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Timeline for Regularization of Zygmund functions

Current License: CC BY-SA 3.0

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May 22, 2014 at 13:49 vote accept CPJ
May 1, 2011 at 14:23 comment added Andrew In terms of the first difference Zygmund functions are generally no better than log-Lipschitz as the example for $f$ above shows. Ok, lets do the math. For this $f$ and small $\epsilon>0$ we have $$ \partial_xf_\epsilon(0)=\int_{-1}^1\rho_\epsilon(y)(\log|y|+1)dy= \int_{-\infty}^\infty\rho(y)(\log|\epsilon y|+1)dy= $$ $$ \int_{-\infty}^\infty\rho(y)\log|y|dy+\log\epsilon+1. $$
May 1, 2011 at 13:05 comment added CPJ But this function is also log-Lipschitz. I know that the estimate on the regularization for such functions is optimal. I would like to know if there is some kernel with which we might get better estimates for Zygmund-functions, just using the Zygmund property.
Apr 30, 2011 at 18:06 history answered Andrew CC BY-SA 3.0