Let $b$ be a function and $W^{1,2}$ the first-order Sobolev space on some Euclidean space (edit: so the whole space, but in arbitrary dimension, although $d=1$ would be interesting for me for the beginning as-well, no subsets or whatsoever). If I know that $b W^{1,2} \subseteq W^{1,2}$, what can I conclude about $b$? I also know that $b$ is bounded from above and below, if that helps.

What is clear is that $b$ is weakly differentiable (by multiplying bump-functions with it). I would like to conclude something in the direction (locally) Lipschitz. Any ideas?

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