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I know this is a special case of Sobolev embedding theorem but I heard there is a simple way to prove this special case. Seems to start with the dense subset $C^\infty [a,b]$. Construct a Cauchy sequence for any function in $H^1$. I'm lost as to what to do next?

Someone suggest that $\forall v\in H^1\cap C^\infty$, $|v(x)-v(y)|=|\int_x^y{v'(t)}dt| \leq |\int_x^y{1^2}dt||\int_x^y{v'(t)^2}dt| \leq \sqrt{y-x}\| v\|_1$. This imply $v$ is Holder continuous, but is that true for general $H^1$ function?

And that is not equicontinuity, $\forall \epsilon ,\exists \delta,\forall x,y \text{ s.t. } |x-y|<\delta,|v(x)-v(y)|<\epsilon ,\forall v$ ,$\delta$ should solely depend on $\epsilon$. However, in our case, $\delta$ has to rely on choice of function.

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  • $\begingroup$ You want to prove equicontinuity of functions with bounded $H^1$ norm. And Hölder continuity in the form you have implies that. $\endgroup$
    – user53153
    Feb 6, 2013 at 3:57
  • $\begingroup$ Read my comment again: for any bounded family of functions in $H^1$ ($\|v\|_{H^1}\le M$) you have equicontinuity by means of $|v(x)-v(y)|\le M\sqrt{x-y}$. This is all you need, because convergent sequences in $H^1$ are bounded. $\endgroup$
    – user53153
    Feb 6, 2013 at 4:23
  • $\begingroup$ I think "uniformly bounded" in Arzela-Ascoli theorem is with regard to $C^0$ norm, not $H^1$ norm. $\endgroup$
    – user33869
    Feb 6, 2013 at 4:41

2 Answers 2

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Given: a sequence of smooth functions $v_j$ that is convergent in $H^1$.

Goal: show that $v_j$ converge in $C^0$.

Method: prove that $\|v_j-v_k\|_{C^0}$ is small when $\|v_j-v_k\|_{H^1}$ is small.

Details: let $u=v_j-v_k$. Note that $|\int_a^b u|\le \sqrt{b-a}\sqrt{\int_a^b u_j^2}\le \sqrt{b-a}\|u\|_{H^1}$. By the mean value theorem for integrals, $u$ attains the value $m=\frac{1}{b-a}\int_a^b u$ at some point $x_0$. Then your continuity estimate gives $$|u(x)-m|\le \sqrt{|x-x_0|}\|u\|_{H^1}\le \sqrt{b-a}\|u\|_{H^1}$$ for all $x\in [a,b]$. Thus, $$ \|u\|_{C^0}\le m+ \sqrt{b-a}\|u\|_{H^1} \le \frac{1}{\sqrt{b-a}}\|u\|_{H^1}+ \sqrt{b-a}\|u\|_{H^1} $$

Mission accomplished: the sequence $v_j$ is Cauchy in $C^0[a,b]$, and therefore converges there.

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Cauchy sequence $\{v_n\}$ in $H^1[a,b] \cap C^{\infty}$[a,b] with norm $||\cdot||_{H_1}$, we have for any small $\epsilon > 0$, there exists p $\in N^*$, such that $||v_n - v_m||_{H_1} < \epsilon$, for any $m, n > p$. Let $\epsilon = \epsilon_0$, concide with $ p = p_0(\epsilon_0)$, and fix $m = m_0 > p_0$, we have $||v_n||_{H_1} \leq ||v_n - v_{m_0}||_{H_1} + ||v_{m_0}||_{H_1} < \epsilon_0 + M_0 = M$, where $M_0 = \sum_{i = 0} ^ {i = m_0} ||v_{i}||_{H_1} $ for any $n > p$. Also $||v_n||_{H_1} < M$ for any $n \in N$.

So $\{v_n\}$ is bounded in norm $H_1$.

Next prove equicontinuous.

For any $n \in N^*$, and any small $\epsilon > 0$, there exists $\delta = (\epsilon /M)^{1/2}$, such that $|v_n(x) - v_n(y)| = |\int_x ^y Dv_n(t) dt | \leq |\int_x ^y 1 ^2 dt |^{1/2} \cdot |\int_x ^y Dv_n(t) ^ 2 dt |^{1/2}\leq \sqrt{\delta} ||v_n||_{H_1} < \sqrt{\delta} M = \epsilon$ , where $|x-y| <\delta = (\epsilon /M)^{1/2}$.

$\{v_n\}$ is equicontinuous and bounded. As by theorem of Arzela-Ascoli $\{v_n\}$ is uniformly convergent, we get that the limiting function $v$ is continuous.

Any function $v \in H^1[a,b]$, we can find Cauchy sequence in $H^1[a,b]\cap C^{\infty}[a,b]$, $\{v_n\}$ converges to v, $(C^{\infty}[a,b]$ is dense in $H^1[a,b]$). And $v$ is continous.

So $H^1[a,b] \subset C[a,b]$

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