norms – What are the functions such that $ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p$?

Let $1 leq p leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g in L_p(mathbb{R})$ such that
$$ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p.$$

For $p = 2$, this relation is satisfied if and only if $langle f, g rangle = 0$. For $p = 1$, it has been shown in this post that the condition is equivalent to $f g geq 0$ almost everywhere.
For a general $p$, the relation is clearly satisfied as soon as the product $fg=0$ almost everywhere. Is this latter sufficient condition also necessary?

If not, then what if we reinforce the condition with
$$ lVert alpha f + beta g rVert_p^p = lVert alpha f rVert_p^p + lVert beta g rVert_p^p$$
for any $alpha, beta in mathbb{R}$?

formal languages – Using pumping lemma to prove that $L = { a^ib^j mid lvert i – j rvert le 2 } $ is irregular

Given the following language:

$L = { a^ib^j mid lvert i – j rvert le 2 } $

I am trying to prove that it is not regular. On the one hand my intuition tells me that the language is non-regular as there is no way of tracking $a^{i’s}$ and $b^{j’s}$.

However, when I try to prove that it is irregular using the pumping lemma I have trouble finding which word I should use to arrive at a contradiction.

Any suggestions?

functional analysis – If $f_nrightarrow f$ in $L^2$ and $lVert f_nrVert_inftyle C$ then why is $fin L^infty$?

This exercise from the $L^p$-theory section of my measure theory course is giving me some trouble.

Let $(f_n)_nsubset L^2cap L^infty$ be a sequence such that $f_nrightarrow fin L^2$ in $L^2$ and $lVert f_nrVert_inftyle C$ for all $n$. I want to show that

  1. $fin L^infty$ and
  2. $f_nrightarrow f$ in $L^p$ for any $2le p<infty$

My most promising attempt at 1. was
$$lVert frVert_infty=lVert f-f_n+f_nrVert_inftyle lVert f-f_nrVert_infty+C,$$
but how can we bound $lVert f-f_nrVert_infty$?

I’m having similar problems with 2., except of course there we want to bound $lVert f-f_nrVert_p$.

complex analysis – Analytic continuation of a function on the right half-plane to a region enclosing the circle ${ lvert z rvert = 3}$

I am trying to solve the following question:

Show that there exists an analytic function $f$ in the open right half-plane such that $(f(z))^2 + 2f(z) equiv z^2$. Show that your function $f$ can be continued analytically to a region
containing the set ${z in mathbb{C} , lvert z rvert = 3}$.

I can solve the first part. Since $z^2+1$ is a non-vanishing holomorphic function in the open right half-plane, which is simply connected, the function admits an analytic square root $h(z)$ in this region; it then suffices to choose $h(z)-1$ as our required function $f(z)$.

I don’t know how to proceed with the analytic continuation part though. Any suggestions?

Functional analysis – Uniform binding to $ lVert chi _ { {u_n = 0 }} rVert_ {W ^ {s, p} ( Omega)} $ for a limited sequence $ u_n $ in $ H ^ 1_0 ( Omega) $?

Suppose I have a sequence $ u_n to u $ in the $ H ^ 1_0 ( Omega) $ on a smooth and limited domain. For some $ p> 1 $ and $ s in (0, frac 12) $it is possible to estimate the norm of the characteristic function of the zero level set $ u_n $, $$ lVert chi _ { {u_n = 0 }} rVert_ {W ^ {s, p} ( Omega)} $$
in relation to norms of $ u_n $? In particular, I am looking for a uniform that is bound for the expression above.

We know that it belongs, for example $ W ^ { epsilon, 2} ( Omega) $ to the $ epsilon < frac 12 $I just want to know if it can be consistently limited.