0
$\begingroup$

Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold). From what I understand, the derivative of $f$ in the sense of distributions is in the Besov space $B_{\infty,\infty}^{\alpha-1}$.

Let $\beta>1-\alpha$. Do we have $B_{\infty,\infty}^{\alpha-1} \subset B_{1,1}^{\beta}$?

Or is the derivative of $f$ an element of the dual of $C^{\beta}$? (I don't know whether this question is weaker than the previous one or not)

In particular, if $\alpha>1/2$, can we pair $f$ with its derivative?

I would be happy to just have the answer to the last question if the other ones are harder. From what I saw, this is not directly stated in Triebel's classical books.

$\endgroup$
3
  • $\begingroup$ Something excapes me. You can embed a Besov space in another if and only if there is the usual inequality between the indices (see e.g. the first chapter of the treatise by Runst and Sickel) $\endgroup$ Mar 5, 2014 at 12:36
  • $\begingroup$ Thanks. So, the answer to the first question about the embedding of Besov spaces is known to be no. But do you happen to know whether the answer to the second and third questions are also no? Because the third statement in particular seems (to me) to be fairly natural and hopefully true. $\endgroup$ Mar 5, 2014 at 14:29
  • $\begingroup$ Certainly you have no control on the behaviour of the functions for large x, so the usual pairing may not be defined. As to the possibility of defining a pointwise product, I think the answer is yes, check the R-S book where this question is analyzed in detail $\endgroup$ Mar 5, 2014 at 17:59

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.