Timeline for An inequality in harmonic analysis with the BMO flavour
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| S Mar 22, 2020 at 8:58 | history | bounty started | J.Mayol | ||
| S Mar 22, 2020 at 8:58 | history | notice added | J.Mayol | Draw attention | |
| Mar 20, 2020 at 9:42 | history | edited | YCor | CC BY-SA 4.0 |
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| Mar 20, 2020 at 8:51 | history | edited | J.Mayol | CC BY-SA 4.0 |
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| Mar 20, 2020 at 8:10 | history | edited | J.Mayol |
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| Mar 20, 2020 at 7:06 | history | edited | J.Mayol | CC BY-SA 4.0 |
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| Mar 20, 2020 at 6:57 | comment | added | J.Mayol | @fedja you are right. However with the family $[1/2,1/2+2^{-j}]$ we have $\|f\|_{L^p} < \infty$ for any finite $p$. So we should not expect $f$ to lie in BMO but I expect that $\alpha _n \lesssim e^{-cn}$ anyway. I could not find any counterexample to that estimate ... I really feel that the condition on any interval $J$ constrained the measure of sets on which $f$ is large, but I do not find any way of making that a rigorous statement. | |
| Mar 19, 2020 at 18:32 | comment | added | fedja | The BMO understood literally is still out of question: Just consider the family $[0.5,0.5+2^{-j}]$ and look at a very short interval centered at $0.5$. So, you need to be a bit more inventive. Hint: replace each interval by a trapezoid function of comparable width; then the sum will be in BMO, indeed. | |
| Mar 19, 2020 at 18:22 | comment | added | J.Mayol | @fedja I edited the question so that the hypothesis concerns any interval $J$, thus ruling out your example (in your case we can take $J=[1/2-\eta,1/2+\eta]$... | |
| Mar 19, 2020 at 18:20 | history | edited | J.Mayol | CC BY-SA 4.0 |
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| Mar 19, 2020 at 18:11 | comment | added | J.Mayol | But then there is something I do not understand with the Remark 4.4 in the attached lecture notes. If any interval involved are dyadic then by Lebesgue differentiation theorem we get that $f$ is bounded and thus in any $L^p$ ... which is strange. | |
| Mar 19, 2020 at 18:07 | comment | added | J.Mayol | Well, I realize that it is not clear. In fact in the lecture notes I just attached a 'tiling' is defined with dyadic intervals so all the intervals involved are in fact dyadic. However we can cahnge the question to make it more interesting by letting $J$ be any interval! | |
| Mar 19, 2020 at 16:21 | comment | added | fedja | Something is fishy: consider the collection of open intervals centered at $1/2$. Then the condition is almost void. Are you sure those are not dyadic intervals or that the condition is not imposed with $J$ being a union of 2 adjacent dyadic intervals? | |
| Mar 19, 2020 at 16:13 | history | asked | J.Mayol | CC BY-SA 4.0 |