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Consider the following two norms:

The interpolation norm: 1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x u\|_{L^{\infty}}$.

The homogeneous Besov norm: 2) $\|u\|_{\dot B_{3,3}^{1/3}}^3 := $ $\int_0^{+\infty} \int_{\mathbb{R}} \frac{|u(x+h)-u(x)|^3}{h} dx \frac{dh}{h}$'

We work with the class of functions $u:\mathbb{R} \to \mathbb{R}$ with mean zero. I strongly believe that $B_{3,3}^{1/3}$ is not a subset of $[L_2,\dot H_1^{\infty}]_{1/3,\infty}$. This would mean that there exists a sequence of functions $u_n$ with

$\frac{\|u_n; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\|}{\|u_n\|_{\dot B_{3,3}^{1/3}}^3} \to +\infty$ as $n \to +\infty$.

All of the examples I have thus far tried however suggest that this is not possible..that in fact it may be true that $B_{3,3}^{1/3}$ is contained in $[L_2,\dot H_1^{\infty}]_{1/3,\infty}$. For instance if one considers a sequence of n adjacent triangles of fixed width and height, one can show that both norms scale like $n^{1/3}$. Although the $L^2$ norm scales like $n^{1/2}$, by spreading enough of $u$ into $u_1$, you can show that the norm blows up like $n^{1/3}$ in fact.

I don't have much of a feeling for these abstract spaces, and so was hoping for the opinion of an expert in the field.

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Idid not have time to check the exponents but if they are fine, the real interrelation space is the same as the Besov space. See Bergh and Lofstrom. There is a paper by Johnen and Scherer (1977) that establishes the equivalence between the K-functional and moduli of smoothness. –  timur Jul 11 '12 at 14:28
    
I have this book and there is nothing in it that establishes the equivalency between these two norms. –  user25042 Jul 11 '12 at 14:50
    
Also the paper you suggested doesn't deal with this type of interpolation norm. In particular it assumes the $L^p$ norm is the same for $u_0$ and $u_1$ which it isn't in this case. –  user25042 Jul 11 '12 at 15:10
    
The main problem about your question is that the spaces of type $W^{1,\infty}$ are not in the standard interpolation scales. I vaguely recall I've seen something about their interpolation properties but I'm not sure where –  Piero D'Ancona Jul 11 '12 at 16:51
    
Ok not an answer but here you can find real interpolation of $W^{1,p}$ spaces: math.univ-lyon1.fr/~badr/new_interp_version.pdf. You might probably ask Nadin Badr directly, she might know the answer to your question –  Piero D'Ancona Jul 11 '12 at 16:57
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