a_{f}(n) = 0 \text{ if } (-1)^{k}n \equiv 2, 3 \mod 4\}$ where $N$ is odd and square-free. Why must he hone in on the plus space to develop this theory? Others such as Ueda and Manickam further develop theories of newforms on the more general space $S_{k+1/2}(4N)$, and they even go further to $S_{k+1/2}(2^{\alpha}N)$ for $2 \leq \alpha \leq 5$. But setting up the oldform space looks odd to me. For instance, for level $4$ we have
$$S_{k+1/2}^{old}(4) = S_{k+1/2}^{+}(4) \oplus S_{k+1/2}(4) \mid U(4).$$
In general, people like Manickam seem to like to define the oldform spaces like
$$S_{k+1/2}^{old}(4N) = \sum_{\substack{d \mid 2N \\ d < 2N}} \left(S_{k+1/2}^{+}(4d) + S_{k+1/2}^{+}(4d)\mid U(4N^{2}/d^{2})\right) + \sum_{\substack{d \mid 2N \\ d \text{ odd}}} \left(S_{k+1/2}(4d) + S_{k+1/2}(4d) \mid U(4N^{2}/d^{2}) \right)$$
(there is an easier way to write this by defining $S_{k+1/2}(N)$ to be either plus space of level $4N$ when $N$ odd or non-plus space of level $2N$ when $N$ even).
Why are $U$-operators used in this definition? Why can we not just define oldforms just as in the integer-weight case? Why is it difficult to get a "theory of newforms" for these half-weight spaces? What makes higher orders of $2^{\alpha}$ dividing $N$ so difficult?