proving an interpolation inequality in $L^p$ norms

Let $$1\le p \le \infty$$. Prove that for all $$\epsilon >0$$ there exists a constant $$C>0$$ such that $$\|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \; \forall u\in W^{2,p}(\mathbb{R}).$$ This is a special case of an interpolation inequality, see here http://conteudo.icmc.usp.br/pessoas/andcarva/sobolew.pdf , theorem 2.

I need help to prove this. In any case the first step is to choose a continuous representative function for u. Then, I have the following 2 ideas:

1) To do it similarly as to prove theorem 2. Apply the mean value theorem to u to receive an estimation for $$|u'|$$. But the first problem I have here is, that the mean value theorem is local. But here the domain is $$\mathbb{R}$$. Therefore, i don't know how to proceed.

2) To use the fundamental theorem of anlysis applied to $$u'$$, to receive something like $$u'(\delta+x)=u'(x)+\int_x^{x+\delta} u''(z)dz.$$ Furthermore, convexity of $$|\cdot |^p$$ and Jensens inequality https://de.wikipedia.org/wiki/Jensensche_Ungleichung might be important, but I am not completely sure how to math that.

The second idea may be more promising. I appreciant any help.

Edit: There are other variants of this inequality available and meanwhile I have discovered Interpolation inequality and Proving an interpolation inequality for $$C^2_b$$ functions which seem to be a variant of the inequality above (however, the norms are different).

• Should it not rather be $W^{2,p}$ otherwise the summands don’t Need to exist? – Idun E. Jan 16 at 2:59
• thank you. I fixed it – TheAppliedTheoreticalOne Jan 16 at 10:32
• The first pdf you link gives a proof (they even prove it first for the one-dimensional case). Is there anything you don't understand? – MaoWao Jan 16 at 15:42
• @MaoWao in the meantime I have been succesful proving it with idea 2! I am going to post my solution on MSE as soon as possible – TheAppliedTheoreticalOne Jan 16 at 16:15

In fact, I think you only need to prove for $$u\in C_c^{\infty}(\mathbb{R})$$ which is a dense subset of $$W^{1,2}(\mathbb{R})$$ (But NOT dense in $$W^{1,2}[0,100]$$). Then the desired result holds by pushing limit. I hope I remembered it right. :)
• Could you explain how you would push the Limits? Since $C^\infty_c(\mathbb{R})$ is dense, we finde a converving subsequence with regards to $W^{2,k}$. But Since $\mathbb{R}$ is not bounded, this does not imply convergence in $L^p$, if I remeber it correctly. So how does ist work? – Idun E. Jan 16 at 11:51
• @IdunE. $C_c^\infty(\mathbb{R})$ is dense in $W^{2,p}(\mathbb{R})$. Since both sides of the inequality are continuous w.r.t. to the $W^{2,p}$-norm, it suffices to prove it on a dense subset. The case $p=\infty$ has to be treated separately, I think (if it's even true at all). – MaoWao Jan 16 at 15:44