# Norm Inequality for 1 Dimensional Sobolev Space

Let $$\Omega \subset \mathbb{R}$$ be an unbounded domain and $$u \in H_{0}^{1}(\Omega)$$. By Sobolev Embedding Theorem for 1 dimensional space, we can obtain $$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)}^{2}$$ Here, $$S := \inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{0\}}\frac{||u||_{H^{1}_{0}(\Omega)}^{2}}{||u||_{p}^{2}}$$

So, my question is how to show that $$S^{\frac{p}{p-2}}\geq \inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{0\}}\bigg(\frac{||u||_{H^{1}_{0}(\Omega)}^{2}}{||u||_{p}^{2}} \bigg)^{\frac{p}{p-2}}$$

I apologize for my elementary question but I am not sure how to proceed. Any hint is much appreciated! Thank you very much!

• If $p>2$ this follows from monotonicity if $x^{\frac 2 {p-2}}$. – Kavi Rama Murthy Feb 18 at 5:27
• Thank you very much, I will try! – Evan William Chandra Feb 18 at 6:02
• From your hint, I obtain "less or equal" but what I need is "greater or equal" – Evan William Chandra Feb 18 at 6:12
• You need both continuity and monotonicity. I have posted an answer. – Kavi Rama Murthy Feb 18 at 6:30

Suppose $$S$$ is infimum of a set $$A$$ of real numbers and suppose $$f$$ is continuous and increasing. Then, for any $$a \in A$$ we have $$f(a) \geq f(S)$$ because $$a \geq S$$ and $$f$$ is increasing. Hence $$\inf f(A) \geq f(S)$$. Also there exists a sequencce $$\{a_n\} \in S$$ converging to $$S$$. Since $$f(a_n) \to f(S)$$ and $$\inf f(A) \leq f(a_n)$$ for all $$n$$ we get $$\inf f(A) \leq f(S)$$. Thus $$\inf f(A) =f(S)$$.