Timeline for Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 28, 2023 at 20:47 | comment | added | Iosif Pinelis | I don't think it is in Folland. | |
| Jul 28, 2023 at 20:33 | comment | added | Isaac | If I remember correctly, one of them is "Real Analysis" by Folland. | |
| Jul 28, 2023 at 20:26 | comment | added | Iosif Pinelis | Which textbooks? | |
| Jul 28, 2023 at 19:02 | comment | added | Isaac | In fact, I believe that the convolution is smooth on any open ball contained in $X \times Y$. | |
| Jul 28, 2023 at 19:00 | comment | added | Isaac | Ok, I think I corrected everything... I ran into $L^{p,q}$ spaces in standard real analysis textbooks. Also, I believe at least the convolution is jointly continuous, so that it belongs to $L^{p,q}$ for the compact underlying regions $X$ and $Y$. | |
| Jul 28, 2023 at 18:55 | history | edited | Isaac | CC BY-SA 4.0 |
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| Jul 28, 2023 at 18:19 | comment | added | Iosif Pinelis | Where did you see the $L^{p,q}$ spaces? Also, do you have a proof that $f * [\phi^n_\epsilon \otimes \phi^m_\epsilon]\in L^{p,q}$ if $f\in L^{p,q}$? | |
| Jul 28, 2023 at 14:52 | comment | added | Iosif Pinelis | In your definition of the convolution, you probably wanted $f(x',y')$ instead of $f(x,y)$. Also, my point (iii) is still not addressed. | |
| Jul 27, 2023 at 23:49 | history | edited | Isaac | CC BY-SA 4.0 |
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| Jul 27, 2023 at 14:28 | history | edited | Isaac | CC BY-SA 4.0 |
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| Jul 27, 2023 at 14:16 | comment | added | Isaac | OK, I will specify all the details. Thank you. | |
| Jul 27, 2023 at 13:52 | comment | added | Iosif Pinelis | (i) What is the "$L^{p'q}$ space"? (ii) "converges to $f$" or to $0$? What if $f=1$ on $X\times Y$? (iii) How is the convolution of $f$ with the mollifier defined, given that $f$ is only defined on $X\times Y$? (iv) What is "the standard mollifier"? | |
| Jul 27, 2023 at 12:35 | history | asked | Isaac | CC BY-SA 4.0 |