Timeline for $L^p$ domination of mixed partial derivatives by the unmixed ones?
Current License: CC BY-SA 4.0
11 events
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| Feb 27 at 15:36 | comment | added | Iosif Pinelis | Thank you for all this additional information. | |
| Feb 27 at 6:13 | history | edited | Willie Wong | CC BY-SA 4.0 |
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| Feb 27 at 5:31 | comment | added | Willie Wong | Additionally, Ornstein's theorem is a lot stronger than just disproving elliptic regularity. It in fact proves that the $L^1$ case of the weakened version must fail. The first part of the paper actually gives a fairly explicit counterexample through a sequence of smooth functions supported in the unit cube of the plane. | |
| Feb 27 at 5:25 | comment | added | Willie Wong | Re (ii): actually, looking at my answer here the example I list in the end (with the Harmonic polynomial chosen to be $P(x,y) = xy$) would be an example of a function for which both $\partial^2_{xx} u$ and $\partial^2_{yy} u$ remain continuous and bounded, but $\partial^2_{xy} u$ is unbounded. This shows that the weakened version also fails for $L^\infty$. @IosifPinelis | |
| Feb 27 at 5:13 | comment | added | Willie Wong | Re (iii): in the $p = 2$ case, you have for $C^\infty_0$ functions by Plancherel $$ \|D_iD_jD_k u\|_2^2 = \int |\xi_i|^2 |\xi_j|^2 |\xi_k|^2 |\hat{u}(\xi)|^2 \leq \int (|\xi_i|^6 + |\xi_j|^6 + |\xi_k|^6) |\hat{u}(\xi)|^2 $$ by AM-GM. So in that case it still holds. | |
| Feb 27 at 4:48 | comment | added | Willie Wong | ... however, this doesn't address whether a slightly weaker inequality of the form $\|D_i D_j u\|_1 \leq C \sum_i \|D_i^2 u\|_1$ is possible. In the G-T example mentioned above, we also have $D_1 D_1 u\not\in C^0$, and the fact that $\Delta u\in C^0$ is due to convenient cancellations. (iii) I have no idea; certainly I have not seen something like that before. // And yes, the $C_p$ is implicitly dependent on $\Omega$, and hence also on $n$. | |
| Feb 27 at 4:33 | comment | added | Willie Wong | (i) Yes and no. Calderon-Zygmund inequality is usually proven using the Calderon-Zygmund decomposition; I would typically classify this as a "harmonic analysis" result, but it does not require taking Fourier transform per se. (ii) There are some well-known counterexamples to "elliptic regularity" in the endpoint cases. For example, Gilbarg-Trudinger list a function $u$ with $\Delta u\in C^0$ but $u\not\in C^2$. And $L^1$ elliptic regularity is known to be false. ... | |
| Feb 27 at 3:44 | comment | added | Iosif Pinelis | Thank you for this answer. I think $C_p$ must also depend on $n$. I would appreciate it if you can respond to the following questions, however briefly. (i) This approach does not seem to use Fourier analysis, does it? (ii) Do you know of counterexamples for $p=1$ and/or $p=\infty$? (iii) Is there a bound of the form $\|D^3u\|_p\le C_{n,p}\sum_{j=1}^n\|D_j^3u\|_p$ with a real $C_{n,p}$ depending only on $n$ and $p$ (even just for $n=2$)? | |
| Feb 27 at 3:38 | vote | accept | Iosif Pinelis | ||
| Feb 27 at 2:25 | history | edited | Willie Wong | CC BY-SA 4.0 |
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| Feb 27 at 2:01 | history | answered | Willie Wong | CC BY-SA 4.0 |