I want to show that for 0 < µ < ß ≤ 1,
[f]µ ≤ 2 sup|f|1 - µ/ß [f]ßµ/ß
where [f]µ := supx≠y |f(y) - f(x)|/|y-x|µ
I don't know how to proceed with this question. Any help would be welcome!
Proving an interpolation inequality on Holder Space
0
$\begingroup$
real-analysis
interpolation
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1Hint: Write $|f(y)-f(x)|=|f(y)-f(x)|^{1-\mu/\beta}|f(y)-f(x)|^{\mu/\beta}$. – 2017-02-27
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0Thanks, that helps! just a follow up, [f]_µ ≤ M sup|f|1 - µ/ß [f]Hßµ/ß would use the same approach? – 2017-02-27
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0Dude I have no idea what you have written there. If you want to do math, it would be to your greatest benefit to learn how to write in LaTeX soon. I think that this website even has a tutorial on how to use it, you just have to search it. – 2017-02-27
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0All i was asking is that in the first equation, on RHS, the second element has subscript H_ß instead of ß? – 2017-02-27