Timeline for Controlling convolutions with maximal functions
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 30 at 9:09 | vote | accept | Ma Joad | ||
| Jan 30 at 2:06 | answer | added | Willie Wong | timeline score: 1 | |
| Jan 30 at 1:39 | comment | added | Willie Wong | @ChristianRemling: the OP means spherical maximal as in "supremum of spherical means". So on $\mathbb{R}$ we are looking at $Mf(x) = \sup_{r > 0} \frac12 \Big( |f(x+r)| + |f(x-r)| \Big)$. If you have a bounded function $f$, we then have $|Mf(x)| \in [\frac12 \|f\|_\infty, \|f\|_\infty ]$ and hence what the OP demands hold. But it fails for (deliberately poorly chosen representatives of) $L^\infty$ functions. | |
| Jan 30 at 1:33 | comment | added | Willie Wong | Incidentally, you probably don't want to consider the spherical maximal function anyway on $L^1(\mathbb{R})$, as it isn't even well-defined. (Take a random $L^1(\mathbb{R})$ function $f$, modify it so that $f(n) = |n|$; this new function differs only on a measure zero set from the original, so is still $L^1$. But at every point you find $Mf(x) = +\infty$.) | |
| Jan 30 at 1:26 | comment | added | Willie Wong | When $n = 1$ what you are looking at holds essentially by accident. When $n \geq 2$ take $\varphi(x) = f(-x)$ to be the characteristic function of the ball of radius 1 centered at then point $(n,0,0,0,\ldots,0)$. Then even with the "spherical" maximal function this extension of Christian's counterexample works. | |
| Jan 29 at 18:06 | comment | added | Christian Remling | The spherical (= symmetric) maximal function is what I had in mind. | |
| Jan 29 at 10:39 | comment | added | Ma Joad | @ChristianRemling Thank you. Does spherical maximal function work, though? | |
| Jan 29 at 10:39 | history | edited | Ma Joad | CC BY-SA 4.0 |
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| Jan 28 at 23:46 | comment | added | Christian Remling | No. If $\varphi(x)=f(-x)=\chi_{(n,n+1)}(x)$, then $(\varphi *f)(0)=\|\varphi\|_1=1$, but $(Mf)(0)\simeq 1/n$. | |
| Jan 28 at 23:26 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jan 28 at 22:56 | history | asked | Ma Joad | CC BY-SA 4.0 |