Timeline for Can Hölder's Inequality be strengthened for smooth functions?
Current License: CC BY-SA 2.5
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| Apr 7, 2011 at 13:46 | history | edited | Denis Serre | CC BY-SA 2.5 |
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| Mar 31, 2011 at 21:18 | comment | added | Kevin O'Bryant | Let $A$ be Sidon set with $|A|$, and take $f(x)=\sum_{a\in A} 1_{(a-1/2,a+1/2)(x)$, so that $f*f$ is piecewise linear with knuckles at $(k,r(k))$, where $r(k)$ is the number of reps of the integer $k$ as a sum of two elements of $A$. Then $\|f*f\|_{\infty}=2$ (by Sidon-ness), and $\|f*f\|_1=n$ (by $|A|=\sqrt{n}$). But $\|f*f\|_2^2$ depends on the number of intervals in $A+A$, or (equivalently) on the sum $\sum_k r(k)r(k+1)$. This is a difference between the continuous and discrete settings: in the discrete setting we always have $\|f*f\|_2^2=2|A|^2-|A|$, and norm ratio would tend to 1. | |
| Mar 31, 2011 at 20:58 | comment | added | Ben Green | Hmm, it's not so clear. The problem is that it seems to be hard to say much about f. For example, f could consist of spikes of height $N$ and width $1/N^2$ about a Sidon set of size about $N$; then $f \ast f$ will look a bit like the characteristic function of a union of $N$ intervals of width $\sim 1/N$; call this set $E$. Unfortunately I can't conclude anything useful about $\Vert \hat{1}_E \Vert_1 = \Vert f \Vert_2^2$ being small. | |
| Mar 31, 2011 at 20:21 | comment | added | Ben Green | Terry, instead of using the idempotent theorem I think a result of Rudin, improved by Saeki is more relevant: MR0225102 (37 #697) Saeki, Sadahiro On norms of idempotent measures. Proc. Amer. Math. Soc. 19 1968 600–602. In R, where there are no interesting compact subgroups, this will tell you that $\|\hat{1}_E\| > 1.2$. WIll have to think about the details. | |
| Mar 31, 2011 at 17:19 | history | answered | Terry Tao | CC BY-SA 2.5 |