In which dimensions is it true that every topological ball embedded by a smoothly embedded sphere is a smoothly embedded ball? I asked a question on MSE with no answer. Here is my question in the generalized version.

Question 1: Suppose we are given a connected three-manifold $M$ (possibly non-compact, or non-orientable) and a smoothly embedded
submanifold $S$ of $M$ such that the following hold:

*

*$S$ is
homeomorphic to $\Bbb S^2$,

*There is a subspace $B$ of $M$ so
that $B$ is homeomorphic to $\Bbb B^3$ with $\partial B=S$.

Is $B$ a
smoothly embedded submanifold $M$?


Question 2: If one asks a similar question in other dimensions, is it still valid? In other words, if a codimension-one smoothly embedded
sphere bounds a topological $n$-ball in a smooth $n$-manifold, can
this topological ball be smoothly embedded?

I hope, for $n=2$ it is true. I don't know in other dimensions. Any help, references, etc., will be appreciated.
 A: Amended answer: It seems to me that Question 1 asks if a codimension 0 submanifold $B$ (not necessarily a ball) of a smooth submanifold $M$ whose boundary is smoothly embedded is a smooth submanifold of $M$. I think this is true in any dimension: smoothness of the interior is automatic, and smoothness near the boundary follows from the existence of a collar neighborhood (which follows from the smoothness of the embedding of the boundary.)
Taken literally, I think this answers Question 2, but I'm not sure what that question is actually asking. For instance, the (still unsolved) smooth Schönflies problem in dimension $4$ asks if a smoothly embedded $3$-sphere $\Sigma$ in $S^4$ bounds a smooth $4$-ball. By the topological Schönflies theorem (Mazur-Brown) $\Sigma$ bounds a topological ball. This topological ball is a smooth submanifold of $S^4$ as remarked above. But it might not be the standard smooth structure on $B^4$. In dimensions $n \neq 4$, you would conclude that the topological ball $B$ is diffeomorphic to to $B^n$ and is smoothly embedded.
(Old answer to a misreading of the question: The Alexander horned sphere (see Rolfsen, Knots & Links) is a counter example in dimension 3.  There are generalizations in all higher dimensions. In each case the sphere fails to have a product neighborhood so is not smooth.
By the way, the non-orientable and noncompact versions reduce immediately to the compact orientable version.)
