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This is somehow related to one of my previous questions. Let us consider $\theta,s\in\mathbb{R}$ with $\theta<s$, and the standard Sobolev spaces $H^s(\mathbb{R})$ and $H^\theta(\mathbb{R})$. Let $T>0$ and consider a function $$ u\in L^\infty\big((0,T);H^s(\mathbb{R})\big) \quad \hbox{such that} \quad \dfrac{du}{dt}\in L^\infty\big((0,T); H^\theta(\mathbb{R})\big) $$ I am wondering if from these hypothesis it is possible to ensure that, for any $\theta\leq s'<s$ we have $$ u\in C\big([0,T]; H^{s'}(\mathbb{R})\big)? $$ I have the feeling that from these hypothesis it should be "easy" (but I do not know how to do it) to prove that $u\in C\big([0,T], H^\theta(\mathbb{R})\big)$, and then the idea would be to try to improve this strong continuity to all $H^{s'}$ spaces for $s'<s$, but I am completely clueless about how to do it. Does anyone has any thought or reference suggestion?

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