Consider $f\in B_{p,q}^s(\Omega)$, $\Omega$ compact in $\mathbb{R}^d$, with $p,q\geq 1$ and $s>d/p$ (so the elements in the space are regular enough to be continuous functions), such that $\|f\|_{B_{p,q}^s}\leq R$.

Since $f$ is continuous and $\Omega$ is bounded, $e^f$ is a continuous mapping (with $\|\cdot\|_\infty$).

Do we have more general results with Besov norms, such as $\|e^f\| \leq K \|f\|^*$, where $K$ is a positive constant possibly depending on $R$ and $\Omega$, and $\|\cdot\|,\|\cdot\|^*$ are some Besov norms?

  • $\begingroup$ As stated the answer is clearly negative: the left hand side scales exponentially while the right hand side scales linearly. (Consider the constant function $f\equiv \lambda$ which ought to be in your Besov spaces. $\|f\|^* \approx \lambda$ while $\|e^f\| \approx e^\lambda$.) $\endgroup$ – Willie Wong Aug 30 '17 at 15:09
  • $\begingroup$ To get the scaling right you want to look at something like (but maybe not precisely) $\ln \|e^{|f|}\|$ on the left hand side instead. And the type of inequality you are looking for would very likely be called a "Moser-Trudinger inequality", if it is known. $\endgroup$ – Willie Wong Aug 30 '17 at 15:13
  • $\begingroup$ @WillieWong Thanks for the reply. Sorry for being sloppy, the condition I am searching for is not necessarily applicable to the whole space, on some ellipsoid is good enough, see the edit please. $\endgroup$ – newbie Aug 30 '17 at 15:48

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