We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:

$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$ where $U(t)= e^{it\Delta} $(free Schrödinger equation), and initial data $u_{0} \in L^{2}(\mathbb R)$[For more details, see this ]. It is well-known (see Theorem 1.1) that for initial data $u_{0}$ in $L^{2}(\mathbb R), $ we have the conservation law in $L^{2}-$ norm: $\|u(t)\|_{L^{2}(\mathbb R)}= \|u_{0}\|_{L^{2}(\mathbb R)}$ for all time $t\in \mathbb R.$

Suppose that $X$ is a Banch space with norm $\|\cdot\|_{X}$ and $X$ is a proper subset of $L^{2}(\mathbb R).$

**My Questions** :

(1) If we start with initial data in $X,$ then can we expect conservation law associated to the above NLS in the norm of $X,$ that is, $\|u(t)\|_{X}=\|u_{0}\|_{X}$ ?(I feel, this may be difficult, I don't know)

(2)Of course, the answer of (1), depends on $X:$ it therefore I am looking for few examples.Can we find few example of $X$ where conservation law hold in the norm of $X$?

(3) Is it true that the space $L^{2}(\mathbb R)$ is the smallest space where one can have the conservation law hold in the space ?

**Motivation**: To prove the globe well poseness results for NLS conservation law is useful.

**Edit**: My motivation comes through this consideration:"If we have a local well posedness in $X$ for cubic NLS, then we can expect global well posedness in $X"$ if we have a conservation law in $X$(I think). Also I am looking for explicitly $H^{1}\subset X \subset L^{2}$ where conservation law hold in $X-$norm. Kindly would you suggest me some $X$ where conservation law hold(So far papers I have been seeing every one has consider conservation in $L^{2}-$norm, I am couriers to know if we can come below $L^{2}$ )