Timeline for Controlling the tensor product of functions in $H^1$ with lower derivatives
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 31, 2023 at 15:47 | vote | accept | Víctor | ||
| Jul 31, 2023 at 15:42 | comment | added | Willie Wong | The tensor product $\phi\otimes \phi: [0,2\pi]^2\ni(x,y) \mapsto \phi(x) \phi(y)$. So $\partial_x (\phi\otimes \phi)(x,y) = \phi'(x)\phi(y)$, whose $L^2$ norm is just the product of the (one dimensional) $L^2$ norms. | |
| Jul 31, 2023 at 15:36 | comment | added | Víctor | Thank you very much. Could you please elaborate a bit on the last estimte for the $\dot{H}^1$ norm of the tensor product? If I try to compute that myself with your example function I still obtain something of order $N$. If it's related to your first comment under my question, how do you know that $\|\phi\otimes\phi\|_{\dot{H}^1} \sim \|\phi\|_{L^2}\|\phi\|_{\dot{H}^1}$? I try to derive an estimate like that I always need to use some higher $L^p$ norms for $\phi$ and/or $\nabla\phi$. | |
| Jul 31, 2023 at 14:02 | history | answered | Willie Wong | CC BY-SA 4.0 |