Let $0< \beta < \gamma <1$. Show that the interpolation inequality holds.
$$||U||_{C^{0,\gamma}(U)} \le ||U||^{\frac{1-\gamma}{1-\beta}}_{C^{0,\beta}(U)} ||U||^{\frac{\gamma-\beta}{1-\beta}}_{C^{0,1}(U)}.$$
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3This is not a juke-box. Have you tried anything? Any ideas or difficulties in applying the definition of the norms? – Siminore Mar 13 '14 at 10:27
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1what I need is a clue because am having difficulty with applying the definition of the norms – Jamin Mar 13 '14 at 10:35
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1Detailed proof here. – Pedro Sep 02 '15 at 03:23
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Hint: express $\gamma$ as a convex combination of $\beta$ and $1$, say $\gamma=a+b\beta$ where $a+b=1$. Then notice that $$|x-y|^\gamma=|x-y|\cdot |x-y|^{b\beta}$$ and use the property $\sup_{z\in I}|g(z)h(z)|\leqslant \sup_{z\in I}|g(z)|\sup_{z\in I}|h(z)|$.
Davide Giraudo
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