Consider a bounded domain $\Omega \subset \mathbb R^d$ with a Lipschitz boundary (could also be a smooth boundary). Is the trace space $H^{1/2}(\partial\Omega)$ embedded in $L^\infty(\partial\Omega)$?
2026-03-30 20:57:25.1774904245
Are trace function embedded in $L^\infty$?
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I don't know how to make a link in comment so I write it here, you don't need to take this as an answer.
@Fundamental's answer is good enough, the answer is no. If you are looking for good reference for Sobolev embedding, especially focus on Trace operator, I would recommend you read Leoni's book, it explains embedding and trace in a very details way.