Timeline for using the M. Riesz Interpolation Theorem
Current License: CC BY-SA 3.0
7 events
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| Mar 28, 2013 at 10:49 | comment | added | Patch | The value $p=4$ might not be special, but it is the only instance in which we are able to prove the $\limsup$ convergence explicitly. | |
| Mar 28, 2013 at 10:45 | history | edited | gerw | CC BY-SA 3.0 |
added 519 characters in body
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| Mar 28, 2013 at 10:44 | vote | accept | Patch | ||
| Mar 28, 2013 at 10:44 | comment | added | Patch | Oh, of course! Sorry. Thanks again. | |
| Mar 28, 2013 at 10:41 | comment | added | gerw | You can use $\lVert e_k \rVert_6 \le C \, \lambda_k^{\delta(6)} \, \lVert e_k \rVert_2$. | |
| Mar 28, 2013 at 10:19 | comment | added | Patch |
Thanks. That really helps. This is the [Hölder interpolation][1] theorem you're referencing, correct? In the case when $p_t\in(4,6)$ though, we end up with $$\lambda_k^{-\delta(p_t)}||e_{\lambda_k}||_{p_t} \leq \left( \lambda_k^{-\delta(4)}||e_{\lambda_k}||_4 \right)^{1-t}\left( \lambda_k^{-\delta(6)}||e_{\lambda_k}||_6 \right)^{t}$$ and we can no longer rely on the unit-norm in $L^{2}(M)$ nor the fact that $\delta(2)=0$. How do we proceed from here? [1]: en.wikipedia.org/wiki/…
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| Mar 28, 2013 at 8:00 | history | answered | gerw | CC BY-SA 3.0 |