Timeline for How to prove that the L-infinity norm is smaller than the Besov norm?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2021 at 21:31 | vote | accept | Inuyasha | ||
| Dec 2, 2021 at 21:18 | comment | added | Willie Wong | @Inuyasha Your final sentence seems to be confused. The inequality shows that $B^{\alpha}_{\infty,\infty}$ embeds in $L^\infty$, and not the other way around. Indeed functions that are $B^{\alpha}_{\infty,\infty}$ are continuous (in fact uniformly continuous, since $B^{\alpha}_{\infty,\infty} = C^\alpha$ [Holder space] when $\alpha \in (0,1)$) | |
| Dec 2, 2021 at 21:08 | comment | added | Willie Wong | Let $K = \|u\|_{B^\alpha_{\infty,\infty}}$, then by definition $\| \phi_\mu(D)u\|_{\infty} \leq 2^{-\mu\alpha} K$. So $\|u_M - u_N\|_\infty \leq 2^{(1-\min(M,N))\alpha} K$. | |
| Dec 2, 2021 at 21:07 | comment | added | Inuyasha | @WillieWong I still do not understand it, sure each $\phi_i(D)u$ is smooth and in $L^\infty$, but how do you show $u_M$ is a Cauchy sequence? Even that is right, then it is strange since a Cauchy sequence of continuous functions in $L^\infty$ should converges to a continuous function, so clearly for general $u\in L^\infty$ the sequence $u_M$ does not converges to $u$. | |
| Dec 2, 2021 at 17:47 | comment | added | Willie Wong | Each $\phi_i(D) u$ is a smooth $L^\infty$ function. So the partial sums $u_M := \sum_{\nu = 0}^M \phi_\nu(D) u$ is a Cauchy sequence in $L^\infty$. (In other words, the series actually converges absolutely in $L^\infty$.) | |
| Dec 2, 2021 at 16:38 | comment | added | Inuyasha | But how to show the inequality $\|u\|_{L^\infty} \leqslant \sum_{i\geqslant0}\|\phi_i(D)u\|_{L^\infty}$? Actually I got stuck here, since convergence of $\sum_{i\geqslant0}\phi_i(D)u$ is in the space of tempered distribution, not $L^\infty$. So I do not think we can get $\|u\|_{L^\infty} =\|\sum_{i\geqslant0}\phi_i(D)u\|_{L^\infty} $ and then use triangular inequality. | |
| Dec 2, 2021 at 16:16 | history | answered | Bazin | CC BY-SA 4.0 |