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Assume that $f$ is a $L^p$ integrable function for $1\le p\le P_0$, with $P_0$ a positive constant. L is a smooth compactly supported function. Define $L_\epsilon(x) = 1/\epsilon^n L(x/\epsilon)$. Is it possible to give an estimate of the measure of the set S, where $S=\{x\in R^n: L_\epsilon*f(x)\ge\lambda f(x)\}$, in term of $\epsilon$ and $\lambda$?

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    $\begingroup$ Would $(L_{\epsilon}*f)(x)$ be better notation than $L_{\epsilon}*f(x)$? $\endgroup$ – Acccumulation Dec 3 '18 at 23:51
  • $\begingroup$ When there is no confusion, I would prefer the second one... $\endgroup$ – Fei Wang Dec 4 '18 at 22:37

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