New answers tagged convolution
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Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let me add some general hints. You want a bound $[f-g]_\alpha\le C\|f-g\|_\infty$ for all non-negative $f$ and $g$ in $L^1(\mathbb R^n)\cap L^\infty(\mathbb R^n)$ with $\int_{\mathbb R^n}fdx= \int_{\...
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2
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Accepted
Lower bound the best $\alpha$-Hölder constant of a convolution
No. Take $f_1$ an $\alpha$-Hölder compactly supported convolution kernel (non-negative, with $\int_{\mathbb R}^nf_1dx=1$), and consider the usual approximation of identity by convolution with $f_\...
- 56.6k
2
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Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Not really, because the estimate "sees" the $L^1$ norm rather than the $L^\infty$ one.
Notice first that $f\mapsto \sigma(\cdot,f)$ is linear, so you question amounts to asking whether $[\...
- 5,163
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Examining the Hilbert transform of functions over the positive real line
The answer to the second question is negative as well. Take for example $g$ supported in $(-\infty,-1)$ and discontinuous in some point. If $f$ is supported in $\mathbb{R}_+$ and $y,z<-1$ it holds ...
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