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Question

Let $X$ be a smooth, non-empty, open subset of $\mathbb{R}^d$ and suppose that $f \in H^{2,k}(X)$ where $k>\frac{1+d}{2}+1$. Then by the Morrey-Sobolev embedding theorem, it follows that $f$ can be viewed as a once continuously differentiable function. In this case, does it hold that $$ \lim\limits_{p \rightarrow \infty} \|f\|_{L^p(X)} = \sup_{x \in X}|f(x)|? $$

Instead of having the essential supremum on the right-hand side as in this question?

Intuition/?:

My intuition here is that, in this case, there is no-longer a problem of ambiguity with the right-hand side's representative due to the Morrey-Sobolev theorem...

AB_IM
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1 Answers1

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Yes, this is true. In general, if $f$ is a continuous function and $f$ is bounded then the essential sup and the sup are the same thing if you are considering this sup and esssup on this open set. But, take care, it doesn't hold in general if you replace $X$ (an open set) by an arbitrary $E \subset X$ measurable set. Take a look at this question: Essential Supremum with the continuous function? . Edit: This question will also help you, I guess: Infinity norm of continuous function.

ABP
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