fa.functional analysis – Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line

Let $newcommand{ptp}{widehat{otimes}}ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, suggest to me that the following should be true:
$newcommand{Real}{{bf R}}$

for some $k > 1$, every compactly supported $C^k$-function $Real^2toReal$ belongs to $L^2(Real)ptp L^2(Real)$.

(Recall that in contrast, there are continuous functions on $(0,1)^2$ that do not belong to $C(0,1)ptp C(0,1)$.)

If the claim above is true, I would like to know if there are standard references, perhaps from the world of integral kernel operators or Sobolev spaces, which I could cite, rather than reinventing the wheel (and probably getting suboptimal values of $k$).

In a slightly different direction, I would also be interested to know of references which prove analogous resuts for $C^k$ functions (suitably interpreted) on compact connected Lie groups.

"Are you sure?" Request for confirmation modal usability research

  1. A user is about to delete a crucial object with no undo.
  2. The designer puts an “Are you sure?” modal confirmation.
  3. The user says “Yeah yeah yeah do it”
  4. Immediately the user regrets his/her decision and panics.

Clearly Undo would be great. But sometimes that is impractical.

Question:

  • What public usability studies are there on this topic?
  • Do users REALLY think about the question or do they just click Yes without thinking?

Thank you.

Request for Feedback: Envy Forums | Forum Promotion

Hi there. I am working on starting a new forum and wanted some initial feedback on what I’ve created so far.

Forum Title: Envy Forums
Forum URL: https://envyforums.net/

  • Setup server
  • Created a domain name
  • Installed XenForo & a Premium Theme (expensive but the site looks pretty at least)
  • Created a few categories/forums and wrote a few posts in them.

I know there’s so much still to do before the site is a thriving forum but what else should I do to get my community seeded with content?

Package Request

Site URL: https://envyforums.net/
Package: Garnet
Total posts your site: 8 posts
Packager Preference: N/A
Preferred posting location: Anywhere but I’d prefer empty categories first if possible.
Would you like to upgrade all replies to threads?: No
Do you permit our packagers to promote Forum Promotion on your site?: Yes
Extra Notes: Member less than a month so I think the package is free :]

reference request – A bounded extension operator

Let $Omegasubsetmathbb{R}^n$ be a bounded domain with smooth boundary $partialOmega$. Consider the harmonic extension operator $E :L^2(partial Omega) rightarrow H^{1/2}(Omega)$ which assigns to a prescribed boundary value $g$ a function $f$ with $frvert_{partialOmega}=g$ and $Delta_Omega f=0$.

I have two questions about this operator:

Can $E$ be bounded from $L^2(partial Omega)$ into $L^2(Omega)$ as a right inverse of the trace operator? (or possibly another modification of $E$ or another extension operator).

Is there any explicit characterization of the range of $E$: $mathrm{ran}(E):=E,L^2(partial Omega)$?

Finally, any reference on some properties of such operator would be helpful.

facebook – I deleted a friend request and now I need to retrieve it again

I did not except a friend request by mistake. Now when I go to his page, it does not all me to add him as a friend. When he goes to my page, he can no longer add me as a friend either. Both of our settings are set for Everyone on & off Facebook can request our friendship. When I go into “Friend suggestions” and click “All”, we can no longer find each other under Friend suggestions. My question is, how can we friend each other again? I should have excepted his friend request but I clicked the wrong button and denied his request. Now we can no longer add each other as friends on Facebook. Help. Thank you

reference request – a square root inequality for symmetric matrices?

In this post all my matrices will be $mathbb R^{Ntimes N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{frac 12}$ of a psd matrix $A$ is defined unambigusouly via the spectral theorem.
Also, I use the conventional Frobenius scalar product and norm
$$
<A,B>:=Tr(A^tB),
qquad
|A|^2:=<A,A>
$$

Question: is the folowing inequality true
$$
|A^{frac 12}-B^{frac 12}|^2leq C_N |A-B|quad ???
$$

for all psd matrices $A,B$ and a positive constant $C_N$ depending on the dimension only.

For non-negative scalar number (i-e $N=1$) this amounts to asking whether $|sqrt a-sqrt b|^2leq C|a-b|$, which of course is true due to $|sqrt a-sqrt b|^2=|sqrt a-sqrt b|times |sqrt a-sqrt b|leq |sqrt a-sqrt b| times |sqrt a+sqrt b|=|a-b|$.

If $A$ and $B$ commute then by simultaneous diagonalisation we can assume that $A=diag(a_i)$ and $B=diag(b_i)$, hence from the scalar case
$$
|A^frac 12-B^frac 12|^2
=sumlimits_{i=1}^N |sqrt a_i-sqrt b_i|^2
leq sumlimits_{i=1}^N |a_i-b_i|
leq sqrt N left(sumlimits_{i=1}^N |a_i-b_i|^2right)^frac 12=sqrt N |A-B|
$$

Some hidden convexity seems to be involved, but in the general (non diagonal) case I am embarrasingly not even sure that the statement holds true and I cannot even get started. Since I am pretty sure that this is either blatantly false, or otherwise well-known and referenced, I would like to avoid wasting more time reinventing the wheel than I already have.


This post and that post seem to be related but do not quite get me where I want (unless I missed something?)


Context: this question arises for technical purposes in a problem I’m currently working on, related to the Bures distance between psd matrices, defined as
$$
d(A,B)=minlimits_U |A^frac 12-B^frac 12U|
$$

(the infimum runs over unitary matrices $UU^t=Id$)