Sobolev space inequality If $f\in H^2(\mathbb R^2)$, I want to show that   $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$  How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?  It is actually one of my Real variables's project. So how can I do this inequality in an relativly "elementary" way?
Sobolev inequality If $f\in H^2(\mathbb R^2)$, I want to show that    $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$   For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb R^2} |f|\le\int_{\mathbb R^2} |\hat f(\xi)|d\xi = \int(1+|\xi|^2)^{-1}(1+|\xi|^2)|\hat f(\xi)|d\xi \le (\int (1+|\xi|^2)^{-2}d\xi)^2(\text {which is integrable in this case})||f||_{H^2}\to \text{is this correct?}$   But for 2, I am stucked. How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?
Is there an invalid pthread_t id? Do Thorncaster Sliver's triggered abilities go on the stack? I was guessing, since you actually target things instead of just dealing damage (similar to ), 's triggered abilities would actually go on the stack.  This came up in a separate game this past Friday when the Sliver deck attacked with two s, one , and two s while his opponent had a  as well as some 2/2.  The reason this even came up was because, if the triggers went on the stack, you could target things so they resolve in the following order:   Predatory Sliver targets Blood Artist, killing it. Predatory Sliver targets the 2/2, dealing one damage to it. Thorncaster Sliver targets the 2/2, killing it. and 5. Both Manaweft Slivers target the player, dealing one damage each to him. How to add programs to the Unity Launcher or Ubuntu Dock? How can I add new programs to the launcher (or the dock in Ubuntu 17.10 and later) in Ubuntu? How does the "this" keyword work? How to distribute objects in a circular pattern on a surface (Radial Array, Polar Array) How can I avoid Java code in JSP files, using JSP 2? proving logical equivalence $(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg P \wedge \neg Q)$ Here's a situation that came up during a back lot league softball game a few weeks ago. (Note: While our league has some of its own rules we try to defer to MLB rules in all causes that aren't specifically addressed.)  There is a runner on second (R2) and a runner on first, with no outs. A ground ball is hit to the third baseman (3B) who cleanly fields the ball away from third base. Seeing that R2 is running toward him, 3B does not go to third and instead waits to tag the incoming R2. R2, in an attempt to buy some time for the other runners, stops advancing to third base. As soon as R2 stops, 3B declares R2 is out (without making a tag) and throws the ball to first to make a play.  During the discussion after the play, the 3B claimed that any player advancing on a force cannot stop moving toward their next bag or they are automatically out (the same way a runner leaving the baseline to avoid a tag is considered out). Our rules don't have a specific clause that addresses this situation and I couldn't find anything in official MLB rules (though it's possible I missed something, as 3B claims it is an MLB rule).  So, is 3B’s claim correct? That if a runner who is forced to advance to their next base stops (in particular to avoid/delay a tag) they are automatically out? Dual boot with windows 7! I'm using windows 7 right now. I want to install Ubuntu with dual boot. Is is safe? And does the process go safely? How to change default action for file types? There is a certain text editor that I would like to set as the default editor for certain file types but if I try selecting it on the “Open With…” menu, that does not work. However, it has put an explorer context menu item in, which does work. How could I make this menu item the default double-click action, for certain file types?  Thank you SSRT and Josh R. Both good answers. It's a pity I can't give you both best answer.  For what it's worth, I found the answer today (twenty-four hours after asking this question). Maximal order of an element in a symmetric group How do we "restore" notepad after crash? Retrieve Force.com registered domain MacTeX - TeX Live Utility broken after 2015 update I have the following strange message.  The repository at ftp://ftp.oleane.net/pub/CTAN/systems/texlive/tlnet  has TeX Live 2015, but you have TeX Live 18446744073709551615  installed. You need to switch repositories in order to continue.   How can I fix this ? Calculate a determinant with pattern Direction of friction for object that rolls with slipping Consider this situation:  A ball is moving forward and undergoing rotation. Assume that it is not slipping. Eventually, the velocity and rate of rotation of the ball decrease, and it comes to a halt. But if you observe the direction of friction (when the ball is rotating clockwise), you will see that the friction should have provided a clockwise torque to the ball and the angular velocity of the body should have increased. But this doesn't happen. Why? When writing large numbers, should a comma be inserted? I know that it is common to write the number 'ten thousand' as a numeral with a comma delimiting the 10 and the 000 like so: 10, 000  However, I have never seen a comma used for numbers less than 10,000, such as 8000 where, I assume using a comma would look like: 8,000.  Are there any rules relating to when a comma should be inserted into a numeral?  N.b. I live in the UK, so answers pertaining to UK English are preferred. Prove that $371\cdots 1$ is not prime Prove that $371\cdots 1$ is not prime.   I tried mathematical induction in order to prove this, but I am stuck.  My partial answer:  To be proved is that $37\underbrace{111\cdots 1}_{n\text{ ones}}$ is never prime for $n\geq 1$. Let $P(n)$ be the statement that $37\underbrace{111\cdots 1}_{n\text{ enen}}$ is not prime.  For $n=1$, we can write $371=7\cdot 53$, and therefore $P(1)$ is true. Let $P(k)$ be true for $k&gt;1$. Then we now have to prove that $P(k+1)$ is true.  I found that $37\underbrace{111\cdots 1}_{k\text{ ones}}$ can be written as $37\cdot 10^k+10^{k-1}+10^{k-2}+\cdots +10^0$ and that $37\underbrace{1111\cdots 1}_{k+1\text{ ones}}$ can be written as $37\cdot 10^{k+1}+10^{k}+10^{k-1}+\cdots+10^0$.  I think I'm pretty close to the answer know, but I don't know how to proceed. Mask and inherit the same systemd unit Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this?